The Multiplier (origin of the algorithm)

The Learning Objects

The learning object, The Multiplier, can be accessed from the link below, using your school's username and password:

• The multiplier

Click for more information about the learning objects and who can register to use them.

The Multiplier consists of a series of 5 learning objects. Two forms of The Multiplier are used in this unit: The Multiplier: make your own hard multiplications and The Multiplier: generate your own hard multiplications.

Working with the learning object with students

Notes:

• This unit is presented as a series of four teaching sessions but it is not anticipated that these sessions would run consecutively. Allow the pace of the sessions to be dictated by the students, working through the material in each session progressively and allowing time for students to become confident before moving on.
• As students calculate answers it is important for them to develop strategies to estimate approximate answers and check whether the answers they are obtaining are reasonable. Encourage them to do this by asking questions such as
  Is that answer reasonable? How do you know?
  Does that answer seem about right? How could you check?

Session 1: Using The Multiplier

Students who have been involved in the previous unit, The Multiplier (2-digit by 2-digit numbers), miss the first session as it repeats the material covered there.

1. Show students The Multiplier: make your own hard 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 43 and 24 may be a good first choice. Then show the students that they need to press 'Solve' next.
3. Show them how moving the top diamond down splits the 43 and brings up two multiplications by 24. Ask them what they can tell you about the two other numbers in the boxes with the twenty-fours (they always add up to 43). Point out that it may be easier to multiply these two numbers by 24 than to multiply 43 by 24.
   What would be a good way to split the 43 to help us work out the answer? Use one of the splits given by the students as an example, 30 x 24 and 13 x 24.
4. Ask them what 30 x 24 is and type the answer, 720, in the box. Ask what has happened in The Multiplier. Make sure that they realise that 720 has been entered beneath the 43 by 24 rectangle.
   
5. Repeat this calculation with the 13 x 24 part of the calculation.
6. Get the students to complete these calculations: 30 x 24 = 720; 13 x 20 = 260; and 13 x 4 = 52. By discussion convince them that 720 + 260 + 52 is the same as 43 x 24.
   What has happened to the two numbers from the original problem? Why?
   How did splitting the numbers make the problems easier to solve?
   

   Enter the answer 1032 in the place provided. Show them that pressing the return key gets a response from Maddy.

7. Ask the students to check their answer using estimation. Note that this problem could be checked by multiplying 40 x 25 (to get 1000) and allowing for the answer to be a larger number due to the 3 in the number 43.
   Is that answer reasonable? How do you know?
   Does that answer seem about right? How could you check?
8. Using 43 x 24, experiment with a few more splits to illustrate the use of the top diamond and side diamonds, ensuring that the tens and ones split is included. 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 43 leads to the easiest calculation? ( 43 = 40 + 3.)
   Why is this the easiest?
   Which way of splitting the 24 leads to the easiest calculation? (24 = 20 + 4.)
   Why is this the easiest?
   
9. Now give them the chance to do several examples for themselves by using The Multiplier: generate your own hard multiplications. 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 and to check their answers by estimation.
10. 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 24 = 30 - 6. So by using the right half-diamond we can get two calculations 43 x 30 and 43 x 6. By working these out to give 2190 and 438, the answer to 73 x 24 can be written as 1290 - 258 = 1032 (emphasise the negative here and ask them why they need to subtract rather than add). (These calculations might be too hard to use every time - see later.)
    If we move this bar to the right what is happening to the number 24?
    Can we use this to find multiplications that are easy to solve?
    What do we need to do with the extra 6 we have added on? Why?
11. Now go on to the bottom half-diamond. Note that here, moving the bottom half-diamond allows us to change 43 to 80 - 7 so 43 x 24 becomes 50 x 24 - 7 x 24. This is relatively easy to do. Let them experiment with this half-diamond.
    If we move this bar down what is happening to the number 43?
    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?
12. Now let them work for a while by doing several examples using The Multiplier: make your own hard multiplications. Remember to encourage them to check the reasonableness of their answers by estimation. After they have worked on these for a while, ask them which is the best strategy in these problems and why.

Now give the students some questions about Maddy and her domestic problems. The answers to these should be calculated using The Multiplier.

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 lounge where she needs 33 carpet tiles in one direction and 16 in the other. How many carpet tiles will she have to buy to cover the floor of the lounge?
2. Draw a rectangular picture with the 33 vertically and the 16 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 3 split as one way to solve the problem. Which is the best method for solving this problem?
4. Let them begin to see that splitting the 33 into 30 and 3 is a helpful thing to do as it makes the multiplication easier. They might also see that changing 16 to 10 + 6 is helpful. Which is the best method for solving this problem?
5. Vary the number of tiles in Maddy's lounge 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?
   How did you know your answer was reasonable?

Session 2: Vertical Format for Multiplication (2 line)

1. Pose the multiplication problem: 23 x 35. Ask students how they would use The Multiplier to solve this.
2. Have the students use The Multiplier to partition the numbers in a variety of ways to find the answer to the problem.
   How can we split these numbers to make the multiplication problem easier?
   Are there any other ways we can split these numbers to make the problem easier?
3. Ask students which splits provide the quickest way to solve the problem. Emphasise that the smaller the number of splits, the quicker the problem can be solved.
   Which split is the quickest way to find an answer? Why?
4. Explain to the students that it's helpful to be able to solve multiplication problems quickly, and tell them that a vertical format is one way to do this. Using the problem 25 x 35 and The Multiplier introduce them to a 2 line algorithm:
   

   Show students the vertical format and how to record the 2 areas from the multiplier under the original problem then add them together vertically: 25 x 35 = 125 [from 25 x 5] + 750 [from 25 x 30) = 875.

   Encourage students to link the vertical format with the splits given by The Multiplier and the numbers in the original problem. Ask:
   Which part of the vertical format shows the blue region of The Multiplier?
   Which numbers in the vertical format show the red region of The Multiplier?
   What has happened to the 35 in this problem? Why?

5. Complete several more examples in this way. The problems 40 x 15, 55 x 22 and 20 x 15 are good examples to use. For each problem encourage the students to link the vertical format with the splits given by The Multiplier.
6. As they work encourage the students to check their answers using estimation.
   Is that answer reasonable? How do you know?
   Does that answer seem about right? How could you check?

Session 3: Vertical Format for Multiplication (4 line)

1. Ask the students to use The Multiplier: make your own hard multiplications to work out the answer to 67 by 35. Experiment with a variety of different ways to split the numbers then ask:
   What would it look like if we split both of the 2-digit numbers into tens and ones?
   Would this make the problem easier to solve? Why?
2. Make these splits on the Multiplier.
   
   

   Ask the students to add the four numbers to work out the answer to the problem: 2345.

3. Remind the students that a vertical format provides a way to solve multiplication problems quickly.
   Can this problem be easily solved with two splits? Why not
   Is the two line algorithm simple to use with this problem? Why not?
4. Explain that a four line algorithm can be used to solve multiplication problems that are most easily solved with 4 splits. Show the students how to lay this problem out as a four row algorithm.
   

   In this algorithm each of the 4 areas from The Multiplier corresponds to one of the lines of the algorithm. Encourage students to link the vertical format with the splits given by The Multiplier and the numbers in the original problem. Ask:
   Which part of the vertical format shows the blue region of The Multiplier?
   Which numbers in the vertical format show the red region of The Multiplier?
   What has happened to the 35 in this problem? Why?
   What has happened to the 67 in this problem? Why?

   Clarify that the order of the lines beneath the problem in the vertical algorithm does not affect the answer to the problems:

   Does it matter which order the 4 parts of the problem are written? Why? Why not?
   Which way do you find easy to list the lines in the vertical format? Why?

5. Provide students with several examples to work through in this way, using The Multiplier to split each of the numbers into tens and ones and recording the four row algorithm for each. Problems to try include: 25 x 63, 31 x 68 and 78 x 54.

   For each problem encourage the students to link the vertical format with the splits given by The Multiplier and the numbers form the original problem.

6. As they work encourage the students to check their answers using estimation.
   Is that answer reasonable? How do you know? Does that answer seem about right? How could you check?

Session Four: Which Format is Best?

1. Explain to the students that it's not always convenient to use The Multiplier or draw a diagram to solve multiplication problems. A quicker way can be to use just the vertical format, splitting the numbers into parts to help solve the problem.
2. Provide the students with a variety of multiplication problems to solve, some that would be most simply solved using a 2 line algorithm and some that would need a 4 line algorithm. Have students complete the examples, using The Multiplier or drawing a diagram only when necessary. As students work, discuss their use of the vertical formats.
   How many lines did you use to work out the answer to this problem? Why?
3. Encourage the students to use the 2 line format where the numbers are more simply multiplied (generally when there is 0, 5 or 2 in the ones place) and the four line format when the numbers are more complex and need to be split into tens and ones.
   Which kinds of numbers is it easy to solve using 2 splits? Why?
   Which kinds of numbers need more then 2 splits? Why?
   Which problems are most easily solved using the 2 line format? Why?
   Which problems are most easily solved using the 4 line format? Why?
4. As they work encourage the students to check their answers using estimation.
   Is that answer reasonable? How do you know?
   Does that answer seem about right? How could you check?

Students working independently with the learning object

Problems in Context

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 58 cars. How many cars will fit in the car park?
• The Sea Tower has 38 floors and each floor has 47 windows. How many windows has the Sea Tower?
• 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 23 toys in each sock, how many toys will he need?
• Gill can save $34 a week packing shelves at the supermarket. She wants to earn $2,000 for a holiday. How many days must she work to save up that much money?

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 good 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, Book 3, Level 3, page 12, What a View!
• Figure it Out, Number, Book 3, Level 3, page 13, Easy Nines
• Figure it Out, Number, Book 2, Level 3, page 7, Singing Up a Storm
• Figure it Out, Number, Book 2, Level 3, page 10, Multiple Methods
• 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.

For students who need reinforcement of basic facts, the activities in Material Masters 6-4, Fly Flip; Material Master 6-6(a), Four in a Row Multiplication; Material Masters 6-6(b), Multiplication Roundabout; and 6-10 Factor Leapfrog can be used.