Remainders

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Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Specific Learning Outcomes

Solve division problems that involve remainders.

Description of Mathematics

Number Framework Stage 7

Required Resource Materials
The Happy Hundred (Material Masters 6-5)

Unifix cubes, beans or counters, place value materials

Calculators

Activity

Using Materials

Problem: Mere (student’s name) has made forty-five cookies for the gala. She wants to put them in boxes of six. How many boxes can she fill?

Allow students to use materials and any mental strategies they wish to solve the problem. Discuss their strategies.

Look for efficient methods and use the materials to illustrate them.For example, Using 6 × ? = 45 (reversibility),  six times five is thirty leaving fifteen to be shared (distributive property),  six times two is twelve leaving three cookies left (compensation).

This could be illustrated on the Happy Hundred array:
 happyhundreds.

 Discuss how the remainder of three could be recorded:

45 ÷ 6 = 7 r 3 (meaning three left over), or, 45 ÷ 6 = 7 3/6 = 7 ½ (meaning what fraction of a whole set remains).

Invite students to key the division into a calculator. This will give the result 7.5. Ask the students how this relates to the other ways of recording the remainder, that is 0.5 is the decimal for one half.

Ask, Which way of recording the remainder is the best in solving Mere’s problem? In this case 7 ½  gives the number of boxes Mere can make.

Pose a different problem to help students appreciate usefulness of the “left over” way of recording the remainder.

Problem: Tony (Student’s name) has $43. Videotapes cost $5 each. What is the largest number of tapes Tony can buy and how much money will he have left?

Allow students to solve the problem using materials, where necessary, and discuss their strategies. In this case recording the division as, 43 ÷ 5 = 8 r 3, gives both the number of tapes bought (8), and the remaining money ($3).

Provide other examples for students to solve using materials and ask them to express the remainder in ways that match the demands of the problem. For example:

(1) Henk has fifty-five stamps. He needs six stamps to post each parcel. How many parcels can he post? How many stamps will he have left? (55 ÷ 6 = 9 r 1)

(2) Salena can wash a car in eight minutes at the fundraising carwash. How many cars can she wash every hour? (60 ÷ 8 = 7 4/8 = 7 ½)

(3) Trent has used twenty-six metres of rope to make eight skipping ropes. Each one is the same length. How long is each skipping rope? (26 ÷ 8 = 3.25m)

Using Imaging

Role playing: Provide similar examples in which a student from the class goes away from view to solve the problem with materials. The other students image what he or she is doing and what the remainder will be.

Depending on number size, the Happy Hundred array or place value materials like beans or play money, are the easiest materials to visualise.

Suitable examples might be:

(1) Charlie’s Chocolates come in boxes of four. He has 39 chocolates to put in boxes. How many boxes can he fill?

(2) Coach Casie needs seven people to make each netball team. She has forty six players. How many teams can she make? How many players will be reserves?

(3) Romi ran six lengths of the court in seventy-six seconds. What was his average time per length in seconds?

Using Number Properties

Provide the students with division problems written as equations. Ask them to solve each problem, express the quotient and remainder in one of the three ways given, then make up a word story to match their recording. For example, 34 ÷ 6 = 5 r 4.

Word story: Thirty-four people wanted to play volleyball. There were six players in each team. How many teams were there and how many players were reserves?

Useful equations to use are:

84 ÷ 9 =         47 ÷ 3 =          35 ÷ 4 =            51 ÷ 6 =        472 ÷ 5 =

 

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Level Four