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Level Four > Number and Algebra

Multiplication and Division Pick n Mix 1

Purpose: 

In this unit we look at a range of strategies for solving multiplication and division problems with whole numbers with a view to students anticipating from the structure of a problem which strategies might be best suited. This unit builds on the ideas presented in the Multiplication Smorgasbord session in Book 6: Teaching Multiplication and Divsion.

Achievement Objectives:

Achievement Objective: NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
AO elaboration and other teaching resources
Achievement Objective: NA4-8: Generalise properties of multiplication and division with whole numbers.
AO elaboration and other teaching resources

Specific Learning Outcomes: 

mentally solve whole number multiplication and division problems using:

  • proportional adjustment
  • place value partitioning
  • rounding and compensation
  • factorisation

use appropriate recording techniques

predict the usefulness of strategies for given problems

evaluate the effectiveness of selected strategies

generalise the types of problems that are connected with particular strategies

Description of mathematics: 

The strategy section of the New Zealand Number Framework consists of a sequence of global stages that students use to solve mental number problems. On this framework students working at different strategy stages use characteristic ways to solve problems. This unit of work is useful for students working at Stage 7-Advanced Multiplicative of the Number Framework. Students at this stage select from a broad range of strategies to estimate and solve multiplication and division problems, subdividing and recombining numbers to simplify problems either additively or multiplicatively. The Number Framework also includes a knowledge component which details the knowledge students will need to develop in order to progress through the strategy stages of the framework. This unit draws on students' knowledge of multiplication and related division facts to 10, whole number place value and connection of basic facts to multiplying powers of ten.

The key teaching points in this unit are:

  • Some problems are easier to solve in certain ways. Teachers should elicit strategy discussion around problems in order to get students to justify their decisions about strategy selection in terms of the usefulness and efficiency of the strategy for the problem situation.
  • Useful strategies for multiplication include place value partitioning, rounding and compensating, proportional adjustment and factorisation.
  • Useful strategies for division include proportional adjustment (with factorisation), rounding and compensating, and partitioning or ‘chunking’.
  • Tidy number strategies (rounding and compensating) are useful when number(s) in an equation are close to an easier number to work from.
  • When applying tidy numbers in multiplication and division it is important to keep track of what has been changed in a problem in order to compensate (rounding and compensating).
  • Place value strategies are most useful when little or no renaming is needed.
  • Proportional adjustment is useful when there is a connection between the numbers that can be used to simplify the problem such as doubling and halving or quadrupling and quartering. Division with factorisation can be viewed as a form of proportional reasoning. In division both of the numbers must be reduced by the same factor.
  • Factorisation is useful for multiplication when one of the factors can be reduced.
Required Resource Materials: 
Large dotty array - Material Master 6-9 (128KB)
Place value equipment- beans, place value blocks
Activity: 

Getting Started

The purpose of this session is to explore the range of strategies that students have to solve multiplication and division problems. This will enable you to elicit the strategies that students currently use and evaluate which strategies need to be focused on in greater depth as well as identifying students in your group as "expert" in particular strategies.

Problem 1:
Craig bikes 38 kilometres each day for five days. How many kilometres has he travelled by the end of the five days?

Ask students to work out the answer in their head. Give the students 2-3 minutes thinking time. Then ask them to share their solutions and how they solved it with their learning partner. The following are possible responses:

Rounding and compensating:
38 x 5
38 is rounded to 40 so the equation becomes 40 x 5 then 10 which is (2 x 5) is subtracted

Proportional adjustment:
38 x 5
Solve instead 19 x 10 using doubling and halving (by halving 38 and doubling 5)

Place value partitioning:
38 x 5
Solve 30 x 5, add 8 x 5

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book).

Problem 2:
There were 136 rowers entered in the eights rowing champs at the Maadi Cup. How many teams are entered?

Ask the students to work out the answer in their heads. Give the students 2-3 minutes thinking time. Then ask them to share their solutions and how they solved it with their learning partner. The following are possible responses:

Place value partitioning (chunking):
136 ÷ 8
I know that 36 ÷ 8 = 4 with a remainder of 4
I now have 104 ÷ 8, I know that 10 x 8 = 80 and that leaves me 24 which is 3 groups of 8 so the answer is 13 and the first four which is 17.

Factorisation (proportional adjustment):
Dividing by 8 is like dividing by 2 then 2 then 2 so half 136 is 68 and half 68 is 34 and divide by 2 again leaves me 17 so the answer is 17.

Rounding and compensating:
160 would be the same as 8 times 20. 136 is 24 less than that, or three teams less than 20… it’s 17.

As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit (perhaps in a modelling book).

Ask students to reflect on the strategies that have been discussed in the session and evaluate which strategies that they personally need further work on, perhaps using thumb signals - thumbs up - confident and competent with the strategy, thumbs sideways - semi confident, thumbs down - not yet confident. Use this information to plan for your subsequent teaching from the exploring section outlined below.

Exploring

Over the next two to three days, explore the following strategies making explicit the strategy you are concentrating on as the teacher and the reason for using the selected strategy.

e.g. In the problem 29 x 7 tidy numbers would be a useful strategy as 29 is close to 30.

The following questions are provided as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs.

Rounding and Compensating (Multiplication)
The Sting netball fans are going to Christchurch to watch a netball game. Each bus has 48 people on it and there are 14 buses travelling altogether. How many Sting fans are heading to Christchurch?

The rounding and compensating strategy involves rounding a number in a question to make it easier to solve. In the above question 48 can be rounded to 50 (by adding 2). The problem then becomes 50 x 14, or 700. In order to compensate for the rounding, two lots of 14 (28) must be subtracted from the ‘rounded’ equation.

The following questions can be used to elicit discussion about the strategy:

  • What tidy number could you use that is close to one of the numbers in the problem?
  • What do you need to do if you tidy up this number? Why?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not seem to understand the tidy numbers concept, use place value equipment or a large dotty array to show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:
69 x 9
148 x 7
398 x 6
26 x 32
7 x 9998
548 x 3

Note that the problems posed here are using a tidying up strategy rather than tidying down. If one of the factors is just over a tidy number (such as 203) then place value tends to be a more useful strategy.

Rounding and compensating (Division)
Sarah uses eight bus tickets every week to travel around town. She wins 152 tickets in a radio competition. How long will they last her?
Rounding and compensating for division involves finding a number that is close to the total, and working from that number to find an answer. For the question above, a student might say:
I know that 8 times 20 would be 160. 152 is 8 less than 160, so the tickets would last her 19 weeks.

If the students do not seem to understand the rounding and compensating concept, use a large dotty array to show the problems physically. Some students may find it useful to record and keep track of their thinking.
343 ÷ 7
198 ÷ 9
1194 ÷ 6
686 ÷ 7
1764 ÷ 18

Proportional Adjustment (Multiplication)
At the music festival there are 32 schools with 25 students in each choir, how many students are there altogether in the choirs?
Proportional adjustment involves using knowledge of multiples to create equivalent equations. Factors are proprtionally adjusted to make one (or both) factors easier to work from. In the above problem the factors could be adjusted as follows:
32 x 25 arrows devide by 4 = 8 x 100
Or, using doubling and halving:
(32 devided by 2 = 16) x (25 x 2 = 50)

The following questions can be used to elicit discussion about the strategy:

  • What could you multiply one of these numbers by to make it easier to work with?
  • What would you then need to do to the other number?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not seem to understand the proportional adjustment concept, use a large dotty array to show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:
333 x 18 (thirding and trebling)
60 x 750
300 x 180 (thirding and trebling)
120 x 225
24 x 125

Proportional Adjustment (Division)
Volunteers from the Southland Ornithological Society tagged 1680 sooty shearwaters in a 12 month period. How many on average did they tag per month?
In division, proportional adjustment involves changing both numbers in the equation by the same factor. Therefore, the numbers used to proportionally adjust the problem must be factors of both numbers in the equation. For example:
If I divide the 1680 and the 12 by 2 my equation becomes 840 ÷ 6 and I can divide them both by 2 again to get 420 ÷ 3 which is 140. Or I could divide them both by 4 to get the same equation..

The following questions can be used to elicit discussion about the strategy:

  • What could you divide both of these numbers by to make an easier equation?
  • Why is this strategy useful for this problem?
  • What knowledge helps you to solve a problem like this?

If the students do not seem to understand the proportional adjustment concept, use equipment to show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:
1800 ÷ 15 (→ 3600 ÷ 30)
1962 ÷ 18 (→ 981 ÷ 9)
1498 ÷ 14 (→ 749 ÷ 7)
1728 ÷ 16 (→ 864 ÷ 8)

Place Value Partitioning (Multiplication)
Nick has $3121, and needs 8 times this amount to buy the new four wheel drive he wants. How much money does the four wheel drive cost?
The place value strategy involves multiplying the ones, tens and hundreds. In the above problem the student might say the following:
I multiplied 3000 x 8 and got 24 000 then I added the $800 (100 x 8) and 160 (20 x 8) then added the 8 (1 x 8) to get 24 968

The following questions can be used to elicit discussion about the strategy:

  • How can you use your knowledge of place value to solve this problem?
  • Why is this strategy useful for this problem?

If the students do not seem to understand the partitioning concept, show the problems physically. Some students may find it useful to record and keep track of their thinking. An extension of the place value strategy involves the use of standard written form for multiplication.

Use the following questions for further practice if required:
61 323 x 30
7 x 4110
1020 x 40
342 x 11

Place value partitioning (division)
Pisi has an after school job at the market, bagging pawpaw into lots of 6. If there are 864 pawpaw to be bagged, how many bags can he make?
The place value partitioning strategy for division involves ‘chunking’ known facts and subtracting them from the answer. The long division written form will be familiar to most teachers. In the case above, a student might think:
100 lots would be 600. That leaves me with 264. I can take 120 away from that, which is twenty lots of 6. That leaves 144. If I take another 120 I get 24, which is 4 lots of 6. So I’ve taken away 100 lots, then 20 then 20, then 4… the answer’s 144.

This thinking could be recorded as:
equation  table

If the students do not seem to understand the partitioning concept, show the problems physically. Some students may find it useful to record and keep track of their thinking. An extension of the place value strategy involves the use of standard written form for division.

Use the following questions for further practice if required:
676 ÷ 4
9760 ÷ 8
3808 ÷ 7
3472 ÷ 15
2546 ÷ 18

Factorisation (Multiplication and Division)
Stephanie has 486 marbles to share evenly amongst eighteen of her friends. How many marbles will each person get?

The factorisation strategy involves using factors to simplify the problem. In this instance eighteen can be factorised as 2 x 3 x 3. This means dividing by two, then three, then three has the same net effect as dividing by 18. Likewise, multiplying by two, then three, then three has the same net effect as multiplying by 18. In applying factorisation to the above problem, a student might think:

18 is the same as 2 x 3 x 3. So I have to halve 486, then third, then third. If I divide 486 by 2 I get 243. 240 divided by 3 is 80, so 243 divided by 3 will be 81. 81 divided by 3 is 27. The answer is 27.

The following questions can be used to elicit discussion about the strategy:

  • How can you use your knowledge of factors to solve this problem?
  • Why is this strategy useful for this problem?

If the students do not understand the factorisation concept, show the problems physically. Some students may find it useful to record and keep track of their thinking.

Use the following questions for further practice if required:
532 ÷ 8 (÷2, ÷2, ÷2)
348 ÷ 12 (÷2, ÷2, ÷3)
4320 ÷ 27 (÷3, ÷3, ÷3)
135 x 12 (x2, x2, x3)
43 x 8 (x2, x2, x2)
27 x 16 (x2, x2, x2, x2)

Each day follow a similar lesson structure to the introductory session, with students sharing their solutions to the initial questions and discuss why these questions lend themselves to the strategy being explicitly taught. Conclude each session by having students make some statements about when this strategy would be useful and why (e.g. "place value is useful when there is limited renaming required" or "factorisation is useful when one of the factors is able to be renamed as a series of smaller factors"). It is important to record these key ideas as they will be used for reflection at the end of the unit.

Reflecting

As a conclusion to the weeks work, give the students the following five problems to solve asking them to predict which strategy they think will be useful for each problem and why they think this is the most useful strategy before they solve them. After they have solved the problems engage in discussion about the effectiveness of their selected strategies for the problems.

When discussing there may be a few students who do not concur with the group about the usefulness of a particular strategy in a given problem. This is perfectly acceptable as long as they are able to provide a reasonable justification for their thinking.

Problems for discussion (more than one strategy might be suitable for these)
48 x 50 (proportional adjustment)
559 ÷ 13 (place value partitioning)
29 x 16 (rounding and compensating)
1926 ÷ 18 (proportional adjustment)
212 x 11 (place value partitioning)
704 ÷ 8 (factorisation)
153 ÷ 17 (rounding and compensating)
421 x 8 (factorisation)

Ask the students to create problems for a partner where one of the strategies covered in this unit is the most useful.

Conclude the unit by showing the students the questions asked in the initial session again and discuss whether they would solve them in a different way now, why or why not. Review the modelling book or record of statements or generalisations about the strategies and make any changes.