Solve It! K-1 - Gateway Academy

The AIMS Education Foundation is a research and development organization dedicated to the improvement of the teaching and learning of mathematics and science through a meaningful integrated approach.

Sensational Springtime

Counting on Coins

It’s About Time

It Must Be a Bird

Under Construction

Winter Wonders

Cycles of Knowing and Growing

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Solve It! K-1: Problem-Solving Strategies is a collection of activities designed to introduce young children to eight problem-solving strategies. The tasks included will engage students in active, hands-on investigations that allow them to apply their number, computation, geometry, data organization, and algebra skills in problem-solving settings. The strategies addressed are: guess and check, look for patterns, use manipulatives, draw out the problem, use logical thinking, write a number sentence, work backwards, and organize the information. Also included are practice problems that can be used for assessment. These activities give students the opportunity to develop a toolbox of problem-solving strategies that they can draw from when approaching problems. These skills will serve them well not only in mathematics, but in other academic subjects and their everyday lives.

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ISBN 978-1-932093-14-8

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SOLVE IT! K-1 i © 2007 AIMS Education Foundation

Solve It! K-1:

Developed and Published by

AIMS Education Foundation

SOLVE IT! K-1 ii © 2007 AIMS Education Foundation

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Developed and Publishedby

AIMS Education Foundation

This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding. The foundation also sponsors a national program of professional development through which educators may gain expertise in teaching math and science.

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ISBN 978-1-932093-14-8

Printed in the United States of America

SOLVE IT! K-1 iii © 2007 AIMS Education Foundation

Solve It! K-1:

Write a Number Sentence Introduction .109Number Story Theater .........................110My Place in Space ...............................116Hidden Numbers ..................................121Sweet Sums ........................................125

Work Backwards Introduction .............131Trail Totals ...........................................132Balancing Equations ............................139Calendar Clues ....................................142

Organize the Information Introduction ..147Circus Cage Count ..............................148Glyph Hangers .....................................153Pizza Possibilities .................................160

Practice Problems ................................167

Introduction ........................................... viiProblem-Solving Strategies .................. viiiStrategy Reference ................................. ix

Guess and Check Introduction .................1No Room in the Tin .................................2Combination Clues ..................................7Shuttle Shuffl e .......................................11Messing With Mass ................................17Sea Shapes ...........................................20

Look for Patterns Introduction ................29Snap, Clap Patterns ...............................30Button Holes ..........................................33Party Patterns ........................................41Patterns That Grow on You ....................43

Use Manipulatives Introduction ..............49Seashore Stories ....................................50Fiddling with Fiddlers ............................55Shifty Shapes ........................................61

Draw out the Problem Introduction ........65Pocket Problems ....................................66Super Safari Scrapbook .........................71Through the Window Numbers ..............79

Use Logical Thinking Introduction .........83Bag a Bear ............................................84Get the Picture .......................................86Bear Logic .............................................96The McGregors’ Garden .......................102

KEEP GOINGKEEP GOING

SOLVE IT! K-1 v © 2007 AIMS Education Foundation

I Hear and I Forget,

I See and I Remember,

I Do and

I Understand. -Chinese Proverb

SOLVE IT! K-1 vii © 2007 AIMS Education Foundation

Solve It! K-1: Problem-Solving Strategies is a collection of activities designed to introduce young children to eight problem-solving strategies. The tasks included will engage students in active, hands-on investigations that allow them to apply their number, computation, geometry, data organization, and algebra skills in problem-solving settings. It can be diffi cult for teachers to shift from teachingmath facts and procedures to teaching with an emphasis on

mathematical processes and thinking skills. One might ask why problemsolving should be taught at all. The most obvious reason is that it is part of most mathematics curricula. However, it is also an interesting and enjoyable way to learn mathematics; it encourages cooperative learning, and it is a great way for students to practice mathematical skills. This in turn leads to better conceptual understanding—an understanding that allows students to remember skills and be able to apply them in different contexts. Introducing students to the eight strategies included in this book gives them a toolbox of problem-solving methods that they can draw from when approaching problems. Different students might approach the same problem in a variety of ways, some more sophisticated than others. Hopefully, every child can fi nd one approach that he or she can use to solve the problems that you present. Over time, and from discussing what other children have done, students will develop and extend the range of strategies at their disposal. It is our hope that you will use the problems in this book to enrich your classroom environment by allowing your students to truly experience problem solving. This means resisting the urge to give answers; allowing your students to struggle, and even be frustrated; focusing on the process rather than the product; and providing multiple, repeated opportunities to practice different strategies. Doing this can develop a classroom full of confi dent problem solvers well equipped to solve problems, both in and out of mathematics for yearsto come.

SOLVE IT! K-1 ix © 2007 AIMS Education Foundation

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No Room in the Tin

Combination Clues

Shuttle Shuffle

Messing With Mass

Sea Shapes

Snap, Clap Patterns

Button Holes

Party Patterns

Patterns That Grow on You

Seashore Stories

Shifty Shapes

Pocket Problems

Super Safari Scrapbook

Through the Window Numbers

Bag a Bear

Get the Picture

Bear Logic

The McGregors’ Garden

Number Story Theater

My Place in Space

Hidden Numbers

Sweet Sums

Trail Totals

Balancing Equations

Calendar Clues

Circus Cage Count

Glyph Hangers

Pizza Possibilities

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SOLVE IT! K-1 1 © 2007 AIMS Education Foundation

The guess and check strategy is help-ful when a problem is complicated, has large numbers or a lot of data, or when the problem requires fi nding one of many possible solutions. This strategy involves guessing the answer, testing to see if it is correct, and using what you have learned to make another guess if the fi rst one is not correct. As students guess and eliminate options, they get closer to the correct answer.


TopicProblem solving

Key Questions1. How many markers will fi t in the tin can?2. How many sardines will fi t in the tin? 3. How many crackers will fi t in the tin?

Learning GoalsStudents will: 1. estimate how many objects will fi t in a container, and2. check their estimates.

Guiding DocumentNCTM Standards 2000*• Apply and adapt a variety of appropriate strategies

to solve problems• Build new mathematical knowledge through prob-

lem solving• Solve problems that arise in mathematics and in

other contexts

MathProblem solvingEstimation

Integrated ProcessesObservingRecording

Problem-Solving StrategiesGuess and checkUse manipulatives

MaterialsTin cans (see Management 1)Markers (see Management 1)Sardine cut outs (see Management 2)Round snack crackers (see Management 3)Student pages

Background Information This activity gives students an opportunity to practice making reasonable estimates by using the problem-solving strategy of guess and check. They are given a tin can containing one marker and asked to estimate the number of markers it would take to fi ll the can. They are then allowed to check their estimate by putting markers in the can until it is full.

Once they have had the three-dimensional experi-ence, they are challenged to apply what they learned to two dimensions. They are given a page showing a tin holding one sardine and are asked how many sardines the tin could hold. After checking their estimates using cut-outs of the sardine, they see how accurate they were. They are then given a second opportunity to estimate, this time using crackers in a tin. Hopefully, by the third problem, they will have gained suffi cient practice to make reasonable estimates.

Management 1. For the fi rst part of the activity, clean out an iden-

tical tin can, such as a soup can, for each group of students, and select uniform objects such as markers, crayons, or pencils to put in the can. Check to see if there are any sharp edges on the cans. If there are, cover them with masking tape.

2. For the second part of the activity, you will need to copy the sardine page and cut out enough so that each student can have at least eight sardines. (Six sardines will fi t, but students should be given more than that.)

3. For the third part of the activity, you will need to buy round snack crackers the same size as the one illustrated on the student page. Each student will need at least 10 crackers. If necessary, groups of students can share crackers.

Procedure 1. Have students get into groups and give each

group an identical tin can. Give each group a set of markers, crayons, pencils, or whatever you would like them to place in the can.

2. Have them place one of the objects in the can and come up with a group estimate of how many it will take to fi ll the can.

3. Once they have recorded their estimates, have them check to see if they were correct. Discuss the strategies each group used to make their estimates.

4. Tell students that they are now going to try to do some similar problems individually.

5. Give each student a copy of the student page showing the sardine tin.

6. Ask them to study the sardine in the tin and estimate how many sardines it would take to fi ll the tin, including the one already there.


7. Have them record their guesses in the space pro-vided on the student page.

8. Distribute at least eight cut-out sardines to each student and have them check their estimates by placing the sardines in the tin. Tell them that the sardines cannot overlap, but that they should be as close together as possible.

9. Ask students to share their answers with the class. If there are discrepancies, discuss the reasons for those. Discuss the different strategies used by the students to make their estimates.

10. Repeat this process with the round tin of crack-ers. Remind students to count the cracker that is already in the tin when they make their estimates. Each student will need at least 10 crackers.

Connecting Learning 1. How many markers [crayons, pencils, etc.] did

you estimate would fi t in the can? 2. How many actually fi t in the can? 3. Was this more than you estimated or less than

you estimated? Why? 4. How many sardines did you estimate would fi t in

the tin? How did you decide on this number? 5. How many sardines actually fi t in the tin? 6. Was this more than you estimated or less than

you estimated? Why?

7. How many crackers did you estimate would fi t in the tin? How did you decide on this number?

8. How many crackers actually fi t in the tin? 9. Was this more than you estimated or less than

you estimated? Why?10. Was it easier for you to estimate how many sar-

dines would fi t in the tin, or how many crackers would fi t in the tin? Why?

11. If you had to do another similar problem, do you think your estimate would be closer? Why or why not?

Extensions 1. Allow students to create their own problems using

real objects. For example, how many erasers would it take to go all the way around the edge of my paper?

2. Repeat this experience with other three-dimensional objects. For example, how many Unifi x cubes will fi t in a baseball cap, or how many Teddy Bear Counters will fi t in a mug.

* Reprinted with permission from Principles and Standards for School Mathematics, 2000 by the National Council of Teachers of Mathematics. All rights reserved.


I counted:

I think:


I think: ______________ I counted: ___________



Key QuestionWhat is the combination on the lock?

Learning GoalsStudents will:1. guess the combination on a three-number lock by

asking yes or no questions, and 2. develop a strategy for determining the combination

in the fewest number of guesses possible.

Guiding DocumentNCTM Standards 2000*• Use multiple models to develop initial understand-

ings of place value and the base-ten number system • Build new mathematical knowledge through

problem solving • Apply and adapt a variety of appropriate strategies

to solve problems

MathPlace valueProblem solvingLogic

Integrated ProcessesObservingRecordingAnalyzingApplying

Problem-Solving StrategiesGuess and checkUse logical thinking

MaterialsSmall sticky notesTransparency fi lmStudent pages

Background Information This activity provides an opportunity for using the language of mathematics and reinforcing understand-ing of place value in the context of a strategy game. The object of the game is to guess the combination on a three-number lock by asking yes or no questions about the individual digits. Emphasis should be placed

on developing the language of mathematics and giv-ing careful thought to strategic questions that will eliminate several possibilities at once. The goal is for students to learn to rely more on logical thinking and less on guess and check as they play multiple rounds of the game. Once this happens, students can be challenged to discover the combination in the fewest number of guesses possible.

Management1. Before presenting the activity, make an overhead

transparency of the fi rst student page, enter a combination in the lock, and cover each number with a sticky note or piece of paper.

2. Each student will need his or her own copy of the second student page.

ProcedurePart One1. Put the transparency on the overhead projector

and tell students that you have a combination lock that you need their help to open. Explain that they need to ask yes or no questions about any of the three numbers in the combination to determine what the combination is.

2. Allow students to ask questions without making any suggestions as to strategy or language. After each question, have a student cross off the number, or numbers, which were eliminated by that guess. (This ensures that they understand the numbers eliminated by their guesses.) When a number in the combination is identifi ed, remove the paper covering it so that it is visible to students.

3. Once a round or two has been played, tell students that you will only answer their questions if they identify the number they are asking about using place value (i.e., ones place, tens place, hundreds place).

4. If they haven’t already, suggest that students think of questions that will eliminate several numbers at once. Some sample questions could include: Is the number in the ones place odd? Is the number in the tens place greater than fi ve? Is the number in the hundreds place less than the number in the ones place?

5. As students grow comfortable with the game, challenge them to focus on their strategies and to try and discover the combination in the fewest number of guesses.


Part Two1. After playing the game several times as a class,

have students get into pairs or small groups and dis-tribute one copy of the second student sheet to each student.

2. Have one student from each group begin by choos-ing a combination and writing it in the lock.

3. Provide students with sticky notes or small pieces of paper to tape over their combinations. Have them keep a tally of the number of guesses it takes to discover each combination and try to improve their strategies as they continue to play.

4. After students have played several rounds in groups, close the activity with a time of class discussion in which students share about their experiences.

Connecting Learning1. Did the kinds of questions you asked change as you

played the game more? Why or why not?2. What were the best kinds of questions to ask? [those

that eliminated multiple numbers] Why?3. What kinds of questions were the least helpful?

[those that eliminated only one number] Why?4. What was the fewest number of guesses you were

able to use to fi nd the combination? 5. Describe the strategy you used.6. What hints would you give to someone playing this

game for the fi rst time?

ExtensionPlay a version of the game that has a four-number combination.



Number of Guesses Tally

Number of Guesses Tally

It took guessesto open the lock.

It took guesses to open the lock.



Key QuestionHow can you use astronaut counters to help you solve word problems?

Learning GoalStudents will guess and check using manipulatives to help them solve word problems.


to solve problems• Build new mathematical knowledge through

problem solving

MathProblem solvingLogic

Integrated ProcessesObservingApplying

Problem-Solving StrategiesGuess and checkUse manipulativesUse logical thinkingAct out the problem

MaterialsAstronaut counters in three colorsLarge astronaut necklaces (see Management 4)Copy paper in three colorsYarnStudent page

Background Information Guess and check can be a very effective problem-solving tool for primary students. In this activity, students are able to solve word problems by manipulating astronauts in a shuttle. They are able to try out different combinations until they fi nd the one that correctly meets all of the clues.

Management1. Each student will need a collection of astronaut

counters in three different colors. The number of counters needed by each child depends on which problems you are doing. The colors used in the word problems are red, blue, and purple, but you can change the colors to refl ect what you have available.

2. Astronaut counters can be purchased from AIMS in packages of 100 (#1929), 300 (#1930), and 500 (#1931). If astronaut counters are not avail-able, copy the page of small astronauts on three different colors of paper.

3. There are problem situations provided for totals of fi ve, six, seven, and eight astronauts. Select those problems that are most appropriate for your students, or make up your own.

4. Make 12 copies of the large astronaut page on three different colors of paper (four pages of each color). Laminate the pages, if possible. Tape the ends of a length of yarn to the top corners of each page so that it can be worn like a necklace by students.

ProcedurePart One1. Tell students that they will be acting out some

problems that you give them by pretending to be astronauts in a space shuttle directed by mission control.

2. Select 12 students to come to the front of the class and put the large astronaut pages around their necks. Tell them that they are the astronauts in the shuttle, and their classmates are mission control. As you read clues, mission control is going to tell them what order to stand in. (Stu-dents should be told that there will always be at least one astronaut of each color on the shuttle.)

3. Read the fi rst clue from one of the word problems. Call on one of the seated students to give the astronauts instructions. For example, if the fi rst clue says that there are three purple astronauts, mission control might tell three of the students with the purple astronauts to step forward into the “shuttle” area.


4. After each clue, have a different student from mission control give instructions to the astro-nauts. When all of the clues have been read and the astronauts are in order, read through all of the clues a second time to verify that the order matches. Make any necessary adjustments.

5. Repeat this process with additional word problems, allowing different students to be the astronauts.

Part Two1. Give each student a copy of the student page and

a suffi cient number of astronaut counters in three colors.

2. Explain that they will be acting as mission control for their own set of astronauts now. Remind them that there will always be at least one astronaut of each color in the shuttle.

3. Read several different word problems out loud to the class. You may also wish to write the key parts of the clues on the chalkboard.

4. Read the clues as many times as children need. Once everyone has arrived at a solution, go through the clues together as a class and deter-mine the correct answer.

5. If students have incorrect answers, try to help them determine where they went wrong in their thinking.

Connecting Learning1. How many times did you have to guess before you

got the right solution?2. Did you have to guess more or fewer times as you

had more practice with the problems?3. How did having the astronauts help you solve the

word problems?4. Do you think you would have been able to get the

right answers if you hadn’t had the astronauts? Why or why not?

5. Was it easier to solve the problems with your own astronauts or with the astronauts at the front of the class? Explain.

6. What other ways could you have solved these problems?

ExtensionHave students work in groups to make up their own problems. Have an adult record the clues (if neces-sary). Trade the riddles between groups and see if they are written in a way that makes them possible to solve.



Word Problems for Five Astronauts:• There are three purple astronauts. • The number of red and blue astronauts is equal. • There are fi ve astronauts on the shuttle.

• There are fi ve astronauts on the shuttle. • The number of purple astronauts is equal to the

number of blue astronauts.• There are fewer red astronauts than there are

purple astronauts.

• The number of purple astronauts is less than the number of blue astronauts.

• There is one red astronaut.• There are fi ve astronauts on the shuttle.

• There are fi ve astronauts on the shuttle.• There are two blue astronauts. • There are more red astronauts than purple astronauts.

Word Problems for Six Astronauts:• There are two more blue astronauts than purple

astronauts. • There is one more red astronaut than there are

purple astronauts.• There are six astronauts on the shuttle.

• There are six astronauts on the shuttle. • The number of purple astronauts is double the

number of red and blue astronauts together.

• There are even numbers of each color astronaut.• There are two red astronauts.• There are six astronauts on the shuttle.

• There are six astronauts on the shuttle.• There is an odd number of purple astronauts.• There is one blue astronaut.• The number of red astronauts is more than the

number of blue astronauts, but less than the number of purple astronauts.

Word Problems for Seven Astronauts:• The number of red and purple astronauts is equal. • There are three blue astronauts.• There are seven astronauts on the shuttle.

• There are seven astronauts on the shuttle.• There are more blue astronauts than purple astronauts.• The number of red astronauts is equal to the

number of blue astronauts.

• There is a different number of each color of astronauts.• There are more purple astronauts than any other

color. • There is only one blue astronaut.• There are seven astronauts on the shuttle.

Word Problems for Eight Astronauts:• There are eight astronauts on the shuttle.• There are more red astronauts than any other color. • The number of blue astronauts is odd.• There are fewer purple astronauts than blue astronauts.

• There is an even number of blue astronauts.• The number of purple and red astronauts equal.• There are fewer blue astronauts than purple astronauts.• There are eight astronauts on the shuttle.

Each shuttle has three different colors of astronauts. How many of each color are there?


TopicMeasurement

Key QuestionHow can we use the problem-solving strategy of guess and check to compare the mass of a set of objects?

Learning GoalsStudents will:1. predict which items are greater than, less than, and

equal to the mass of a select set of objects; and2. properly use a balance to check their predictions.

Guiding DocumentNCTM Standards 2000*• Use tools to measure• Recognize the attributes of length, volume, weight,

area, and time• Compare and order objects according to these

attributes• Understand how to measure using nonstandard

and standard units• Build new mathematical knowledge through

problem solving• Solve problems that arise in mathematics and in

other contexts• Apply and adapt a variety of appropriate strategies

to solve problems

MathMeasurement massEstimatingProblem solving

Integrated ProcessesObservingComparing and contrastingClassifyingPredictingRecording


MaterialsBalances, one per groupA variety of classroom objectsStudent recording page

Background Information In this activity, students will have an opportunity to practice the correct use of a measuring tool, the balance, while utilizing the problem-solving strategy of guessing and checking. Guessing and checking is a problem-solving strategy that is helpful when a problem has many possible answers or when the problem asks the solver to fi nd one solution but not all possible solutions to a problem. When problem solvers use this strategy, they guess the answer, check to see if it is correct, and make another guess if the fi rst one was incorrect. In this way, they gradu-ally come closer and closer to the correct answer by making increasingly more reasonable guesses. Problem solvers often use this strategy to get started on a problem and then fi nd another strategy that will better suit the problem.

Management1. Divide the students into groups of three or four.

Have them work together to solve the problems, but stress that each student needs to use the balance.

2. During the guessing stage of the activity, help stu-dents see that it is okay if the guesses are not all the same in the group.

3. Demonstrate how to equalize a balance by using the slide. This is a skill that students should use every time they use a balance.

4. Prior to teaching this lesson, gather board erasers, scissors, and markers for each group. Substitutions can be made for these suggested items; however, there should be enough of the selected items so that each group will have identical objects.

5. If this is the students’ fi rst experience with a bal-ance, it will be necessary to discuss what the balance will look like when two objects are bal-anced, when the object in the left pan is heavier, and when the object in the right pan is heavier. One way to see if the students understand how a balance works is to have them hold out their arms and demonstrate what the balanced will look like


when the objects have an equal mass, when the object on the right is heavier, and when the object on the left is heavier.

6. If students are struggling to fi nd a single object to solve each challenge, allow them to use combina-tions of objects.

Procedure1. Distribute one balance, one pair of scissors, one

board eraser, and one marker to each group. Dem-onstrate how to make sure the balance is equalized before placing items in the balance pan.

2. Select an item such as a board eraser and ask the students what will balance the eraser. Allow the students to make predictions. Record the predic-tions on the board.

3. Ask each group to place their eraser in the left pan of the balance and challenge them to fi nd an object(s) that will balance the eraser. Have the students record their results in pictures or words in the first box on the student page.

4. Bring the class back together and discuss what object(s) each group found that had a mass equal to the eraser. Question the students about the strategies they used to solve the problem.

5. Display a pair of scissors. Ask the students what object(s) would have a greater mass than the scissors. Record their predictions on the board. Allow time for each group to test their predic-tions and record their results in the second box on the student page. Discuss their results.

6. Repeat the process using another object such as a marker. Have the students predict, test, and record their results in the third box on the student page. Discuss their fi ndings.

7. Draw the students’ attention to the fourth box on the student page. Ask them what the drawing represents. Challenge them to fi nd two different objects with equal masses. Remind them to test and record their results.

8. Discuss how the guess and check strategy helped them solve their mass challenges.

Connecting Learning 1. How close were your predictions?2. Explain how to use a balance.3. How did you use the guess and check strategy in

this activity?4. What classroom objects did you fi nd that were

heavier than the board eraser? 5. What classroom objects did you fi nd that had

equal masses?6. How could we fi nd the actual mass of each of the

objects that we used?7. Name another time that using the guess and check

strategy would help you.



TopicGeometry

Key QuestionHow can you put together a set of geometric shapes to make different animals?

Learning GoalsStudents will:1. name and recognize the geometric shapes that

make up a set of tangrams, and2. use problem solving to assemble a set of tangrams

in a variety of ways.

Guiding DocumentNCTM Standards 2000*• Recognize, name, build, draw, compare, and sort

two- and three-dimensional shapes• Investigate and predict the results of putting together

and taking apart two- and three-dimensional shapes• Build new mathematical knowledge through

problem solving• Apply and adapt a variety of appropriate strategies

to solve problems

MathGeometry two-dimensional shapesProblem solving

Integrated ProcessesObservingIdentifyingRecording


MaterialsStudent pagesTangram puzzle pieces (see Management 1)

Background Information The tangram puzzle is one that has been around for a very long time, with a multitude of variations. The puzzle consists of fi ve isosceles right triangles (two large, one medium, and two small), a square, and a parallelogram. All of the shapes (except the two large triangles) can be combined with others in some way to make one or more of the larger shapes. For example, the two small triangles can be put together to make the medium-sized triangle, the square, or the parallelogram. The traditional challenge is to use all of the shapes to make a square, but the pieces lend themselves to creating many other fi gures as well. This activity gives young learners the opportu-nity to use their problem-solving skills as they try to put the tangram pieces together to form a variety of shapes. The strategy of guessing and checking is very appropriate here, and is the strategy most often used by adults and children alike when confronted with problems of this nature. There are many different challenges presented, some more diffi cult than others, so that students at every level can be given appropriate problems. There is also the opportunity to build and develop geometric vocabulary and spatial sense as the stu-dents name the shapes they are working with and learn the results of moving them around and putting them in different orientations.

Management1. Each student will need his or her own set of tangram

puzzle pieces. If you have access to an Ellison die-cut machine, you can use the tangram die to make a class set of puzzle pieces. If not, copy the page provided and cut out the puzzles for the students. Whichever method you use, it is recom-mended that you use card stock and/or laminate the pieces for durability and ease of handling.

2. There are a total of seven challenges given, all at different levels of diffi culty. All students should begin with the fi rst page, and only move on to the remaining pages if appropriate.

3. To record their solutions, have students trace around their tangram pieces inside the frame of the shape.


Procedure1. Give each student a set of tangram puzzle pieces.

Have them name and describe each kind of shape in the set. Ask them what they can tell you about each of the shapes.

2. Allow time for some free exploration with the shapes so that students can see how they fi t to-gether and the relationship between the sizes of the different pieces.

3. Distribute the fi rst student page and be sure that everyone understands the challenges. Allow time for students to solve each of the three puzzles.

4. Have students share their solutions with the class. Encourage them to make use of geometric vocabulary as they describe how they put the pieces together.

5. As appropriate, distribute the remaining student pages. For this portion, students may be less frus-trated if they work in groups to solve the puzzles.

6. Close with a time of class discussion and sharing.

Connecting Learning1. What shapes are your puzzle pieces? [triangles,

square, parallelogram]2. What do you notice about all of the triangle pieces?

[Various. They are all the same shape, just different sizes; two little ones make one medium one; etc.]

3. What else do you notice about the puzzle piece shapes? [Various. Two small triangles make a square, the square and two small triangles make a big triangle, etc.]

4. How did you solve the puzzles?5. Was this problem-solving strategy one you would

use again on a similar problem? Why or why not?6. How did you decide which puzzle pieces to use for

each puzzle? Could you ever use different pieces to solve the same puzzle?

7. Which puzzle was the easiest for you to solve? Why?8. Which puzzle was the most diffi cult for you to

solve? Why?

Extensions1. Have students create their own tangram puzzles

and trade them with other classmates to solve. 2. Have students work in groups and use two or

three sets of tangrams to see what shapes can be created.

3. Use the tangram pieces to explore concepts of symmetry, rotation, fl ips, slides, turns, etc.



Copy this page onto cardstock and cut out the puzzle pieces. Each student will need one set.

Tangram Pieces


Make this fi sh using two shapes.

Make the same fi sh using three shapes.Show how you did it.

Now use four shapes. Show how you did it.


Make this sailboat usingfour shapes.

Show how you did it.

Make this crab claw using four shapes.



Make this seagull using six shapes.



Make this fi sh using fi ve shapes.


Make this pelican using all seven shapes.



Make this seahorse using all seven shapes.



Patterns are ever-present in our world. At an early age, children begin to recognize and identify patterns all around them. Looking for patterns is a problem-solving strategy that involves examining relationships between objects, pictures, and numbers. Mathematical problems can often be solved by identifying, extending, and creating patterns.


TopicPatterns

Key QuestionHow can we represent patterns in different ways?

Learning GoalsStudents will:1. recognize, describe, and extend patterns; and2. translate the patterns from one representation to

another.

Guiding DocumentNCTM Standards 2000*• Recognize, describe, and extend patterns such as

sequences of sounds and shapes or simple numeric patterns and translate from one representation to another

• Analyze how both repeating and growing patterns are generated

• Build new mathematical knowledge through problem solving

MathPatternsProblem solving

Integrated ProcessesObservingComparing and contrastingClassifyingPredicting

Problem-Solving StrategiesLook for patternsUse manipulatives

MaterialsPattern cards (see Management 1)Pocket chart or display area100 Unifi x cubes in a variety of colorsMr. Noisy’s Book of Patterns (see Curriculum Correlation)

Background Information Our number system is based on patterns and relationships. Understanding patterns is essential for young children to become problem solvers and abstract thinkers. This activity provides an opportunity for students to explore relationships, find connections, make generalizations, and translate patterns from one representation to another.

Management1. Copy several sets of the pattern cards onto card

stock and laminate for extended use.2. It is expected that the children have had prior

experience identifying and extending patterns.

Procedure1. Read Mr. Noisy’s Book of Patterns to the class. As

each page is read, have the children identify the pattern and discuss how many parts each pat-tern has. For example, When Mr. Noisy laughs, he goes, Hee-Hee-Haw, Hee-Hee-Haw. The children should identify this as a three-part pattern.

2. Display one set of the pattern cards. Draw the students’ attention to the cards. Tell them that like Mr. Noisy, they are going to make patterns using body movement. Invite several students to demon-strate the movement represented on each card.

3. Create an AB pattern in the pocket chart using the pattern cards. For example, snap, clap, snap, clap. Invite a student to lead the class in the pat-tern. Discuss what would come next if they ex-tend the pattern. Allow a student to extend the pattern one complete cycle by placing additional cards in the pocket chart and then lead the class in the extended pattern.

4. When the class has identifi ed and extended the pat-tern correctly, encourage the students to translate the patterns into letters, Unifi x cube trains of color, etc. Discuss what the AB pattern might look like on the playground. [swing, space, swing, space, etc.]

5. Repeat procedures 3 and 4 using the three- and four-part pattern cards. Discuss which type of pattern they see most often in the real world.

Connecting Learning1. What is a pattern?2. Where do we see patterns?3. What shape comes next in this pattern? Circle,

square, circle, _______4. What letter comes next in the pattern ABBABB?5. What might an AB pattern look like if you used

sounds instead of letters? [snap, clap, snap, clap]

ExtensionUse a digital camera to take photos of the students demonstrating different movements and have the class create patterns using their pictures.

Curriculum CorrelationWilliams, Rozanne Lanczak. Mr. Noisy’s Book of Pat-terns. Creative Teaching Press, Cypress, CA. 1996.



Flap Arms Tap Tummy

Clap Hands Tap Head

Slap Knees Snap Fingers


Tap

He

ad

Sla

p K

nee

sC

lap

Ha

nds

Tap

Tum

my

Cla

p H

and

sSn

ap

Fin

ge

rsFl

ap

Arm

s


TopicPatterns

Key QuestionWhich buttons are needed to complete the patterns?

Learning GoalsStudents will:1. recognize, describe, and extend patterns using

buttons; and2. translate the patterns from one representation to

another.

Guiding DocumentsProject 2061 Benchmark• Patterns can be made by putting different shapes

together or taking them apart.

NCTM Standards 2000*• Recognize, describe, and extend patterns such as



• Sort, classify, and order objects by size, number, and other properties


MathPatternsProblem solving

Integrated ProcessesObservingComparing and contrastingClassifyingPredicting


MaterialsButtons (see Management 2)Pattern cardsTransparencies of button mats (see Management 5)Button mats

Background Information Looking for patterns is a problem-solving strat-egy that involves examining relationships between objects, pictures, and numbers. We can often solve mathematical problems by looking for a pattern and then continuing that pattern. In this activity, students will fi rst need to identify the pattern and then extend the pattern in order to determine what buttons are hidden in the pocket.

Management 1. It is expected that the children have had prior

experience identifying and extending patterns. 2. This activity was written to be used with specifi c

buttons that can be purchased from The Oriental Trading Company (1-800-627-2829, http://www.oriental.com). If these buttons are not available, copy the page provided and cut out the buttons for students to use.

3. Copy several sets of the pattern cards onto card stock and laminate for extended use.

4. When your students are comfortable with identi-fying and extending shape and design patterns, ask them questions that focus on number pat-terns. For example, “What type of button would appear in the sixth position? …ninth position? …12th position?”

5. Copy the button mats onto transparencies for large group work. Several button mats can be prepared and placed at a center for additional practice. Use the blank button mat to create additional patterns.

Procedure 1. Explain to the students that they are going to be

using buttons to make patterns. 2. Display one of each of the buttons on the over-

head. Draw the students’ attention to the buttons. Invite several students to identify adjectives that could be used to describe the buttons.

3. Create an AB pattern using buttons on the over-head. For example, star, circle, star, circle. Invite a student to identify the pattern. Discuss what would come next if the pattern were extended. Allow a student to extend the pattern one complete cycle by placing additional buttons onto the overhead.

4. When the students are confi dent in their ability to correctly extend the button patterns, encourage them to translate the button patterns into letters.


5. Invite several students to create patterns using the buttons on the overhead. Encourage the class to identify the patterns.

6. When the students have demonstrated their ability to recognize, extend, and create patterns, encourage them to help you solve some prob-lems involving buttons and patterns.

7. Place one of the button mats onto the overhead. Explain to the students that the person pictured is wearing a necklace made of buttons. Discuss the pattern that is exposed. Invite a student to place the correct buttons onto the pictures of but-tons. Question the class about the importance of matching the number of holes, size, and shape of the buttons. (Students should recognize that color does not matter.)

8. Draw the students’ attention to the pattern that extends beyond the pocket. Tell the students that you would like for them to look at the pattern and predict what button (or buttons) are in the pocket. Ask for a volunteer to help you place the missing button(s) into the pocket. Remind them to position the buttons in the correct order.

9. Ask the students what strategy they used to fi gure out which buttons were missing.

10. Repeat Procedures 8 and 9 using the additional button mats.

Connecting Learning 1. What color cube comes next in this pattern? Red,

blue, yellow, red, blue, _______ 2. If the pattern is small button, large button, small

button, large button, what size button would be in the sixth position of the pattern? How do you know?

3. What might an AB pattern look like on the play-ground? [bar, space, bar, space, or swing, space, swing, space, etc.]

4. What type of button is missing from this pattern: star, star, circle, square, star, star, _____, square?

5. How would you describe the following pattern using letters? Two-holes, two-holes, four-holes, two-holes, two-holes, four-holes.

6. Describe a pattern that you could make using buttons.



The

pa

ttern

is…

The

pa

ttern

is…

The

pa

ttern

is…

The

pa

ttern

is…


Butto

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at


TopicPatterns

Key QuestionWhat patterns can be made with party hats?

Learning GoalStudents will identify, extend, and create growing and repeating patterns using party hats.



NCTM Standards 2000*• Sort, classify, and order objects by size, number,

and other properties• Recognize, describe, and extend patterns such as




MathNumber senseOrdinal numbersCountingProblem solvingPatterns

Integrated ProcessesObservingPredictingClassifyingComparing and contrastingRelating


MaterialsParty hats (see Management 1)

Background Information Identifying, extending, exploring, and creating patterns are basic processes of mathematical think-ing. They involve examining relationships between objects, pictures, numbers, etc., and are often viewed as a form of problem solving. Patterning experiences provide students with opportunities to develop their logical reasoning abilities and vocabulary (if, then; all, some; etc.). Patterns are the basis of our number system; there-fore, it is important for young learners to experience patterns in concrete, representational, and abstract formats. This activity will provide an opportunity for students to identify, extend, and create repeating and growing patterns in an exciting real-world setting.

Management 1. Gather enough party hats for each of your stu-

dents plus a few extras. The assortment of party hats should include a variety of designs and col-ors. There should be several of each type of hat so that it is possible to create and extend a variety of patterns several times.

2. When your students are comfortable with iden-tifying and extending color and design patterns, ask them questions that focus on number pat-terns. For example, “What type of hat would appear in the sixth position? …ninth position? …12th position?”

ProcedurePart One—Repeating Patterns 1. Give each student a party hat. 2. Using four party hats, create a two-part pattern

on the fl oor or chalk tray. 3. Ask the students if they can identify the pattern.

They may say red, blue, red, blue, etc. Assist them in translating the pattern into letters, ABAB, etc.

4. Have the students look at their own party hats and ask, “Who has a hat that would come next in the pattern?” Allow a student with the correct hat type to place his/her hat in the fi fth position. Encourage the class to extend the pattern as far as possible with the selection of hats available.

5. Ask the class what the AB pattern might look like if they used balloons instead of hats.

6. Question the students about what the AB pattern might look like if they used movements instead of hats.


9. Have the class compare the two parts of the pat-tern and identify how the pattern is growing. [It is growing by one vertically and one horizontally.]

10. Ask the students what the third piece of the growing pattern would look like and why.

11. Discuss similarities and differences between repeating and growing patterns.

Connecting Learning 1. How did you decide what came next in each

pattern? 2. How might the fi rst pattern look using body

movements? 3. If the pattern is blue hat, red hat, blue hat, red

hat, what color hat would be in the sixth position of the pattern? How do you know?

4. What is the difference between a growing pattern and a repeating (AB) pattern?

Extensions 1. Repeat the process using birthday candles. 2. Using a digital camera, make a playground pat-

tern book. Take pictures of things like the monkey bars and ask the children to identify the patterns that they see.

Curriculum CorrelationWilliams, Rozanne, Lanczak. Mr. Noisy’s Book of Pat-terns. Creative Teaching Press. Huntington Beach, CA. 1996.


7. Create other patterns using the hats on the chalk tray, such as an ABB pattern. Encourage the stu-dents to identify the patterns and extend them.

8. Ask the students to look around the room at the variety of hats. Invite a student to create a pat-tern. Have him/her call specifi c students to the front of the room and arrange their hats to form a pattern. Ask the rest of the class to identify and extend the pattern chosen.

Part Two—Growing Patterns 1. Distribute the party hats used in Part One of this

activity. 2. Review repeating patterns with the class. 3. Using six party hats, create a growing pattern by

placing one hat in the fi rst position, a stack of two in the second position, and a stack of three in the third position. Ask the students what they notice about the pattern. Discuss the fact that the color and/or picture are not important in this pattern. Guide their explanations to in-clude the fact that the stack is growing each time.

4. Invite a student to fi ll in the fourth position of the pattern. Discuss how each position is grow-ing by one hat in this particular pattern.

5. Have the students extend the growing pattern as far as possible with the number of hats available.

6. Invite one student to stand at the front of the room with two hats on his/her head. Have an-other student stand to the right of the fi rst student with four hats on his/her head. Continue with a third student that has six hats on his/her head. Encourage the class to identify and extend the pattern. (At some point the students may need to remove the hats and place them in a line on the fl oor or chalk tray.)

7. Gather the class into a large open area. 8. On the fl oor, place a row of three hats. Above the

fi rst hat, place a fourth hat. Tell the class that this is the fi rst piece of a new growing pattern. Leave a bit of space on the fl oor and create the second piece of the growing pattern by placing four hats in a row and two hats above the row.

Part One Part Two


TopicPatterns

Key QuestionWhat is a growing pattern?

Learning GoalStudents will identify, extend, and create growing patterns.



NCTM Standards 2000*• Recognize, describe, and extend patterns such

as sequences of sounds and shapes or simple numeric patterns and translate from one repre-sentation to another



MathNumber senseCountingPatternsProblem solving

Integrated ProcessesObservingPredictingClassifyingComparing and contrastingRelating


MaterialsGrowing pattern strips (see Management 1)Area tiles (see Management 1)Overhead transparencies

Background Information Patterns are a way for young children to recog-nize, order, and organize their world. They are also an important aspect of mathematics. Although most patterning experiences for young children focus on repeating patterns, students can also begin to talk about growing patterns in the early grades. This activity encourages students to analyze growing patterns and make predictions about what will come next based on their analyses. The pattern mats are designed to lead students into the exploration of number patterns. They encourage students to build designs that grow geometrically and numerically in an orderly, predict-able sequence.

Management 1. Each child will need a set of 10 area tiles or col-

ored squares and copies of the pattern strips. These can be copied on card stock and laminated for extended use. Area tiles are available from AIMS.

2. It is recommended that students be introduced to growing patterns prior to doing this activity.

3. Copy the fi rst page of pattern strips onto an over-head transparency for demonstration purposes.

4. Pattern strips fi ve and six may be harder for your students than the others. On strips one through four, the numeric equivalents of the patterns are either consecutive odd, even, or counting numbers. This is not the case for patterns fi ve and six. In these patterns, you have to look at the differences between consecutive numbers of tiles to fi nd the recognizable patterns. [Pattern fi ve uses the square numbers. The differences between the numbers of tiles are the consecutive odd numbers—3, 5, 7, 9, etc. Pattern six uses the triangular numbers (minus one, which is the fi rst triangular number). The differences between the numbers of tiles are the consecutive counting numbers starting with three—3, 4, 5, 6, 7, etc.]

Procedure 1. Distribute the sets of area tiles. 2. Tell the students that they are going to look for

patterns. 3. Using area tiles and a transparency of the fi rst

pattern strip on the overhead projector, create the growing pattern by placing one tile in the fi rst position on the left, a column of two in the second


position, and a column of three in the third posi-tion. Ask the students what they notice about the pattern. Discuss the fact that the colors are not important in this pattern. Guide their explanations to include the fact that the column is growing each time.

4. Invite a student to fi ll in the fourth position of the pattern. Discuss how each position is growing up by one tile in this particular pattern.

5. Have the students extend the growing pattern as far as possible with a handful of additional area tiles.

6. Place the second pattern strip onto the overhead and invite a student to place the correct number of area tiles in the fi rst and second positions. Have the class compare the two parts of the pattern and identify how the pattern is growing. [It is growing by two vertically.]

7. Have the class predict what should go in the third position. Ask for a volunteer to place the correct number of tiles into the appropriate arrangement in the third position. Discuss how the pattern grew.

8. Give each child a copy of the third pattern strip. Ask students to look at the fi rst two pictures on the strip and use their area tiles to build what would come next. Walk around the room and observe them as they build. When everyone has complet-ed the pattern, discuss how the pattern grew and what the fourth position would look like. Repeat this process with the additional pattern strips.

9. As the students begin to gain confi dence in their abilities to identify and extend growing patterns, transition them from objects such as the area tiles to numbers by building the pattern with the area tiles and writing the number pattern below each position.

10. Revisit each of the pattern strips and have the students write the numbers below the pictures and predict the number of tiles that would be used in the next position rather than building it. Eventually, transition the students to growing patterns that have only numbers, such as 2, 4, 6, ____, etc.

Connecting Learning 1. What is a growing pattern? 2. How did you decide what came next in the pattern? 3. If the pattern is two tiles, three tiles, four tiles,

what would come next? 4. If the pattern starts with two tiles and grows by

two each time, how many tiles will be used in the fourth position? How do you know?

5. What is the difference between a growing pattern and a repeating pattern?

6. What is another way that we could show a grow-ing pattern?


1, 2, 3, ____


1 2

Wh

at’

s N

ext?

Wh

at’

s N

ext?


3 4

Wh

at’

s N

ext?

Wh

at’

s N

ext?


5 6

Wh

at’

s N

ext?

Wh

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s N

ext?

It is often helpful for young children to manipulate objects or act out a problem to arrive at a solution. Using concrete objects to represent the parts of the problem allows them to have all the necessary information in front of them at once. By taking an active role in fi nding the solution, students are also more likely to remember the process they used and be able to use it again for solving similar problems. This strategy is useful when students need to picture the parts of a problem.



Background Information Sometimes one of the most useful problem-solving strategies is to simply use manipulatives to represent the parts of a problem. This will help you organize the information so that you can see the results of a variety of actions. For young children to truly understand the effects of adding and subtracting whole numbers, it is important for them to act out a problem. By moving objects around while they are trying to solve a problem, they develop visual images of both the data in the problem and the solution process. By taking an active role in fi nding the solution, students are more likely to remember the process they used and be able to use it again for solving similar problems. In this activity, the children will be using goldfi sh crackers as the manipulative to represent stories read to them by the teacher.

Management1. The storyboard can be enlarged, colored, and

laminated for extended use. Enlarged, they are ideal to use in a fl annel board format in front of the class. They can also be copied onto over head trans par en cies and displayed using an overhead projector.

2. It is suggested that you use colorful goldfi sh crackers as the manipulative. A page of fi sh has been included for you to use as an alternative to the goldfi sh crackers.

3. A second option for a manipulative is shell pasta. Macaroni shells can be purchased in three sizes at most local grocery stories. The pasta can be dyed by placing it in a container with a small amount of rubbing alcohol and a few drops of food coloring. Stir the shells until they reach the color intensity desired, then place them on newspaper to dry.

Procedure1. Give each child a storyboard and 15-20 goldfi sh

in a variety of colors, or invite the students to come to the front of the class where you have an enlarged ver sion.

2. Read one of the following problems aloud to your students. Then read it again while they act it out with the goldfi sh either on their boards or in front of the class with the enlarged version. Some students may need to listen to the story several times while they solve the prob lem.

TopicWhole number operations

Key QuestionHow can we use a storyboard to act out math e mat i cal problems?

Learning GoalsStudents will:1. use their own words to describe experiences in a

math e mat i cal setting, and2. use manipulatives to represent number stories.

Guiding DocumentNCTM Standards 2000*• Develop a sense of whole numbers and represent

and use them in fl exible ways, including re lat ing, composing, and decomposing numbers

• Count with understanding and recognize “how many” in sets of objects

• Understand the effects of adding and sub tract ing whole numbers


• Solve problems that arise in mathematics and in other contexts

MathCountingWhole number operations addition subtractionProblem solving

Integrated ProcessesObservingComparing and contrastingRelating

MaterialsStoryboard (see Management 1)Goldfi sh crackers (see Management 2)

Problem-Solving StrategiesUse manipulativesAct out the problem


3. Tell the students to clear their boards after they fi nish each question.

4. Repeat this process with as many problems as desired.

Kindergarten Problems• How many goldfi sh do you have? Put fi ve goldfi sh

in a bucket. (Creation/Rec og ni tion of a Set)• Put some of your goldfi sh in the water and leave

some on the sand. Are there more on the sand or in the water? Tell us about your picture. (Comparison of Sets)

• If you put six goldfi sh in the water and four in the small bucket, will there be more goldfi sh in the water or in the bucket? How do you know? (Comparison of Sets)

• Five goldfi sh are on the sand, but fewer are in the water. How many are on the sand and how many in the water? (Comparison of Sets)

• Juan had two goldfi sh on the sand and three in his large bucket. How many goldfi sh did he have al to geth er? (Addition, Combining)

• Four purple goldfi sh are in Amy’s buck et. Amy gives two away. How many goldfi sh are left in the bucket? (Subtraction, Take-Away Model)

• Chhay had fi ve goldfi sh. Two were in the medium sized bucket. The rest were in the small bucket. How many were in the small bucket? (Missing Addend)

• Charlie placed four green goldfi sh at the bottom of the ocean, two purple goldfi sh in the waves, and three red goldfi sh in a bucket. How many goldfi sh did he have al to geth er? (Addition, Combining)

First Grade ProblemsYou may use all of the kindergarten questions and may increase the diffi culty by adding more goldfi sh and using the following set of in struc tions and questions:• Makenzie has eight goldfi sh in her pail and

Shaketa gives her three more. How many fi sh does Makenzie have? Tell us about your picture.

• Three goldfi sh are in the water, two more are on the sand, and one is in a bucket. How many goldfi sh are there al to geth er? (Addition, Combining)

• Chase has fi ve orange goldfi sh in the small bucket. Rita has seven yellow goldfi sh in the large bucket. Are there more goldfi sh in Chase’s bucket or in Rita’s bucket? How many more? (Subtraction, Comparison Model)

• There were four goldfi sh in the water and fi ve goldfi sh on the beach. A very large wave came on shore and washed three of the goldfi sh off the beach and into the water. How many goldfi sh were left on the shore? How many are now in the water? (Subtraction, Take-away Model)

Connecting Learning1. Did the storyboards help you solve problems?

Explain.2. How did you know when to add or subtract the

goldfi sh?3. How did you solve Chhay’s problem with fi ve

goldfi sh where two were in the me di um sized bucket and the rest were in the small bucket? Explain.

4. How did you solve the problem with three goldfi sh in the water and two more goldfi sh on the sand than in the water? Explain.

ExtensionHave your students create their own storyboards and math questions to go with them.

Curriculum CorrelationButler, Christina M. Too Many Eggs. David R. Godine Publisher. Boston. 1988.

Crews, Donald. Ten Black Dots. Greenwillow Books. New York. 1986.

Sis, Peter. Waving. Greenwillow Books. New York. 1988.

* Reprinted with permission from Principles and Standards for School Mathematics, 2000 by the Na tion al Council of Teachers of Mathematics. All rights reserved.


Exploring Operations

Through Storyboards

Creation/Recognition of a Set Given a number, the child will build a set to represent the num ber, or given a group of objects, the child will count the members of the set.

Comparison of Sets The concepts of more, less, and the same as are basic re la tion ships that are es sen tial to develop mean-ing in operations. The child should con struct sets to show these re la tion ships using manipulatives, as well as make comparisons or choices be tween two given sets.

Composition/Decomposition of Sets (Part-Part-Whole) Children either build a designated quantity in two or more parts, or they begin with a designated amount and separate it into two or more parts.

Addition of Sets: Combining (Part-Part-Whole) To learn what joining a set of two (a part) and a set of three (a part) means, the child must manipulate two objects and three objects and combine to make a set of fi ve ob jects (the whole).

Addition: Comparison Model If the smaller of two sets and the difference between them are known, then ad di tion tells how many are in the larger set. A real-world example would be Sally saying to Patty, “I have three more pennies than you have,” and Patty knows that she is hold ing two pennies. Addition tells how many Sally has.

Subtraction: Take-away Model (Whole-Part-Part) The child creates a set of a designated amount (the whole) and is asked to take away an amount (a part) and identify what is left (a part).

Subtraction: Comparison Model The child creates two sets, compares them, and then identifi es the dif fer ence between them.

Missing Addend (Whole-Part-Part) The child creates a set (the whole) and must cover up or hide some of the set (a part). The child isencouraged to “think ad di tion,” or to answer the ques tion: “What goes with the re main ing pieces (a part) to make the whole amount?”


Problem-Solving StrategyUse manipulatives

MaterialsArea tiles or pennies, 55 per student transparent tapeScissors self-adhesive dots, 55 per studentStudent pages

Background Information Fiddlers are an easy-to-make math manipulative that help students develop number sense. A Fiddler is a sequence of one or more square or round tiles taped together. Fiddlers lend themselves to explor-ing the numbers they represent. The fl exible unit construction makes it easy to model whether a given Fiddler is odd or even, what its divisors are, and the family of addition facts that sum to the number on the Fiddler. In this activity, students use Fiddlers to count, defi ne and explore odd and even numbers, and to practice addition and subtraction facts through 10 in a story problem context. Webster’s dictionary defi nes fi ddle as “to move the hands or fi ngers restlessly.” When a Fiddler is held in the hand, it’s hard not to fi ddle with it and, in the process, learn many “facts” about the number the Fiddler models.

Management1. Make a set of Fiddlers (see Making A Set of Fiddlers)

to familiarize yourself with the process. With your set of Fiddlers in hand, review the content of the student pages and Background Information.

2. Each student should have a set of Fiddlers. Making your own set of Fiddlers will help you decide whether you want students to make their own set or whether you need to enlist volunteer help.

3. Students should have had some prior manipulative experiences with even and odd numbers such as pairing the plastic counters in a set to fi nd out if the number of counters in the set is odd or even. If each counter has a partner, the number of counters in the set is even. If a single counter is left unpaired, the number of counters in the set is odd.

4. If appropriate for your students, you can use the last two student pages for them to record their responses. Otherwise, read the questions out loud and have the class respond verbally.

12"

34"

TopicNumber sense

Key QuestionWhat can you learn about numbers by playing with Fiddlers?

Learning GoalsStudents will:1. use a set of manipulatives to model the counting

numbers one through 10, 2. use the manipulatives to explore the concepts of

odd and even numbers,3. use the manipulatives to review and practice

addition and subtraction number facts, and4. use the manipulatives to solve story problems.

Guiding DocumentsProject 2061 Benchmarks• Numbers can be used to count any collection of

things.• Numbers and shapes can be used to tell about

things.

NCTM Standards 2000*• Develop a sense of whole numbers and represent

and use them in fl exible ways, including relating, composing, and decomposing numbers

• Connect number words and numerals to the quantities they represent, using various physical models and representations

• Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections



MathNumber sense and numerationCountingWhole number operations addition subtractionProblem solving

Integrated ProcessesObservingComparing and contrastingAnalyzing Generalizing


Procedure1. Distribute a set of Fiddlers to each student (or the

materials necessary to make a set of Fiddlers). Pair the students so that they can help each other.

2. Give each student a copy of the fi rst student page.3. Show students how the 2-Fiddler folds together

so that each tile has a partner. Explain that the Fiddlers that can be folded so that every tile has a partner are called even Fiddlers.

4. Show students that no matter how one tries to fold the 3-Fiddler, there is always one tile left without a partner. Explain that Fiddlers that always have one tile left over when folded are called odd Fiddlers.

5. Tell the students to sort the remaining Fiddlers into sets of even and odd Fiddlers. Have them record their fi ndings in the spaces provided.

6. Distribute the fi nal two student pages, if appropriate. Help the students use their Fiddlers to work through the story problems.

Connecting Learning1. Which Fiddlers have an even number of tiles? [2,

4, 6, 8, 10] How do you know? [Every tile has a partner.]

2. Which Fiddlers have an odd number of tiles? [1, 3, 5, 7, 9] How do you know? [One tile was always unpaired.]

3. Did you notice any patterns in which Fiddlers are odd and which are even? [Odd and even numbers alternate.]

4. How did you solve the story problems? Did it help to have the Fiddlers when you were trying to fi nd the answers?



Fiddler 1

Fiddler 2

Fiddler 3

Fiddler 4

Fiddler 5

Fiddler 6

Fiddler 7

Fiddler 8

Fiddler 9

Fiddler 10

Set of Fiddlers

Making a Set of FiddlersMaterials• Plastic AIMS Area Tiles or pennies • Scissors• transparent tape • Ruler• self-adhesive dots

Procedure1. Tape the ends of a ruler to a fl at surface.

2. Align the tiles or pennies against the edge of the ruler.

3. Cover the tiles with a single strip of transparent tape.

4. Trim away any excess tape with a pair of scissors.

5. Turn the tiles over and put a self-adhesive dot in the center of each one. Write the number of tiles in the fi ddler on one of the end dots.

6. Repeat this process to make a complete set of Fiddlers.

34"

12"


Odd and Even Fiddlers

2 Even3 Odd OR

Sort your set of Fiddlers into even and odd Fiddlers.

Even2

Odd3


Sandra has 4 tiles. Juan has 7 tiles. How many more tiles does Juan have than Sandra?

Nick has 10 tiles. Chameka has 5 tiles. How many fewer tiles does Chameka have

than Nick?

T.J. has 6 tiles. Elise has 4 tiles. How many tiles do they have altogether?

Andy has 5 tiles. Ming has 3 tiles. How many tiles are there in all?

1

2

3

4


Diego has 4 tiles. How many more does he need to make 10?

Raylene had 7 tiles. She lost 4 of her tiles. How many tiles does she have now?

Chris has 2 tiles. How many tiles will he have to get to make 8?

5 Chhay has 8 tiles. Does he have an odd or even

number of tiles? How do you know?

Charlie has 10 tiles. If he gives away 3 tiles, how many will he have left?

Isadora has 9 tiles. Is that an odd or even number? How do you know?

6

7

9

8

10


TopicGeometry

Key QuestionHow many different ways can you make the hexagon pattern block shape using smaller pattern blocks?

Learning GoalsStudents will: 1. determine all of the ways to make the hexagon pat-

tern block shape using smaller pattern blocks, and2. make a record of those solutions.



NCTM Standards 2000*• Investigate and predict the results of putting to-

gether and taking apart two- and three-dimensional shapes

• Describe attributes and parts of two- and three-dimensional shapes

• Recognize, name, build, draw, compare, and sort two- and three-dimensional shapes


MathGeometry 2-D shapesProblem solving

Integrated ProcessesObservingCollecting and recording dataComparing and contrasting

Problem-Solving StrategiesUse manipulativesGuess and check

MaterialsPattern blocksStudent pageCrayons or colored pencils, optional

Background Information Pattern blocks are a common manipulative in the primary classroom and can provide valuable opportunities for students to develop spatial skills and explore concepts of shape composition. In this activity, students will look at the hexagon pattern block and use their problem-solving skills to discover each way that this shape can be made using any combina-tion of smaller pattern blocks. This experience will help young learners to develop a deeper understanding of shape composition, and will also provide a setting in which they can practice their problem-solving skills and geometric vocabu-lary. Understanding that larger shapes are composed of combinations of smaller shapes is foundational to later learning about the properties and characteristics of shapes. For example, when young learners know that a hexagon can be composed of six equilateral triangles, it will be easier when they are older for them to understand why the interior angles of a hexagon all measure 120°.

Management 1. Students should work together in small groups of

three or four. Each group will need a handful of pattern blocks with enough of each shape so that every student can work on fi nding solutions at the same time.

2. Even though the square and the tan rhombus pattern block pieces cannot be used to cover the hexagon, they should still be included so that students can recognize this fact.

Procedure 1. Have students get into groups and give each

group a handful of pattern blocks. 2. Begin by going through each of the pattern block

shapes and naming them. Discuss characteris-tics of each shape and how to identify them.

3. Have students fi nd a hexagon pattern block and put it on the desk in front of them. Tell them that they are going to see how many different ways they can cover the hexagon using combinations of smaller pieces.

4. Be sure that students understand that the hexagon must be covered completely, without any holes or


any parts of pieces sticking over the edge. By having them cover the actual hexagon piece instead of placing the pieces into a hexagon outline, you provide students with the oppor-tunity to gain a concrete, visual understanding of how the smaller pieces go together to make a hexagon.

5. After students become comfortable with the pro-cess of covering the hexagon completely, hand out the recording sheet. Challenge each group to record all of the different solutions that they can fi nd in the spaces provided. Students can record their solutions by tracing around the pieces or by using a ruler to draw in the lines. If desired, students can color their solutions to correspond to the colors of the pieces.

6. Discuss how students solved the problem, and what solutions they found.

Connecting Learning 1. How many different ways are there to cover the

hexagon using smaller shapes? [There are seven unique piece combinations that can be used to cover the hexagon, but your students may say that there are more if they consider two different arrangements or orientations of the same pieces to be different solutions.]

2. Which pattern block shapes can you use to cover the hexagon? [the triangle, the blue rhombus, and the trapezoid]

3. Which pattern block shapes cannot be used to cover the hexagon? [the square and the tan rhombus]

4. How many solutions use triangles? [5] 5. How many solutions use the blue rhombuses? [4] 6. How many solutions use trapezoids? [3] 7. How many solutions use one kind of shape? [3] 8. How many solutions use two kinds of shapes? [3] 9. How many solutions use three kinds of shapes? [1]10. Can you fi nd a solution that uses four kinds of

shapes? [No] Why not? [Since the square and the tan rhombus cannot be used to cover a hexagon, there are only three shapes left besides the hexagon itself.]

ExtensionFor those students who would like an extra challenge, you can develop other shapes for them to cover that have more possible solutions. (Try creating shapes that use the square and/or the tan rhombus.) Students can also be challenged to come up with their own shapes and fi nd as many ways to cover them as possible.

Solutions There are seven unique piece combinations that will make a hexagon, as shown below. The solution that uses two rhombuses and two triangles has two different arrangements, which are both given.


Two rhombuses,two triangles

Two rhombuses,two triangles

One trapezoid,three triangles

One rhombus,four triangles

Six triangles

One trapezoid,one rhombus,one triangle

Three rhombuses

Two trapezoids


Record every solution you fi nd by drawing it in one of the hexagons below.


Drawing a picture is an effective problem-solving tool for young children. This strategy allows them to “see” the problem as they represent relevant information, and it gives them a concrete way to justify their responses. However, it is impor-tant to stress to students that the pictures do not need to be elaborate. This strategy is useful when the problem requires all of the necessary information to be represented in order for a solution to be determined.



Key QuestionHow can drawing pictures help us solve mathematical problems?

Learning GoalStudents will draw pictures to solve word problems.



• Understand the effects of adding and subtracting whole numbers



MathWhole number operations addition subtractionProblem solving

Integrated ProcessesObservingComparing and contrastingRelatingRecording

Problem-Solving StrategyDraw out the problem

MaterialsCrayonsScratch paperSet of pocket problems (see Management 1)Pocket chart

Background Information Drawing a picture is an effective problem-solving tool for young children. This strategy allows them to represent all of the necessary information, and gives

them a concrete way to justify their responses. This activity provides the students with the type of prob-lems where they naturally want to draw a picture in order to fi nd an accurate solution.

Management1. Copy one set of the pocket problems onto card

stock and laminate for extended use.2. The pocket problems can be used one at a time

over several days as a “bright beginning” to start math for the day, or used as a complete set for an entire class period devoted to the strategy of making a drawing to solve a problem.

Procedure1. Distribute crayons and scratch paper to each

student.2. Explain to the students that you will be placing

some problems in the pocket chart for them to solve. Remind them that they are allowed to use scratch paper to help them solve the problems if they choose.

3. Place one pocket problem into the pocket chart. Allow time for the class to fi nd a solution.

4. When the students have had ample time to solve the problem, discuss their solutions. Question them about the strategies they used to fi nd their answer. If students say that they drew pictures, suggest that drawing a picture is also how you would solve the problem. Repeat this process each time you present pocket problems.

Connecting Learning1. How did you solve the pocket problems?2. How can drawing pictures help you in math?3. Which problems were the easiest to solve? Why?4. Which problems were the hardest to solve? Why?

ExtensionHave students write their own set of pocket problems and add them to the collection of problems used each day in the pocket chart.



Thre

e

how

ma

ny

?

Two

ho

w m

any

?

Six

how

ma

ny

?


Five

ho

w m

any

sid

es?

Four

ho

w m

any

co

rne

rs?

Two

ho

w m

any

sid

es?


Four

ho

w m

any

?

Two

ho

w m

any

?

Thre

e

how

ma

ny

?


One

a

nd tw

o

how

ma

ny

?

Two

a

nd th

ree

ho

w m

any

?

Four

a

nd tw

o

how

ma

ny

?



Key QuestionHow can you fi gure out what Sam and Sadie saw on their safari?

Learning GoalStudents will draw pictures to help them solve problems.

Guiding DocumentNCTM Standards 2000*• Connect number words and numerals to the

quantities they represent, using various physical models and representations



MathCountingProblem solving

Integrated ProcessesObservingRecording


MaterialsSafari scrapbook, one per studentColored pencils or crayonsTransparencies (see Management 3)

Background Information This activity provides a scenario in which draw-ing a picture provides a simple and effi cient way to solve a problem. Students are given a “scrapbook” of a safari that includes descriptions of the animals seen. However, the descriptions leave out some key information that students are challenged to deter-mine and record. By drawing pictures, the answers to the problems become obvious, and can be easily verifi ed.

Management1. Each student will need his or her own copy of Sam

and Sadie’s Super Safari Scrapbook. To make each book, copy the pages front to back so that they can be folded in half, nested together, and sta-pled along the edge. If you do not wish to copy the pages front to back, you can cut the pages in half, put them in order, and staple along the left edge.

2. You may want to spread this activity over several days because of the number of problems that are given in the book.

3. Make an overhead of pages two and three of the scrapbook to facilitate the explanation process.

Procedure1. Distribute one copy of Sam and Sadie’s Super

Safari Scrapbook to each student. 2. Tell them that they will be helping Sam and Sadie

fi nish their scrapbook. Put up the transparency of page two and read the text together. Make sure that everyone understands the challenge.

3. Put up a transparency of page three and solve the fi rst challenge together as a class.

4. Provide crayons and colored pencils for students. Tell them that they may want to use these to help them answer the questions.

5. Allow time for students to draw pictures and com-plete the scrapbook.

6. Close with a time of class discussion.

Connecting Learning1. How did you answer the questions in the scrapbook?2. Why did drawing pictures help you? 3. What other ways could you have used to fi nd the

answers? 4. Name some other times when it helps to draw a

picture.


Super SafariScrapbook


Sam

an

d S

ad

iew

en

t o

n a

sa

fari.

The

y sa

w lo

tso

f an

ima

ls.

This

is th

eir

scra

pb

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k.

He

lp t

he

mre

me

mb

er w

ha

tth

ey

saw

.

Sam

sa

w 2

che

eta

hs. H

ow

m

any

leg

s d

id S

am

se

e?

Sam

sa

w _

____

leg

s.

112


Sam

saw

4 ea

rs. Ho

w m

an

y e

lep

ha

nts d

id Sa

m se

e?

Sam

saw

_____ ele

ph

an

ts.

Sad

ie sa

w 3 h

ipp

os. H

ow

m

an

y ea

rs did

Sad

ie se

e?

Sad

ie sa

w _____ e

ars.

103


Sad

ie s

aw

6 w

ing

s. H

ow

m

an

y b

irds

did

Sa

die

se

e?

Sad

ie s

aw

___

__ b

irds.

Sam

sa

w 6

fl a

min

go

s. H

ow

m

an

y fe

et

did

Sa

m s

ee?

Sam

sa

w _

____

fee

t.4

9


Sam

saw

5 tails. H

ow

ma

ny

mo

nke

ys did

Sam

see?

Sam

saw

_____ mo

nke

ys. 8

5

Sad

ie sa

w 4 p

aw

s. Ho

w

ma

ny lio

ns d

id Sa

die

see?

Sad

ie sa

w _____ lio

n.


Sad

ie s

aw

12

leg

s. H

ow

m

any

gira

ffe

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ad

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Sad

ie s

aw

___

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Sam

sa

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s. H

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Sam

sa

w _

____

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67



Key QuestionHow can drawing pictures help us solve mathematical problems?

Learning GoalStudents will draw pictures to solve math problems.










MaterialsCrayonsTransparency fi lmStudent pages

Background Information Drawing a picture is an effective problem-solving tool for young children. This strategy allows them to “see” the problem as they represent relevant informa-tion, and it gives them a concrete way to justify their responses. However, it is important to stress to the stu-dents that the pictures do not need to be elaborate.

Management1. Copy one set of the window pages for each student.2. The window pages can be used one at a time over

several days as a “bright beginning” to start math for the day, or be used as a complete book.

3. Copy the fi rst window page onto transparency fi lm.

Procedure1. Distribute crayons and the fi rst window page to

each student.2. Place the transparency page on the overhead with

nothing in the window space. Explain to the students that Kim looked out her window and wrote a number sentence about what she saw. Tell the class that you would like them to draw what they think Kim saw out of her window and solve the problem that Kim wrote. Allow time for the class to fi nd a solution.

3. When the students have had ample time to solve the problem, discuss their solutions. Allow the students to share their solutions. Discuss whether each solution is a reasonable answer or not.

4. Tell the class that they will now get a chance to make an entire book of window pages. Distribute the rest of the pages and provide ample time for students to complete the problems.

5. When students have completed their books, dis-cuss the variety of drawings that provided correct solutions to each problem.

Connecting Learning 1. How did you solve the problems?2. How can drawing pictures help you in math?3. Which problems were the easiest to solve? Why?4. Which problems were the hardest to solve? Why?5. How were the window drawings for each problem

similar and how where they different?

ExtensionHave students write their own window problems and exchange them with their classmates.



Kim

wro

te 2

+ 1

= 3

.W

ha

t d

id K

im s

ee?

Min

g w

rote

3 +

1 =

4.

Wh

at

did

Min

g s

ee?


Ca

rlos

wro

te 3

+ 4

= 7

.W

ha

t d

id C

arlo

s se

e?

Keya

un

na

wro

te 1

+ 1

+ 2

= 4

.W

ha

t d

id K

eya

un

na

se

e?


Ma

ria w

rote

3 +

1 +

2 =

6.

Wh

at

did

Ma

ria s

ee?

Jac

ob

wro

te 2

+ 2

= 4

.W

ha

t d

id J

ac

ob

se

e?


Using logical thinking as a problem-solving strategy involves combin-ing information and using deductive and inductive reasoning to explain

why a solution is valid. In many cases, inferences must be made to fi ll in missing information. Sometimes a grid is used to organize

the clues. Other times manipulatives represent the parts of the problem. This strategy is used when multiple

pieces of information need to be combined to arrive at an answer.


which of 10 bags contains a bear. As they guess, they will receive clues telling them that the bear is either before or after the number they guessed. It is diffi cult for young students to remember all of the information that has been given in this kind of sit-uation, especially if it takes fi ve, six, or seven guesses to fi nd the bear. Unless they have highly-developed logical thinking skills already, they will likely make a series of random guesses until the bear is discovered, not taking into account any information they might have received along the way. If, however, students are able to somehow keep track of the information they have been given, they will be better able to use logic to solve the problem in fewer guesses.

Management 1. Label the fronts of the lunch sacks with the ordinal

numbers 1st through 10th. To reinforce multiple rep-resentations of words, you may also wish to write out “fi rst,” “second,” “third,” etc. on the bags. On one side of each bag, draw an arrow pointing to the left with the word “before” written beneath it. On the opposite side, draw an arrow pointing to the right labeled “after.”

2. If you don’t have Teddy Bear Counters, any other small manipulative will work.

Procedure 1. Put a Teddy Bear Counter in one of the bags.

(Do not let the students see which bag you have chosen.)

TopicLogic

Key QuestionHow can you use clues to help fi nd which bag the bear is hiding in?

Learning GoalStudents will play a game to practice their logical thinking skills.

Guiding DocumentNCTM Standards 2000*• Apply and adapt a variety of appropriate strate-

gies to solve problems• Build new mathematical knowledge through

problem solving• Develop understanding of the relative position and

magnitude of whole numbers and of ordinal and cardinal numbers and their connections

MathLogicOrdinal numbersProblem solving

Integrated ProcessesObservingApplying

Problem-Solving StrategyUse logical thinking

Materials10 paper lunch sacks (see Management 1)1 Teddy Bear Counter

Background Information For an adult, using logical thinking to make good guesses and narrow down options is an easy task. For primary students, however, this skill must be learned. This activity provides an opportunity for young learn-ers to practice logical thinking skills in the context of an engaging game. They will be trying to discover


2. Place the bags in order at the front of the class so that everyone can see them. Tell students that they are going to try and guess which bag con-tains the hidden bear. To reinforce good language skills and practice the ordinal number vocabulary, inform students that you will only accept guesses that are complete sentences, such as, “I think the bear is in the sixth bag.”

3. Allow individual students to take turns guessing which bag contains the bear. After each guess say, “The bear is in a bag [before or after] the [bag selected] bag.”

4. Once the bear has been discovered, place it in a different bag and play the game one or two more times. Do not yet make any comments or sug-gestions about the nature of students’ guesses.

5. After playing several games, tell students that you would like to help them fi nd the bear faster. See if anyone has ideas about how you might do this.

6. Discuss what the before and after information given after each guess tells students. (For exam-ple, if Cheng guesses that the bear is in the third bag, and he is told that the bear is after the third bag, you know the bear cannot be in the fi rst bag or the second bag.)

7. Show students the arrows on the sides of the bags, and explain that you will use these arrows to help them remember the information you give them after each guess.

8. Play the game again, this time turning each bag that students guess to show the arrow pointing in the appropriate direction. (If the bear is in the 6th bag and a student guesses the 2nd bag, turn the 2nd bag sideways to show the after arrow.)

9. After each guess, poll the class to see if every-one agrees that the guess is not redundant (e.g., guessing the 8th bag when you already know that the bear is before the 7th bag). Have stu-dents justify why a guess should or shouldn’t be made. Allow students to change redundant guesses.

10. Once students are comfortable with this new for-mat, let them play the game themselves, hiding the bear and giving the clues after each guess.

Connecting Learning 1. How long did it take the class to fi nd the bear in

the fi rst game? …the second game? 2. Did the clues your teacher gave help you find

the bear? 3. Was it easier to fi nd the bear once you had the arrows

reminding you of the clues? Why or why not? 4. How long did it take you to fi nd the bear when

you had the arrows helping you? 5. Was this faster or slower than before you had the

arrows? Why do you think that is?

ExtensionFor a more diffi cult game, add to the number of bags.



TopicLogic

Key QuestionHow can you use photograph clues to identify a specifi c painting?

Learning GoalStudents will use “photograph” clues to fi nd the correct “painting” from a set of similar paintings.



problem solving


Integrated ProcessesObservingComparing and contrastingApplying

Problem-Solving StrategiesUse logical thinkingUse manipulatives

MaterialsGlue sticksStudent pages

Background Information Logic problems come in all shapes and sizes. This activity challenges students to use deductive reason-ing and their powers of observation to help the owner of an art gallery solve a problem. The context of the problem makes it playful and engaging, and students will enjoy working together to solve the mystery of the mixed-up paintings.

Management1. There are fi ve logic problems presented here. Three

have three clues and four pictures, and two have four clues and six pictures. There are three different sets of clues for each of the six-pictures sets. Select the problems most appropriate for your students.

2. Students need to work together in groups of three for the three-clue problems and in groups of four for the four-clue problems.

3. Cut out the “photographs” and “paintings” for stu-dents ahead of time. Each group of students will need one set of photographs with the same letter and the corresponding paintings.

4. There are two different student pages, one for the three-clue problems, and one for the four-clue prob-lems. Copy the appropriate page for each group.

Procedure1. Set up the problem using the following scenario:

There is an art gallery in town that has paintings by local artists. The paintings have gotten all mixed up. Someone took them out of the frames and add-ed fake paintings (forgeries) that look similar to the originals. The owner of the gallery can’t remem-ber what the original paintings looked like. Luckily, she did take some photographs of the paintings before they were removed from the frames. But, she only took pictures of small parts of each paint-ing at one time. Your job is to help the owner fi gure out which paintings are originals by comparing her photographs to all the paintings. The painting that matches all of the photos will be the original.

2. Distribute a set of photographs with the same letter and the corresponding paintings to each group of students. (Ideally, each group will have a different set of photographs to work with.)

3. Instruct each student within the group to take one of the clue photographs. Tell the students that they must work together within their group to fi nd the painting that matches all of the photographs.

4. Allow time for students to determine the original painting.

5. When groups are fi nished, distribute glue sticks and the appropriate student page to each group. Instruct them to glue the correct painting in the frame and to glue the clue photographs in the spaces below the frame.

6. Have groups share what they discovered. Be sure they go through all the photographs and demon-strate how they match the painting selected.

Connecting Learning1. How did you use the photographs to help you fi nd

the right painting?2. Was it better to look at certain photographs before

others? Why or why not? 3. Would you have been able to fi gure out the right

painting if any of your clues had been missing? Why or why not?

Extensions1. Have students choose their own clues to describe

one of the paintings.2. Allow groups to create their own paintings and

photograph clues.



3-Clue Photographs


3-Clue Paintings-A


3-Clue Paintings-B


3-Clue Paintings-C


4-C

lue P

ho

tog

rap

hs


4-C

lue P

ain

tin

gs-

D,

E,

F


4-C

lue P

ain

tin

gs-

G,

H,

I


TopicLogic

Key Questions1. Which bear lives in which cave?2. Which bear has which job?3. How does each bear get to school?

Learning GoalStudents will use Teddy Bear Counters and logical thinking skills to solve situational problems.

Guiding DocumentNCTM Standards 2000*• Build new mathematical knowledge through



to solve problems• Monitor and refl ect on the process of mathematical

problem solving


Integrated ProcessesObservingComparing and contrastingRecording dataInferringDrawing conclusions


MaterialsTeddy Bear CountersCrayons or colored pencilsStudent pages

Background Information Youngsters in the early grades should have many opportunities to make reasonable or logical conjec-tures about situations with concrete materials. Every child should be involved in hands-on activities that

allow him or her to verbalize and describe or act out the situation described (the problem). Through discus-sion and careful questioning on the part of the teacher, students may see relationships and implications and make observations, generalizations, predictions, and interpretations. Students may use Teddy Bear Counters to assist them in solving the problems. A bear is placed in each and every square in the grid. Removing bears from the grid is a way of recording a “no” response to a given clue. Bears left in position at the end of the problem indicate a “yes” response. Only one bear should remain in each row and column. The questions posed in the Connecting Learning section are general in nature and may be appropriate for all of the logic in this series. It is important that teachers use applicable questions for the specifi cs of each activity. For instance, in the Bear Caves prob-lem, what does the clue that states “Gus does not live in Creaky Cave” tell us? What does that clue tell us about Sam or Allie? These questions are more specifi c to the problem.

Management1. Students may work in pairs or small groups. First-

time exposure to simple logic problems may require a highly structured, teacher-directed experience.

2. It may be helpful to use a transparency of each page on the overhead when discussing questions and possible solutions.

3. If Teddy Bear Counters are not available, use some other small manipulative that will fi t inside the grids.

Procedure1. Distribute one of the activity pages and the ap-

propriate number of Teddy Bear Counters to each student.

2. Have students place one Teddy Bear Counter in each space on the grid. All bears in a single row represent the named bear on the left. For instance, in Bear Caves, the top row of bears represents Bonnie Bear.

3. Read the situation together.4. Have one student in each group re-tell the situation

or problem to his or her group.


5. Discuss each clue with the students and the infor-mation it provides.

6. Invite students to remove bears as appropriate.7. Have them record their conclusions.8. Repeat this procedure with the other scenarios as

desired.

Connecting Learning1. What does a sentence or clue with the word not

tell you? If not tells you that something is not true, how might it also tell you something that is true?

2. What kinds of words signal a yes response?3. What kinds of words signal a no response?4. After discovering a yes response, how does that

affect the remaining choices in the row? …in the column? …in the rest of the grid?

SolutionsBear Caves One Since Carla does not live in Crystal Cave, the Ted-dy Bear under Crystal Cave and across from Carla is removed. This leaves only the Whispering Cavern as an option for Carla. Since Carla and Daniel live next door to each other and not in the same cave, Daniel must live in Crystal Cave.

Bear Caves Two The key clue is the final one, which introduces the concept of between. It places Allie in Creaky Cave. Since Sam does not live in Boulder Bluff (clue 3) and cannot live in Creaky Cave with Allie (clue 1), he must live in Honey Comb Cavern. This leaves Boulder Bluff as the residence of Gus.

Bear Jobs• Sally must be the dentist (clue 3).• Gina is not the librarian (clue 2) and cannot be the

dentist. Therefore, Gina must be the astronaut.• This leaves Carlos as the librarian.

Bears Come to School• Mai must be the bear that rides the bus (clue 2).• Corey rides to school. We know he can’t ride the

bus because Mai rides the bus, so he must ride his bike.

• This leaves Tom as the one who walks.



Clues:1. Carla and Daniel live in caves next door to each other. 2. Carla does not live in Crystal Cave.

Where do the bears live?

Carla lives in ______________________ .

Daniel lives in _____________________ .

Carla

Daniel


Clues:1. Gus, Sam, and Allie all live in different caves.2. Gus does not live in Creaky Cave.3. Sam does not live in Boulder Bluff.4. Allie lives between Gus and Sam.

Where do the bears live?

Gus lives in _________________________.

Sam lives in ________________________.

Allie lives in _________________________.

Gus

Sam

Allie


Clues:1. Carlos is afraid of heights. He does not like to fl y.2. Gina likes to read. She does not work at the library.3. Sally tells people to “brush three times a day.”

Who has each job?

Sally is the ___________________________.

Gina is the __________________________.

Carlos is the _________________________.

Sally

Gina

Carlos


Clues:1. Each bear comes to school a different way. One rides the bus,

one rides a bike, and one walks.2. Mai lives too far from school to walk. She does not ride a bike.3. Tom does not ride a bus.4. Corey and his friend ride to school.How does each bear get to school?

_______________ rides the bus.

_______________ walks.

_______________ rides a bike.

Corey

Mai

Tom


TopicLogic

Key QuestionIn what order are the vegetables planted in the McGregors’ garden?

Learning GoalStudents will use logical thinking skills and positional and ordinal words to describe the McGregors’ garden.

Guiding DocumentsProject 2061 Benchmarks• Numbers and shapes can be used to tell about

things.• Letters and numbers can be used to put things in

a useful order.• Numbers can be used to count things, place them

in order, or name them.

NCTM Standards 2000*• Count with understanding and recognize “how

many” in sets of objects• Understand the effects of adding and subtracting

whole numbers• Describe, name, and interpret relative positions in

space and apply ideas about relative position• Build new mathematical knowledge through



to solve problems• Monitor and refl ect on the process of mathematical

problem solving

MathSpatial relations positional and directional wordsOrdinal numbersProblem solvingLogicWhole number operations addition subtractionCounting

Integrated ProcessesObservingClassifyingComparing and contrastingCommunicating


MaterialsFor each group: 12" x 18" brown construction paper tape set of 24 vegetable pictures Mr. and Mrs. McGregor fi gures (see Management 2) berry basket pennies crayons or colored pencils scissors

Background Information Young children should have multiple opportuni-ties to make reasonable or logical conjectures about situations using concrete materials. This activity incorporates the use of logical thinking skills as well as ordinal and positional words in a real-world setting.

Management1. This activity is divided into two parts. Part One

in volves students in constructing a garden using positional words, directional words, and ordinal numbers. Part Two in cor po rates whole number addition and sub trac tion as the stu dents harvest the vegetables.

2. To construct the McGregor fi gures, copy them onto card stock, cut them out, and tape a penny to the tab on each fi gure. Fold the tab under so that the fi gures will stand.

ProcedurePart One1. Distribute the McGregor fi gures, vegetables,

scissors, and crayons or colored pencils to each group. Have the groups color and cut out the fi g-ures and vegetables. Be sure that the vegetables are colored correctly—lettuce is green, corn is yellow, tomatoes are red, and carrots are orange. Assist them in assembling the fi gures as described in Management 2.


5. Continue reading the recipes and asking ques-tions that deal with the addition and subtraction of the veg e ta bles.

Connecting Learning1. Mr. and Mrs. McGregor begin to walk around their

garden. They want to stop to rest on the right side of the garden. Place the McGregors where they want to pause to rest.

2. Place Mr. McGregor in back of Mrs. McGregor. De scribe Mrs. McGregor’s position. [She’s in front of Mr. McGregor.]

3. If you planted one of each of the four different veg- e ta bles, how many plants would you have? [4] If you planted two of each of the different veg e ta bles, how many plants would you have? [8] …three of each? [12]

4. What other vegetable would you like the McGregors to plant? If you planted six of these in your gar-den, how many plants would you have in all? [30] How many dif fer ent kinds of veg e ta bles would you have? [5]

5. If two rabbits came along and ate all the lettuce and carrots, how many veg e ta bles would they have eat en? [12]

Extensions1. Allow students to ar range the vegetables in their

gar dens fol low ing pat terns such as AABBCCDD, ABCDABCD, etc.

2. Ask students to con struct story prob lems about the har vest ing of the veg e ta bles. If ap pro pri ate, have them write the num ber sen tenc es.


2. Distribute the construction paper and the vege-table pictures to each group. Direct students to place the brown construction paper on the table in front of them. Tell them that this paper will repre-sent the ground that they will plant a garden in. The front of the garden is the side of the paper closest to them, and the back is the side farthest from them.

3. Have them place Mrs. McGregor to the right side of their garden and Mr. McGregor behind the garden.

4. Invite the stu dents to name each of the veg e ta bles the McGregors decided to plant. Tell them that you will select a “seed pack et” with di rec tions that the McGregors will use for plant ing the four rows of their garden. Di rect them to place their vegetables in their gar dens ac cord ing to the di rec tions. Read the di rec tions for Seed Pack et No. 1.

5. After the stu dents have had time to arrange their veg e ta bles in the garden, tell them that you will read the directions again so that they can check their work.

6. Ask the students where they planted the row of to ma -toes. Discuss the placement of the vegetables using ordinal numbers (fi rst, second, third, and fourth).

7. Ask the students what vegetable is behind the green vegetable… between the orange and green veg e ta bles… in front of the red vegetable, etc.

8. Direct the students to move Mrs. McGregor to stand in front of the garden and Mr. McGregor to stand on the left side of the garden. Select another seed packet and continue the procedure of garden plant ing, emphasizing the use of positional words and or di nal numbers.

9. Allow time for the students to construct and de scribe their gardens to each other using positional words and ordinal numbers.

Part Two1. Tell the students that it is time to harvest their veg-

etables. Give each group a plastic berry basket and explain that they will be using it to harvest the vegetables. Have them position Mr. And Mrs. McGregor next to each other, behind the garden.

2. Allow them time to set up the vegetables in their garden in any order they choose. Ask them how many vegetables of each kind they have [6], and how many vegetables there are altogether [24]. Begin a story line about Mr. and Mrs. McGregor trying out various recipes for veg e ta ble soup.

3. Inform them that they are to pick the vegetables and put them in the basket according to the direc-tions you will read from the recipe cards.

4. Read the directions for Recipe No. 1. Ask students questions such as: How many tomatoes did you be gin with? How many did you harvest? How many are still in the garden? How many vegeta-bles are still in the gar den after you picked all you needed for the recipe?


Tape penny hereand fold tab back

Tape penny hereand fold tab back


The red vegetable is in the front of the garden.

The orange vegetable is behind the red vegetable.

The green vegetable is in the back of the garden.

The yellow vegetable is between the orange and the green vegetables.

The yellow vegetable is planted in the fi rst row at the front of the garden.

The green vegetable is in the third row.

The red vegetable is planted be tween the fi rst and third row.

Plant the orange vegetable where you think it should go.

The two roundish vegetables are found between the other two vegetables.

The corn is in the front of the garden.

The tomatoes are next to the corn.

Plant the lettuce and carrots where you think they go.

The carrots are in a row at the front of the gar den.

The corn is plant ed in a row at the back of the garden.

The row of let tuce is right behind the carrots.

Plant the row of tomatoes where you think they should go.


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When dealing with word problems or similar challenges, it is often benefi cial to simply write down the numbers and operation symbols needed to solve the problem. Writing the number sentence helps students organize the information so that they can see if the numbers and operations used correspond to the information given and if the answer makes sense.


Background Information Using drama to develop number sense in count ing, operations, and problem solving is a child-cen tered application to real-world ex pe ri enc es. As the children act out number stories, they begin to build a number sense vocabulary. Through the telling of these stories, the children can be presented with com par ing/counting, op er a tion al, and problem-solving ex pe ri enc es in a bodily ki nes thet ic way. By recording the numbers and operation symbols needed to solve a problem, students can see if the operation they selected makes sense with the numbers in the problem and the question being asked. The completed number sentence allows them to have an additional, more abstract representation of the solution.

Management1. Using scratch paper, prepare one set of large

num ber cards from zero to 10.2. Prior to this lesson, prepare several slips of paper

with number story scenarios on them. The stu dents can then choose problems to present to the class. See Pro ce dure 5 for examples.

3. If desired, make several copies of the animal headbands on card stock to use as costumes.

Procedure1. Give one student a number card and ask him or her

to create a set of brown-eyed children that is equal to that number. (As a variation, you can give a group of students a set of num ber cards 0-10 and ask them to fi nd the card that represents the set of chil dren in their group.)

2. Invite two boys and three girls to come to the front of the room. Ask the class if there are more girls or boys in the set and how many more. Com pare the


Key QuestionHow can we act out our number problems?

Learning GoalsStudents will:1. create and recognize a set of num bers, 2. compose and decompose sets of numbers, and3. create number sentences that represent the stories

they act out.

Guiding DocumentNCTM Standards 2000*• Develop a sense of whole numbers and rep re sent





• Monitor and refl ect on the process of mathematical problem solving

MathCountingWhole number operations addition subtractionProblem solving


Problem-Solving StrategiesWrite a number sentenceAct out the problem

MaterialsNumber cards 0-10 (see Management 1)Number story scenarios (see Management 2)Animal headbands, optional (see Management 3)


heights of two or three students in the class. Ask the class who is taller? …shorter? …the tallest? …the short est?

3. Choose fi ve students to act out the fi rst num ber story theater production. Ex plain to them that they are going to be birds and that three will be sitting and fl apping their wings on a tree branch, then two more birds will fl y in and join them on the branch.

4. Tell the story as the actors present it in front of the class. After the presentation, ask the students who were not involved in the production to identify the number sen tence that was represented. [3 + 2 = 5] Have a stu dent write the number sen tence on the board as the students re-enact the story.

5. Have small groups choose a slip of paper con tain ing a number story to present to the class. Here are some possible scenarios:

• Four elephants were walking in a line. Two went off in search of food.

• Four little speckled frogs were sitting on a speck led log eating the most delicious bugs. One fell into the pond.

• Three bears were playing in the woods. One bear lay down to take a nap. How many bears were left to play?

6. When the students are comfortable with acting out the stories, ask them to make up their own stories and present them to the class.

Connecting Learning1. How did the number story theater help you write

number sentences?2. When we had a set of birds and another set joined

them, were we adding or subtracting the sets?3. When we had a set of frogs and one jumped into

the pond, were we adding or subtracting the sets?4. If you see a story problem that asks how many are

left, will you be adding or subtracting to fi nd the answer?

5. Make up a story in which you add (subtract) two sets of numbers.

Curriculum CorrelationMurphy, Stuart. Animals on Board. HarperCollins. New York. 1998.

Murphy, Stuart. Elevator Magic. HarperCollins. New York. 1997.

Murphy, Stuart. Monster Musical Chairs. HarperCollins. New York. 2000.

* Reprinted with permission from Principles and Stan dards for School Mathematics, 2000 by the National Council of Teachers of Mathematics. All rights reserved.


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Key QuestionHow can we use a space storyboard and astronauts to act out mathematical problems?

Learning GoalsStudents will:1. use manipulatives to represent number stories, and 2. write number sentences that describe what is

being acted out on their storyboards.







• Apply and adapt a variety of appropriate strategies to solve problems



Problem-Solving StrategiesWrite a number sentenceUse manipulativesAct out the problem

MaterialsStoryboards (see Management 2)Astronauts (see Management 3)Number cards

Background Information Many problems are straightforward and require nothing more than the application of arithmetic rules. However, for young children to truly understand the effects of adding and subtracting whole numbers, it is important for them to act out a problem or to move objects around while they are trying to solve a prob-lem. This allows them to develop visual images of both the data in the problem and the solution pro-cess. By taking an active role in fi nding the solution, students are more likely to remember the process they used and be able to use it again for solving simi-lar problems. In this activity, the children will be using astronauts as the manipulative and will be taking the problem solving to an abstract level by using the set of number cards to write the number sentences that represent the dramatizations.

Management1. Each student will need his or her own copy of

the storyboard and one set of number cards. The number cards should be copied and cut out for students ahead of time.

2. The storyboard can be enlarged, colored, and laminated for extended use. Enlarged, it can be used as a fl annel board. It can also be copied onto trans-parency fi lm and used on the overhead projector.

3. Astronaut counters can be purchased from AIMS in sets of 100 (#1929), 300 (#1930), and 500 (#1931). If astronaut counters are not available, make enough copies of the page of astronauts so that each student can have 10 astronauts. Unifi x cubes or other counters could also be substituted for the astronauts.

4. Remind students to clear their boards after solving each problem.

5. Depending on your students’ reading abilities, you may choose to write some of the questions on the board or on sentences strips. This reinforces their reading and is most like the way that they would see similar problems in a testing situation.

6. It is suggested that students fi rst manipulate the astronauts and number cards until they are com-fortable with the process and that they later record the number sentences in a math journal.

Procedure1. Give each student a space storyboard, a set of

number cards, and 10 astronauts, or bring the stu-dents to the front of the class where you have an enlarged version.


2. Read the following instructions and question aloud to your students. Then repeat them and ask the student to act out the problems using the space storyboards and astronauts. After each problem, discuss what they did on the storyboard and what number sentence they made using the number cards.

• Place two astronauts beside the rocket and three astronauts on the Earth. How many astronauts do you have on your storyboard altogether?

• There were fi ve astronauts on the surface of the moon. One climbed back into the rocket. How many astronauts were left on the moon’s surface?

• There was one astronaut in the round window of the rocket, two astronauts in the square window, and three astronauts in the rectangu-lar window. How many astronauts were there in all?

• There were four astronauts on the Earth and six astronauts on the moon. How many more astronauts were on the moon than on the Earth?

• There were four astronauts on the Earth. Three fl ew to the moon. How many are on the Earth now?

• There are nine astronauts on the moon. Three astronauts are to the left of the rocket. The others are to the right. How many astronauts are to the right of the rocket?

• There were three astronauts fl oating below the Earth. How many would have to join them to make eight?

• There are six astronauts that needed to be evenly spread out among the two bottom rocket windows. How many astronauts should be in each window?

• There was one astronaut in the top, or fi rst window, one astronaut in the middle, or sec-ond window, and three astronauts in the last, or bottom window. How many astronauts were in the rocket altogether?

• There were two astronauts in the window that is between the round window and the rectan-gular window. One fell out of the window onto the surface of the moon. How many were left in the rocket?

• There were six astronauts visiting the moon. Some were in the rocket. Two were on the surface of the moon. How many were in the rocket?

Connecting Learning 1. How did the storyboard help you solve problems?2. How did you know whether the number sentence

should be addition or subtraction?3. Write the number sentence for this problem: There

were two astronauts in the rocket and three astro-nauts outside of the rocket. How many astronauts are there altogether? [2 + 3 = 5]

4. Show me what 1 + 5 = 6 might look like on the storyboard.

ExtensionHave the students create their own storyboards and questions to go with them.



12

34

58

+–

–=

=1

23

+7

69

12

34

58

+–

–=

=1

23

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69


TopicNumber combinations

Key QuestionHow can you determine the number of frogs (or lady-bugs) that are hidden?

Learning GoalStudents will investigate the relationship between the whole and its parts by informally exploring number families.

Guiding DocumentNCTM Standards 2000*• Count with understanding and recognize “how

many” in sets of objects• Understand various meanings of addition and

subtraction of whole numbers and the relation-ship between the two operations


MathCountingAlgebraic thinkingMath patternsProblem solving

Integrated ProcessesObservingComparing and contrasting

Problem-Solving StrategiesWrite a number sentenceUse logical thinkingUse manipulatives

MaterialsFor each group: plastic bowl copy of lily pad or leaf counters (see Management 4) blue construction paper, optional

Background Information This activity gives students an opportunity to work with number combinations to 10. As children repeat the activity, they will learn the number combinations within number families. The amount of time each child will work with each number varies. However, a child should be comfortable with combinations for the numbers 4, 5, and 6 before moving on to 7, 8, and 9. NCTM’s Principles and Standards for School Mathematics, 2000 tells us, “Two central themes of alge-braic thinking are appropriate for young students. The fi rst involves making generaliza-tions and using symbols to represent mathematical ideas, and the second is representing and solving problems” (page 93). The hidden numbers in this experience can be determined by adding (part + missing part = whole) or by subtract-ing (whole – part = missing part). For young learners, determining the missing part is the goal. For older learners, a record of the number sentences should be made so that students can analyze the combinations in number families and form generalizations which later address the commutative property of addition. Frogs and ladybugs have been provided as a con-text for this exploration. Other suggestions can be found in the Extension section of this activity.

Management 1. Prior to teaching this lesson, copy, color, and

laminate one of the game sets for each pair of students. Turn the plastic bowl over and glue or tape the lily pad (leaf) to its bottom.

2. If desired, create a “pond of water” by rounding the edges of one piece of blue construction paper for each group of students. The bowl with the lily pad can be placed on top of the pond.

3. For very young learners, or if students have dif-fi culty with fi ve counters, start with four or three. Continue until they are successful with these quantities and then progress to larger numbers.


4. Three-dimensional objects such as plastic frogs or insects work well for this activity. If these are not available, use the picture representations included in this lesson.

Procedure 1. Ask a student to count out fi ve frog (ladybug)

counters. 2. Tell him/her to place the lily pad (leaf) over these

counters. 3. Ask the students how many counters are under

the lily pad (leaf). 4. Reach under the lily pad (leaf) and pull out some of

the frogs (ladybugs), leaving some underneath. 5. Place the frog (ladybug) counters that you removed

on top of the lily pad (leaf). 6. Ask the student how many counters are under

the lily pad (leaf). 7. Remove the lily pad (leaf) and ask the student to

check his/her guess. 8. Tell the student to place the counters under the

lily pad (leaf) again. 9. Ask the student how many counters are under

the lily pad (leaf). 10. Repeat steps four through eight allowing the stu-

dents to explore all of the combinations for the family of fi ve. Discuss the combinations that they encounter.

11. Continue until the students are successful work-ing with you, then allow them to play in pairs. Encourage them to use more counters when they are successful with fi ve.

Connecting Learning 1. If you were playing with fi ve frog (ladybug) coun-

ters and there were three on top of the lily pad (leaf), how many would have jumped off the lily pad (leaf)?

2. If you were playing with 10 frogs (ladybugs) and there were six under the lily pad (leaf), how many should you have on top of your lily pad (leaf)?

3. How did you know how many counters were underneath the lily pad (leaf) just by looking at the frogs (ladybugs) on top?

4. How could you solve for a missing number if you were dealing with only numbers and not frogs (ladybugs)?

5. Name a number combination for fi ve, six, etc.

ExtensionOther possible contexts that could be used to reinforce this concept might be: a box representing a garage with toy cars, or a cup representing a spaceship with astronaut counters, or a sock representing a fi sh with gummy worms.



TopicAddition

Key QuestionWhat are all of the ways you can make the numbers from one to six using candies divided in two bags?

Learning GoalsStudents will:1. use manipulatives to model addition problems, and 2. explore the various combinations of numbers that

will produce the same sum.

Guiding DocumentNCTM Standards 2000*• Illustrate general principles and properties of

operations, such as commutativity, using spe-cific numbers

• Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols

• Develop a sense of whole numbers and represent and use them in fl exible ways, including relating, composing, and decomposing numbers

• Gain new mathematical knowledge through problem solving

MathWhole number operations additionProblem solving

Integrated ProcessesObservingRecordingComparing and contrastingApplying

Problem-Solving StrategiesWrite a number sentenceUse manipulativesOrganize the information

MaterialsFor each student: six candies or other small manipulative colored pencils or crayons student pages

For each group: chart paper

Background Information This activity provides an engaging way for students to learn all of the two-number combinations that pro-duce sums from three to six. In the process, they will use a variety of problem-solving strategies, including using manipulatives, organizing the information, and writing a number sentence. Sweet Sums is divided into three sections. In the fi rst section, students will use manipulatives to explore various ways in which objects can be arranged between two spaces to total a particular sum. In the second section, they will make a record of one possibility for each sum, and in the third sec-tion, they will fi nd all possible ways to make each sum and record those solutions abstractly. An important distinction to be aware of while doing this activity is the difference between com-binations and permutations. A combination is a grouping of objects, numbers, etc., without regards to order. In this case, 1 + 2 = 3 would be seen as the same as 2 + 1 = 3. A permutation is a grouping of objects, numbers, etc., in which order matters. In this case, 1 + 2 = 3 would be seen as different from 2 + 1 = 3. In the third section of the activity, be sure that students are aware that they are fi nding all of the possible combinations, not permutations.

Management1. In order to do this activity, each student will need

six candies or other small manipulatives. If you wish to have them make a more concrete record of their solutions by gluing the candies to a piece of paper, you will need more than six per student.

2. As written, the activity asks students to deal with numbers from zero to six. You may wish to increase or decrease this range to make it more appropriate for your students.

Procedure1. Distribute the fi rst student page and six candies to

each student. Ask students to put candies on the pictures of the bags so that there are a total of three candies. (Tell them that bags may be empty.)

2. Have several students share their combinations. Compare the various ways that candies can be placed in the bags to total three (0 + 3, 1 + 2). Repeat this process for several other numbers until students are comfortable with making sums.


3. Once this initial time of exploration is over, tell students that they are now going to record one way to make each number from one to six using candies divided between two bags.

4. Hand out the second and third student pages and colored pencils or crayons.

5. Have students work individually to record one solution to each problem by drawing candies in the bags. If this is too abstract for your students, have them glue candies onto the student sheets or another paper to represent one version of each addition problem.

6. After all students have recorded their solutions, have them get together into small groups of four or fi ve. Instruct students to share their solutions and compare the different solutions discovered within the group.

7. Tell students that their challenge is to discover and record every way that each sum can be reached by putting candies in two bags. Hand out a sheet of chart paper for each group to use as they record their solutions. Each solution should be recorded as a drawing showing the number of candies in each bag, as well as the total. (This activity is intended to be a study of combina-tions. Be sure that students are aware that order does not matter.)

8. Once groups have had suffi cient time to discover all of the solutions, hand out the fi nal student page. Challenge each group to record their solutions in the numeric (abstract) form in the appropriate space on the student page.

9. Come together for a fi nal time of class discussion where groups share their solutions and compare with other groups. If there are differences, discuss why some groups got answers that others did not.

Connecting Learning1. How many ways did you fi nd to make one? …two?

…three? …etc.? 2. How do you know you have found all of the

solutions? 3. How do your solutions compare to other groups? 4. Was it helpful to have the candy to work with when

you were fi nding your solutions? Why or why not?5. Did you organize your solutions in any way? How?

Extensions1. Look for patterns in the number of combinations

possible for each sum. Zero and one both have one combination (0 + 0 = 0, 0 + 1 = 1), two and three both have two combinations (0 + 2 = 2, 1 + 1 = 2; 0 + 3 = 3, 1 + 2 = 3), etc.

2. Use three bags instead of two and look for all of the ways to make the numbers from one to 10.



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Draw candies in the bags to make the number sentences true. (Bags may be empty.)


How many ways can you make six? Write each way.

How many ways can you make fi ve? Write each way.

How many ways can you make four? Write each way.

How many ways can you make three? Write each way.

How many ways can you make two? Write each way.

How many ways can you make one? Write each way.


The problem-solving strategy of working backwards is simply taking a regular problem and doing it in reverse. This strategy is appropriate to use when the goal or fi nal destination is known, but there is a variety of ways to get to that goal or destination. In problems like these, working backwards allows you to determine the best steps to take. It is also appropriate to work backwards when you know the end result or answer and need to determine the starting point. For example, if you want to know how much money you started with at the beginning of the week,

and you know how much money you ended with and all of the transactions you made during the

week, you can work backwards to fi gure out the starting amount.



Key QuestionHow can you go through a maze and end up with the target number of points at the end?

Learning GoalStudents will experience a setting in which working backwards is the best way to solve a problem by going through mazes and collecting the correct number of points along the way.

Guiding DocumentNCTM Standards 2000*• Develop fl uency with basic number combinations

for addition and subtraction• Understand the effects of adding and subtracting

whole numbers • Apply and adapt a variety of appropriate strate-

gies to solve problems• Build new mathematical knowledge through

problem solving

MathProblem solvingWhole number operations addition

Integrated ProcessesObservingComparing and contrastingRecording

Problem-Solving StrategiesWork backwardsGuess and checkUse manipulatives, optional

MaterialsMazesCrayonsArea tiles or math chips, optional

Background Information The strategy of working backwards is one that can be diffi cult for young learners to use. This experi-ence provides a setting in which working backwards is the most advantageous way to solve the problem, although that will probably not be the strategy that children fi rst use to approach the problem. The activ-ity consists of mazes that children must traverse while collecting a certain number of points. In each of the mazes given, there are multiple ways to get from the start to the fi nish. Only one of these ways, however, gives you the correct number of points. In each case, the most effective method for solving the maze is to start at the fi nish and work backwards. This is because the mazes are designed to have only one route leading to the fi nish, while there are multiple routes at the start. If you start from the fi nish, you are guaranteed to cross at least two sets of objects before the path splits, giving you the ability to more easily eliminate incorrect paths.

Management1. This activity needs to be completed by students

individually, although a time of whole-class dis-cussion after all children have been exposed to the mazes may be benefi cial.

2. Have students use a different color of crayon each time they try to solve the maze. This will allow them to see their own attempts and will also give you insight into their thought processes.

3. To cut down on the amount of copying, you may wish to laminate a few copies of the mazes and have students use dry erase markers, transpar-ency pens, or wax pencils.

4. A selection of several different mazes is given. Choose those that are most appropriate to the needs of your students. Each student will need his or her own copy of the mazes.

5. For very young learners, you may wish to have them stack area tiles or math chips on top of the objects in the maze. When they cross over a set of objects, they can collect them. This will allow them to always know how many they have with-out keeping track mentally.


Procedure1. Select the appropriate maze(s) and give each

student a copy.2. Explain that the goal is to go through the maze

and only collect the number of points written at the top of the paper.

3. Give students a selection of crayons and tell them to use a different color each time they go through the maze.

4. Allow students time to complete the maze(s).5. Discuss their thinking and how they solved the

maze(s).

Connecting Learning1. Where did you start when you went through the

maze(s)? Why?2. Do you think it would have been easier if you had

gone the other direction? Why or why not?3. How did you keep track of the number of points

you had?



5


8


10


Integrated ProcessesObservingComparing and contrastingCommunicatingCollecting and recording data

Problem-Solving StrategiesWork backwardsWrite a number sentenceUse manipulativesGuess and check

MaterialsFor each group: one fi lm canister (see Management 2) balance (see Management 4) Teddy Bear Counters in two colors crayons to match the colors of bears 10 teddy bear strips

Background Information Young learners need many hands-on ex pe ri enc es to develop a strong sense of number and num ber operations. Children gain greater insight into num bers and op er a tions on numbers when the numbers have meaning built on real-world ex pe ri enc es. In this activity, students construct a strong sense of addition by counting two sets of objects that will balance (equal) a target number (10). The fact family of 10 is the focus for this experience, but any target number could be utilized. Students will record the numbers at a rep re sen ta tion al level on strips of paper with bears drawn on them (i.e., six bears col ored blue and four bears colored yellow). They will then phys i cal ly connect the two sets to form a combined set. The fi nal step will be to record the numbers and symbols (6 + 4 = 10). Throughout this process, students will make use of multiple problem-solving strategies, including working backwards, guess and check, and writing a number sentence.

Management 1. This activity works well in groups of three or four. 2. Each group will need two copies of the Teddy

Bear Strips page for a total of 10 strips.

TopicFact families of 10

Key QuestionHow can we use a balance to learn about the 10s family?

Learning GoalStudents will identify fact families for the number 10.

Guiding DocumentsProject 2061 Benchmark• Numbers and shapes can be used to tell about

things.

NRC Standard• Employ simple equipment and tools to gather data

and extend the senses.

NCTM Standards 2000*• Count with understanding and recognize “how

many” in sets of objects• Develop a sense of whole numbers and rep re sent


• Understand various meanings of addition and subtraction of whole numbers and the re la tion ship between the two op er a tions


• Develop and use strategies for whole-number computations, with a focus on addition and sub trac tion

• Develop fl uency with basic number com bi na tions for addition and subtraction

• Apply and adapt a variety of appropriate strategies to solve problems


MathNumber sense additionProblem solving


3. Use transparent tape to affi x a number 10 card to each fi lm canister. Pre pare one number “10” canister per group by fi lling the empty fi lm canis-ters with salt so that the canister with the lid will balance 10 Teddy Bear Counters.

4. Place the equal symbol on the center of the balance.

5. Check to see that the students use the bal ance correctly.

6. Using two colors of Teddy Bear Counters will make the comparison of number combinations easier.

Procedure 1. Tell the students that they will be using a balance

to fi nd out as much information as they can about “my friend 10.”

2. Demonstrate or review the correct pro ce dures for using a balance.

3. Give each group 10 recording strips, crayons, a fi lm canister, 20 bears of two different colors, and a balance.

4. Direct each group to place the fi lm canister “10” into the yellow pan of their balance. Instruct them to place one counting bear in the red pan. Draw students attention to the equal sign attached to the balance. Explain that they will be fi nding combinations that equal 10. Ask students how many more bears of another color they think it would take to “balance the equation.” Allow them to add bears until it is balanced. [9 bears]

5. Have them take the bears out of the pan and group them by the color.

6. Direct the students to line up both sets of bears along the recording strip.

7. Have them record the number of bears by color-ing one bear picture on the strip to correspond to each counting bear.

8. Place the bear strip that represents the combina-tion 1 + 9 = 10 on the board and below it record the number sentence 1 + 9 = 10. Explain that they now know that two of “my friend 10’s” fam-ily members are 1 and 9.

9. Instruct the students to fi nd all the different ways to make 10 using two dif fer ent colors of bears.

10. Have each group compare their recording strips. Discuss any patterns the students discover. Once again demonstrate how to write the number sen-tences that each strip represents. Discuss the relationships between the number sentences [1 + 9 = 10, 9 + 1 = 10, etc.] Explain that they have discovered fact families for 10.

Connecting Learning 1. How many different ways can you make 10? 2. How is a balance like an equal sign? 3. How would the problem change if you could use

three colors of bears instead of two? 4. What are some things that come in 10s? 5. How would you solve the problem if the target

number were 15 instead of 10? 6. What can you tell me about the fact family of

2, 8, 10?



10=Copy and cut out 10 strips for each group.



Key QuestionHow can you fi gure out a date by counting backwards on a calendar?

Learning GoalStudents will use a calendar to work backwards and fi nd the dates described by clues.




other contexts

MathProblem solvingNumber and operations addition subtractionPatterns and relationships number patterns

Integrated ProcessesObservingComparingRecordingApplying

Problem-Solving StrategiesWork backwardsLook for patterns

MaterialsCalendar page, one per student

Background Information The problem-solving skill of working backwards is simply taking a regular problem and doing it in reverse. When you know the end result and have to determine the starting point, you are working back-wards. One simple context in which primary students can easily work backwards is with a calendar. In this activity, they are given clues that tell them one date,

and challenge them to work back from that date to fi nd a second date. This provides the opportunity for plenty of practice working backwards, while simultaneously working on students’ addition and subtraction skills. They will also have the opportunity to observe number patterns in the calendar.

Management1. Each student will need a copy of the calendar page.

Before copying the page, determine which month you are going to use and put in the appropriate title, dates, and holidays. A list of the holidays used in this activity and when they occur is given at the end of this teacher’s manual. (The holidays selected are those that occur on a fi xed day or date each year. Those that depend on lunar cycles or other calculations are not included.)

2. For long-term use, laminate a blank copy of the calendar for each student. Before using each time, write in the correct month, holidays, and dates with a dry-erase marker.

3. A few questions are suggested for each month, but you are encouraged to tailor the questions to the students in your class. Include their birthdays, life events, school celebrations, cultural holidays, and other things that will make the content rel-evant to them.

Procedure1. Give each student a copy of the calendar page.

Ask them to identify all of the holidays that are marked to be sure that everyone knows what they are.

2. Read a clue to the class. If appropriate, write the key components of the clue on the board or over-head projector for students to reference.

3. Allow time for each student to arrive at an answer. 4. Have them share their answer with the students sit-

ting around them. If there are differences, have each group arrive at a single answer they all agree upon.

5. Ask the groups to share their solutions with the class. If there are differences, decide as a class which answer is correct.

6. Repeat this process with as many clues as time permits.

7. Once students have become comfortable with the process, discuss the techniques they are using and any patterns they notice in the calendar.


Connecting Learning1. How did you work out the answers to the questions?2. Would you have been able to solve any of the

clues without using a calendar? Why or why not?3. Which kinds of clues did you need a calendar for?4. Which kinds of clues could be answered without a

calendar?5. What number patterns did you notice in your

calendar?

Extensions1. Allow students to make up their own clues for

each month and tell them to their classmates.2. Invent a new calendar for an imaginary planet.

Change the number of days in a week and the length of a month. Include the holidays celebrated and come up with clues for this new calendar.

Holidays UsedJanuaryMartin Luther King Jr. Day—Third Monday

FebruaryGroundhog Day—February 2Valentine’s Day—February 14President’s Day—Third Monday

MarchSt. Patrick’s Day—March 17

AprilApril Fool’s Day—April 1Earth Day—April 22

MayCinco de Mayo—May 5Mother’s Day—Second SundayMemorial Day—Last Monday

JuneFlag Day—June 14Father’s Day—Third Sunday

SeptemberLabor Day—First Monday

OctoberColumbus Day—Second MondayHalloween—October 31

NovemberVeteran’s Day—November 11Thanksgiving—Fourth Thursday

DecemberChristmas—December 25Kwanzaa—December 26-January 1



Clues for JanuaryJamal’s birthday is eight days before Martin Luther King Jr. Day. What day is Jamal’s birthday?

Arisa’s family leaves for vacation on January 2. They come home January 12. Four days before coming home, they will visit the Grand Canyon. What day will that be?

Brittney was born six days after Jose. Brittney’s birthday is January 28. When is Jose’s birthday?

Clues for FebruaryJosh’s birthday is a week before Valentine’s Day. When is his birthday?

Groundhog Day is February 2. How many days is that before President’s Day?

Mrs. Mendez was sick for three days. She was well again on February 19. What day did she get sick?

Clues for MarchSt. Patrick’s Day is four days after Maria’s birthday. What is her birthday?

Mark gets his allowance fi ve days before the end of the month. What day will he get his allowance?

Paul went to his grandma’s house the third Satur-day in March. Three days before that, he had a test in school. What day was the test?

Clues for AprilSamantha was born eight days after April Fool’s Day. Her party was two days before her birthday. When was her party?

The cherry trees started to bloom fi ve days before the second Saturday in April. What day did they start to bloom?

Jared’s family took a 12-day vacation that ended on Earth Day. What day did it begin?

Clues for MayFour days before Cinco de Mayo, Lorena’s dad bought hamburgers to barbecue. What day did he buy the hamburgers?

Jamie got a Mother’s Day present for his Mom six days before Mother’s Day. What day did he get the present?

Chhay’s family went to visit his grandpa nine days before Memorial Day. What day did they visit?

Clues for JuneEmma’s aunt bought a new fl ag 13 days before Flag Day. What day did she buy the fl ag?

Amajou’s birthday is three days before the end of the month. What day is his birthday?

The last day of school is one day before the sec-ond Friday in June. What day is the last day of school?

Clues for SeptemberTodd’s birthday is September 15. Wendy’s birthday is 9 days before Todd’s. When is Wendy’s birthday?

School starts the fi rst Wednesday after Labor Day. What is the date the day before school starts?

One week before the last Friday in September, Mr. Mitchell had a substitute teacher. What day was there a substitute?

Clues for OctoberHow many days before Columbus Day is October 5?

Veronica got her Halloween costume 15 days be-fore Halloween. When did she get her costume?

Alyssa’s birthday is 10 days before Halloween. What day was she born?

Clues for DecemberJoel’s big brother comes home from college 12 days before Christmas. What day does he come home?

Vacation starts fi ve days before Christmas. What day does it start?

How many days before the end of the year does Kwanzaa begin?

Clues for NovemberKeesha got a puppy fi ve days before Veteran’s Day. What day did she get the puppy?

It rained all day on November 19. How many days before Thanksgiving is that?

How many days are there between Veteran’s Day and Thanksgiving?


Sunday Monday Tuesday Wednesday Thursday Friday Saturday


Graphs, tables, lists, and Venn diagrams are all organizational tools that are appropriate for young children. However, for organizing information to be rel-evant at the primary grades, there needs to be a purpose to organization. Students need to see that by organizing information they are organizing their thinking about the problem. This allows them to see what information they have and what is still needed to solve the problem.



Key QuestionHow many cages will the circus need to buy so that all their animals will fi t?

Learning GoalStudents will determine the number of small, medium, and large cages that the circus needs to buy in order for all of their animals to have homes.

Guiding DocumentNCTM Standards 2000*• Count with understanding and recognize “how

many” in sets of objects • Apply and adapt a variety of appropriate strate-

gies to solve problems• Solve problems that arise in mathematics and in

other contexts • Build new mathematical knowledge through

problem solving

MathProblem solvingNumber sense grouping

Integrated ProcessesObservingCollecting and recording dataOrganizing data

Problem-Solving StrategiesOrganize the informationUse manipulatives

MaterialsAnimal crackersCages (see Management 3)Student pages

Background Information For organizing information to be relevant at the primary grades, there needs to be a purpose to the organization. In this activity, students are challenged to fi nd the number of cages that need to be purchased to hold all of the animals owned by a circus. Since

they are too young to simply divide the number of animals by the number that will fi t in each cage, they need to have a way to organize the information so that they can come up with an answer. This is provid-ed by giving them animals and cages to manipulate, and then a way to record that information so that it can be used later.

Management1. Each group will need animal crackers, fruit snack

animals, or something about that size to represent the circus animals.

2. The number of animals given to each group needs to be a multiple of one, two, and three (6, 12, 18, etc.). Give each group the number of animals appropriate for their ability level. Different groups can have different numbers of animals.

3. Groups will need something to represent small, medium, and large cages. Clear plastic cups in three sizes work well. (Bars can be drawn on the cups to make them look more like cages.) You can also use plastic lids in three sizes (for exam-ple, yogurt, cottage cheese, and butter container lids). Each group will need multiple cages of each size. (The number of cages actually needed will depend on the number of animals, but groups should always have more cages than they really need so that the solution is not given away.)

4. The student pages have enough room to record up to 18 animals. If you give your students more than that, they will need to have multiple copies of each page, otherwise one copy per student will be enough.

5. This activity has the possibility for many exten-sions, and can be done at a variety of levels. Select the portions of the activity that are appropriate for your students.

Procedure1. Have students get into groups and distribute the

student pages and the appropriate number of ani-mals and cages to each group.

2. Tell students that the circus has received some new animals and cages need to be bought for them all.

3. Explain that there are three sizes of cages that they can buy—small, medium, and large. Only one animal will fi t in each small cage. Two animals will fi t in each medium cage, and three animals will fi t in each large cage.


4. Challenge students to determine how many cages the circus would need to buy for their animals if they bought all small cages, all medium cages, or all large cages.

5. Instruct them to use the cages and animals to organize the information. Once they have deter-mined the total number of cages necessary, have them record that information on the appropriate student page by drawing in the animals and writ-ing the total number of cages needed.

6. Discuss how students solved the problem and what solutions they got. If desired, repeat this pro-cess with more animals or move on to some of the extensions.

Connecting Learning1. How many small cages would your group need to

buy? How did you get that answer?2. How many medium cages would your group need

to buy? How did you get that answer?3. How many large cages would your group need to

buy? How did you get that answer?4. How could you have solved the problem without

the animals and cages?5. How many cages would you need if each cage

could hold six animals? How do you know?

Extensions1. Make bar graphs of the number of small, medium,

and large cages that would need to be purchased for different numbers of animals.

2. Allow students to buy a combination of small, me-dium, and large cages. Challenge them to see how many different combinations of one-, two-, and three-animal cages they can use with the animals.

3. Assign specifi c cage sizes to the animals avail-able. Give students a variety of animals and have them fi gure out what combination of cages would need to be purchased. For example, elephants can only go in the large cages, giraffes can only go in the medium cages, and tigers can only go in the small cages.



TopicData organization

Key QuestionWhat do the glyphs tell us about the students in our class?

Learning GoalStudents will organize the class data to gain informa-tion about the class.

Guiding DocumentNCTM Standards 2000*• Sort and classify objects according to their attri-

butes and organize data about the objects• Represent data using concrete objects, pictures,

and graphs• Describe parts of the data and the set of data as a

whole to determine what the data show• Count with understanding and recognize “how

many” in sets of objects• Build new mathematical knowledge through


other contexts• Apply and adapt a variety of appropriate strate-

gies to solve problems• Monitor and refl ect on the process of mathematical

problem solving

MathGraphing bar graph, circle graph Venn diagramProblem solving

Integrated ProcessesObservingComparing and contrastingClassifyingCollecting and recording dataOrganizing dataInterpreting dataDrawing conclusionsCommunicating


MaterialsStudent pagesCrayonsScissorsYarn (see Management 7)Glue sticksCard stock

Background Information For organizing information to be relevant to pri-mary students, there needs to be a purpose for the organization. In Part One of this activity, the students will create glyphs, which will consist of pictures or symbols that give detailed information about the stu-dents that created them. In Part Two of the activity, students will use the legend and organize the glyphs in various ways in order to familiarize themselves with various data organizers such as bar graphs, Venn diagrams, etc. In Part Three they will use the legend, glyphs, and their organization skills to solve a variety of problems. Using the glyphs makes data collection and analysis fun while allowing students to practice using a legend.

Management 1. Have several examples of picture writing avail-

able for your students to examine. Include such things as Egyptian hieroglyphics, Native Amer-ican picture writing, and present-day symbols such as the handicap accessible symbol. (See Internet Connections.)

2. Display the legend where it can be easily seen by all students. This can be done by enlarging it onto chart paper or by making an overhead transparency of it.

3. The data collection process will be determined by the ability of the students. For younger stu-dents, you will want to present the survey/legend one step at a time. With older students, all the material can be presented in one step, with the survey/legend on one sheet of paper that each student receives. As you go through the glyph, emphasize the meaning of the features. This will help the students to focus on what they mean and not what they are.

4. Prior to teaching this lesson, complete a glyph that represents your responses to the legend questions.


5. It is assumed that the students have had prior experience with bar graphs, circle graphs, and Venn diagrams.

6. This activity should to be spread out over several days.

7. Several three-meter lengths of yarn in two colors will be used for Part Two of this activity.

8. Make one copy of the backpack on card stock for each student.

ProcedurePart One—Making the Glyph 1. Lead your students in a discussion about picture

writing and hieroglyphics. Show the students several examples and have them infer what each picture represents. Tell your students that they will be doing some picture writing to communicate information about themselves.

2. Show the students your completed glyph. Ask them what they notice about the book bag. [There is a red apple in it, there is a glue stick in it, etc.] Tell the students that the pictures represent things about you.

3. Draw the students’ attention to the legend. Show them how the glue stick in the book bag means that this book bag belongs to a girl or how the white glue in the book bag means the bag belongs to a boy. Continue using the legend to interpret your sample book bag. To reinforce writing, you may ask the students to help you write sentences that describe you, based on your picture. For example, I am a girl that walks to school. I bring my lunch to school, etc.

4. Distribute a set of student pages, crayons, scis-sors, and a glue stick to each student.

5. Ask students to think about how they get to school most mornings. Explain that the legend will tell them what color to color their apple based on how they get to school.

6. Ask all students who walk to school most morn-ings to raise their hands. Refer the students to the fi rst section of their survey page where the word yellow is written below the picture of the student walking. Tell those students to take their yellow crayon and circle the picture of the stu-dent walking to remind them later to color their apple yellow.

7. Ask the students what color they should color their apple if they ride to school in a car most mornings. [red] Tell those students to make a red mark with their crayon around the picture of a car to remind them later to color their apple red. Follow the same procedure for those students who ride a bus to school most mornings. Allow the class time to color their apples and cut them out before moving on.

8. Draw the students’ attention back to the survey page. Read question two aloud. Have the stu-dents circle their response. Allow them time to go to the page of pictures and cut out the glue container that matches their response. Wait as they glue the picture on the book bag.

9. Ask the students to look at question three on the student survey page. Explain that they are to write their ages on the box of crayons that is drawn on the book bag. Tell them to circle the fi ve if they are fi ve, six if they are six, etc. Give them time to write the correct number on the crayon box that is on the book bag.

10. Read question four aloud. Ask the students to respond by circling either the open or closed scissors. Instruct them to cut the correct pair of scissors out from the picture page.

11. Allow time for the students to color and glue the pictures onto their book bags. Collect the book bag glyphs.

12. Display a glyph and ask the students to use the legend to help you describe the owner of the book bag. Repeat the process several times until the students are comfortable with using a legend to “read” the glyphs.

Part Two—Data Collection, Organization, and Analysis 1. Gather the students around the collection of

glyphs. Ask students to think about different ways the glyphs could be sorted.

2. Invite a student to sort the glyphs using a rule. [Girls/boys, walkers/bus riders/car riders, etc.] Ask the students if they can identify the rule.

3. On the fl oor, assist students in placing the glyphs so that they create a bar graph. Analyze the data by asking questions such as, Which do we have more of? …less of? Why do you think we have more ______? Do you think that we would get the same results if we made glyphs in a different grade level? Etc.

4. Return the glyphs to a sorting area. Encourage students to sort the glyphs using a new rule. Allow a student to demonstrate the sort.

5. Organize and display the glyphs in a circle graph. Separate the sections with pieces of yarn. Be sure to label the sections.

6. Discuss the results of the sort. Compare the circle graph display to the bar graph display.

7. Continue sorting using a new rule. 8. Place two different colored yarn circles on the

fl oor. Label one set girls and the other bus riders. Place the class glyphs in the correct sets. For those glyphs that represent girls that are bus rid-ers, overlap the circles to create a set with both characteristics. Analyze the class data. Discuss commonalties among the students and other possible ways to sort and display the glyphs.


Part Three—Problem Solving with the Glyphs 1. Display the legend and collection of glyphs on

the chalk tray. 2. Explain to the class that you will be asking them

to solve several problems and that they may use the glyphs to organize their information.

3. Over the next several days, ask the following questions:

• If we need enough chairs for all of the walkers and bicycle riders to sit in while they wait for the bus riders to be released, how many chairs do we need?

• If we order fl owers for all of the girls in the class, how many do we need to order?

• If we make teams of one boy and one girl, how many teams will we have? Will there be any boys or girls left over?

• What is the most common age in this class? • How many lunches does the cafeteria sell to

our class in an average week? 4. After each problem, discuss how students went

about solving the problems and what solutions they got.

Connecting Learning 1. How many girls do we have in our class? How

many boys do we have? 2. How do you know what each picture means? 3. How are our glyphs like hieroglyphics? 4. Choose a glyph. Based on the glyph, what do you

know about the student who created it? 5. Name several rules that we could use to sort the

glyphs. 6. How did you solve the chair problem? …the fl ower

problem?

Internet ConnectionsExamples of hieroglyphics and pictographs:http://www.seaworld.org/fun-zone/fun-guides/egypt/hieroglyphics.htm

http://net.indra.com/~dheyser/rockart.html



==== Glyph Hangers ==== Legend

= Boy

= Girl

= Age

red = car yellow = walk/bike/skate green = bus

= Buy lunch

= Bring lunch


==== Glyph Hangers ==== Student Survey

1. How do you get to school?

2. Are you a boy or a girl?

3. How old are you?

4. Do you bring or buy your lunch?

Color apple

Red Yellow Green

Boy Girl

4 yearsold

4 5 6 75 years

old6 years

old7 years

old

Buy Bring


TopicData collection

Key QuestionWhat kind of pizzas do people like?

Learning GoalsStudents will:1. collect data from their friends and family, and2. draw conclusions about pizza preferences based

on the data.

Guiding DocumentsProject 2061 Benchmarks• Simple graphs can help to tell about observations.• Numbers can be used to count things, place them

in order, or name them.

NCTM Standards 2000*• Pose questions and gather data about themselves

and their surroundings• Represent data using concrete objects, pictures,

and graphs• Describe parts of the data and the set of data as a

whole to determine what the data show• Build new mathematical knowledge through


other contexts

MathGraphingCountingOne-to-one correspondenceVenn diagramProblem solving

Integrated ProcessesObservingComparing and contrastingCollecting and recording dataInterpreting dataCommunicating


MaterialsFor each student: Pizza Possibilities survey scissors

For the class: The Best Vacation Ever (see Curriculum Correlation) pizza station cards refrigerator biscuits, (see Management 5) pizza sauce (see Management 5) pizza toppings (see Management 5) toaster oven (see Management 2) construction paper (see Management 3)

Background Information Graphs, tables, lists, and Venn diagrams are all organizational tools that are appropriate for young children. However, for organizing information to be relevant at the primary grades, there needs to be a purpose to organization. In this activity, students will fi nd out what kind of pizzas people like. By organiz-ing the information they gather through the use of a survey, they can then draw conclusions about pizza preferences based on data. It is important for students to be actively involved in the process of collecting, organizing, and interpret-ing data. Young children are often asked to respond to a variety of graphs and charts when, more often than not, the questions are generated by the teacher. The children’s only participation in the data collection process then becomes fi lling in their responses to the question. While it is important to offer these experi-ences to students, it is equally important for students to generate the questions and collect, organize, and interpret the data. This data collection activity allows the children to be actively involved in the entire data organiza-tion process. The children will ask a question about pizza—a topic of interest to most of them, gather data through the use of surveys, organize the data using charts, and analyze the data to make an informed decision.

Management1. This activity will take several days to complete. 2. Prior to this lesson you will need to arrange to use

the school cafeteria in order to cook the individual pizzas made in Part Three of this activity, or bring in a toaster oven.

3. Children will be making models of their favorite piz-zas in Part Three of this activity. Because of varied paper cutting abilities, it will be best to establish a color-coded system to identify mushrooms, pepper-oni, and/or cheese. [Crust, white; sauce, red; mush-rooms, gray; pepperoni, brown; cheese, yellow]


4. Copy the pizza station cards on cardstock and laminate for extended use.

5. Purchase pizza ingredients. Each child will need one refrigerator biscuit, one tablespoon of pizza sauce, and their desired toppings of cheese, mush-rooms, and/or pepperoni.

6. Cut the fi rst student page in half along the center line. Students will use the fi rst half on the fi rst day and the second half on the second day.

ProcedurePart One1. Read The Best Vacation Ever. Talk about the use of

surveying as a way of collecting data. Discuss the questions asked by the girl in the story and her fami-ly’s responses to the questions. Ask the students what the girl learned from each of the charts she made.

2. Tell students that they will be doing some survey-ing like the little girl in the story. Tell them that a local pizzeria is ordering toppings and needs to know what to order. Ask the Key Question. Discuss the fact that there are so many different ways to order a pizza—thick or thin crust; stuffed or not stuffed crust; toppings galore: pepperoni, mush-rooms, onions, peppers, etc.

3. Give each child the fi rst half of the fi rst Pizza Pos-sibilities page. Guide the students through the survey process. Instruct them to ask fi ve of their friends and/or family members what kind of crust they like on their pizza. Tell students to record their responses by using check marks on the chart of their handout. Allow students to take the survey home to gather the information needed.

4. When the surveys are returned, compile the class results, and discuss them.

Part Two1. Instruct students to look at the second half of the

fi rst student sheet—Survey Two. Tell students that this time they will survey their families about pizza toppings. Explain that they will need to write a question that will generate information about what their families and/or friends like best on pizza—mushrooms, cheese, and/or pepperoni.

2. Brainstorm some possible questions. Allow the students to write their questions in the Survey Two section. Discuss how many columns and rows would be needed for their charts. Have students make the charts and then survey the same friends and/or family members surveyed before.

3. When the results come back the next day, distrib-ute the second student page and ask the students to use the results from both surveys to create a paper model of the pizza that most of their families and/or friends surveyed said they liked.

4. Instruct students to cut the recording page in half. When students are fi nished constructing their paper

models, invite the students to bring their models to an open area. Spend several minutes sorting the pizzas in different ways.

5. Create a class graph using one of the sorting schemes. Discuss the results.

Part Three1. Survey your students by asking what toppings

they like on their pizza. Allow them to make a construction paper model of their favorite kind of pizza toppings using the color code suggested in Management 3.

2. Invite the class to come to an open area and form a circle. Ask each student to place his or her model pizza in the center of the circle. Sort the pizzas and create a graph to compare the toppings cho-sen. Encourage the students to look closely at the graph. Ask them to tell you what they observe. Extend the thinking by asking questions that allow students to make inferences based on what the data show. For example, “Why do you think that more people like cheese pizza than mushroom pizza?”

3. Use yarn to create a three-circle Venn diagram. Label one circle cheese, the second circle mush-rooms, and the third pepperoni. Choose one model pizza and ask the students which circle it belongs in. If it has two of the characteristics labeled, place it in the area where the two circles overlap. Dis-cuss the placement. Continue until everyone’s pizza is placed into the Venn. Discuss the results.

4. Allow students to read and follow the directions on the pizza station cards to build their favorite pizzas.

Connecting Learning 1. Name one way to collect data.2. What did we learn from our pizza surveys?3. Why would we survey people?4. What might be some good survey questions to

ask our class?5. Name one time that you or your family was sur-

veyed for information.6. What did we fi nd out about our class when we

compared our pizzas?

Extensions1. Have students develop a list of questions on a

topic of their choice. Allow them to survey other classes or grade levels to compare results.

2. Collect data in other ways such as using a ballot box, observations, etc.

Curriculum CorrelationMurphy, Stuart. The Best Vacation Ever. HarperCollins. New York. 1997.



Surv

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One

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?

Thic

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Thin

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1 2 3 4 5 Tota

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Surv

ey

Two

____

____

____

____

____

____

____

____

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Make the pizza that most of your friends and family want.

Thin Crust

Thick Crust


Wash your hands.

Write your name onthe foil.


Flatten your biscuit.

Put one tablespoon of sauce on the dough.


Bake at 350° for 15 minutes or until done.

Add the toppings.


The problems on the following pages are provided for additional practice with the problem-solving strategies covered in this book. No strategies have been recommended for the individual problems, and they do not follow any particular order. Students must decide which strategy to use based on the individual problem. It is suggested that the problems be copied onto transparencies and cut apart. A problem can then be placed on the overhead as a “bright beginning” to start math class or at any time during the day when a few minutes are available for review. To receive maximum benefi t from the problems, be sure to have a time of discussion after each one where the emphasis is on the process and strategies used rather than arriving at the correct answer.


Ann has 7 cookies. She has 3 more cookies than Ben and 1 cookie less than Drew. How many cookies do Ben and Drew each have?

I had 10 cats. Five ran away. How many cats do I have left?

There was 1 dog at the park. Then 5 more dogs came. How many dogs are in the park now?

There were 10 rats. Two rats had no tails. How many rats had tails?

I have 6 coins worth 51 cents. What coins do you think I have?

How many coins do you need to make 7 cents?

What are some ways that you can make 25 cents?

How many groups of 5 can you make with 15?


How many different ways can you make 8 by adding 2 numbers?

There are 3 red bears and 7 green bears in a bag. What color bear are you most likely to pull out of the bag?

You have 3 Unifix cubes in one pocket and 4 in another pocket. How many do you have all together?

Draw 3 orange balls and 2 red balls. Count them. How many balls are there in all?

Serena saw a frog with 4 spots. How many spots would Serena see on 3 similar frogs?

How many sides do 5 triangles have?

There are 3 people sitting at the lunch table. How many feet are under the table?

On Monday, 2 children share at show-and-tell. On Tuesday, 3 share. On Wednesday, 4 share. If this pattern continues, how many will share on Friday?


What comes next?1, 4, 7, 10, _____

What comes next?Monday, Wednesday, Friday, __________

The answer is 10. What is the question?

Sam’s birthday is 3 days before Christmas. What day is his birthday?

Count down from 20.

Charlie has 10 cents left. He spent 5 cents on an eraser and 10 cents on a pencil. How much money did he start with?

Tyrese’s birthday is July 10th. His party will be 3 days before his birthday. What date is his party?

I have 3 coins that total 16 cents. What coins do I have?


On our zoo trip we saw 3 elephants, 5 bears, 2 seals, and 4 giraffes. Make a graph that shows all of the animals we saw at the zoo.

Flip a penny 10 times. Record the number of times it landed heads up and the number of times it landed tails up.

Four people had soup. Two people had salad. Three people had soup and salad. Was more soup or salad eaten?

Write a number sentence that describes the picture. It rained 10 inches this

month. That’s 4 inches more than last month. How much did it rain last month?

Which candy is most popular?How many people like it best?

Sugar Bites Choco-Treats

Which number sentence describes the picture?3 + 3 + 2 = 8 2 + 3 = 54 + 3 + 1 = 8 2 + 2 = 4

How many feet on two cows and three chickens?


There are 3 bears in a line. One is yellow, one is red, and one is blue.• The blue bear is in

the middle.• The red bear is not first.

What is the order of the bears?

Celina spent $3.00 at a book fair. What did she buy?• The Three Little Pigs

$1.75• Stickers $0.50• Ugly Slugs and

Yucky Bugs $2.00• Fairy Tales $1.25

Muhammad hid some colored eggs. He hid more than 6. He hid less than 9. He hid a number that 2 people can share equally. How many eggs did he hide?

After rolling a number cube 20 times, Taylor has collected this data. Help her make a graph with it.

1

2

3

4

5

6


Sue, Jerome, and Tim were standing in line. Sue was not fi rst. Tim was third. What order were the three friends standing in?

There are 3 bears in a row. One is yellow, one is red, and one is blue.• The red bear is to

the left of the yellow bear.

• The blue bear is fi rst.What order are the bears in?

Take three Unifi x cubes: blue, black, and yellow. Build a tower using these clues: • The blue cube is

second.• The blue cube is on

top of the yellow.

Take three Unifi x cubes: blue, red, and white. Build a tower using these clues: • The white cube is in

the middle.• The blue cube is

not on top.

SOLVE IT! K-1 © 2007 AIMS Education Foundation

The AIMS Program AIMS is the acronym for “Activities Integrating Mathematics and Science.” Such integration enriches learning and makes it meaningful and holistic. AIMS began as a project of Fresno Pacific University to integrate the study of mathematics and science in grades K-9, but has since expanded to include language arts, social studies, and other disciplines. AIMS is a continuing program of the non-profit AIMS Education Foundation. It had its inception in a National Science Foundation funded program whose purpose was to explore the effectiveness of integrating mathematics and science. The project directors, in cooperation with 80 elementary classroom teachers, devoted two years to a thorough field-testing of the results and implications of integration. The approach met with such positive results that the decision was made to launch a program to create instructional materials incorporating this concept. Despite the fact that thoughtful educators have long recommended an integrative approach, very little appropriate material was available in 1981 when the project began. A series of writing projects ensued, and today the AIMS Education Foundation is committed to continuing the creation of new integrated activities on a permanent basis. The AIMS program is funded through the sale of books, products, and professional-development workshops, and through proceeds from the Foundation’s endowment. All net income from programs and products flows into a trust fund administered by the AIMS Education Foundation. Use of these funds is restricted to support of research, development, and publication of new materials. Writers donate all their rights to the Foundation to support its ongoing program. No royalties are paid to the writers. The rationale for integration lies in the fact that science, mathematics, language arts, social studies, etc., are integrally interwoven in the real world, from which it follows that they should be similarly treated in the classroom where students are being prepared to live in that world. Teachers who use the AIMS program give enthusiastic endorsement to the effectiveness of this approach. Science encompasses the art of questioning, investigating, hypothesizing, discovering, and communicating. Mathematics is a language that provides clarity, objectivity, and understanding. The language arts provide us with powerful tools of communication. Many of the major contemporary societal issues stem from advancements in science and must be studied in the context of the social sciences. Therefore, it is timely that all of us take seriously a more holistic method of educating our students. This goal motivates all who are associated with the AIMS Program. We invite you to join us in this effort. Meaningful integration of knowledge is a major recommendation coming from the nation’s professional science and mathematics associations. The American Association for the Advancement of Science in Science for All Americans strongly recommends the integration of mathematics, science, and technology. The National Council of Teachers of Mathematics places strong emphasis on applications of mathematics found in science investigations. AIMS is fully aligned with these recommendations. Extensive field testing of AIMS investigations confirms these beneficial results: 1. Mathematics becomes more meaningful, hence more useful, when it is

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Wind,4-5WeatherSense:Moisture,4-5What on Earth? K-1What’sNext,Volume1,4-12What’sNext,Volume2,4-12What’sNext,Volume3,4-12Winter Wonders, K-2

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Sensational Springtime

Counting on Coins

It’s About Time

It Must Be a Bird

Under Construction

Winter Wonders

Cycles of Knowing and Growing

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Solve It! K-1: Problem-Solving Strategies is a collection of activities designed to introduce young children to eight problem-solving strategies. The tasks included will engage students in active, hands-on investigations that allow them to apply their number, computation, geometry, data organization, and algebra skills in problem-solving settings. The strategies addressed are: guess and check, look for patterns, use manipulatives, draw out the problem, use logical thinking, write a number sentence, work backwards, and organize the information. Also included are practice problems that can be used for assessment. These activities give students the opportunity to develop a toolbox of problem-solving strategies that they can draw from when approaching problems. These skills will serve them well not only in mathematics, but in other academic subjects and their everyday lives.

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ISBN 978-1-932093-14-8

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Solve It! K-1 - Gateway Academy

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