This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.
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find a proportion between two other proportions
Number Framework Links
Use this activity to:
• develop confidence in students who are beginning to use advanced proportional strategies (stage 8)
• help students consolidate and apply their knowledge of equivalent fractions (stage 8).
Like the previous activity, this one requires students to compare proportions in the context of drink flavours. But while the previous activity asks them to rank two proportions, this one asks them to find a proportion that is between two others.
If your students have worked through the previous activity, you may wish to give them this one with little introduction and challenge them to find a strategy that will work. Draw their attention to the comment in the speech bubble.
Alternatively, you could discuss the strategy suggested by the speech bubble and ask “How will this idea help us compare the proportions?” The strategy involves converting both proportions to unit fractions (fractions with a numerator of 1) and then using the denominator to compare their sizes.
In Terry’s recipe in question 1, the proportion of tablespoons of powder to millilitres of milk is 4:250, which can be written as 4/250 or 1/62.5. Your students may need help with this last step. In Tracey’s recipe, the proportion is 6:300 or 1/50. Once both proportions are expressed as unit fractions, students should see that an acceptable ratio is somewhere between them. There are many possibilities, but one that is somewhere near the middle of the two unit fractions would best fit the context, for example, 1/55. This would equate to a drink made with 4 tbs of powder and 4 x 55 = 220 mL of milk, or 5 tbs and 5 x 55 = 275 mL of milk, or 6 tbs and 6 x 55 = 330 mL of milk.
Students can use the unit fraction strategy again for question 2a. They should find that a suitable recipe for Tiana will have a cordial:water ratio of between 1:5 and 1 : 5.5. This number line shows the range of potential values for the denominator:
A suitable mix for Tiana would be 1 : 5.2, 1 : 5.25, or 1 : 5.3. The middle ratio would translate into a drink made from 2 parts of cordial and 10.5 parts of water.
The challenge in question 2b is to find a suitable unit against which the two recipes can be compared; one that doesn’t involve working with unfriendly fractions. The best is 1 full glass. Here is a possible line of reasoning based on this unit: “Terry uses 6 tsp for three-quarters of a glass, so each quarter has 2 spoons. This means she would need 8 spoons for a whole glass. Tracey would use 10 spoons for 1 whole glass because 2 times 5 is 10. So Tracey’s mix is stronger.”
The most obvious ratio for a drink that has a “teaspoons of powder : glass of milk” ratio that lies between 8:1 and 10:1 is 9:1. This translates into 9 tsp of powder in a full glass of milk.
Answers to Activity
1. Terry’s recipe is equivalent to 1 tbs of powder for just over 60 mL of milk, and Tracey’s recipe is 1 tbs for 50 mL of milk. So Tiana’s drink should
be made using a ratio of 1 tbs of powder for about 55 mL of milk. For a similar sized drink, the recipe might be 5 tbs of powder for 275 mL of milk.
2. a. Terry’s recipe is 2:11 or 1 part orange cordial to 5 1/2 parts of water. Tracey’s recipe is 1:5 or 1 part orange cordial to 5 parts of water. So Tiana’s drink should be between 5 and 5 1/2parts of water per part of orange cordial. A suitable recipe would be 5 1/4 parts of water per part of orange cordial (or 21 parts of water per 4 parts of orange cordial).
b. Terry’s recipe is 6 tsp of powder for 3/4 glass of milk, which is the same as 8 tsp of powder for one full glass of milk. Tracey’s recipe is 5 tsp of powder for 1/2 glass of milk, which is the same as 10 tsp for a full glass of milk. A suitable recipe would be 9 tsp of powder for a full glass of milk.