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Level Three > Number and Algebra

# The Multiplier (small 2-digit by 1-digit numbers)

Keywords:
Purpose:

This unit uses The Multiplier, one of the digital learning objects that can be found in Te Pataka Matihiko, Our Digital Storehouse, to multiply 2-digit numbers by 1-digit numbers. It is designed for students transitioning from Stage 5-Early Additive to Stage 6-Advanced Additive of the Number Framework. This unit is the first in a series of four units that focus on using the Multiplier as an aid to learning multiplication. It includes examples of word problems that use multiplication, students are also encouraged to make up problems of their own.

Achievement Objectives:

Specific Learning Outcomes:

split a 2-digit number less than 20 into tens and ones.

use The Multiplier to partition and solve 2-digit by 1-digit multiplications.

describe how to solve 2-digit by 1-digit multiplications using a diagram.

Description of mathematics:

This learning object is suitable for students transitioning from Stage 5-Early Additive to Stage 6-Advanced Additive of the Number Framework who can find the product of two numbers by repeated addition; skip counting or similar method but who have a limited number of basic multiplicative number facts. For this reason the numbers in this unit should be kept to 2-digit numbers less than 20 and 1-digit numbers up to 5.

Students at this stage are learning to use part-whole strategies to solve problems. Here we see them using these strategies to simplify products of 2-digit and 1-digit numbers. They should see that an efficient way to do such multiplications is to split the 2-digit number into tens and ones and then multiply each of these by the original 1-digit number.

Activity:

### Working with the learning object with students

1. Show students the Multiplier: make your own easy multiplications and introduce them to Maddy, who will help them to break numbers up so that they can be more easily multiplied.
2. Enter two numbers into the learning object. Depending on the students' ability, 12 and 3 may be a good first choice. Then tell them that they need to press 'Solve' next.
3. Show them how moving the top diamond down splits the 12 and brings up two multiplications by 3. Ask them what they can tell you about the two other numbers in the boxes with the threes (they always add up to 12). Point out that it may be easier to multiply these two numbers by 3 than to multiply 12 by 3. Ask them
What would be a good way to split the 12 to help us work out the answer? Use one of the splits given by the students as an example, say 4 x 3 and 8 x 3.
4. Ask them what 4 x 3 is and type the answer, 12, in the box. (This can be done by repeated addition, skip counting, or it may be a known fact) Ask what has happened. Make sure that they realise that 12 has been recorded beneath the 12 by 3 rectangle.
5. Repeat this with the 8 x 3 part of the calculation. Get them to see for themselves that 12 + 24 is now recorded below.
6. By discussion convince them that 12 + 24 = 36 is the same as 12 x 3.
What has happened to the 12 from the original problem? Why did we split the number that way?
Enter the answer 36 in the place provided. Show them that pressing the return key gets a response from Maddy.
7. Click the 'Reset' button to redo the 12 x 3 problem and show how the numbers can be partitioned in several different ways to make the problem easier. Involve the students in each calculation. If possible let the students do this by themselves or in pairs at their own computer.
Which way of splitting the 12 leads to the easiest calculation? (Lead them to 12 = 10 + 2.)
Why is this the easiest?
8. Now ask them what they think the side diamond might do. Take them through the use of the side diamond in a similar way to the way you did the top diamond.
What happens when we move this diamond?
How would that help us solve the problem?
9. Consider which diamond is more useful and which way of splitting 12 x 3 gives the simplest calculation overall. Encourage the students to look for known facts as they experiment with different number splits.
10. Now give them the chance to do several examples for themselves. If you only have a limited number of computers, let the students work in front of the group so that all the other students can be involved. Encourage them to think about the best ways of splitting up the numbers in each case. At this stage, keep the single digit number to one that is less than or equal to 5 and the 2-digit number to one that is less than 20.

Once the students are confidently using the learning object give them examples to solve involving Maddy and her domestic problems. Ask them to use the Multiplier to find a solution.

Maddy is tiling the floors in her house and she needs to know how many tiles to buy.

1. Tell them that Maddy has a kitchen where she needs 13 tiles in one direction and 4 in the other. How many tiles will she have to buy to cover the floor of the kitchen?
2. Draw a rectangular picture with the 13 vertically and the 4 horizontally. How can Maddy find out how many tiles she'll need? Lead them to use The Multiplier to do this.
3. Encourage students to include the 10 and 3 split as one way to solve the problem.
4. Vary the number of tiles in Maddy's kitchen and find out how many are required.
5. As students work, question them about the ways they are splitting the numbers to solve the problems.
How did you use The Multiplier? What numbers did you use in the problem? How did you split up these numbers? Why did you split the number in this way? Why was that the easiest way to split the number? How could you show that in a diagram?

### Students working independently with the learning object

Use one of the contexts below to set problems for the students to solve independently, by themselves or in pairs, with the aid of The Multiplier. You can vary the numbers in each context to provide more problems.

• Mum is going to cover part of the garden with concrete blocks. She needs to use 17 in one direction and 5 in the other. How many blocks will she have to buy?
• Dev is being paid \$5 an hour for a job that will last 16 hours. How much will he get paid?
• Marti is making up bags of lollies for the school fare. He is going to put 4 lollies in each bag. If he plans to make up 18 bags, how many lollies will he need?
• Milly and her mates are going to get a hamburger each. Hamburgers cost \$4 and there are 13 girls. How much will all the hamburgers cost?

When the students have solved a problem, ask them to draw a diagram to show how they used the Multiplier. Using a rectangle in their drawing would be helpful but they may think of other ways to represent the problems. Ask them the following questions:

How did you use The Multiplier?
What numbers did you use in the problem?
How did you split up these numbers?
Why did you split the number in this way?
Why was that the easiest way to split the number?
How could you show that in a diagram?

When all of the students have described their solution diagrammatically, get the entire group together and let everyone describe their solutions using their diagrams. Ask them if they could now do these problems without using Maddy's Multiplier.

### Students working independently without the learning object

Independent activities that develop the same concepts as the learning object include:

• Figure It Out, Number, Levels 2-3, page 14 High Flyers
• Number Sense and Algebraic Thinking,, Book 1, Levels 2-3, page 18, Clean Cars
• Students can make up problems of their own to solve and pass on to another member of the group or pair to solve. The whole group can then be brought back to check answers and discuss strategies.

Independent activities that develop the knowledge required at this stage include:

• Material Master 6-2 Multiplication or Out
• Material Master 6-3 Multiplication Loopy
• Material Masters 6-4 Fly Flip;
• and Material Masters 6-6(a) Four in a Row Multiplication

## The Number Partner

This unit uses one of the digital learning objects, the number partner, to support students as they investigate possible pairings for numbers from 10 to 30. It is suitable for students working with Advanced Counting and Early Additive strategies (Stage 4-5 of the Number Framework). It includes problems and questions that can be used by the teacher when working with a group of students on the learning object, and ideas for independent student work.

## Multiplication and Division Pick 'n' Mix 2

In this unit we look at a range of strategies for solving multiplication and division problems with whole numbers and decimal fractions, 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 Multiplication and Division Pick ‘n’ Mix 1.

## Squirt Level 4

This unit explores how the suite of learning objects, "Squirt", can be used to support students’ development of multiplicative thinking. Squirt encourages students to anticipate multiplicative measurement relationships, e.g. three measures of A fit in B, by partially filling a container and imaging how many more squirts will be needed. Since the containers are not always cylindrical it also develops ideas about conservation of volume.

## Multiplication and Division Pick n Mix 1

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.

## The Multiplier (origin of the algorithm)

This unit uses The Multiplier, one of the digital learning objects that can be found in Te Pataka Matihiko, Our Digital Storehouse, to multiply 2-digit numbers by 2-digit numbers. It is designed for students working at Stage 7-Advanced Multiplicative of the Number Framework and is the fourth in a series of four units that focus on using the Multiplier as an aid to learning multiplication. In this unit, The Multiplier is used as a tool to outline the origins of the standard algorithm for multiplication.