Comparing ratios with a common whole
The purpose of this activity is to support students to understand that expressing the part-whole relationships in ratios is useful for comparing the characteristics of the ratios, such as taste, colour, density, etc.
About this resource
New Zealand Curriculum: Level 3–4
Learning Progression Frameworks: Multiplicative thinking, Signpost 6 to Signpost 7
These activities are intended for students who have some previous experience with treating fractions as numbers. This should include a well-developed understanding of the meaning of the numerator and denominator. Students should also have well-developed knowledge of multiplication facts.
Comparing ratios with a common whole
Achievement objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-5: Know fractions and percentages in everyday use.
Required materials
- connecting cubes
- calculators
1.
Pose a ratio comparison problem using coloured cubes, where the total number of cubes in each ratio is the same.
- Here are the ratios: 2:3 and 3:2.
- Which mixture has the strongest taste of blueberry?
Expressing the part to whole relationships as fractions confirms the result that 3:2 has a stronger blueberry taste than 2:3.
Blueberry |
Apple |
||||
|---|---|---|---|---|---|
3⁄5 |
3 |
: |
2 |
2⁄5 |
|
2⁄5 |
2 |
: |
3 |
3⁄5 |
|
2.
Pose further examples with the common numbers of cubes given in the whole. Ask students to find the part-whole fractions for each ratio and use the fractions to compare the ratios for strength of blueberry flavour. Good examples might be:
a.
- Here are the ratios: 4:8 and 3:9.
- Which mixture has the strongest taste of blueberry?
Blueberry |
Apple |
||||
|---|---|---|---|---|---|
4⁄12 |
4 |
: |
8 |
8⁄12 |
|
3⁄12 |
3 |
: |
9 |
9⁄12 |
|
b.
- Here are the ratios: 7:3 and 5:5.
- Which mixture has the strongest taste of blueberry?
Blueberry |
Apple |
||||
|---|---|---|---|---|---|
7⁄10 |
7 |
: |
3 |
3⁄10 |
|
5⁄10 |
5 |
: |
5 |
5⁄10 |
|
c.
- Here are the ratios: 5:4 and 7:2.
- Which mixture has the strongest taste of blueberry?
d.
- Here are the ratios: 4:3 and 2:5.
- Which mixture has the strongest taste of blueberry?
3.
Revisit each ratio above and ask this question:
- Which ratio has the strongest taste of apple?
Look for students to recognise that a stronger taste of blueberry corresponds to a weaker taste of apple, while a weaker taste of blueberry corresponds to a stronger taste of apple.
For example, consider the ratios 4:3 and 2:5 above. The part-whole fractions for blueberry are 4/7 and 2/7, respectively.
- If we know the fractions for blueberry are 4/7 and 2/7, what are the fractions for apple?
Do students recognise that the fractions for a given ratio add to one (the whole)?
- Since 4/7 + 3/7 = 1, if four sevenths of the mixture is blueberry, then three sevenths are apple.
- Since 2/7 + 5/7 = 1, if two sevenths of the mixture is blueberry, then five sevenths are apple.
1.
Progress from making cube models of ratios to working with the symbols. You might fold back to the physical model if answers need verification or if you want students to demonstrate their results by using symbols.
- For example, Recipe A has a 3:5 mix of blueberry to apple, and Recipe B has a 6:2 ratio. Which recipe tastes most strongly of blueberry?
2.
Use examples where the ratios do not have the same number of parts, but one ratio can be altered to allow comparison.
For example,
- Which ratio, 2:3 or 3:7, has the strongest taste of blueberry?
- Since 2:3 can be copied twice to make 4:6 with the same strength of blueberry, 2:3 has a stronger blueberry taste compared to 3:7.
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