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Topic
3
Looking for Patterns in
a Decimal Chart
Lesson
1
Understand it!
There are patterns
in decimal number
charts. Continue
the pattern to
label the other
squares.
Learning how and when
to look for a pattern
can help when solving
problems.
0·01 0·02 0·03
0·08
0·15 0·16
0·1
0·19
0·29
0·32
0·34
0·37
Another Example
Using the same system as above, you could fill in the
diagonals of a decimal number chart.
ge
s
In this decimal number chart, what
is the pattern in the diagonals?
0·55
pa
0·55
m
pl
2 more tenths
than in 0·75
Sa
Guided Practice
Find the pattern for each of the
following and then complete.
1
2
1 more tenth
1 more hundredth than in 0·55
0·86
1 more tenth
1 more hundredth than in 0.75
0·75
e
2 more tenths
than in 0·55
0·66
0·95
Independent Practice
Find the pattern for each of the following and then complete.
3
0·12
0·14
4
0·61
0·34
0·81
0·42
0·16
0·84
5 0·55
0·67
46
forty-six
ACMNA124: Investigate everyday situations that use integers. Locate and represent these numbers on a number line. ACMNA133: Continue
and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence.
What are the missing decimals?
0·01
As you work with
vertical columns,
you will see the
tenths increase by 1
and the hundredths
stay the same as you
move down.
What are the missing decimals?
0·01
0·29
0·11
Moving from left to right, tenths are the
same in each row except for the last
number; the hundredths increase by 1.
0·21
0·31
0·26 0·27 0·28 0·29 0·30
Continue each pattern.
70·39, 0·40 ,
???
,
???
,
???
,
, 2·931, 2·932,
???
???
8
???
10
,
???
???
,
,
???
???
, 1·306, 1·406
, 3·43, 3·23,
???
Problem Solving
m
pl
e
9
???
pa
ge
s
6Dane drew a grid of five cells in a row. The number 0·075 was in the middle cell.
What did Dane’s grid look like?
Sa
11a Some of the numbers are hidden. Fill in the missing numbers.
0·10
0·28
b
Describe the pattern.
forty-seven
47
Topic
3
Estimating Sums and Differences
Lesson
2
How can you estimate with decimals?
Understand it!
To estimate means to find an approximate answer or solution.
Changing numbers to
other numbers that are
easy to compute can
help you estimate the
sums or differences of
decimals.
The 10-second barrier in the 100-metre sprint was broken in
the 1968 Olympics, when the winning time was 9·95 seconds.
Mrs Johnson, the physical education teacher, ran the 100 metres
in 14·7 seconds. About how much
faster was the 1968 Olympic
time than Mrs Johnson’s time?
14·7
seconds
9·95
seconds
Guided Practice
11·769 + 0·686 = 1·
???
???
220·45 – 13·15 =
m START
plFINISH
e
pa
ge
s
Complete each estimate by rounding to the nearest tenth.
+ 0·
???
???
=
·5
???
– 13·2 =
Round each number to the nearest whole number to estimate the answer.
31·456 + 5·4 + 14·08 =
472·43 – 59·8 =
???
???
–
Reasoning
???
+
???
=
???
+
=
???
???
Sa
5Kirra and Jeremy want to go to a movie and have popcorn and a drink.
A movie ticket costs $7.75 and the popcorn and drink combo costs $7.85.
Kirra says if they bring $15, it will be enough. Jeremy says they each
need more than $15. Who is correct? Why?
Independent Practice
Round each number to the nearest whole number to estimate each answer.
620·791 + 5·25 + 3·84 =
7$10.10 – $3.65 =
48
forty-eight
???
???
–
+
???
+
=
???
=
???
???
ACMNA128: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the
reasonableness of answers
One Way
Another way
Use rounding to quickly estimate
sums and differences. Round each
number to the same place value.
Use just the whole numbers to make an
estimate, and then adjust the estimate using
the remaining digits.
Round each number to the
nearest whole number.
14·7
14
2 9·95
29
5
14·7
15
2 9·95
2 10
5
The difference is about 5 seconds.
Since 0·7 , 0·95, the
difference is less than 5.
The 1968 Olympic time is less than 5
seconds faster than Mrs Johnson’s time.
Use just the whole numbers to estimate each answer.
???
–
+
???
???
=
+
???
10 Circle the best estimate for 0·375 + 2·46.
A 2·5
B3
???
=
???
ge
s
991·26 – 30·463 =
???
pa
87·12 + 2·501 + 9·2 =
C3·5
D4
Sa
m
pl
e
11Rachel is shopping and needs to buy bread, ham and
apples to make lunch. She has a ten-dollar note. Will she
have enough money for her purchases? Use estimation to find
whether she will have enough money. Explain your answer.
Problem Solving
12a The difference between the heights of two sisters is about 0·25 m.
How tall might each sister be?
b
Measure the difference in height between you and a classmate.
Express this as both a fraction and a decimal of a metre.
forty-nine
49
Topic
3
Multiplying Decimals
by 10, 100 or 1 000
Lesson
3
Understand it!
$0.45 per kg
What is the pattern when multiplying
decimals by 10, 100 or 1 000?
Patterns can be used
when mentally
multiplying decimals
by multiples of 10,
100 and 1 000.
$2.89 per kg
A baker buys some of the ingredients he uses in bulk.
He needs to purchase 10 kg of pecans and 100 kg of
flour. How much will the baker spend for each amount?
Choose an Operation Multiply to join equal groups.
Mental Computation
???
2 3·1
???
???
Multiply each number by 10.
50·006
???
6 2·3
7 3·08
m
pl
9In the problem at the top of the page, how much
would the baker spend if he buys 10 kg of flour?
???
4 7·4
8 1·34
???
???
???
100 kg of flour?
???
Sa
Reasoning
???
e
Guided Practice
3 0·45
pa
10·009
ge
s
Multiply each number by 100.
10 Marcy and David each multiplied 5·6 × 10 and 0·721 × 100. Marcy got 0·56 and 7·21
for her products. David got 56 and 72·1 for his products. Which student multiplied correctly?
Explain your answer.
Independent Practice
Type of coin
Number Saved
The table shows the coins saved by Tina and her sister for one
year. Use it to answer each of the following.
5c
1 000
11Find the total value of each type of coin the girls have saved.
10c
100
20c
1 000
$2
10
5c
???
10c
???
20c
???
$2
???
50
fifty
ACMNA130: Multiply and divide decimals by powers of 10
Cost of the pecans
$2.89 per kg
$0.45 per kg
Cost of the flour
$2.89 × 10 = $28.90
$0.45 × 100 = $45
When multiplying by 10,
each number moves to the
next place value.
When multiplying by 100,
each number increases
100 times, therefore it
moves 2 place values.
12 Find the total value of the coins saved by the sisters.
???
Complete the following.
???
× 34·2 = 342
17
???
× 68 = 68 000 15 90·3 ×
Sa
14
m
pl
e
pa
ge
s
13 The principal of Middleton School has a big
glass jar of marbles. The empty jar weighs about
400 grams, and each of the 1 000 marbles weighs 6·7 grams.
Find the total weight in grams of the jar of marbles.
???
18 347·2 × 1 000 =
???
= 9 030
???
16 7·04 × 100 =
19 603·2 ×
???
???
= 6 032
Problem Solving
20a Marian multiplied a number by 100 and got an answer between 280 and 289.
What might the number have been?
???
b
What is the smallest and largest number it could have been, to two decimal places?
smallest
largest
???
???
fifty-one
51
Topic
3
vv
Multiplying Decimals
Lesson
4
How can you multiply whole numbers and decimals?
Understand it!
Barry displayed four paintings side-by-side in one row.
Each painting has the same width. What is the total
width of the four paintings?
Choose an Operation Multiply to find the total
width of the four paintings.
Multiplying decimals is
similar to multiplying
whole numbers. Placevalue relationships help
you determine where to
place the decimal point
in the product.
ge
s
Each is 0.36
metres wide.
Another Example
How can you multiply a decimal by a decimal?
pa
Find 0·5 × 0·3. Use what you know about the numbers to help you calculate the answer.
The same operation can be said in different ways.
Another Way
e
One Way
0·5 is the same as one half.
0·5 =
m
pl
Use a model to show 0·5 by shading
the first 5 columns.
1
2
0·3 = 0·30 or 30 hundredths
The product is the area where the
shading overlaps.
Half of 0·3 or half of thirty
hundredths is 15 hundredths.
Sa
Show 0·3 by shading the first 3 rows.
0·5 × 0·3 = 0·15
0·5 × 0·3 = 0·15
Guided Practice
Multiply the parts and then add the parts to calculate the answer for each of the following.
14 × 0·32 = (4 × 0·3 + 4 × 0·02) =
27 × 2·1 = (7 ×
???
+7×
???
???
)=
+
???
???
+
=
???
???
=
???
Independent Practice
Mentally calculate the answer for each of the following.
310 × 0·32 =
52
fifty-two
???
4 100 × 0·129 =
???
5 2·36 × 1 000 =
???
ACMNA129: Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating
decimals, with and without digital technologies
What You Think
Find 0·36 3 4.
Multiplying 0·36 m 3 4 is like adding 0·36
four times on a decimal model.
Unit
n
n
Unit
Ud
n
d
Frac
M
Simp
What
Write
Fac
FYou
D Clear
d
Fix
Multiplying
the Op
parts
is one way to calculate
Op1
2
%
the
answer.
Int
Mode
?
MR/MC
1000.
n
100.
%
Taking
notes helps keep track of the parts.
n
n
πd
Ud
Frac(
)
0·3 × 4 = 1·2 M
F D
7 × 4Fac=80·24
9
0·06
MR/MC
Simp
d
Fix
%
1.
1000.
%
0.1
π
0.01
Op1
Op2
Add
1·2
4 the parts:
5 Int
6 + 0·24.
–
The total width is 1·44 metres.
2 multiply
3
( 1 can) also
+ a calculator.
You
using
7
0.001
80
9.
Press: 0·36
4
5
6
Display: 10.
1.
0.1
0.01
10.
1
2
(–)
–
4
ENTER
=
1,44
TI-15
3 keys
+
ge
s
100.
The product is the total area shaded.
0·36 m 3 4 5 1·44
?
Mode
75
TI-15 display
Multiply the parts and then add the parts to calculate the. answer for ENTER
each of the following.
(–)
0.001
0
=
???
7 6 × 0·28 =
???
8 3 × 0·134 =
???
pa
60·7 × 12 =
m
pl
e
9Mary is making a patchwork quilt. Each panel is TI-15 keys
12·5 cm wide. She has 8 panels for the first row.
How wide is her quilt so far?
???
Sa
10If Mary’s quilt is to be 1·25 m wide, how many
more panels does she need for the first row?
???
Problem Solving
11a Jen solved this number sentence: 9 × 0·989 = 89·01. How can you use estimation
to show that Jen’s answer is incorrect? What is the correct answer?
???
b
Explain where Jen went wrong.
fifty-three
53
Topic
3
Dividing Decimals by
10, 100 or 1 000
Lesson
5
Understand it!
How can you divide decimals by 10, 100 and 1 000?
Patterns can be used
when mentally dividing
decimals by 10, 100 or
1 000.
Sandra wants to cut a cloth into 10 strips. All the strips
should be exactly the same size. How wide will each strip be?
Choose an Operation Divide to find equal parts of a whole.
Mental Computation
Mentally calculate the answer for each of the following.
472·5 ÷ 10 =
???
???
2 126·4 ÷ 100 =
???
pa
1370·2 ÷ 10 =
ge
s
89·5 cm
5 28·14 ÷ 100 =
???
6 42·5 ÷ 1 000 =
???
e
Guided Practice
???
3 684·5 ÷ 1 000 =
m
pl
7If Sandra wanted to cut the 89·5 cm cloth into 100 strips,
how wide would each strip be? ???
Sa
8Using the table at the top of the page, why was it necessary to place a zero in the tenths place
when dividing by 1 000?
Independent Practice
Pacific School posted the winning swimming times.
Use the table to answer each of the following.
9What was the time per metre of
the swimmer who swam the butterfly?
50-metre freestyle
22·17 seconds
100-metre backstroke
53·83 seconds
100-metre butterfly
58·49 seconds
???
10 If the 50-metre freestyle swimmer could swim the 100-metre freestyle
in exactly double the 50-metre time, what would be the time per metre?
11What was the time per metre of the swimmer who swam the backstroke?
54
fifty-four
ACMNA130: Multiply and divide decimals by powers of 10
???
???
Notice the patterns in the table.
Find 89·5 4 10.
The quotient of a number
divided by 10, 100 or 1 000
is less than the number.
Divide
by
Examples
Numbers move
to the right
1
12·5 4 1 5 12·5
0 places
Dividing by 10 decreases the
value of each digit.
10
12·5 4 10 5 1·25
1 place
100
12·5 4 100 5 0·125
2 places
Since place value is based
on 10, dividing by 10, 100 or
1 000 gives the same result
as moving the numbers to
the right.
1 000
12·5 4 1 000 5 0·0125
3 places
So, 89·5 4 10 5 8·95
Each cloth strip will be 8·95 centimetres wide.
pa
ge
s
12 How is dividing 360 by 10 similar to dividing 3 600 by 100? Explain.
Problem Solving
Sa
m
pl
e
13a Helen saved for 10 weeks to buy a gym membership.
The membership fee was $315.00.
How much did she save each week?
???
b
If she goes to the gym every day for 100 days, how much has her membership
cost her per day?
???
fifty-five
55