The Multiplier (larger 2-digit by 1-digit numbers)

Working with the learning object with students

Students who have been involved in the previous unit, The Multiplier (small 2-digit by 1-digit numbers), may be able to skip most of steps 1 to 10.

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; the numbers 73 and 8 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 73 and brings up two multiplications by 8. Ask them what they can tell you about the two other numbers in the boxes with the eights (they always add up to 73). Point out that it may be easier to multiply these two numbers by 8 than to multiply 73 by 8.
   What would be a good way to split the 73 to help us work out the answer? Use one of the splits given by the students as an example, say 50 x 8 and 23 x 8.
4. Ask them what 50 x 8 is and type the answer, 400, in the box. Ask what has happened in The Multiplier. Make sure that they realise that 400 has been entered beneath the 73 by 8 rectangle.
5. Repeat this with the 23 x 8 part of the calculation. (Actually this is not that easy. There may be an easier way.) Get them to see for themselves that 400 + 184 is below the rectangle.
   
6. By discussion convince them that 400 + 184 = 584 is the same as 73 x 8.
   What has happened to the 73 from the original problem?
   Why did we split the number that way?
   Enter the answer 584 in the place provided. Show them that pressing the return key gets a response from Maddy.
7. Click the 'Reset' button to redo the 73 x 8 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 73 leads to the easiest calculation? (Lead them to 73 = 70 + 3.)
   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 73 x 8 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 in turns 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.
11. Ensure the strategies used by students include compensation. Note that there are half-diamonds on the right and bottom of the rectangle. Experiment with moving the right one to see what it does. It should soon be clear that these allow for compensation to be used. Clearly 8 = 10 - 2. So by using the right half-diamond we can get two calculations 73 x 10 and 73 x 2. By working these out to give 730 and 146, the answer to 73 x 8 can be written as 730 - 146 = 584 (emphasise the negative here and ask them why we need to subtract rather than add).
    If we move this bar to the right what is happening to the number 8?
    Can we use this to find multiplications that are easy to solve?
    What do we need to do with the extra 2 we have added on? Why?
12. Now go on to the bottom-half diamond. Note that here, moving the bottom-half diamond allows us to change 73 to 80 - 7 so 73 x 8 becomes 80 x 8 - 7 x 8. Again this is relatively easy to do. Let the students experiment with this half-diamond.
    If we move this bar down what is happening to the number 73?
    Can we use this to find multiplications that are easy to solve?
    What do we need to do with the extra 7 we have added on? Why?
13. Now let the students work for a while with 'generate your own easy multiplications'. Here they will be given random numbers to multiply. After they have tried several of these, ask them which is the best strategy in these problems and why.

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 bathroom floor where she needs 32 tiles in one direction and 9 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 32 vertically and the 9 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 30 and 2 split as one way to solve the problem. They might see that changing the 9 to 10 - 1 is also good. Which is the best method for solving this problem?
4. Vary the number of tiles in Maddy's bathroom and find out how many are required.
   How did you do this?
   What did you do with the numbers? Why?
   What was the best way to split the numbers? Why?

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.

• A floor of the Earth Tower has a car park with 46 rows for cars and each row can fit 8 cars. How many cars will fit in the car park?
• The Sea Tower has 8 floors and each floor has 32 windows. How many windows does the Sea Tower have?
• Even Park has a stand that has 7 rows and each row has 56 seats. How many people can be seated in that stand?
• Marti is making up Christmas Socks to sell in the local dairy. If he has 89 socks and is going to put 6 toys in each sock, how many toys will he need?
• Two soccer teams and the referee all sign their autographs 6 times. How many signatures were made? (Assume that no reserves signed their names.)

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 by 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, Level 3, page 22, High-powered Thinking
• Figure It Out, Number, Book 2, Level 3, page 6 Movie Maths
• Figure It Out, Number Sense and Algebraic Thinking, Book2, Level 3, page 11, Horsing Around
• 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 consolidate 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.