Multiplication and Division Pick n Mix 1

Purpose

This unit presents a range of strategies for solving multiplication and division problems with multi-digit whole numbers. Students are encouraged to notice the structure of problems, and to anticipate which strategies might be best suited to solving them. This unit builds on the ideas presented in the Multiplication Smorgasbord session in Book 6: Teaching Multiplication and Division.

Specific Learning Outcomes
  • Mentally solve whole number multiplication and division problems using:
    1. proportional adjustment
    2. place value partitioning
    3. rounding and compensation
    4. 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 New Zealand Curriculum requires students to understand and use a range of mental, written and digital calculation strategies to multiply and divide multi-digit whole numbers. This unit of work is useful for students working at or towards Level Four Stage 7-Advanced Multiplicative of the Number Framework). Students at this stage partition and recombine numbers to simplify calculations and draw on their knowledge of multiplication facts and related division facts with factors up to ten. Understanding whole number place value underpins all strategies in this unit.

The key teaching points are:

  • Features of problems, particularly the numbers involved, privilege the efficiency of particular strategies. Teachers should elicit strategy discussion about problems in order to encourage students to justify their decisions about strategy selection in terms of the usefulness and efficiency of the strategy for the given 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. For example, 6 x 48 might be solved using 6 x 50 = 300.
  • 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). For example, 6 x 2 must be subtracted from 300 to get the product of 6 x 48.
  • Standard place value partitioning is always a trustworthy strategy that is particularly appropriate where one or both factors are not easily rounded up. For example, 6 x 43 is best solved as 6 x 40 + 6 x 3.
  • Proportional adjustment is useful when there is a ‘common factor’ connection between the factors in multiplication, or dividend and divisor in division, 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 the dividend and divisor must be adjusted by the same factor.
Required Resource Materials
Activity

Getting Started

The purpose of this session is to explore the range of strategies that your students already use to solve multiplication and division problems. This will enable you to 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 (Copymaster Two):
Vanessa bikes 38 kilometres each day for five days. How many kilometres has she travelled by the end of the five days?

Ask students to work out the answer in their head if they can and record their strategy on paper. Give the students an appropriate amount of thinking time. Then ask them to share their solutions with their learning partner. The following are possible responses:

Note that the convention is to record the multiplier first so equations should be written as 5 x 38 =  .

Rounding and compensating:
5 x 38
38 is rounded to 40 so the problem becomes 5 x 40 = 200, then 10 (5 x 2) is subtracted from the product to get 5 x 38 = 190.

In full the strategy might be written as 5 x 38 = 5 x 40 – 5 x 2.

View how to do this using Video One.

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

In full the strategy might be written as 5 x 38 = 10 x 19

View how to do this using Video Two.

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

The strategy can be written as 5 x 38 = 5 x 30 + 5 x 8 or in working form:

View how to do this using Video Three.

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). Be aware that some students may elect to add rather than multiply.

For example:

                      

You might like to discuss the efficiency of multiplication versus repeated addition.

Problem 2 (Copymaster Three):
There are 256 rowers entered in the eights rowing champs at the Maadi Cup, not including the drivers (coxwains).

How many crews of eight rowers can be made?

Ask students to work out the answer in their head if they can and record their strategy on paper. Give the students an appropriate amount of thinking time. Then ask them to share their solutions with their learning partner. The following are possible responses:

Place value partitioning (chunking): 
184 ÷ 8
I know that 160 ÷ 8 = 20. That is 20 crews.
There are 24 rowers left. 24 ÷ 8 =3

The answer is 20 + 3 = 23.

View how to do this using Video Four.

Factorisation (proportional adjustment): 

Dividing by 8 is like dividing by 2 then 2 then 2 so half 184 is 92 and half 92 is 46 and divide by 2 again leaves me 23 so the answer is 23.

View how to do this using Video Five.

Rounding and compensating: 
If there were 200 rowers, that would be 25 crews because 4 x 25 =100 so 8 x 25 = 200.  184 is 16 rowers less. That is two crews. So, the answer is 25 – 2 = 23.

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). Watch for repeated subtraction or partial use of multiplication, such as:

10 x 8 = 80

80 + 80 = 160, 20 x 8 = 160

160 + 8 = 168, 168 + 8 = 176, 176 + 8 = 184

So 23 crews can be made

View how to do this using Video Six.

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. For example, In the problem 7 x 29 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.

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

  • What tidy number or numbers could you use that are close to one of the factors 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?

Place Value Partitioning (Multiplication) 
Nick has $54, but he needs 7 times this amount to buy the new mountain bike he wants. How much money does the bike cost?
The place value strategy involves multiplying the ones, and tens separately then combining the partial products. This strategy applies the distributive property of multiplication, as 54 is distributed into 50 + 4.  In the above problem the student might say the following:
I multiplied 7 x 50 and got 350, then I multiplied 7 x 4 and got 28. I added 350 and 28 to get 378.

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:

  • 3 x 135
  • 9 x 66
  • 6 x 132
  • 8 x 79

Place value partitioning (division)
Pisi has an after-school job at the market, bagging pawpaw into bags 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 x 6 = 60 so100 bags would be 600. 864 – 600 = 264. That leaves me with 264.

I can take 120 away from that, which is 20 x 6. That leaves 144. If I take another 120 pawpaw away I get 24, which is 4 lots of 6. So, I’ve taken away 100 lots, then 20 then 20, then 4… the answer is 144.

This thinking could be recorded as:

If the students do not seem to understand the partitioning concept, show the problems physically, e.g. using place value blocks. 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:

  • 760 ÷ 5
  • 516 ÷ 4
  • 992 ÷ 8
  • 3808 ÷ 7
  • 522 ÷ 3
  • 2505 ÷ 9

Rounding and Compensating (Multiplication) 
The Southern Sting netball fans are going to Christchurch to watch a netball game against the Canterbury Tactix.

Each bus is full, with 48 people in it, and there are 9 buses.

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 9 x 50 = 450. In order to compensate for the rounding, two lots of 9 people (18) must be subtracted from the ‘rounded’ equation.

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. Recording might look like this:

9 x 48 = 9 x 50 – 9 x 24                              9 x 40 = 10 x 48 – 1 x 48
= 450 – 18                                                     = 480 - 48
= 432                                                              = 432
 

View how to do this using Videos Seven and Eight.

Use the following questions for further practice if required, still using the same bus context:

  • 7 x 29
  • 6 x 38
  • 5 x 57
  • 3 x 69
  • 4 x 97
  • 8 x 36

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 42) then standard place value partitioning tends to be a more useful strategy.

Rounding and compensating (Division) 

Sarah uses nine bus tickets every week to travel around town. She wins 162 tickets in a radio competition. How many weeks will the tickets last her?
Rounding and compensating for division involves finding a number that is close to the dividend (starting amount) and working from that number to find an answer. For the question above, a student might say:

I know that 20 multiplied by 9 equals 180. 162 is 18 less than 180, that’s 2 x 9.

The tickets would last her 20 – 2 = 18 weeks.

If the students do not seem to understand the rounding and compensating concept, use place value materials, or a large dotty array, to represent the problems physically. Students may find it useful to record and keep track of their thinking, especially if they partially divide the dividend at first.

View how to do this using Video Seven.

Use the same context of buses to pose problems where rounding and compensating is a sensible strategy.

  • 343 ÷ 7
  • 224 ÷ 8
  • 597 ÷ 3
  • 392 ÷ 4
  • 1764 ÷ 18

Proportional Adjustment (Factorisation) 
At the Kapa Haka 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 factors and multiples to create easier equations that have the same answer. Factors are proportionally 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.

View how to do this using Video Eight.

Use the following questions for further practice if required:

  • 25 x 200 (multiply and divide by five)
  • 27 x 3 (thirding and trebling)
  • 33 x 18 (thirding and trebling)
  • 44 x 25 (multiply and divide by four)
  • 24 x 125 (multiply and divide by eight)

Proportional Adjustment (Division) 
Tim delivered the same number of pamphlets each month.

In his first six months the delivers a total of 912 pamphlets?

How many did he deliver each 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 (dividend and divisor). For example:
If I divide the 912 by 3 and 6 by 3, my equation becomes 304 ÷ 2 which has the same answer. Half of 304 is 152. So, Tim delivers 152 pamphlets.

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

  • What is a common factor of both numbers that could be used to make the problem easier?
  • 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:

  • 636 ÷ 12 = 212 ÷ 4 = 106 ÷ 2 = 53
  • 480 ÷ 15 = 96 ÷ 3 = 32
  • 1962 ÷ 18 = 981 ÷ 9 = 109
  • 1498 ÷ 14 = 749 ÷ 7 = 107
  • 1728 ÷ 16 = 864 ÷ 8 = 108

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 halve 486, then third, then third. If I divide 486 by 2 I get 243. 243 divided by 3 is 81. 81 divided by 3 equals 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 a factorisation 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.

There may be problems for which two or more multiplication and division strategies are equally efficient. However, using additive strategies with these problems will not be efficient.

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.


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