1: Stage 8: Advanced Proportional
Welcome to Module Twelve
This is one of seven modules describing effective teaching practices.
Once you have finished you will be able to select and implement learning activities appropriate for students working
at stage 8: advanced proportional.
2: General principles for the teaching and learning activities presented:
o
o
o
o
o
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Activities are designed to strengthen students’ understandings within the ACTUAL STAGE that they are
linked to. Teachers wishing to move students into the next strategy stage need to select activities linked to
that NEXT STAGE.
Each of the teaching modules is structured under the following headings:
o Strategy stage review
o Knowledge being developed
o Activities to develop knowledge
o Strategy being developed
o Activities to develop strategy.
The knowledge component is described first as the knowledge outlined provides a foundation for the
development of strategy at each stage.
The activities assume that students understand the ideas from earlier stages so it is important that teachers
check that students have these understandings.
The activities do not necessarily require the purchase of specialised materials. In general they use
resources readily available within New Zealand schools or easily accessible from the Project Materials
section of the nzmaths website.
The activities presented here are provided as a starting point. Teachers are encouraged to develop other
activities using the methods presented here as the basis of an approach. Further ideas for activities can also
be found from the number section of nzmaths, from the Numeracy Development Project Books, and by
using the Numeracy Planning Assistant.
3: Strategy stage review
Stage 8: Advanced proportional
Students at this stage are able to choose appropriately from a range of part-whole strategies to solve problems with
whole numbers, integers, and decimals. At stage 8 students are extending their ability to add and subtract factors by
learning to operate on fractions with unlike denominators. They are learning to combine ratios and proportions.
Students at this stage use a range of multiplicative strategies to solve problems with whole numbers. At stage 8
students learn to solve multiplication problems involving fractions and decimals. Strategies developed at this stage
include partitioning of fractions and conversions between fractions and decimals. Students are also learning to solve
simple division problems with decimals.
Students at this stage are able to use basic multiplication and division facts and apply their knowledge of factors to
solve a range of problems that include fractions and ratios. At stage 8 students are learning to use proportional
thinking strategies to solve problems involving percentages, ratios and rates.
The next 3 slides show video clips of students solving addition and subtraction, multiplication and division and
proportional problems.
The next 3 slides show video clips of students solving addition and subtraction, multiplication and division and
proportional problems.
4: Strategy stage review
Stage 8: Advanced proportional, addition and subtraction domain
Video Clip
5: Strategy stage review
Stage 8: Advanced proportional, multiplication and division domain
Video Clip
6: Strategy stage review
Stage 8: Advanced proportional, proportional domain
Video Clip
7: Knowledge being developed
At this stage it is important to continue developing students’ knowledge in the Number Sequence and Order, and
Grouping / Place Value domains. This knowledge will enable them to use a range of proportional strategies to solve
problems involving fractions, decimals, percentages and ratios which is characteristic of students working at the
advanced proportional stage.
In the Number Sequence and Order domain students need to be able to:
o
o
know the number of tenths, hundredths, and thousands that are in numbers up to three decimal places
know what happens when a whole number or decimal is multiplied or divided by a power of 10
In the Grouping and Place Value domain students need to be able to:
o
order fractions, decimals, and percentages
Students will need to know these things in order to use advanced proportional strategies.
8: Activities to develop knowledge
There are other areas of knowledge that students will continue to develop at Stage 8. More specifically students need
to develop the ability to:
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o
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say word number sequences by thousandths, hundredths, tenths, ones etc starting from any decimal
number (Number Sequence and Order)
recall fraction ↔ decimal ↔ percentage conversions for given fractions and decimals (Basic Facts)
identify common factors of numbers to 100, including the highest common factor (Basic Facts)
9: Knowledge being developed: number sequence and order
Students at this stage are learning to order fractions, decimals and percentages. Students should already be able to
order sets of fractions and sets of decimals to three decimal places. They will also be able to recall conversions
between benchmark fractions and decimals. At this stage students are learning to use this knowledge to order sets
that include mixtures of fractions, decimals, and percentages.
10: Activities to develop knowledge: number sequence and order
Comparing and ordering fractions, decimals, and percentages
Use a strip of paper to make a triple number line that shows fractions, decimals, and percentages. Fold the strip of
paper in half and ask the students what fraction, decimal and percent are on the half way mark.
Continue folding the paper to show quarters and ask the students what fraction, decimal and percentages are shown
by the folded lines. Mark these numbers on the triple number line.
Ask the students to order some fractions, decimals and percentages. For example, order 60%, 0.4 and 3/4 from
smallest to biggest. Direct students thinking by asking questions such as:
Which number would do you think would be the smallest? (0.3)
Why? (It is the only one less than a half)
How can you compare 60% and 3/4? (Change both to percentages)
Which is bigger 60% or 75%? (75%)
Students can check their ordering by marking the numbers on the number line. More numbers such as thirds, fifths
and tenths can be marked on the number line for reference as required.
11: Activities to develop knowledge: grouping and place value
Students at this stage are extending their knowledge of decimal numbers. They are learning that the base ten nature
of the number system extends to the decimal numbers. A foundational understanding is that the value of each place
is 10 times larger than the one to its right. When you multiply by 10 each digit effectively moves one place to the left,
as this makes the value of each digit ten times larger than it was previously. Just as there are 10 tens in a 100 and 10
hundreds in a 1000, there are 10 hundredths in a tenth and 10 tenths in 1. Students at this stage will learn what
happens to whole and decimals numbers when they are multiplied or divided by 10 and they will be able to name the
number of tenths, hundredths and thousandths in decimal numbers.
12: Activities to develop knowledge: grouping and place value
Multiplying whole numbers by a power of 10
Ask the students: What is 32 x 10?
Encourage the students to think of moving the digits a place value column to the left by asking:
What is 1 ten times ten? (1 hundred)
32 is made of 3 tens, what is 3 tens times ten? (3 hundreds)
What is 2 ones times by ten? (2 tens)
Now combine the 3 hundreds and the 2 tens, what is the answer? (320)
Continue with multiplying by 100 and then 1000.
Comparing decimals with equipment
The Pipe Decimals equipment animation shows how the size of two decimals can be compared using equipment.
Dividing whole numbers by a power of 10
Ask the students to divide $560 by ten. There are 50 tens in 500 and 6 tens in 60. If necessary show this using the
play money. Ensure students understand that dividing by 10 is the same as asking how many tens are in $560.
Continue with dividing by a 100 and then a 1000.
13: Activities to develop knowledge: grouping and place value
Multiplying decimal numbers by a power of 10
Show the students $6.20 using play money from Material Master 4-9 (available from Material Masters). Remind the
students that $0.20 is 20 cents. Ask the students: What is 10 x $6.20? What is 10 x $6? What is 10 x 20c? (If
necessary count this out with coins to check) Show the question on a place value table. Ensure that the students can
relate this to multiplying whole numbers by ten and understand why multiplying by ten results in a shift to the left.
100s
10s
1s
. 0.1s
0.01s
0.001s
10 x 10 10 x 1 10 x 0.1 . 10 x 0.01 10 x 0.001 10 x 0.0001
6
6
. 2
2
.
Ask problems that include numbers with zeros as place holders, for example 4.05 x 10 = 40.5. Continue with
multiplying decimal numbers by 100 and 1000. For example, 3.25 x 100 = 325.
100s
10s
1s
. 0.1s
0.01s
0.001s
100 x 1 100 x 0.1 100 x 0.01 . 100 x 0.001 100 x 0.0001
3
2
3
. 2
5
.
5
14: Activities to develop knowledge: grouping and place value
Multiplying decimal numbers using place value
Use place value equipment to show multiplication with decimals. Explain to the students that in this example the small
cubes will represent tenths, the rods will represent ones and the flat blocks represent tens. Remind students that 10
tenths make 1 one and show this with the place value equipment.
Ask the students: what is 4 tenths times by 10? Write this as 0.4 x 10
Count out 4 tenths and make 10 sets of this. Then rearrange the 40 tenths to show they are the same as 4 rods or 4
ones.
The place value equipment can also be used to demonstrate problems using more than one unit, for example 1.4 x 3.
15: Activities to develop knowledge: grouping and place value
Dividing decimal numbers by a power of 10
Ask the students: What is $35.50 divided by 10?
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What is 30 divided by 10?
What is 5 divided by 10?
What is 50c divided by 10?
Show the question on a place value table and ask the students to notice the pattern of moving digits a column to the
right 35 ÷ 10 = 3.55. Ensure students understand that dividing by ten will make a number ten times smaller. As the
value of each place within a number is ten times smaller with each move to the right dividing a number by ten
effectively moves all digits one column to the right.
Ask problems that include numbers with zeros as place holders, for example, 10.75 ÷ 10 = 1.075. Continue with
dividing decimal numbers by 100 and 1000. For example, 70.3 ÷ 100 = 0.703
Multiplying and dividing with decimals using calculators
Students need to understand that multiplying by ten will make a number ten times bigger and will result in the digits
moving to the left. Dividing by ten will make a number ten times smaller and will result in the digits moving to the right.
Using a calculator students can look for patterns when multiplying and dividing numbers by ten. Ask the students to
use a calculator to solve 3.2 x 10. Ask them to explain the answer they get in terms of where the digits move to.
16: Knowledge being developed: grouping and place value
Relationship between tenths and hundredths
The Decimats 1 equipment animation shows how to use decimats to illustrate the relationship between tenths and
hundredths.
17: Knowledge being developed: grouping and place value
Number of tenths, hundredths, thousandths in whole and decimal numbers
Place value tables can support students learning to name the number of tenths, hundredths, thousandths in whole
and decimal numbers. Ask the students to write the number 4.752 on a place value table.
100s
10s
1s
. 0.1s
0.01s
0.001s
4
. 7
5
2
.
Ask the students how many hundredths are in 4.752
Start at the hundredths column:
How many hundredths are in the hundredths column? (5)
How many hundredths are in 7 tenths? (7 x 10 = 70)
How many hundredths are in 4 ones? (4 x 100 = 400)
How many hundredths are in 4.752? (475)
Ask problems that include numbers with zeros as place holders. For example, how many tenths are in 60.32?
Start at the tenths column:
How many tenths are in the tenths column? (3)
How many tenths are in 0 ones? (0 x 10 = 0)
How many tenths are in 6 tens? (6 x 100 = 600)
How many tenths are in 60.32? (603)
18: Knowledge being developed: grouping and place value
Using arrow cards to find the number of tenths, hundredths, thousandths
The decimal arrow cards from Material Master 7-2 (available from (available from Material Masters) can be used to
help students work out how many tenths, hundredths or thousandths are in decimal number. Use the arrow cards to
show 2.436 with the 2, 0.4, 0.03 and 0.006 cards overlaying each other.
Show the students how the number 2.436 can be expanded out into the separate arrow cards.
Ask the students how many tenths are in 2.436?
Start at the thousandths column:
How many tenths are in the thousandths column? (0)
How many tenths are in 3 hundredths? (0)
How many tenths are in 4 tenths? (4)
How many tenths are in 2 ones? (2 x 10 = 20)
How many tenths are in 2.436? (24)
19: Strategy being developed
Students at this stage are learning to select from a repertoire of part-whole strategies to solve problems involving
fractions and proportions. The skill of estimation is also developed.
Students at this stage need to solve problems in each of the operational domains of the Number Framework.
1.
2.
3.
Addition and Subtraction:
o adding and subtracting fractions and decimals, for example, 1/3 + 1/4 = 7/12 (using equivalent
fractions), 1.8 + 2.5 = 2 + 2.5 - 0.2 = 4.3 (compensation)
o combining ratios, for example, apples and oranges are packed into mixed bags, the bag of 25 has
a ratio of 2:3, the bag of 35 has a ratio of 2:5, so the combined bag has a ratio of 1:2.
Multiplication and Division:
o solve problems that involve multiplication and division of decimals, for example, 1.5 x 4.5 = 3 x 2.25
= 6.75 (doubling and halving)
o solve problems that involve multiplying fractions, for example, 1/2 x 3/4 as half of 1/4 is 1/8 so half
of 3/4 is 3/8 (partitioning of fractions)
Proportions and Ratios:
o solve problems involving fractions, percentages and ratios using a range of strategies. For
example:
o reunitising (restructure the quantity) fractions, percentages and ratios, for example 75% of 120 is
50% +25% of 120;
o simplifying ratios, for example; 36 : 48 is 3:4
o converting ratios to fractions, for example. 1:4 is 1/5
20: Activities to develop strategy: addition and subtraction
By this stage students are able to add and subtract fractions with related denominators (that is, one is a multiple of
the other for example 2/4 + 3/8 = 5/8). At stage 8 students are learning to add and subtract fractions with unlike or
unrelated denominators, and this involves converting the fractions to equivalent fractions with a common denominator
Using fraction strips to add and subtract fractions
The fraction strips from Material Master 7-7 (available from Material Masters) can be used to show how fractions with
unrelated denominators can be added and subtracted. Cut out the 1/3 and the ¼ pieces from one set of the fraction
strips. Ask the students to use the fraction strips to find a fraction strip the same length.
When students have found the twelfths strip ask the students: what is the relationship between twelfths and thirds
and quarters? Students should notice that 12 is the lowest common multiple of 3 and 4 and therefore can be the
common denominator.
Ask the students: What are the equivalent fractions for 1/3 and 1/4 using 1/12s?
Students should be confident at finding equivalent fractions 1/3 = 4/12 and 1/4 = 3/12.
Ask the students: What is 4/12 + 3/ 12? (7/12)
So 1/3 + 1/4 is equivalent to 4/12 + 3/12 = 7/12.
The same understandings can be applied for subtracting fractions. For example, 3/5 - 1/3 becomes 9/15 – 5/15 =
4/15.
21: Activities to develop strategy: addition and subtraction
Combining ratios and proportions
To combine ratios students need to remember what a ratio means.
Ask the students to make a length with 1 blue unifix cube and 2 yellow unifix cubes.
Ask the students: What is the ratio of blue to yellow?
Ask the students to make another length of 1 blue cube and 2 yellow cubes and join this to the first one.
Ask the students: How many blue cubes and yellow cubes are there? (2 blue cubes and 4 yellow cubes). What is the
simplified ratio? (2:4 simplifies to 1:2)
Repeat this by adding a third length.
Ask the students: How many blue cubes and yellow cubes are there? (3 blue cubes and 6 yellow cubes). What is the
simplified ratio? (3:6 simplifies to 1:2)
Ask the students: Why does the ratio stay the same? (Because the proportion of blue to yellow cubes does not
change, it just the number of original lengths that increases.)
Ask the students to use the unifix cubes to make a number of lengths of cubes that all show the ratio of 2:3.
22: Activities to develop strategy: addition and subtraction
Using materials to combine ratios and proportions
Use counters to help students understand how ratios of different amounts can be combined. Pose the problem: one
class has 7 green bouncy balls and 6 tennis balls, another class has 8 bouncy balls and 4 tennis balls. When the
classes combine for sports, what is the ratio of bouncy balls to tennis balls? Show a set of 13 counters with 7 green
counters and 6 yellow counters and another set of 12 counters with 8 green counters and 4 yellow counters.
Ask the students what is the green to yellow ratio in the first set? (7:6)
Ask the students what is the green to yellow ratio in the second set? (8:4 or 2:1)
Combine the two sets to make:
Ask the students how many green counters and how many yellow counters there are in the combined set? (15 green
counters and 10 yellow counters)
Ask the students: what is the simplest form of the ratio 15:10? (2:1)
23: Activities to develop strategy: multiplication and division
At this stage students are solving problems that involve multiplication and division of decimals, and solving problems
that involve multiplying fractions.
Multiplication of fractions – fractions multiplied by a whole number
The Material Master 4-19 (available from Material Masters) has fraction pieces that can be used to help students
understand the multiplication of fractions. It is important that students understand why the word “of” in a problem is
interpreted to mean multiply, for example 1/3 of 6 is the same as 1/3 x 6.
Give the students copies of the fraction pieces and pose a problem. For example, there were 4 cakes at the party and
two thirds of each was eaten, how much cake was eaten? Ask the students to shade in two thirds of each of the 4
cakes.
Ask the students: How many thirds are shaded? (8)
Students can write this as 8/3 remembering that the denominator (bottom number) describes the size of the units
(thirds) and the numerator (top number) counts the number of units (8).
After a few more examples students should recognise that the whole number multiples the number of pieces
(numerator), but does not change the size of the pieces (denominator). For example, 2/5 x 6 = 12/5, 3/4 x 3 = 9/4, 2/3
x 5 = 10/3.
Encourage the students to write their answers as both improper and mixed fractions for example 8/3 or 2 2/3.
24: Activities to develop strategy: multiplication and division
Multiplication of fractions – fractions multiplied by a fraction
The decimats from Material Master 7-3 (available from Material Masters) can be used to help students understand
how to multiply fractions by fractions.
Ask the students to shade in 3 fifths on the decimat. Remind the students that 1/2 x 3/5 is the same as 1/2 of 3/5.
Ask the students to fold the shaded region in half. Half of the shaded region is 3 tenths. 1/2 x 3/5 = 3/10.
25: Activities to develop strategy: multiplication and division
The learning object Fractions of Fractions Tool (available from Learning Objects Number Level 4)provides a visual
representation of why the numerators are multiplied together and why the denominators are multiplied together.
26: Activities to develop strategy: multiplication and division
Multiplication of decimals – decimal multiplied by a whole number
The Decimats 4 equipment animation shows how decimats can be used to help students understand the
multiplication of decimals. Pose the problem: it takes 0.3m of fabric to make one hat, how much fabric is needed to
make 4 hats? Ask the students to colour in 0.3 on the decimat, then change colour and colour in another 0.3 for the
second hat and so on. When 0.3 for each of the 4 hats has been coloured in ask the students to count how much is
shaded and record the answer.
.
27: Activities to develop strategy: multiplication and division
Multiplication of decimals – decimal multiplied by a decimal number
Students should be encouraged to make reasonable estimates by recognising the effect of number size. Discuss with
students that multiplication by a decimal results in an answer smaller than the original number. This is different to
operating with whole numbers where the result is always larger than either number. Multiplying by a decimal, for
example “0.3 of ..” or “1/2 of ...” means looking for part of the original number so the answer has to be smaller.
Pose the question 0.4 x 1.5 = ? Help the students solve this problem by asking:
How can this problem be interpreted? (What is about half of 1.5?)
What do you estimate as the answer? (Half of 1.5m is approximately 0.7m)
What digits will be in the answer? (Multiply 4 and 15 to get 60, so expect 6 in the answer).
Using your estimate what do you think the answer is? 0.6
28: Activities to develop strategy: multiplication and division
Multiplication of decimals – by converting a decimal to a fraction
When one or more of the decimals converts to a benchmark fraction this is a useful strategy to use. For example 0.25
x 0.6 can be interpreted as what is one quarter of 0.6. The solution can be found by finding half of 0.6, then half of 0.3
to give 0.15.
Multiplication of decimals – using arrays
The decimats from Material Master 7-3 (available from Material Masters) can be used to help students understand
how to multiply decimals just as these were used to help students multiply fractions by fractions. Ask the students to
solve 0.5 x 0.2. Ask the students to shade in 0.2 on the decimat. Remind the students that 0.5 x 0.2 is the same as
half of 0.2. Ask the students to fold the shaded region in half. Half of the shaded ration is 1 tenth. .
Give the students problems that involve rearranging the factors (commutativity), for example 0.4 x 0.5 is the same as
0.5 x 0.4 which is the same as half of 0.4.
29: Activities to develop strategy: multiplication and division
Division of decimals - divisor is a decimal
The Decimats 5 equipment animation shows how decimats can be used to help students understand what happens
when a decimal is divided by a decimal number.
Pose the question: Sue had 2.5 kg of fruit, if it takes 0.5 kg of fruit to make 1 jar of jam, how many jars can Sue
make?
Give the students copies of decimats (available from Material Masters)and ask them to shade in 2.5 to represent the
amount of fruit.
Ask them to circle around 0.5 to indicate 1 jar of jam, and so up to 2.5.
How many jars of jam is that? (5)
30: Activities to develop strategy: multiplication and division
Division of decimals - divisor is a whole number
Students also need to explore division problems where the divider is a whole number and the quotient is a decimal.
Again like solving problems that involve multiplying decimals students should recognise the effect of the size.
Pose the problem: 4.2 metres of string is cut into7 equal lengths, how long is each length?
Help students to solve the problem by asking:
What do you estimate as the answer? (4 metres split into 7 lengths is a bit more than half a metre in each length)
What digits will be in the answer? (42 divided by 7 is 6, so expect 6 in the answer).
Using your estimate what do you think the answer is? 0.6
31: Activities to develop strategy: multiplication and division
Division with fractions – when a fraction measures another fraction.
The fraction strips from Material Master 7-7 (available from Material Masters) can be used to show how fractions can
be used to measure other fractions.
Pose the problem: James fills his petrol tank and uses three-fifths of the tank on the trip.
How many trips can he do on a full tank?
Ensure students recognise that three fifths of the tank is the same as 1 trip, and two fifths of the tank is the same as
two thirds of the trip. The question asks how many trips can James do on a full tank, so the answer is 1 and two
thirds. 1 ÷ 5/8 = 8/5 = 1 2/3
32: Activities to develop strategy: multiplication and division
Division with fractions – when a fraction measures another fraction.
Pose the problem: Kate’s petrol tank is three quarters full. Each trip she does uses a sixth of the tank. How many trips
can Kate do? When both numbers are fractions, it is necessary to convert them to a common denominator. In this
case twelfths is the common denominator.
Three quarters full is equivalent to 9/12.
Each trip is equivalent to 2/12.
Four and a half trips are equivalent to three quarters of the petrol in the tank. 3/4 ÷ 1/6 = 4 1/2.
33: Activities to develop strategy: proportions and ratios
At this stage students are learning to solve problems involving fractions, percentages, and ratios. Many of the
problem types involving fractions have been covered in the additive and multiplicative domains in this workshop.
Ratios and proportions involve multiplicative rather additive comparisons. The concepts of proportions, percentages,
ratios and rates overlap, for example percentages are a way to express a ratio, and a rate describes the ratio
between two different measures (for example kilometres per hour).
Percentages – finding percentages using equivalent fractions
Students should come to stage 8 being able to calculate a percentage, for example 36/48 is equivalent to 3/4 which is
75%. The Percentage strips equipment animation how percentage strips can be used to find the percentage. For
example 18 out of 24 is the same as 76 out of 100 or 76%.
34: Activities to develop strategy: proportions and ratios
Percentages – finding percentages using partitioning
Students should also be able to solve percentage problems where the part is unknown. For example, Lara scored
65% on her test. There were 80 questions, how many did Lara get correct?
Ask the students: what would Lara’s score be if she scored 50%? (Half of 80 is 40)
Ask the students: what would Lara’s score be if she scored 10%? (A tenth of 80 is 8)
Using a table will help students keep track of the numbers.
100% 80 questions
50%
40 questions
10%
8 questions
5%
4 questions
Ask the students: how could you work out 65%? (5% is half of 10% so it would be 4, 65% = 50% + 10% + 5%, so 40
+ 8 + 4 = 52).
35: Activities to develop strategy: proportions and ratios
Percentages – combining percentages
The learning object Mix Up: different sized (available from Learning Objects Number Level 4) provides students with
opportunities to investigate finding average percentages.
For example, pose the question: one bag of contains 10% blue counters, another bag has three times as many
counters 50% of which are blue. If the colours were shared equally what percentage of the counters in each bag
would be blue?
Ask the students: How will the size of the bags affect the average? (Because Bag B is three times bigger it has three
times the influence on the combined percentage.) The average percentage is 50% + 50% + 50% + 10% = 160% ÷ 4 =
40%.
36: Activities to develop strategy: proportions and ratios
Students can check this by combining all the counters and sharing them into two bags with the blue counters shared
equally. There will be 40 counters in each bag, 16 of which will be blue.
37: Activities to develop strategy: proportions and ratios
Ratios show the multiplicative relationship between two quantities of the same measure. For example, 2:3 could be 2
cups of flavouring to 3 cups of water. The ratio can also be expressed as a fraction, 2/5 flavouring and 3/5 water or as
a percentage 40% flavouring and 60% water
Ratios – ratios and percentages
The Material Master 7-10 (available from Material Masters) is a spreadsheet that allows students to explore the
connection between ratios and percentages. Students can enter the ratios and the spreadsheet generates the
percentages and shows them on a pie graph.
38: Activities to develop strategy: proportions and ratios
Ratios – simplifying and comparing proportions
At stage 8 problems involving ratios often require students to compare ratios. The learning object Exploring ratios and
proportions (available from Learning Objects Number Level 5) asks students to decide if two ratios are equivalent.
39: Activities to develop strategy: proportions and ratios
Ratios – converting and comparing proportions
Pose the problem: At a school disco three drink mixtures have been made from lemonade and juice. Which of the
drinks mixtures will have the fizziest taste?
Drink A: 5 litres lemonade: 3 litres juice
Drink B: 4 litres lemonade: 2 litres juice
Drink C: 3 litres lemonade: 2 litres juice
By converting these ratios to fractions or percentages it is easier to compare them.
Ask the students: How can the ratio 5:3 be expressed as a fraction?
How much drink is there in Drink A? (5 litres + 3 litres = 8 litres)
How much of Drink A is lemonade? (5 litres)
What fraction of Drink A is lemonade? (5 out of 8 litres, 5/8)
Ask the students to convert Drink B and Drink C to a fraction. (Drink B= 4/6, Drink C= 3/5)
Ask the students to compare the fraction of lemonade in the 3 drinks, either by ordering the fractions or by converting
the fractions to percentages to compare them.
Ask the students which of the drinks has the fizziest taste? (Drink B has 66/6% lemonade compared to Drink A which
has 62.5% and Drink C which has 60% lemonade).
40: Activities to develop strategy: proportions and ratios
Rates
Rates are examples of ratios where the two quantities are of different measures. For example speed is the ratio of
distance per time (50 kilometres per hour). The learning object Drive: easy problems (available from Learning Objects
Number Level 4) allows students to work out ratios involving speed and distance.
Students can find the missing time quantity by finding the relationship between12 kilometres and 36 minutes (3 times)
and applying it to the 20 kilometres to get 60 minutes as the missing time quantity. Or by finding the relationship
between 12 and 20 and applying it to 36. 12 + (2/3 of 12) = 20, so 36 + (2/3 of 36) = 60.
41: Activities to develop strategy: proportions and ratios
Rates – with more complex proportions
The learning object Drive: hard problems (available from Learning Objects Number Level 5) is the same format as
Drive: easy problems but the ratios require more proportional thinking. For example, on the problem shown ask the
students: How can you simplify the ratio 90:54? (divide both by 9 to get 10mins:6 km)
Using the simplified ratio what is the relationship between 35 minutes and 10 minutes? (35 is 3.5 times bigger)
Apply this relationship to find the missing distance quantity. (3.5 times 6 is 21)
The missing quantity is 21.
42: End of Module Twelve
This module has covered selecting and implementing learning activities appropriate for students working at stage 7:
advanced multiplicative. It is intended as an introduction only. Further information and additional activity ideas can
also be found in:
o
o
o
o
The teaching units and activities available from the number section of the nzmaths website. Select
appropriate strategies and knowledge activities from those listed under Level Four or Level Five.
The Numeracy Planning Assistant . Find appropriate activities for this stage under "advanced proportional".
The Equipment Animations. Find appropriate activities for this stage under the "proportional" heading.
The Numeracy Development Project Books (PDF versions available from Numeracy Development Project
Books). Relevant activities for this stage can be found in:
Book 4: Teaching Number Knowledge
Book 5: Teaching Addition, Subtraction, and Place Value
Book 6: Teaching Multiplication and Division
Book 7: Teaching Fractions, Decimals, and Percentages