What's Best?

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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
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Student Activity

 

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Specific Learning Outcomes

use different strategies to solve multiplication problems

Description of Mathematics

Number Framework Links
These activities will help students to consolidate advanced multiplicative part–whole thinking (stage 7) by comparing a range of strategies for multiplying whole numbers.
 

Required Resource Materials

FIO, Level 3, Number Sense and Algebraic Thinking, Book One, What's Best? pages 6-7

Classmates

Activity

Students need to be able to choose appropriately from a broad range of mental strategies when solving problems. These activities are designed to help them develop that ability by having them compare and identify the types of problems and numbers that would best suit each multiplication strategy.
Students need to be at least progressing towards being advanced multiplicative part–whole thinkers (stage 7) to complete these activities. They need to understand how to use each of the strategies outlined below and have good recall of their multiplication basic facts.
These activities give students lots of opportunities to come to a shared understanding of terminology such as “adding on”, “using place value”, and “doubling and halving”, which they need when talking about their problem-solving strategies.

Activity One

Before the students look at the activity, pose the problem: There are 4 teams of 18 players going to the rugby championships. How many players are going altogether?
Record 4 x 18 =  on the board or in the class modelling book (a scrapbook for recording outcomes and strategies). Encourage the students to describe their solution path with a classmate or small group before the range of strategies is shared in the main group. The students may need to use materials to help them understand each strategy.
Record all the different ways of calculating the answer on the board as they are described and reflect back what each student said, using phrases such as: So you went … and then you … Is that how you did it? This gives the listeners visual support as well as another chance to hear each explanation.
Now ask the students to read the strategies used by Ashleigh’s group in Activity One.
Then ask:
Which strategies are the same as ours?
Which ones are different? (Record any new ones on the board.)
Can you model these strategies with the place value materials?
The students can approach the questions in the activity orally as think-pair-share or group discussion starters. They should record their thinking so that they can report back. The chart in the Answers provides a suitable structure for recording their responses to question 1.

Activity Two

Before starting this activity, draw a table with three columns on the board:
Then ask:
What are some examples of problems from Activity One that can be solved quickly and easily with these strategies?
(Some problems may be suited to more than one strategy. Note that 27 x 3 best suits a thirding and trebling strategy.)
What do you notice about the numbers in the tidy number column? (One number in each problem is close to a tidy number, for example, 49 and 19.)
What do you notice about the numbers in the doubling strategies column? (One number is 4 or 8; both numbers are even.)
This activity provides opportunities for promoting algebraic thinking. Question 4 in Activity Two asks students to identify similarities between problems that are best solved using tidy number strategies or doubling and halving. This takes them away from specific examples to properties that are always true for all numbers.
At the end of the activity, the students make up their own problems, and you could use these to assess who has understood and generalised the characteristics of the problems that suit each strategy. Students should be able to explain why they chose particular numbers. Note that doubling and halving is one of a family of strategies that includes thirding and trebling, for example, 18 x 33 = 6 x 99. Doubling repeatedly is one example of a family that uses factors of a factor, for example, 6 x  = 3 x 2 x  or 2 x 3 x . Place value works on any problem, but renaming sometimes makes it difficult. Student reasons could include:
• For a tidy number strategy:
“I want one of the numbers in the problem to be close to a tidy number so that it’s easy to round in my head. If I wanted it to be a 2-digit number, I could choose 1, 2, 3, 7, 8, or 9 for the ones digit, for example, 41 or 49 are close to 40 and 50. If I wanted it to be a 3-digit number, I could make it hundred and ninety , for example, 693 or 697, which are both close to 700. So I could end up with problems like 6 x 49 = 6 x 50 – 6 or 3 x 697 = 3 x 700 – 9.”
• For doubling and halving:
“I want to turn the problem into an easier problem or a fact I know. I want at least one of the numbers in the problem to be even so that it’s easy to halve, for example, 6, 8, 10, 12, 14, 16, 18, or 20. (I need to know my multiplication facts to use these.) I want the other number to end up being a times table I know when it’s doubled, for example, if I chose 4, I could use my 8 times tables. I know how to multiply by tens or hundreds, so I could make the second number half of a tens or hundreds number, for example, 25, 35, 50, or 200. So I’d end up with problems like 16 x 4 = 8 x 8, 18 x 50 = 9 x 100, or 14 x 35 = 7 x 70.”
• For doubling repeatedly:
“I want one of the factors to be 2, 4, or 8 so that I can double for 2, double and double again for 4, or double and double and double for 8. The doubling would be easier if I made the digits in the other number small, for example, 13. So I could end up with problems like 8 x 13 = 4 x 26 = 2 x 52 or 4 x 32 = 2 x 64.”
• For a place value strategy:
It’s easier to use place value partitioning if the digits in the numbers are small, such as 3 x 213, so that there is no renaming. It’s also easier if one number has only one digit or is a multiple of 10 or 100, so that there are only two or three parts to add together, such as 5 x 423 = (5 x 400) + (5 x 20) + (5 x 3), or 50 x 423 = (50 x 400) + (50 x 20) + (50 x 3), or 500 x 423 = (500 x 400) + (500 x 20) + (500 x 3).
Some students are reluctant to use a variety of strategies and prefer to use one trusted method. You could use the following strategies to support those who are having difficulties:
• Try giving them more practice in trying different strategies: Let’s all use Sarah’s method to solve this next problem.
• Have group discussions about why being able to use a variety of strategies may be useful.
• Emphasise the importance of efficiency by asking questions such as: Now that you’ve solved this problem, what would you do differently next time if you were doing a similar problem? or Is there a faster way of solving this problem?
After the students have completed the activity, pose questions for the group to talk about in a think-pair- share discussion:
Which multiplication strategy do you use most often? Why?
Now that you have compared some strategies, is there a strategy that you will try to use more often? What kinds of problems will you use it for?
Which “special numbers” will you keep an eye out for in multiplication problems that will tell you to try a certain strategy?
“Numbers close to tidy numbers, such as 41 or 199, will tell me to try a tidy number strategy.”
“If there is a 4 or an 8 in the question, I can try doubling repeatedly. I can look out for numbers that can be halved and doubled to make a multiplication fact I know.”

Extension

Have the students, in pairs, make a Wanted poster or write a newspaper Situations Vacant advertisement that describes the characteristics of problems that can be solved efficiently using a particular strategy.
Sentence starters might include:
• Wanted! Problems that can be solved using tidy numbers!
• We are looking for …
• Successful applicants will be/have …
• These numbers can be recognised by …
Display the posters or advertisements as a reference to remind the students to choose their strategies wisely.
 

Answers to Activities

Activity One
1. Discussion will vary. Possible comments include:
2. Opinions will vary. If you know 8 x 9, doubling and halving is an efficient strategy. The tidy number strategy is also a smart way of simplifying the problem because 18 is close to 20, so it can be easily solved in your head. Place value strategy is also efficient for this problem if you know 4 x 8 = 32.
3. Discussion will vary.
Adding on: This would take a long time because you’d have to add 9 lots of 24 (or 24 lots of 9).
It would be easy to lose track of how many you’d added on so far.
Double and double again: This is not a sensible strategy to use because 9 is an odd number.
Tidy numbers: This is an efficient strategy because 9 is close to 10, a tidy number:
9 x 24 = (10 x 24) – (1 x 24)
= 240 – 24
= 216
You could also use the tidy number 25, which is close to 24, if you’re good at multiplying by 25:
9 x 24 = (9 x 25) – 9
= 225 – 9
= 216
Place value: This is an efficient strategy because there aren’t too many steps to remember in your head: 9 x 20 = 180; 9 x 4 = 36; 180 + 36 = 216
Doubling and halving: This strategy doesn’t give you an easy fact that you’d probably know straight away, but it might help you make the problem easier to calculate using another strategy. You’d have to double and halve a few times to help:
9 x 24 = 18 x 12 = 36 x 6 = 72 x 3, or you could third and treble: 9 x 24 = 3 x 72
Using known facts: Whether this strategy would be efficient for you would depend on which facts you knew. If you knew 9 x 12 = 108, you might double that to get 9 x 24. Or you could work out (9 x 10) + (9 x 10) + (9 x 4).
4. a. i. Efficient strategies include:
Doubling repeatedly: 2 x 26 = 52; 2 x 52 = 104, 2 x 104 = 208
Double and halve repeatedly: 8 x 26 = 4 x 52 = 2 x 104 = 208
Place value: (8 x 20) + (8 x 6) = 160 + 48 = 208
Using known facts: If you know that 4 x 25 = 100, then 8 x 25 = 200.
Add one more lot of 8 so there are 26 lots of 8: 200 + 8 = 208
ii. Efficient strategies include:
Tidy numbers: (50 x 7) – 7 = 350 – 7 = 343
Place value: (40 x 7) + (9 x 7) = 280 + 63 = 343
b. Reasons will vary.
5. a. Efficient strategies include:
i. Double and halve: 1 000 x 4 = 4 000
Place value: (5 x 8) x 100 = 40 x 100 = 4 000
ii. Tidy numbers (because 19 is close to a tidy number): 20 x 25 = 500.
500 – 25 = 475
iii. Divide by 3 and treble (which turns the problem into a known fact): 9 x 9 = 81
b. Answers will vary.
Activity Two
1. Problems will vary, but each one should have at least 1 factor that is close to a tidy number, for example: 3 x 198, 29 x 5, 15 x 28
2. Problems will vary. For doubling repeatedly, one of the factors should be 4, 8, or 16, for example: 4 x 46, 23 x 8. For halving and doubling, the problem should turn into a known fact when halved and doubled, for example: 4 x 16 can be solved as 8 x 8, or 12 x 25 can be solved as 6 x 50.
3. Problems will vary. Place value strategy is especially useful when there is no renaming, for example: 8 x 12 = 8 x 10 + 8 x 2.
4. a. One of the numbers in the problem ends in 1, 2, or 3 or in 7, 8, or 9.
b. One number is even and can easily be halved. The other number is easy to double.
5. Practical activity

Attachments
WhatsBest.pdf449.43 KB
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Level Four