Place Value
Place value determines the value of a digit in a number, based on thelocation of the digit.
4 3 2 7 . 8 9 5
4 thousand (4000)3 hundred (300)2 tens (20)7 units/ones (7)8 tenths (0.8)9 hundredths (0.09)5 thousandths (0.005)
The decimal point separates whole numbers from decimals (pieces).
TH H T U t h th
whole numbers
decimals
Rounding numbers to the nearest 10, 100 or 1000
6 5 3 9
Number in words: six thousand, five hundred and thirty-nine.
When rounding to the nearest 10 we must look at the tens digit, whenrounding to the nearest 100 we must look at the hundreds digit whilewhen rounding to the nearest 1000, we must look at the thousandsdigit. Then always remember the easy trick!
So,6539to the nearest 10 is 6540 6539to the nearest 100 is 6500 6539to the nearest 1000 is 7000
TH H T U
Rounding numbers to the nearest whole number, tenth or hundredth
4 . 1 7 3
When rounding to the nearest whole number, always look at the unitsdigit. On the other hand, when rounding to the nearest tenth, always look at the tenths digit and when rounding to the nearest hundredth look at the hundredths digit. Then use the rounding trick again to get your answer.
So,4.173 to the nearest whole number is 44.173 to the nearest tenth is 4.24.173 to the nearest hundredth is 4.17
U t h th
Remember
Any zeros that are after the decimal point at the very end can be removed.
Examples:
5.70 = 5.7
8.30 = 8.3
2.340 = 2.34
6.020 = 6.02
9.0 = 9
28.0 = 28
10.002 = 10.002
Dividing by 10
When you divide by a whole number, the answer is smaller than theoriginal number.
Method 1: When you divide a number by 10, the digits move one placevalue to the right (forwards).
736 ÷ 10 = 73.6
H T U t hhundreds tens units tenths hundreds
7 3 6
7 3 6
Dividing by 10
Method 2: Another method to use when dividing by 10 is to move thedecimal point to the left. If you are dividing by 10, move the decimalpoint one place to the left (backwards).
736 ÷ 10 = 73.6
7 3 6 . = 73.6
Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.
Dividing by 100
When you divide by a whole number, the answer is smaller than theoriginal number.
Method 1: When you divide a number by 100, the digits move twoplace value to the right (forwards).
736 ÷ 100 = 7.36
H T U t hhundreds tens units tenths hundreds
7 3 6
7 3 6
Dividing by 100
Method 2: Another method to use when dividing by 100 is to movethe decimal point to the left. If you are dividing by 100, move thedecimal point two places to the left (backwards).
736 ÷ 100 = 7.36
7 3 6 . = 7.36
Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.
Multiplying by 10
When you multiply by a whole number, the answer is bigger than theoriginal number.
Method 1: When you multiply a number by 10, the digits move oneplace value to the left (backwards).
73.6 x 10 = 736
H T U t hhundreds tens units tenths hundreds
7 3 6
7 3 6
Multiplying by 10
Method 2: Another method to use when multiplying by 10 is to movethe decimal point to the right. If you are multiplying by 10, move thedecimal point one place to the right (forwards).
73.6 x 10 = 736
7 3 . 6 = 736
Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.
Multiplying by 100
When you multiply by a whole number, the answer is bigger than theoriginal number.
Method 1: When you multiply a number by 100, the digits move twoplace value to the left (backwards).
7.36 x 100 = 736
H T U t hhundreds tens units tenths hundreds
7 3 6
7 3 6
Multiplying by 100
Method 2: Another method to use when multiplying by 100 is to movethe decimal point to the right. If you are multiplying by 100, move thedecimal point two places to the right (forwards).
7.36 x 100 = 736
7 . 3 6 = 736
Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.
Remember
When dividing a multiple of 10, 100, 1000, etc, we cross out the zeros from the back of the number.
Examples:
54Ø ÷ 1Ø = 54
16ØØ ÷ 1ØØ = 16
831Ø ÷ 1Ø = 831
535ØØ ÷ 1ØØ = 535
9ØØØ ÷ 1ØØØ = 9
8760Ø ÷ 1Ø = 8760
400ØØ ÷ 1ØØ = 400
Remember
Multiplication is the inverse of division and vice versa.
Examples:
5 x 6 = 30 30 ÷ 6 = 5
7 x 8 = 56 56 ÷ 8 = 7
4 x 9 = 36 36 ÷ 4 = 9
8 x 3 = 24 24 ÷ 8 = 3
45 ÷ 9 = 5 9 x 5 = 45
28 ÷ 7 = 4 7 x 4 = 28
36 ÷ 6 = 6 6 x 6 = 36
80 ÷ 10 = 8 10 x 8 = 80
Important: Multiplying and dividing any number by 0 makes 0.15 x 0 = 024 ÷ 0 = 0
Doubling
Different methods for doubling numbers:
double 37
Method 1: 37 + 37 = 74
Method 2: 37 x 2 = 74
Method 3: double 30 = 60double 7 = 1460 + 14 = 74
Method 4: double 35 = 70double 37 = 70 + (double 2) = 74
Doubling
Different methods for doubling numbers:
double 59
Method 1: 59 + 59 = 118
Method 2: 59 x 2 = 118
Method 3: double 50 = 100double 9 = 18100 + 18 = 118
Method 4: double 60 = 120double 59 = 120 - (double 1) = 118
Halving
Different methods for halving numbers:
half of 420
Method 1: 420 ÷ 2 = 210
Method 2: half of 400 = 200half of 20 = 10200 + 10 = 210
Important: Use same methods to double and half numbers with decimal points.Examples: double 2.4 = 4.8 double 4.6 = 9.2
half of 10.6 = 5.3 half of 11.8 = 5.9
2 11.8 5.9
Multiplying
One of the most basic ways to multiply is to add. When we multiplytwo numbers we are simply adding one number to itself a certainnumber of times. For example, if the problem says to multiply 20 x 5,you are adding the number 20 to itself five times.
So, 20 + 20 + 20 + 20 + 20 = 100, or 20 x 5 = 100.
When multiplying larger numbers we can use various methods.
Remember, choose the method that you prefer most!The most important thing is that you get your answers right!
Let’s see some methods together!
Multiplying
37 x 56
Method 1:
Add your answers: 1500 + 350 + 180 + 42 = 2072
30 750 1500 3506 180 42
Multiplying
37 x 56
Method 3:
Multiply both sides: 30 x 50 = 150030 x 6 = 1807 x 50 = 3507 x 6 = 42
Add your answers: 1500 + 180 + 350 + 42 = 2072
30 7 50 6
Multiplying by nearly 50, by nearly 100, by 25
Multiplying by nearly 50: 84 x 4984 x 100 = 840084 x 50 = 420084 x 49 = 4200 – 84 = 4116
Multiplying by nearly 100: 67 x 10167 x 100 = 670067 x 101 = 6700 + 67 = 6767
Multiplying by 25: 46 x 2546 x 100 = 460046 x 50 = 230046 x 25 = 1150
halving
halving
Multiplying by doubling and halvingdoubling
halving
x2 x4 x8 x16 x32
2 4 8 16 32
4 8 16 32 64
6 12 24 48 96
8 16 32 64 128
10 20 40 80 160
12 24 48 96 192
14 28 56 112 224
16 32 64 128 256
18 36 72 144 288
20 40 80 160 320
Multiplying by doubling and halvingdoubling
halving
x6 x12 x24
3 x 6 = 18 3 x 12 = 36 3 x 24 = 72
x7 x14 x28
5 x 7 = 35 5 x 14 = 70 5 x 28 = 140
x8 x16 x32
7 x 8 = 56 7 x 16 = 112 7 x 32 = 224
x9 x18 x36
8 x 9 = 72 8 x 18 = 144 8 x 36 = 288
Finding fractions of quantities
¾ of 96
Step 1: Divide the number by the denominator.
4 96
24
Step 2: Multiply the answer by the numerator.
24 x3
72
Therefore,
¾ of 96 = 72
Finding fractions of quantities
⅝ of 280
Step 1: Divide the number by the denominator.
8 280
35
Step 2: Multiply the answer by the numerator.
35 x5
175
Therefore,
⅝ of 280 = 175
Simplifying a fraction
To simplify a fraction find a common factor of both the numeratorand the denominator and divide the fraction by that factor.
Example: Simplify .
In this case, 5 is a common factor of both 15 and 25.
We can say that is a fraction in its simplest form because there areno common factors for 3 and 5. A fraction in its simplest form is alsocalled a fraction in its lowest terms.
Remember Find the largestpossible number to divide the numerator and the denominator to simplify the fraction at once.
Ordering fractions
Before putting fractions in order, one must first make the fractions with the same denominator so they can be compared.
Example: Put these fractions in order smallest first.
Now that the fractions have a common denominator, one can orderthem starting from the smallest first. Remember in your answer youmust write the fractions given in the question.
Comparing Fractions with different denominators
Watch this:
https://www.youtube.com/watch?v=KNdUJQ_qd4U
Finding the midpoint between two fractions
To find the midpoint between two fractions always start bymultiplying the fraction by 2. If the fraction is a mixed fraction, donot multiply the whole number.
Example: What is the midpoint between ?
Therefore we can say that the midpoint between
Adding to make 100 and 1000
100 1000
66 + 34 = 100 660 + 340 = 1000
53 + 47 = 100 530 + 470 = 1000
88 + 12 = 100 880 + 120 = 1000
71 + 29 = 100 710 + 290 = 1000
25 + 75 = 100 250 + 750 = 1000
16 + 84 = 100 160 + 840 = 1000
52 + 48 = 100 520 + 480 = 1000
33 + 67 = 100 330 + 670 = 1000
Remember: When adding to 100, the units (ones) add up to 10 (pair of 10).
Example: 43 + 57 = 100 (3 + 7 = 10)
Adding to make 10 and €10 (with decimal point)
10 €10
3.7 + 6.3 = 10 €3.26 + €6.74 = €10
4.2 + 5.8 = 10 €7.79 + €2.21 = €10
6.7 + 3.3 = 10 €8.81 + €1.19 = €10
8.5 + 1.5 = 10 €5.27 + €4.73 = €10
9.8 + 0.2 = 10 €4.01 + €5.99 = €10
2.9 + 7.1 = 10 €2.55 + €7.45 = €10
1.4 + 8.6 = 10 €1.33 + €8.67 = €10
5.3 + 4.7 = 10 €9.18 + €0.82 = €10
Remember: When adding to 10, the numbers after the point (tenths) add up to 10.
When adding to €10, the numbers after the point (tenths and hundredths) add up to 100.
What should be added to this number to make 100?
64 + ? = 100
Methods:
100 –6436
Counting on…
64
6 + = 36
OR 70 100
30
What should be added to this number to make 10?
1.5 + ? = 10
Methods:
100 –1585
Counting on…
15
5 + = 85
OR 20 100
80
Let’s pretend that 1.5 is 15 and we want to write the pair to 100.
8.5
What should be added to this number to make 10?
9.1 + ? = 10
Methods:
100 –919
Counting on…
91
9
OR 100
Let’s pretend that 9.1 is 91 and we want to write the pair to 100.
0.9
What should be added to this number to make €10?
€6.23 + ? = €10
Methods:
10.00 –6.233.77
Counting on…
6.23
0.77 + = 3.77
OR 7.00 10.00
3.0 3.77
What should be added to this number to make 300?
140 + ? = 300
Methods:
300 –140160
Counting on…
140
60 + = 160
OR 200 300
100 160
What should be added to this number to make 1200?
750 + ? = 1200
Methods:
1200 –750450
Counting on…
750
50 + = 450
OR 800 1200
400 450
Sam the cat weighs 3.54 kg. How many grams must it gain to weigh 4 kg?
3.54 kg + ? = 4 kg
Methods:
4.00 –3.540.46
Counting on…
3.54
0.46
OR 4.0
0.46
14.2cm + ? = 21cm
Methods:
21.0 –14.27.8
Counting on…
14.2
0.8 + = 6.8
OR 15.0 21.0
6.0 6.8cm
The sun flower is 14.2cm tall.How many cm must it grow to be 21cm tall?
We are going to set out a calculation in a column.
We are going to do
35 + 12T U
3 5+ 1 2
The biggest number
always goes on the top!
Now, we’re ready to
add…
First, we add together the units, and write the total in the units column…
7
Next, we add together the tens, then we write the total in the tens column…
4
€58 - €2.99 = ?
58.00- 2.99
10
.19.
Methods:
Counting on…
2.99
0.01 + = €55.01
3.00 58.00
55.00
OR
.5
7
5
The multiples of anumber are thosenumbers which arefound in the timestable of thatparticular number.
The multiples of 2for example are2, 4, 6, 8, 10,etc. The multiplesare always equalto or larger thanthe number given.
Common MultipleA multiple that 2 or more numbers
have in common.
Example:
Find the first 3 common multiples of 3 and 6.
Common Multiples: 6, 12, 18
3, 6,9,12, 15, 18…3:6, 12, 18, 24, 30,36…6:
Least Common Multiple (LCM)
14, 28, 42 and 70 are allcommon multiples of 2 and 7but the lowest commonmultiple is 14 because it is thesmallest number from all thecommon multiples.To find the common multipleof two or more numbers startby finding the multiples of thebiggest number until you finda number which is a multipleof the other numbers.
Properties of Numbers – Divisibility Rules
÷ 2 Is it even? Is the last digit 0,2,4,6,8?÷ 3 Is its digital total a multiple of 3?÷ 4 Do the last two digits make a multiple of 4?÷ 5 Is its units digit 0 or 5?÷ 6 Is it even and its digit total a multiple of 3?÷ 8 Is half of half of it even?÷ 9 Is its digit total a multiple of 9?÷ 10 Is its unit digit 0?÷ 25 Are the last two digits 00, 25, 50 or 75?
Hundreds Tens Units Tenths HundredthsThousands
To multiply by 10 all the digits move one space to the left.
6 8 9
68.9 x 10 = _____
6 8 9
689
Hundreds Tens Units Tenths HundredthsThousands
To multiply by 10 all the digits move one space to the left.
3 7 2
3.72 x 10 = _____
3 7 2
37.2
Hundreds Tens Units Tenths HundredthsThousands
1 9 5
1.95 x 100 = _____
1 9 5
195
To multiply by 100 all the digits move two spaces to the left.
Hundreds Tens Units Tenths HundredthsThousands
4 1 2
41.23 x 100 = _____
4 1 2
4123
To multiply by 100 all the digits move two spaces to the left.
3
3
Hundreds Tens Units Tenths HundredthsThousands
To divide by 10 all the digits move one space to the right.
7 5 5
755 ÷ 10 = _____
7 5 5
75.5
Hundreds Tens Units Tenths HundredthsThousands
To divide by 10 all the digits move one space to the right.
2 1 7
21.7 ÷ 10 = _____
2 1 7
2.17
Hundreds Tens Units Tenths HundredthsThousands
4 1 8
41.8 ÷ 100 = ______
4 1 8
0.418
To divide by 100 all the digits move two spaces to the right.
Thousandths
0
Hundreds Tens Units Tenths HundredthsThousands
0 6
0.6 ÷ 100 = ______
0 6
0.006
Thousandths
00
To divide by 100 all the digits move two spaces to the right.
To multiply a 1-digit number by a 2 or 3-digit
number (H/TU x U) we partition the number
into separate HUNDREDS, TENS and UNITS.
Example:
4 x 72 = ?
70 + 2
(4 x 70) + (4 x 2) =280 + 8 = 288
Apart from using multiplication method by
partition, we can use the vertical method.
Example: 28 x 4 = ?
28x 4
4 x 8 324 x 2080
211
Example: 185 x 29 = ?
185x 29
9 x 5 459 x 80
2000
55
9 x 100720
6
2 20 x 5
20 x 80
20 x 100
900100
1600
3
Example: 344 x 64 = ?
344x 64
4 x 4 164 x 40
1800062
4 x 300160
1
1
60 x 4
60 x 40
60 x 300
1200240
2400
0
1
2
1
Long Multiplication
Watch these:
https://www.youtube.com/watch?v=FJ5qLWP3Fqo
https://www.youtube.com/watch?v=RVYwunbpMHA
Factors & Square Numbers
Factors are the numbers you multiply together to get another number.
When you multiply a whole number times itself, the resulting product is called a Square number, or a perfect square or simply "a square.“
So 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 are all square numbers.
Multiplication & Division are inverse operations.
Example:
2 x 4 = 8
So,
8 ÷ 2 = 4
8 ÷ 4 = 2
Use the same three
numbers to find
division facts.
Multiplication:
1.2 x 6 =
Division:
7.2 ÷ 1.2 = 6
7.2 ÷ 6 = 1.2
Use the same three numbers
to find division facts.
1.2
x 66 x 0.26 x 1.0
1.26.07.2
7.2
Multiplication and Division Fact Family Triangles
Multiply the bottom two numbers to give the top number.
8 3x
24
Multiplication and Division Fact Family Triangles
Multiply the bottom two numbers to give the top number.
5 30x
150
7 25
175
x
Multiplication and Division Fact Family Triangles
When one of the bottom numbers is unknown, divide the
top number by the one at the bottom.
6÷120
20
120 ÷ 6 = ?
Now let’s have a look at all the Multiplication and Division Facts
6÷120
20Multiplication:
6 x 20 = 120
20 x 6 = 120
Division:
120 ÷ 6 = 20
120 ÷ 20 = 6
÷
x
Multiplication and Division Fact Family Triangles
When one of the bottom numbers is unknown, divide the
top number by the one at the bottom.
4÷
200
50
200 ÷ 4 = ?
Multiplication and Division Facts
Multiplication:
4 x 50 = 200
50 x 4 = 200
Division:
200 ÷ 4 = 50
200 ÷ 50 = 4
4
200
50÷ ÷
x
What is a Remainder?
• Sometimes when dividing there is something left over.
• It is called the remainder.
Example 1:
There are 7 bones to share with 2 puppies.
But 7 cannot be divided exactly into 2 groups,so each puppy gets 3 bones,and there is 1 left over:
Remainder as a fraction
• We say:
• "7 divided by 2 equals 3 with a remainder of 1"
• And we write:
• 7 ÷ 2 = 3 R 1
It is also possible to cut the remaining bone in half.
7 ÷ 2 = 3 R 1 = 3 1
2bones
1 out of 2
NOTE: The remainder on top,
and the number you are dividing
by at the bottom.
• Work out 19 ÷ 5
• 19 cannot be divided exactly by 5. The closest we can get (without going over) is:
• 3 x 5 = 15 which is 4 less than 19.
Example 2:
= 3 𝟒
𝟓
4 out of 5
Remainder as a fraction should be done with:
• the remainder on top, and• the number you are dividing by at the bottom.
So the answer of 19 ÷ 5 = 3 R 4
• Work out 55 ÷ 4
Example 3:
= 13 𝟑
𝟒
The fraction should be done with:
• the remainder on top, and• the number you are dividing by at the bottom.
55 ÷ 4 = 13 R 3
86 cookies are shared among ten children.
How many cookies do they get each?
Example 4:
= 8 𝟔
𝟏𝟎𝐜𝐨𝐨𝐤𝐢𝐞𝐬86 ÷ 10 = 8 R 6
Remainder as a fraction
Remainder as a decimal
8 𝟔
𝟏𝟎= 8.6
The confectioner has 789 cupcakes. Boxes hold 100 cupcakes.
How many boxes do I need to hold all the cupcakes?
Example 5:
= 7 𝟖𝟗
𝟏𝟎𝟎𝐜𝐮𝐩𝐜𝐚𝐤𝐞𝐬789 ÷ 100 = 7 R 89
Remainder as a fraction
Remainder as a decimal= 7.897 𝟖𝟗
𝟏𝟎𝟎
44 children are going on a picnic, by car. Each car holds 5 children.
How many trips must the car make?
Example 6:
= 8 𝟒
𝟓𝐭𝐫𝐢𝐩𝐬44 ÷ 5 = 8 R 4
Remainder as a fraction
Remainder as a decimal= 8.8= 8 𝟖
𝟏𝟎8 𝟒
𝟓
x2
x2
Converting Fractions to Decimals
Watch these:
https://www.youtube.com/watch?v=WV5VY76Pf5U
https://www.youtube.com/watch?v=do_IbHId2Os
These numbers are called
Improper fractions
4
2
25
15 7
6
9
310
7
30
22
An improper fraction has a top number larger than (or equal to) the bottom number.
It is "top-heavy"
When a number includes both whole numbersand fractions.
1
What is it called?
It is called a
mixed number
1
2
Changing fractions from Mixed to Improper fractions
To change fractions from mixed to improper, get M.A.D.!!
Step 1: Multiply the whole number by the denominator.
= 3 x 4 = 12
Step 2: Add the answer to the numerator. 12 + 1 = 13
Step 3: Denominator remains the same as before. 134
234
4 x 5 = 20
20 + 3 =3
4=
Keep the denominator the
same.
Convert mixed number to improper fraction
Convert improper fractions to mixed fractions
• Divide the numerator by the denominator.
• Write down the whole number answer.
• Then, write down any remainder above the denominator.
Example:
14 ÷ 5 =𝟏𝟒
𝟓= 2
𝟒
𝟓divide by 5
This is the same as this...
Convert improper fractions to mixed fractions
30 ÷ 8 =𝟑𝟎
𝟖= 3
𝟔
𝟖divide by 8
This is the same as this...
Convert improper fractions to mixed fractions
56 ÷ 10 =𝟓𝟔
𝟏𝟎= 5
𝟔
𝟏𝟎divide by 10
This is the same as this...
Convert improper fractions to mixed fractions
34 ÷ 6 =𝟑𝟒
𝟔= 5
𝟒
𝟔divide by 6
This is the same as this...
Good to know that:
1cm (centimetre) = 10mm
1dm (decimetre) = 10cm = 100mm
1m (meter) = 10dm = 100cm = 1000mm
The smallest unit of measurement
Write the number of metres.
4dm, 5cm and 7mm
Note: The smallest unit of measurement is mm
1m = 1000mm
1dm = 10cm = 100mm4dm = 400mm
So, convert every measurement to mm
5cm = 50mm
7mm
400 + 50 + 7 = 457mm
Answer: 0.457m
dm cm mm
457 ÷ 1000 = 0.457m
Write the number of metres.
6dm, 4cm and 2mm
Note: The smallest unit of measurement is mm
1m = 1000mm
1dm = 10cm = 100mm6dm = 600mm
So, convert every measurement to mm
4cm = 40mm
2mm
600 + 40 + 2 = 642mm
Answer: 0.642m
642 ÷ 1000 = 0.642m
Write the number of metres.
9dm and 1mm
Note: The smallest unit of measurement is mm
1m = 1000mm
1dm = 10cm = 100mm9dm = 900mm
So, convert every measurement to mm
1mm
900 + 1 = 901mm
Answer: 0.901m
901 ÷ 1000 = 0.901m
Write the number of metres.
8cm and 8mm
Note: The smallest unit of measurement is mm
1m = 1000mm
1cm = 10mm8cm = 80mm
So, convert every measurement to mm
8mm
80 + 8 = 88mm
Answer: 0.088m
88 ÷ 1000 = 0.88m
Good to know that:
1cl (centilitre) = 10ml
1dl (decilitre) = 10cl = 100ml
1l (litre) = 10dl = 100cl = 1000ml
re
re
re
re
Write the value of 4.
0.460l 0.842l
l mldl cl l mldl cl
400
1000l =
4
10l 4dlor
40
1000l =
4
100l 4clor
Ordering decimalsWhen we order decimals we are going to compare digit
by digit.
0. 4 5 6 7
0. 4 8 1
For example, let’s compare these two numbers:
First we compare the whole number in front of
the decimal
These are the same, so we move on.We now compare the first number after the
decimal point.
These are again the same, so we move on to the
second decimal place.
We now have one decimal bigger than the other
so we can say that the whole number is bigger
We can say that 0.481 is larger than 0.4567
We now have one decimal bigger than the other
so we can say that the whole number is bigger.
Order these from the smallest to the largest?
0.0456 0.123
0.04569 0.4
0.0456, 0.04569, 0.123, 0.4
In order from smallest to largest
Order these from the smallest to the largest?
5.5 5.455.504
5.45, 5.5, 5.504
In order from smallest to largest
Order these from the smallest to the largest?
3.91 3.193.091
3.091, 3.19, 3.91
In order from smallest to largest
Order these from the smallest to the largest?
1.003 1.031.303
1.003, 1.03, 1.303
In order from smallest to largest
Ordering Decimals
Watch these:
https://www.youtube.com/watch?v=lnHvuxp5Q9c
https://www.youtube.com/watch?v=cw9RCCx9Rs8
Measuring Length
When we measure length, we find the size of the dimensions ofobjects. Some tools which measure length include ruler and measuringtape. Use a table to convert from one unit of length to another.
Remember well:
¼km = 0.25km = 250m
½km = 0.5km = 500m
¾km = 0.75km = 750m
¼m = 0.25m = 25cm
½m = 0.5m = 50cm
¾m = 0.75m = 75cm
½cm = 0.5cm = 5mm
Converting Units of Measure
Multiply or divide the measurements accordingly to convert from one unit to the other.
Measuring Weight
When we measure weight, we find out how heavy an object is. A scaleis used to measure weight. Use a table to convert from one unit ofweight to the other.
Remember well:
¼kg = 0.25kg = 250g
½kg = 0.5kg = 500g
¾kg = 0.75kg = 750g
Converting Units of Measure
Multiply or divide the measurements accordingly to convertfrom one unit to the other.
Bar-line graphs
A bar-line graph is similar to a bar graph but the bars are replacedby lines instead. Bar-line graphs are used to represent different data.
bar-line graphbar graph
Favourite sport Eye colour
sports colours
Bar-line graphs
Bar graphs and bar-line graphs (or charts) must have a title. The titleoffers a short explanation of what is in your graph. This helps thereader identify what they are about to look at. It can be creative orsimple as long as it tells what is in the graph.
Bar graphs and bar-line graphs have anx-axis and a y-axis. In most bar graphs,the x-axis runs horizontally, whereas,the y-axis runs vertically. The y-axisusually starts counting at 0 and can bedivided into as many equal parts as youwant to.
Line graphs
A line chart or line graph is a type of chart which displaysinformation as a series of data points called 'markers' connected bystraight line segments. It is a basic type of chart common in manyfields. It is used to show change over time.
A line graph always has a title. Ithas two different axes; onehorizontal and one vertical. It isimportant to read what informationis on each axis. There must be alabel on each axis that describesthe information.Much like a bar graph, firstly welook at the horizontal axis, to plotour points.
Line graphs
Learn to draw and interpret line graphs.
Miles traveled during the week
0
10
20
30
40
50
60
70
Mon. Tues. Wed. Thurs. Fri. Sat. Sun.
Days
Mile
s T
rav
ele
d
Conversion graphs
A conversion graph can be used to change one quantity to anotherwhen the units are changed. Here are some examples:
Converts pounds and francs Converts inches and centimetres
Convert 50% to Marks
A conversion graph
010
20304050
607080
90100
0 5 10 15 20 25 30 35 40 45 50 55 60
Marks
Pe
rce
nta
ge
Step 1: Find 50% on the vertical axis and draw a straight line to meet the conversion line on the graph.
Step 2: Draw a straight line to the other axis. This position gives the conversion value.
A conversion graph
010
20304050
607080
90100
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Inches
Ce
nti
me
tre
s
Convert: 12 inch = ____ centimetres
24 inch = ____ centimetres
32 inch = ____ centimetres
30
60
80
A conversion graph
0
10
20
30
40
50
60
70
80
0 5 10 15 20 25 30 35 40 45 50
Miles
Kilo
me
tre
s
Convert: 35 miles = ____ kilometres
25 miles = ____ kilometres
50 miles = ____ kilometres
55
40
80
The Mean
The mean is the average of the numbers. It is easy to calculate:
Step 1: add up all the numbersStep 2: divide by how many numbers there are
In other words, it is the sum divided by the count.
The Mean
The mean is the average of the numbers. It is easy to calculate:
Step 1: add up all the numbersStep 2: divide by how many numbers there are
In other words, it is the sum divided by the count.
Multi-Digit Addition and Subtraction
Watch these:
https://www.youtube.com/watch?v=mAvuom42NyY
https://www.youtube.com/watch?v=Y6M89-6106I
Area of rectangles
Area is a measure of how much space there is inside ashape. Calculating the area of a shape or surface can be useful ineveryday life, for example you may need to know how much paint tobuy to cover a wall.
Calculating Area Using the Grid MethodWhen a shape is drawn on a scaled grid you can find the area bycounting the number of grid squares inside the shape.
In this example there are 10 grid squaresinside the rectangle. (5 x 2 = 10cm2)
Area of rectangles
In order to find an area value, using the grid method, we need to knowthe size that a grid square represents.The example below uses centimetres but the same method applies forany unit of length or distance. You could, for example be using inches,metres, miles, feet etc. In this example each grid square has a widthof 1cm and a height of 1cm. In other words each grid square is one'square centimetre'. Count the grid squares inside the large square tofind its area.
There are 16 small squares so the area of the largesquare is 16 square centimetres.We abbreviate 'square centimetres' to cm2.The 2 means ‘squared’.
The area of the large square is 16cm2 (4 x 4 = 16cm2)
Areas of squares and rectangles
The simplest (and most commonly used) area calculations are forsquares and rectangles.To find the area of a rectangle multiply its height by its width.
For a square you only need to find the length of one of the sides (aseach side is the same length) and then multiply this by itself to findthe area.
Length x Breadth/Width
Calculating area of Irregular shapes
Often, in real life, shapes can be more complex. For example, imagineyou want to find the area of a floor, so that you can order the rightamount of carpet. A typical floor-plan of a room may not consist of asimple rectangle or square:
In this example, the trick is to split the shape into several rectangles (or squares). Find the area of each shape and then find the total area.
It doesn’t matter how you split the shape - any of the three solutions will result in the same answer.
Finding the length / breadth of a missing side involving Area
If you are given the breadth and the area of a rectangle and areasked to find its length, you should use the formula
Length = Area ÷ Breadth
If you are given the length and the area of a rectangle and are askedto find its breadth, you should use the formula
Breadth = Area ÷ Length
area =72cm2
Length = Area ÷ Breadth
breadth = 8cmarea = 72cm2
length = ?
72cm2 ÷ 8 = 9cm
area =100cm2
Breadth = Area ÷ Length
length = 25cmarea = 100cm2
breadth = ?
100cm2 ÷ 25 = 4cm
25cm
8cm
Area of triangles
It can be useful to think of a triangle as half of a square orparallelogram.
The area of a triangle is (Length × Breadth) ÷ 2.In other words you can work out the area of a triangle in the same way as the area for a square or parallelogram just divide your answer by 2.
Area of triangle3cm x 3cm = 9cm2
9cm2 ÷ 2 = 4.5cm2
The area of this triangle is 4.5cm2.
Finding the Length of a missing side in a triangle
Remember:The area of a rectangle is doublethe area of a triangle.
Area of rectangle = double 33cm2
66cm2
Length = Area ÷ Breadth
66cm2 ÷ 5cm = 16cm
Therefore, the length of side Xof the triangle in 16cm
area =33cm25cm
X
Area
Watch these:
https://www.youtube.com/watch?v=xCdxURXMdFY
https://www.youtube.com/watch?v=d_6XOBpoDz4
Finding the Perimeter of Regular Polygons
A polygon is a plane shape (two-dimensional) with straight sides.Examples include triangles, quadrilaterals, pentagons, hexagons and soon. A regular polygon has sides of equal length, and all its interiorangles are of equal size.
To find the perimeter of a regular polygon simply multiply the value of one of its sides by the total number of sides.
For example:If one of the sides of a regular pentagon is 6cm. To find its perimeter work out: 6cm x 5 (no. of sides) = 30cm
Finding the length of a missing side involving Perimeter
perimeter =48cm
Two lengths = 19cm + 19cm = 38cmTwo breadths = 48cm (perim) – 38cm = 10cmOne breadth = 10cm ÷ 2 = 5cm
Therefore the missing side is 5cm long
19cm
perimeter =80cm
Two breadths = 14cm + 14cm = 28cmTwo lengths = 80cm (perim) – 28cm = 52cmOne breadth = 52cm ÷ 2 = 26cm
Therefore the missing side is 26cm long
14
cm
Perimeter
Watch these:
https://www.youtube.com/watch?v=AAY1bsazcgM
https://www.youtube.com/watch?v=JAy_CETEyUM
Reflective Symmetry
When you draw a line of symmetry of a shape, you divide the shapeinto two identical parts which when folded cover each other in anexact way.
Reflective Symmetry
Important:
To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.
Reflective Symmetry
Important:
To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.
Reflective Symmetry
Important:
To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.
Symmetry
Watch these:
https://www.youtube.com/watch?v=YFzktJNmnPU
https://www.youtube.com/watch?v=MtqtIiJsfiE&t=110s
Division
Watch these:
Basic division:https://www.youtube.com/watch?v=KGMf314LUc0&t=68s
Long division:https://www.youtube.com/watch?v=LGqBQrUYua4
Long division with 2-digit divisors:https://www.youtube.com/watch?v=HdU_rf7eMTI
Capacity
Capacity is the total amount of fluid / liquid that can be contained in acontainer. It is the word we use when we are measuring liquids. Sometools which measure capacity include a measuring jug and cups.
1 litre = 1000 millilitres¼ litre = 250 millilitres½ litre = 500 millilitres¾ litre = 750 millilitres
Converting between litres and centilitres
Remember:
To convert litres to centilitres multiply by 100.To convert centilitres to litres divide by 100.
1 litre = 100 centilitres2 litres = 200 centilitres3 litres = 300 centilitres
4.5 litres = 450 centilitres
Capacity
Watch these:
https://www.youtube.com/watch?v=VpRa5OY-Q-k
https://www.youtube.com/watch?v=TPFQd7ZjErU
Units of time• Weeks in a year: 1 year = 52 weeks
• Days in a normal year: 1 year = 365 days
• Days in a week: 1 week = 7 days
• Days in a fortnight: 2 weeks = 14 days
• Hours in a day: 1 day = 24 hours
• Minutes in an hour: 1 hour = 60 minutes
• Seconds in a minute: 1 minute = 60 seconds
Other terms you should know:
• Days in a leap year: 1 leap year = 366 days
• Days in a fortnight: 1 fortnight = 2 weeks = 14 days
Leap Year
Once in every four years, instead of 28 days,February has 29 days. Hence, it is a Leap Year.
How can I find if a year is a leap year?Divide the year by 4. If you get a remainder, then the year is not aleap year. If the number is exactly divisible by 4 and you get noremainder, then that year is a leap year.
Since in the first one we have a remainder, we can say that 1998 wasnot a leap year. On the other hand, when we divided the year 2000 by4, we did not get any remainder. Therefore we can say that it was aleap year.
4 1998499r2
4 2000500
Days in the months
30 Days has September,April, June and November.
All the rest have 31,Except for February, it’s the one,
Which only has 28 days clear,And 29 in each Leap Year.
Count the months on your knuckles and the grooves between yourknuckles. Leave out your thumb knuckle. Every month that lands on aknuckle is 31 days, every month that lands on a groove betweenknuckles is 30 days (or 28 for February).
12-hour and 24-hour clock times
There are two ways of telling the time: 12-hour or 24-hour clock times.
• In 12-hour clock times we use a.m. and p.m. to distinguish betweenmorning and evening. It runs from 1am to 12 noon and then from1pm to 12 midnight.
• In 24-hour clock times we use 00 to 11 hour digits to show morningtimes (instead of a.m.) and 12 to 23 hour digits to show eveningtimes (instead of p.m.) It uses the numbers 00:00 to 23:59(midnight is 00:00).
Examples: 10:30 a.m. = 10:303:50 p.m. = 15:50 (3 + 12 = 15)
6:15 p.m. = 18:15 (6 + 12 = 18)
9:00 p.m. = 21:00 (9 + 12 = 21)
clock times are converted to 24-hour by adding
12:00 to the time.
24-hour clock 12-hour clock
00:00 12:00 midnight
01:00 1:00 a.m.
02:00 2:00 a.m.
03:00 3:00 a.m.
04:00 4:00 a.m.
05:00 5:00 a.m.
06:00 6:00 a.m.
07:00 7:00 a.m.
08:00 8:00 a.m.
09:00 9:00 a.m.
10:00 10:00 a.m.
11:00 11:00 a.m.
12:00 12:00 noon
24-hour clock 12-hour clock
13:00 1:00 p.m.
14:00 2:00 p.m.
15:00 3:00 p.m.
16:00 4:00 p.m.
17:00 5:00 p.m.
18:00 6:00 p.m.
19:00 7:00 p.m.
20:00 8:00 p.m.
21:00 9:00 p.m.
22:00 10:00 p.m.
23:00 11:00 p.m.
24:00 12:00 midnight• a.m. (ante meridiem, meaning before midday) • p.m. (post meridiem, meaning past midday)
Timetables
A timetable is a detailed plan showing when events or activities willhappen; a list of the times when buses, trains, and planes leave andarrive; a list of the times when lessons in school happen, etc.
Different Types of Angles
Angles are measured in degrees ex. 24º, 90º, 115º, 180º, 230º, 360º.
The following are the six types of angles you should be familiar with:
• Acute angle (from 1º to 89º)
• Right angle (90º)
• Obtuse angle (91º to 179º)
• Angle on a straight line (180º)
• Reflex angle (181º to 359º)
• One complete turn / one whole revolution (360º)
Acute Angles (from 1º to 89º)
Obtuse Angles (from 91º to 179º)
Angles on a straight line (180º)
There are 2 right angles in an angle on a straight line90º + 90º = 180º
One complete turn / one whole revolution (360º)
There are 4 right angles in one complete turn90º + 90º + 90º + 90º = 360º
Angles in Compass Points
Remember:
When you move clockwise you move in the direction in which the handsof the clock move. When you move anticlockwise you move in theopposite direction in which the hands of the clock move.
From N, clockwise to E = 90ºFrom N, clockwise to S = 180ºFrom N, clockwise to W = 270ºFrom N, clockwise to N = 360ºFrom W clockwise to NE = 135ºFrom NW, anticlockwise to NE = 270ºFrom S, clockwise to NE = 225ºFrom NE, anticlockwise to NW = 90º
anticlockwise anticlockwise
Angles in Clocks
In one hour the minute hand (the big hand of the clock) turned anangle of 360º while the hour hand (the small hand of the clock) turnedan angle of 30º. Remember that the angle between any twoconsecutive numbers in a clock is 30º.
Therefore, the angle between 3 and 4 is 30º. The angle between 3and 5 is 60º. The angle between 3 and 6 is 90º (right angle).
1 hour passed
More about Angles
Study these:
⅓ right angle 30º
½ right angle 45º
⅔ right angle 60º
1 right angle 90º
2 right angles 180º
3 right angles 270º
4 right angles 360º
Finding Missing Angles in different Triangles
To find the missing angle:
Add 90º to the given angle.90º + 27º = 117º
Subtract the answer from 180º.180º - 117º = 63º
The missing angle is 63º.
Right-angled Triangle• 1 angle is a right
angle (90º)
?
Finding Missing Angles in different Triangles
To find the missing angle:
Add the two given angles together.80º + 60º = 140º
Subtract the answer from 180º.180º - 140º = 40º
The missing angle is 40º.
Scalene triangle• no equal sides• no equal angles
? 80º
60º
Finding Missing Angles in different Triangles
To find the missing angles:
Subtract the given angle from 180º.180º - 30º = 150º
Half the answer.150º ÷ 2 = 75º
The missing angles are 75º.
Isosceles Triangle• 2 equal sides• 2 equal angles
? ?
30º
Finding Missing Angles in different Triangles
To find the missing angle:
Double the given angle.70º x 2 = 140º
Subtract the answer from 180º.180º - 140º = 40º
The missing angle is 40º.
Isosceles Triangle• 2 equal sides• 2 equal angles
?
70º
Finding Missing Angles in Right Angles
1 right angle = 90ºTherefore the missing angle is90º - 62º = 28º
Finding Missing Angles on a Straight Line
angles on a straight line = 180ºTherefore the missing angle is180º - 77º = 103º
Finding Missing Angles in One Whole Revolution
one whole revolution = 360ºTherefore the missing angle is360º - 230º = 130º
one whole revolution = 360º63º + 50º + 75º + 110º = 298ºTherefore the missing angle is360º - 298º = 62º
Finding Missing Angles in a Perfect Cross
one whole revolution = 360º
Use the RULE: Opposite angles are equal!So I can say that angle x is 55º.
Since angles x and y are on a straight line, I can find angle y.180º - 55º = 125º
Once again use the rule that opposite angles are equal for angle z.I can say that angle z is 125º because it is opposite angle y.
Using a protractor
If you are asked to calculate the angle, you should find the angle bymeans of sums and calculations. But if you are asked to measure theangle you should use a protractor.
Make sure you know how to place the protractor on the angles, so as to measure and draw angles.
50º135º
Percentages (%)
In percentages the whole number is always 100%.
For example, 40% means 40 out of 100 =
You can change 40% into a fraction in its simplest form = =
10% = 80% = = 25% =
15% = 20% = = 50% =
60% = = 7% = 75% =
410
25
110
320
610
35
810
45
210
15
7100
14
12
34
Finding Percentages
To find a percentage of a given number:Example: 25% of 260
Write the percentage as a fraction over 100.Simplify the fraction.
25% = = ¼
Then use the fraction to find the answer
¼ of 260 = 654 260
65 x 1 = 65
When finding fractions of quantities,
remember to dividethe number by the denominator and
multiply the answer by the numerator.
Finding 10% of an amount
Examples:
• 10% of 650 650 ÷ 10 = 65
• 10% of 1150 1150 ÷ 10 = 115
• 10% of 880 880 ÷ 10 = 88
• 10% of 790 790 ÷ 10 = 79
• 10% of 100 100 ÷ 10 = 10
• 10% of 70 70 ÷ 10 = 7
To find 10% of an amount divide the
amount by 10
10100
110
Reduction in Prices including Percentages
Reduction in price = 60% of €20
of €2Ø = €12
The new price is €20 - €12 = €8
Reduction in price = 20% of €70
of €7Ø = €14
The new price is €70 - €14 = €56
6Ø1ØØ
2Ø1ØØ
SALE 20% OFF
SALE 60% OFF
Reduction in Prices including Percentages
Reduction in price = 50% of €40
of €4Ø = €20
The new price is €40 - €20 = €20
Reduction in price = 25% of €60
of €6Ø = €15
The new price is €60 - €15 = €45
5Ø1ØØ
2510Ø
SALE 50% OFF
SALE 25% OFF
Fractions Decimals Percentages
0.5 50%
0.25 25%
0.75 75%= 0.20 20%
0.10 10%
0.70 70%
It is important to study
these fractions
and decimals by heart.
Square Numbers
A square number is the answer of a number multiplied by itself.Examples: 4 x 4 = 16
9 x 9 = 81 Therefore, 16 and 81 are called square numbers.
The square numbers which we use frequently are: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
More examples: The square number of 15 (152) is 225The square number of 30 (302) is 900
If the square number of 16 (162) is 256, what is 16 x 17?
• 16 x 16 = 256• 16 x 17 = 256 + 16 = 272• Therefore, 16 x 17 = 272
Remember 1 is the smallest square number
Sequences
31, 33, ____ , ____ , 39, ____ , 43, ____ .
Before you start writing the missing numbers, you should check if thenumbers are increasing (+) or decreasing (-) and by how much. In thiscase the numbers are increasing by 2.
31, 33, 35, 37, 39, 41, 43, 45.
3, 4, 6, ____ , 13, ____ , ____ , 31.
Pay attention because sometimes there is a pattern as to how numbersin sequences are increasing or decreasing. In the next sequence youhave a pattern. Once you realise what the pattern is, you can fill in themissing numbers.
3, 4, 6, 9, 13, 18, 24, 31.+1 +2 +3 +4 +5 +6 +7
+2 +2 +2 +2 +2 +2 +2
Even and Odd Numbers
An even number is a number that is divisible by 2 (÷2).
Ex. 2, 4, 6, 8, 10, 12, 14, etc.
If a number is not divisible by 2 it is an odd number.
Ex. 3, 5, 7, 9, 11, 13, 15, etc.
To find if a big number is odd or even look at the last digit. If thelast digit is even, the whole number is even and if the last digit is odd,the whole number is odd.
Ex. 19,356 is even because the last digit 6 is even.24,483 is odd because the last digit 3 is odd.67,272 is even because the last digit 2 is even.
even
odd
Proportions as Fractions
When writing proportions as fractions, the fractions should be in theirsimplest form.
red =
blue =
Proportion of the gridwhich is not coloured
yellow
green
1025
1025
25
25
2025
45
=
coloured
525
15
=
not coloured
610
35
=
410
25
=
Proportions - Problems
Problem 1: Chris worked for 10 days in the month of June, whatproportion of the same month did he work for?
Problem 2: Kurt has 40 sweets. 15 are chocolates. What proportionare not chocolate?
40 – 15 = 25 not chocolate
Problem 3: During the months of July and August, Maria had 18 pianolessons. What proportion of the same months she didn’t have lessons?
62 – 18 = 44 no lessons
1030
13
=
2540
=58
=
4462
=2231
=
Problem 4: Colour the proportions shown in the following grid:
orange
purple
pink
orange (colour 1 box orange)
purple (colour 4 boxes purple)
pink (colour 15 boxes pink)
120
15
34
Since in the grid there are 20 small squares, before colouring in the proportions you should
change the fractions to having a denominator 20.
120
15
420
=x4
x4
34
1520
=x5
x5
Three-Dimensional Shapes
Three-dimensional shapes are also known as solid shapes. The followingare the three-dimensional shapes (3-d) which you should be familiar to:
cylinder cube sphere prism
pyramid cuboid cone
Pyramids and Prisms
What is the difference between a pyramid and a prism?• In a pyramid the sides coming out of the base meet at a point (like
the pyramids of Egypt).• In a prism the base and the lid have the same shape.
pyramid prism
Faces, Edges and Vertices
Three-dimensional shapes have faces, edges and vertices.
Remember:
2 vertices, but 1 vertex
Faces, Edges and Vertices
Shapes Names Faces Edges Vertices
cube 6 12 8
cuboid 6 12 8
cylinder 3 2 0
cone 2 1 1
pyramid 5 8 5
sphere 1 0 0
Faces, Edges and Vertices – Pyramids and Prisms
Shape Faces Edges Vertices
Square Pyramid 5 8 5
Triangular Pyramid 4 6 4
Rectangular Pyramid 5 8 5
Pentagonal Pyramid 6 10 6
Hexagonal Pyramid 7 12 7
Octagonal Pyramid 9 16 9
Triangular Prism 5 9 6
Pentagonal Prism 7 15 10
Hexagonal Prism 8 18 12
Octagonal Prism 10 24 16
Nets of Three-Dimensional Shapes
When a three-dimensional shape is ‘opened’ the shape formed by itsfaces is called the net of that particular shape.
cube cuboidcone
cylinder
pyramid prism
Top Related