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Level Five > Number and Algebra

Leftovers

Student Activity: 

  

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Achievement Objectives:

Achievement Objective: NA5-3: Understand operations on fractions, decimals, percentages, and integers.
AO elaboration and other teaching resources

Purpose: 

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

Specific Learning Outcomes: 

interpret decimal answers to division problems

Description of mathematics: 

Number Framework Links
Use this activity with stage 7 students to encourage them to think about the meaning of the fractional parts of quotients and to interpret them sensibly in the context.

Required Resource Materials: 
FIO, Levels 3-4, Multiplicative Thinking, Leftovers, pages 8-9
A calculator
Activity: 

This activity explores how best to express remainders in division problems. Suitable expression depends on the context of the problem, so sometimes the decimal given by a calculator calculation will make sense and sometimes it will not.
Most of the contexts in Leftovers involve division of measurements. For example, the milk quantity is measured in litres, the foodwrap is measured in metres, and the time in hours and minutes.
In most of these cases, the decimal produced in a calculator division will have meaning in terms of the measurement units. 7 kilograms of potatoes shared among 8 people can be calculated as 7 ÷ 8 = 0.875 kg. This is equivalent to 875 grams by conversion of units because 1 kilogram is equivalent to 1 000 grams. The synergy between the metric system of measures and our base 10 number system makes this possible. In measurement contexts such as time, where the base may be 60 or 12, this synergy does not exist.
In many contexts, however, the exact (calculator) answer is not appropriate. For example, for the potatoes in this activity, the best solution would be to get each share as close as possible to 875 grams without cutting potatoes into pieces. For the flying fox time, students need to realise that 2 hours and 45 minutes is not the same as 2.45 hours. 45 minutes is 45/60 of an hour, not 45/100, because there are 60 minutes in an hour. The best approach in this case is to convert 2 hours and 45 minutes into minutes (2 x 60 + 45 = 165 minutes). The calculator now gives 165 ÷ 8 = 20.625
minutes.
Most students are likely to have trouble interpreting 0.625 of a minute because it does not mean 62.5 seconds! In the context, 20 minutes is an acceptable answer to the question. No matter how much the campers value their turn on the flying fox, they could not or would not attempt to equalise time to the nearest second. Encourage your students to discuss the realities of the situation.
The sharing of discrete (separate) objects can present a different problem because it is often inappropriate to divide separate, whole objects into fractions, especially if the objects are living creatures! Biscuits, however, can be partitioned with minimal social consequences. The calculator operation gives 5 x 30 ÷ 8 = 18.75. A decimal to fraction conversion is needed: 0.75 = 3/4. So each group gets 18 biscuits. Practically, this is not likely to be a sensible outcome, so it might be that 6 groups get 19 biscuits each and 2 groups get just 18. Let your students discuss how they would
prefer to see this sharing problem resolved.
In question 3 in Activity Two, students generalise about the situations when a calculator division output is helpful.
From their investigations, they should conclude that the calculator answer is always helpful when the problems involve some kind of metric measurement but that the practicalities of the situation often mean that some rounding is required. When discrete objects are shared, it is only the decimal part of the quotient that needs interpretation. This usually involves either converting the decimal part to a tidy common fraction or concluding that the objects should not be partitioned further and
giving a whole-number answer.
 

Answers to Activities

Activity One
1. i. 10 ÷ 8 = 1.25
ii. 7 ÷ 8 = 0.875
iii. 13 ÷ 8 = 1.625
iv. 2.45 ÷ 8 = 0.30625
v. 5 x 30 ÷ 8 = 18.75
2. Of the calculator answers in question 1, the useful ones are:
i. 10 ÷ 8 = 1.25 means 1.25 L or 1 250 mL.
ii. 7 ÷ 8 = 0.875 means 0.875 kg or 875 g.
iii. 13 ÷ 8 = 1.625 means 1.625 m or 162.5 cm or 1 625 mm.
3. a. Meaningless or not very sensible calculator answers:
iv. 2.45 ÷ 8 = 0.30625 is meaningless because 2.45 doesn’t mean 2 45/100 hrs.
v. 5 x 30 ÷ 8 = 18.75 has some meaning because 18.75 means each group gets
18 biscuits; but 0.75 doesn’t tell you how many biscuits will be left over.
However, the calculator answer of 18.75 could be interpreted as 18 3/4 biscuits.
b. iv. 2 hrs and 45 min needs to be turned into min: (2 x 60) + 45 = 165 min.
165 ÷ 8 = 20.625 min. The 0.625 min can be ignored (for example, to allow
for group changeovers), so each group gets 20 min on the flying fox.
v. 5 x 30 ÷ 8 = 18.75 means each group gets 18 whole biscuits. 0.75 x 8 = 6
means 6 biscuits are left over (or each group gets another 0.75 or 3/4 of a
biscuit!).

Activity Two

1. a. i. 33 ÷ 5 = 6.6. So each group gets 6 whole punnets with 5 x 0.6 = 3 punnets
left over. These 3 punnets could be shared in a variety of ways, for example, 4 groups could get a punnet while the fifth group gets an extra punnet.
ii. 24 ÷ 5 = 4.8 fish. So each group could get 4 whole fish. The remainder of 4
fish could be dealt with in several ways, for example, one group could get the
4 biggest fish while the other 4 groups could get 5 fish each.
iii. 57 ÷ 5 = 11.4 eggs. Each group gets 11 eggs with a remainder of 2 eggs. So 2
of the 5 groups could get 12 eggs.
b. Tessa is right. 0.6 is the same as 3/5. Chenda’s answer of 6.3 means 6 3/10, and
3/10 is not the same as 3/5.
2. a. i. 19 ÷ 3 = 6.3333 bags. So 2 groups could get 6 bags while the other group gets 7 bags, or they could share out the cherries in the extra bag.
ii. 21 ÷ 3 = 7 shellfish for each group
iii. 16 ÷ 3 = 5.3333 buckets. So 2 groups get 5 buckets and the other group gets
6 buckets, or they could share out the mushrooms in the extra bucket.
b. Cherries: 19 ÷ 3 = 6 1/3 or 6.333
Shellfish: 21 ÷ 3 = 7 or 7.0
Mushrooms: 16 ÷ 3 = 5 1/3 or 5.333
3. Fractions are better if the remainders are whole numbers, but decimals are more useful if you are sharing out quantities measured in metric units.
 

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LeftOvers.pdf1.25 MB