Comparing ratios with a common part
The purpose of this activity is to support students in comparing ratios with a common part but a different whole. They do so by creating fractions for the part-whole relationships and comparing the fractions.
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 part
Achievement objectives
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
NA4-4: Apply simple linear proportions, including ordering fractions.
Required materials
- connecting cubes
- calculators
1.
Pose a ratio comparison where the number of cubes in one part is the same but the whole is different.
- Here are the ratios: 2:3 and 2:4. Which mixture has the strongest taste of blueberries?
Expressing part-to-whole relationships as fractions allows the comparison.
Since 2/5 is greater than 2/6 and because fifths are larger than sixths, it can be claimed that 2:3 has a slightly stronger taste of blueberry than 2:4. Write the inequality 2/5 > 2/6.
- Which ratio tastes most strongly of apple?
Students should recognise that a stronger taste of blueberry means a weaker taste of apple. Looking at the fractions confirms that that is true for these ratios: 3/5 is less than 4/6, so 2:4 has a stronger taste of apple than 2:3.
You might use a calculator to confirm the inequality, using the operations 3 ÷ 5 = 0.6 and 4 ÷ 6 = 0.666 to change 3/5 and 4/6 into decimals. Write the inequality 3/5 < 4/6.
2.
Pose further problems with one part of both ratios being the same and the wholes being different. Use familiar fractions to make the inequalities easier to see. If needed, use a calculator to confirm the inequalities. Write down the inequalities as you work through each problem. Good examples might include:
- Which has a stronger taste of blueberry, 3:5 or 3:7?
b. Which has a stronger taste of blueberry, 1:4 or 2:4?
c. Which has a stronger taste of blueberry, 4:8 or 4:6?
d. Which has a stronger taste of blueberry, 4:6 or 6:6?
1.
Increase the level of abstraction by progressing from using fraction strips to diagrams to symbols only. Maintain the types of ratios in the lesson description until students make the comparisons using the symbols alone.
2.
Progress in comparison problems involves a mixture of a common part and a common whole. Support students in discriminating between these types of problems. For example, you might provide this suite of ratio comparison problems, using blueberry and apple juice as a context:
- Which mixture tastes most strongly of blueberry?
Note that the first number represents parts that are blueberry, and the second number represents parts that are apple.
- 3:4 or 6:8? (equivalent ratios so equal flavour)
- 5:3 or 5:2? (equal part)
- 5:7 or 4:8? (equal whole)
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