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Year 9
NUMBER – Part 2
Recall: Adding and Subtracting Decimals
Decimals are added and subtracted in the same way as
whole numbers
e.g.
Sometimes we need to add
extra zeros to show how
decimals line up vertically
Adding and Subtracting Decimals
The reading on a cars odometer at the start of a trip is
shown below. What will the reading be at the end of a
196.3 km journey?
4
+
1
9
6
.3
3
2
9
2
1
ALPHA IWB
Ex 2.03-2.04 pg 37-38
Ex 2.05-2.06 pg 40-42
Starter
The shape below, outlined in
green is 1 whole unit
What fraction of a whole does
1 column represent? _______
1/10
What fraction of a whole does
1/10
1 row represent? __________
What decimal fraction of a
whole does 1 square
0.01
represent? __________
How many small squares have
both kinds of shading
(purple)? __________
Write this as a decimal
fraction of the whole square
__________
Starter
The shape below, outlined in
green is 1 whole unit
What fraction of a whole does
1 column represent? _______
1/10
What fraction of a whole does
1/10
1 row represent? __________
What decimal fraction of a
whole does 1 square
0.01
represent? __________
How many small squares have
both kinds of shading
(purple)? __________
42
Write this as a decimal
fraction of the whole square
__________
0.42
Today we are learning to multiply decimals
Multiplying decimals follows the same rules as
multiplying whole numbers
e.g. Multiply 4.1 x 2.9
4 x 3
= 12
MY CALCULATOR IS BROKEN!! Write the number in
the display column as it should appear with the decimal
in the correct position
Correct
Display
2.73
1.408
1.89
21.6
36.04
12.093
16
0.9702
7.14
1.44
24.867
0
28.50
Remember to
estimate
first!
Note 2: Multiplying Decimals
• Multiplying decimals follows the same rules as
multiplying whole numbers
e.g. 0.3 x 0.22 = 0.066
1 dp + 2 dp
= 3 dp
You might need to
add an extra zero
ALPHA IWB
Ex 2.08 pg 47 no calc
Ex 2.09-2.10 pg 48-49
Fill the 1 litre jug.
Pour out 0.3 litres into
the small jug 3 times.
There is now ___
0.1 L
left in the large jug.
Add 1 small jug (0.3 L)
to the large jug.
Starter
If 5 rabbits eat 12.5 carrots in 5 days, how many
carrots will 10 rabbits eat in 10 days?
Dividing Decimals by a whole number
Dividing Decimals by a whole number
7
.7
42
.42
20
10
125
1.25
= 0.52
= 7.75
= 0.0068
= 0.655
= 0.071
= 0.01937
Dividing Decimals by a whole number
Sometimes you need to add extra zeros to
make division easier
. 34
e.g.
0
.0
0
Note 3: Dividing Decimals
e.g.
Sometimes you need to add extra zeros to
make division easier
e.g.
Note 3: Dividing Decimals by decimals
We can make the question simpler by dividing by a whole
number instead. In order to do this we must move the decimal
point in both number by the same amount
e.g.
has the same answer as
has the same answer as
ALPHA IWB
Ex 2.14 pg 54 - 55
Ex 2.16 pg 57
Ex 2.17 pg 58
Starter
0.22222222222
0.142857142..
0.5
0.125
0.8
1 dp
Terminating Decimals
3 dp
0.09
2 dp
Recurring Decimals
Note 4: Recurring Decimals
Recurring decimals go on forever in a
particular pattern.
e.g.
Sometimes a group of several digits repeat.
e.g.
Note 4: Recurring Decimals
We use dots or a line to show a repeating pattern
e.g.
e.g.
.
= 0.8
_
or
0.8
..
__
or 0.18
= 0.18
Note 4: Recurring Decimals
e.g.
The dot is where the repeating pattern begins
These numbers have an infinite number of decimal places
ALPHA IWB
Ex 2.18 pg 60
Ex 2.19 pg 61
Note 5: Rounding Decimals
Finding an approximate answer is called rounding
Rules for rounding
Decide how many decimal places you will round
to. This is the last digit of your answer.
Look at the next digit (to the right of that place)
• If the digit is 5, 6, 7, 8 or 9, round up.
• If the digit is 0, 1, 2, 3 or 4, leave the digit unchanged
(round down)
e.g. Round the following to 3 dp
0.1841 = 0.184
0.0128 =
0.013
0.4555 = 0.456
e.g.
Nearest whole #
21 cm
1 dp
214 mm
21.4 cm = _____
In mathematics, we often get an answer more accurate
than we need, therefore we must learn to round sensibly.
TIP – don’t give answers that are
more accurate than the question.
e.g.
A sensible answer would be 34.7 m
ALPHA IWB
Ex 2.20 pg 63
Ex 2.21 pg 65-66
In mathematics, we often get an answer more accurate
than we need, therefore we must learn to round sensibly.
TIP – don’t give answers that are
more accurate than the question.
Starter
A seedling on average grows 114 mm per week.
How much does it grow per day?
114 / 7 = 16.28571429
A sensible answer would be 16 mm or 16.3 mm
Starter
Write down the fraction shown
in each of these diagrams
Write these fractions in
order, smallest to largest.
19/20
1/10
3/4
1/5
Draw a diagram
to represent 2/5
Note 6: Sharing & Fractions
We can compare measurements and numbers
of objects using fractions
5
6
3 m
The total length of the post is ____
2
3
Note 6: Fractions on Number Lines
Any number line can be split up into equal lengths to
show fractions
What fraction of a centimetre is
one mm?
Where is
7
10
43
Where is
100
1
10
on this ruler ?
on this ruler ?
ALPHA IWB
Ex 7.02 pg 165-166
Ex 7.03 pg 169
1
10
C
3
10
J
6
10
10
=
10
K
A
1
Starter
3
4
H
4
4
9
4
=0
I
I
Note 7: Equivalent Fractions & Simplifying Fractions
Sometimes two fractions can be represented by the same point on a
number line. These two fractions are equivalent.
4 x2 8
=
5 x 2 10
We can find equivalent fractions by multiplying (or
dividing) the top and bottom by the same number.
Note 7: Equivalent Fractions & Simplifying Fractions
(scaling up)
4 x2 8
=
7 x 2 14
Other possibilities are:
x3
12 16
21 28
x4
20
35
x5
Note 7: Equivalent Fractions & Simplifying Fractions
In math, we always want to give our answers in simplest form.
We often need to change fractions to their equivalent simplest form.
6
The fraction
9
can be simplified to
2
3
Note 7: Equivalent Fractions & Simplifying Fractions
What is the highest common factor of the
numerator and denominator?
4
20
The fraction
can be simplified to
7
35
ALPHA IWB
Ex 7.04 pg 173
Ex 7.05 pg 176-177
Simplify
12
120
=
=
18
180
35
=
63
5
9
2
3
Starter
Write an equivalent fraction for the following
3 =
4
6
8
6
12
=
11
22
20
10
=
13
26
2
1
=
34
17
1
48
720 = 15
Simplify
12
=
20
3
5
Round to 2dp
0.9650
1.3460
2.4319
= 0.97
= 1.35
= 2.43
Write as a recurring decimal
.
= 1.532
1.532222
..
= 4.312. .
4.3121212
3.99012012 = 3.99012
Note 8: Adding and Subtracting Fractions
To add two fractions with the same denominator – add the
two numerators and leave the denominator unchanged
5 2 5 2 7
+ =
=
9 9
9
9
7
5
3
7  3 10
+ =
=
=
12 12
6
12
12
Note 8: Adding and Subtracting Fractions
To subtract two fractions with the same denominator –
subtract the two numerators and leave the denominator
unchanged
5 2 52 3
1
− =
=
=
9 9
9
3
9
7
73 4
3
1
−
=
=
=
12 12
12
12
3
Note 8: Adding and Subtracting Fractions
with different denominators
1. Change the fractions to equivalent fractions with the same
denominator
2. Add the equivalent fractions
3. Simplify if possible
4 1
8
1
7
− = − =
5 10 10 10 10
1
1
3
4
7
+
+
=
=
4
3 12 12 12
ALPHA IWB
Ex 7.06
pg 179
Ex 7.07,7.08 pg 182
Ex7.09
pg 184-185
Puzzle pg 183
Starter
Each half could be cut into 3 equal pieces
What fraction of the pizza is each slice?
x
1
2
1
1
x
=
3
6
x
1
3
Note 9: Multiplying Fractions
This circle is split
into 20 equal parts
e.g.
simplify
Try These!
8
=
15
5
=
24
9
=
20
9
=
16
21
=
40
3
=
20
1
=
6
21
=
100
3
=
50
4
=
15
Multiplying a fraction by a whole number
e.g. 1
x 3= 3
1
4
4
1 x 4= 4
8 1 8
1
=
2
ALPHA IWB
Ex 8.01 pg 189
#1 k-o, #2-4
Ex 8.02 pg 190-191
Extension
3 1 1
3
x x =
4 5 3
60
1
=
20
Starter
1
3
18
5
23
+
=
+
=
6
5
30
30
30
Note 10: Reciprocals of Fractions
Dividing by fractions
To get the reciprocal of a fraction, just turn it
e.g.
What do you get when
you multiply a fraction
by its reciprocal?
3 4 12
x =
4 3 12
= 1 (always)
Note 10: Reciprocals of Fractions
Dividing by fractions
e.g.
Note 10: Reciprocals of Fractions
Dividing by fractions
To divide by a fraction, we multiply it by its reciprocal
e.g.
5 4 20
x =
1 1
1
20 slices of apples
= 20
e.g.
=
3 3 9
x =
5 2 10
Try These
3
2
2
3
10
7
4
=4
1
1
3
=1
3
4
6
5
2
7
1
7
1
100
1
85
ALPHA IWB
Ex 8.04 pg 194
PUZZLE pg 195
IWB pg 194
Starter
1
1
1
x =
4
3
12
There are 12 pieces of
pumpkin
3
x 32 = 24
4
8 belong to girls
24 belong to boys and __
Note 11: Mixed Numbers and Improper
Fractions
A mixed number is a combination of a counting
number and a fraction smaller than a whole.
e.g.
3
What this really means is 4 +
4
An improper fraction is when our numerator is
larger than our denominator.
We can also write this using mixed numbers.
3 7 21
e.g. x =
5 2 10
(improper fraction)
21
20 1
is the same as
+
10
10 10
= 2+
1
1
2
(mixed number)
or
10
10
11
e.g. Change
to a mixed number
5
Try these – Change to Mixed Numbers
1
= 2
3
5
= 1
6
4
= 5
5
7
=1
10
2
=1
17
1
= 12
3
= 24
5
= 10
6
=
4
5
=2
12
2
e.g. Change 3 to an improper fraction
5
You try! 4 5 = 4 + 5
9
9
36 5
= +
9 9
41
=
9
ALPHA IWB
Ex 8.05 pg 197-198 #2-8
Ex 8.06 pg 199
Ex 8.07-08 pg 202
Workbook
Ex D page 30
Ex I page
In problems involving mixed numbers – change to an
improper fraction first, then solve. At the end, turn the
answer into a mixed fraction.
e.g.
Starter - 7 minute challenge!
Starter Using mixed numbers
=8
=
3
-1
4
32
4
-
7
4
1
25
=6
=
4
4
3
1
1 =
2
2
1
3
2
7
=
2
3
7 21
1
x
= =5
2
2
4
4
Note 12: Tenths, hundredth, thousandths
Note 12: Tenths, hundredth, thousandths
e.g. Write as a decimal
7
= 0.7
10
22
0.22
=
100
7
0.007
=
1000
1
1
= 1.1
10
1
4
4.01
=
100
42
75
75.42
=
100
Note 12: Tenths, hundredth, thousandths
e.g. Write as a fraction
3
6
0.6 =
=
5
10
39
=
0.39
100
953
=
0.953
1000
e.g. Write as a mixed number
75
2.75 = 2
100
3
= 24
1
4.01 = 4
100
999
7.999 = 7
1000
ALPHA IWB
Ex 8.10 pg 208
Ex 8.09 pg 203-204
Starter
7
1
10
1
3
19
1 + 2
= 3
5
4
20
Note 13: Changing between Decimals &
Fractions
Decimal → Fraction Decide first what number to write in the
denominator (i.e 10, 100, 100) based on the number of decimal
places. Don’t forget to simplify your fraction!
Try these!
8
0.8 =
10
= 4
5
48
0.48 =
100
12
=
25
125
0.125 =
1000
= 1
8
How could we check that
our answer is correct?
Note 13: Changing between Decimals &
Fractions
Fraction → Decimal Divide the numerator (top) by the
denominator (bottom)
Try these! (round to 3 dp where appropriate)
9
= 0.375
24
2
3 = 3.286
7
5
= 0.185
27
2
7
=
7.08
25
632
= 0.632
1000
1
34
= 34.013
78
Note 13: Comparing Fractions
It is useful to change fractions to decimals when
comparing fractions
9
7
e.g. Which fraction is larger
or ?
20 16
Changing each to a decimal
9
= 0.45
20
7
16 = 0.4375
9
7
Therefore,
>
20 16
Note 13: Comparing Fractions
Write each as a fraction, and then, change each to a decimal
17
= 0.17 at Northbank
100
16
80
= 0.2 at Southridge
Therefore, Southridge had
a worse rate of infection
ALPHA IWB
Ex 8.11 pg 209
Ex 8.12 pg 210
Ex 8.13 pg 212-213
Note 14: Changing between decimals,
fractions & percentages
Per cent means ‘out of 100’
The symbol we use is %
e.g. 50% means
50
Which simplifies to
100
1
2
Therefore, we can also say that half of something is also 50%
Note 14: Changing between decimals,
fractions & percentages
What percentage of these blocks are shaded?
20
100
= 20%
28
100
= 28%
Note 14: Changing between decimals,
fractions & percentages
Match each diagram with the most likely percentage
= 12%
= 85%
= 40%
= 50%
= 20%
Note 14: Changing between decimals,
fractions & percentages
Percentages are like fractions with a
denominator of 100
Change these percentages to a fraction
e.g 27% =
27
100
42
42% =
100
=
21
50
16% =
16
100
=
4
25
Note 14: Changing between decimals,
fractions & percentages
To change a fraction to a percent we
multiply by 100
Change these fraction to percentages
e.g
6
10
=
0.6 x 100
= 60 %
ALPHA IWB
Ex 9.02 pg 220
Ex 9.03 pg 222
Ex 9.04 pg 224
31
= 0.861 x 100
36
= 86.1%
55
= 0.55 x 100 = 55 %
100
17
42
= 0.405 x 100 = 40.5 %
Starter
Round to the nearest dollar
a.) $3.19
b.) $45.50
$3.00
$46.00
Calculate
a.) ¼ of 56
= .25 x 56
= 14
b.) 70% of 85
= 0.7 x 85
= 59.5
c.) $199.99
$200.00
c.) 0.1 of 30
= 0.1 x 30
=3
If there are 25 students in the class and 88% of the students
passed the test, what fraction of the class did not pass?
12% did not pass
0.12 x 25 = 3
3
did not pass
25
How could you work out what the
deposit would be?
Note 15: Percentages of Quantities
x
To work out a percentage of a quantity, we multiply the
quantity by the percentage (as a decimal).
(or a fraction)
e.g. Find 25% of $80
= .25 x $80
1 $80
$80
= x
=
4
1
4
= $20
= $20
What would be the down payment?
0.1 x $168 000
$16 800
Note 15: Percentages of Quantities
e.g.
= 0.6 x $300
= $180
e.g.
= 0.25 x $82.60
= 20.65
Workbook
Ex E pg 78
ALPHA IWB
Ex 9.05 pg 227-228
Note 16: One amount as a percentage of
another amount
Write as a fraction first,
then multiply by 100 to turn it into a percentage
e.g.
= 60 %
Note 16: One amount as a percentage of
another amount
e.g. Write each of the first quantities as a percentage of the second
20 kg, 300 kg
18 m, 72 m
18 m, 7.2 km
7200 m
1 mm, 5 cm
50 mm
20
300
x 100 = 6.67 % (2 dp)
18
x 100
72
18
x 100
7200
1
x
100
50
= 25 %
= 0.25 %
= 2%
45
$45, $900 900 x 100
330
330 mL, 2 L 2000 x 100
2.5 ha, 50 000m2 2.5 x 100
5
= 5%
= 16.5 %
= 50 %
When
calculating
percentages
to compare
quantities,
the units
must be the
same
July
August
24
80
8
25
x 100 =
x 100 =
30 %
32 %
Yes, the rate of speeding has increased from 30% of
cars in July to 32% of cars in August
ALPHA IWB
Ex 9.06 pg 230-231
Note 17: Percentage increases and decreases
We compare the change to the old (original) quantity
e.g. The price of a movie ticket increases from $8 to $10.
What percentage has it increased by?
Change = new price – old price
= $10 – $8
= $2
% Increase = change
original
= $2 x 100
$8
= 25 % increase
Note 17: Percentage increases and decreases
We compare the change to the old (original) quantity
increases
25 % as a decimal
x (1 + 0.25)
$8.00
x
1.25
= $10.00
Note 17: Percentage increases and decreases
What would be the new cost of a movie ticket if we increased the price by:
a.) 30%
b.) 40%
c.) 5%
$ 8 x 1.3 = $ 10.40
$ 8 x 1.4 = $ 11.20
$ 8 x 1.05 = $ 8.40
increases
x (1 + r)
$8.00
x
(1 + r) =
% as a decimal
Note 17: Percentage decreases
We compare the change to the old (original) quantity
e.g.
The cost of a dress has dropped from $35 to $28.
What is the percentage discount?
Change = new price – old price
= $28 – $35
= − $7
% decrease = change
original
= $7 x 100
$35
= 20 % decrease
Note 17: Percentage increases and decreases
We compare the change to the old (original) quantity
decreases
20 % as a decimal
x (1 − 0.2)
$35.00
x
0.8
= $28.00
Note 17: Percentage increases and decreases
What would be the new cost of the dress if we decreased the price by:
a.) 30%
b.) 40%
c.) 5%
$ 35 x 0.7 = $ 24.50
$ 35 x 0.6 = $ 21.00
$ 35 x 0.95 = $ 33.25
decreases
% as a decimal
x (1 − r)
$35.00
x
(1 - r)
=
Note 17: Percentage increases and decreases
x (1 + r) increases
x (1 − r) decreases
= change x 100
original
r as a decimal
ALPHA IWB
Ex 9.07 pg 235-236