Enough Rice?

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Purpose

This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.

A PDF of the student activity is included.

Achievement Objectives
NA4-4: Apply simple linear proportions, including ordering fractions.
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (263 KB)

Specific Learning Outcomes

solve problems involving simple linear proportions

Description of Mathematics

Number Framework Links
Use this activity to:

• help the students to apply and extend their advanced multiplicative part–whole strategies (stage 7) for proportional adjustment

• help the students to extend their ideas in the operational domain of proportions and ratios (stages 7–8)

• encourage transition to advanced proportional thinking (stage 8).

Required Resource Materials

FIO, Level 3, Number Sense and Algebraic Thinking, Book Two, Enough Rice? page 21

A classmate

Activity

In this activity, students explore the relationship between the amount of time a sack of rice would last and the number of people eating. The relationship is an inversely proportional one: when one quantity is increased (for example, rice), the other is decreased (time). By contrast, in a directly proportional relationship, when one quantity is increased, so is the other. (For example, on a trip to the cinema, if the number of people increases, so does the total cost of admission.) The rice consumption is an example of a rate where different measures are related
multiplicatively, that is, 1 sack : 4 people : 24 days.
This activity would be useful as a follow-up after the students have explored the strategy of doubling and halving and have recognised that it can be extended to adjusting by any inversely proportional amount, such as trebling and thirding, for example, 6 x 9 = 3 x 18 = 2 x 27, or multiplying and dividing by 4 or by 10.
The students in independent groups will need to break into pairs or groups of 3 to complete parts of this activity. Remind the students about the strategy of doubling and halving and suggest that they look for opportunities to use it in this activity.
Set the scene with a guided teaching group by asking the students about the foods their family might buy in big packets at the supermarket, such as cereals, potatoes, rice, or non-food items, such as washing powder or shampoo. Show them a big box of cereal and ask them to estimate how many days they think it might last their family. Ask: What would happen if you had relatives come to stay and so you had twice as many people in your home who all liked that same cereal? How long would you expect the cereal to last then?
What about if some of your family went away on holiday and so you had only half the number of people eating the cereal? How long would you expect it to last then?
Divide the group into smaller groups. When they share back with the whole group, listen reflectively and summarise what they say by asking So you … and then you … Is that what you did?, at the same time recording number sentences to show what they did on the board. This ensures that you understand what the student has said, gives other group members another chance to hear the strategy, and provides a model of effective communicative language and recording.
For questions 1 and 2, encourage the students to think about how they could apply the strategy of doubling and halving to work out how long the rice would last for 8 people, 2 people, and then 1 person. Promote the identification of patterns by recording summaries of all the problems as they are solved:
The rice lasts:
8 people for 12 days
4 people for 24 days
2 people for 48 days
1 person for 96 days. (This is called the unit rate. It is one way to solve any rate problem, although it is not always the most efficient.)
Ask the students:
What other patterns besides doubling and halving can you see in the numbers in these statements?
Can you tell me how the numbers relate to each other?
If the students need help to complete the table in question 2b, ask general questions to begin with and then become increasingly focused as needed, for example:
What do you have to work out?
What do you know so far that might help you?
“You could start by filling in the easier cells first and working from there. For  example, we’ve been told 4 people and 24 days, so we can easily do 2 and 48, 8 and 12, and 1 and 96.”
A physical or diagrammatic model will help here:

diagram.
“You could use trebling and thirding to work out how long the rice would last for 6 people based on the information for 2 people, and then you could halve and double that to work out how long it would last for 3 people.”
The students will need to accept closeness in estimating the length of time needed for 5, 7, and 20 people because these calculations must approximate whole numbers of days.
These problems can be used to evaluate whether the students have generalised sufficiently to be able to apply the inverse proportional adjustment strategy even when the numbers are not easy multiples and factors. In fact, most students will tend to think additively, for example:
People 4      6       8
Days   24   16     12
Support the students’ thinking if necessary by asking:
Are there any numbers of people on the table that you can double and halve or treble and third to make 5 or 7?
So you can’t double and halve or treble and third for these ones, but could you find another relationship you could use, such as x 4 and ÷ 4, or x 5 and ÷ 5, or use fractions?
“I could use ‘the rice lasts 1 person for 96 days’ and x 5 and ÷ 5 to work out for 5 people and x 7 and ÷ 7 to work out for 7 people. So 96 ÷ 5 = 19.2 days for 5 people, and 96 ÷ 7 = 13.7 days for 7 people.”
“I could use ‘the rice lasts 4 people for 24 days’ and x 5 and ÷ 5 to work out how long it lasts for 20 people because I know 4 x 5 = 20, so 24 ÷ 5 = 4.8 or 4 days.”
Again, the use of diagrams will be very important.
For question 3, support the students’ thinking if necessary by asking increasingly focused questions, such as:
You know that 1 person takes 96 days to eat the rice. How could knowing that help you?
What fraction of the sack of rice would that 1 person have eaten after 1 day? (1 ninety-sixth, )
How many ninety-sixths are there in a whole sack of rice?
How could knowing that help you?
If lots of people ate each of the rice on the same day, how many people could you feed out of the sack?
Encourage reflective discussion by asking, in a think-pair-share situation:
Does 42 x 67 = 42 x 29 x 67 ÷ 29? Explain your answer. (Yes, it does. Multiplying and dividing by the same number balances out, so 42 x 67 is unchanged.)

Extension

You could get the students to brainstorm examples of direct and inversely proportional relationships, for example, a person’s mass on a see-saw to the distance from the fulcrum or the number of apples in a kilogram versus the mass of each apple.
 

Answers to Activity

1. a. 12 days. (There are now double the people, so the sack of rice will last half as long.  24 ÷ 2 = 12)
b. 48 days. (There are now half the original number of people to feed, so the sack will last twice as long. 24 x 2 = 48)
c. 96 days. (You could double the amount of time it will last 2 people: 2 people for 48 days, 1 person for 96 days. Or you could notice that the sack will last 4 times longer for 1 person than for 4 people. 24 x 4 = 96)
2. a.–b.

table.
Methods will vary. One strategy is to work from the facts you know and adjust the
numbers proportionally. For example, if you know that the sack will last 2 people
48 days, you could multiply the number of people by 3 and divide the number of days by 3 to work out that it would last 6 people for 16 days (2 x 3 = 6, 48 ÷ 3 = 16). Once you know that the sack feeds 6 people for 16 days, you could halve and double to work out that it would last 3 people for 32 days (6 ÷ 2 = 3, 16 x 2 = 32).
Working out how many days the sack will last for 5 and 7 people is harder because 5 and 7 don’t relate easily to any of the facts you already know. You could work from knowing that the rice will last 1 person for 96 days and multiply by 5 and divide by 5 to work out how long it would last for 5 people: 1 x 5 = 5, 96 ÷ 5 = 19.2, which is 19 whole days, with a bit left over. Then multiply by 7 and divide by 7 to work out how long it would last for 7 people:
1 x 7 = 7, 96 ÷ 7 = 13.714285 (using a calculator), so the rice would last 7 people
for 13 whole days, and there would be some left over, but not quite enough for the
14th day.
c. 4.8 days. Methods will vary. If you worked from knowing that the rice will last 1 person for 96 days, you could multiply by 20 and divide by 20 to work out how long it would last 20 people. 1 x 20 = 20, 96 ÷ 20 = 4.8, so it would last 4 whole days,
and there would be some left over but not quite enough for the 5th day. You could
also work from knowing that it will last 4 people for 24 days and multiply by 5 and
divide by 5 to work out how long it would last for 20 people. 4 x 5 = 20, 24 ÷ 5 = 4.8, that is, 4 whole days with some left over.
3. About 96 people. (If the sack lasts 1 person for 96 days, this means that each day they eat 1 ninety-sixth [1/96] of the sack. To use a whole sack in 1 day, 96 people would need to eat 96 ninety-sixths [96/96 ] in 1 day.)
4. a.–b. Problems will vary.

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