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

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 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.

• Enter two numbers into the learning object; the numbers 73 and 24 may be a good first choice. Then show the students that they need to press 'Solve' next.
• Show them how moving the top diamond down splits the 73 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 73). Point out that it may be easier to multiply these two numbers by 24 than to multiply 73 by 24. 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, 50 x 24 and 23 x 24.
• Ask them what 50 x 24 is and type the answer, 1200, in the box. Ask what has happened in The Multiplier. Make sure that they realise that 1200 has been entered beneath the 73 by 24 rectangle.
  
• Repeat this calculation with the 23 x 24 part of the calculation. Actually this is not that easy. It's probably a good idea to break up the 24 at this stage. Ask them how they might do this to make the multiplication easier. This probably requires dragging the left diamond to give, for example, the split 23 x 20 and 23 x 4. (Note that if 23 x 4 is too hard then you may like to break 23 x 4 into 20 x 4).
• Get the students to complete these calculations: 50 x 24 = 1200; 23 x 20 = 460; and 23 x 4 = 92. By discussion show them that 1200 + 460 + 92 = 1752 is the same as 73 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 1752 in the place provided. Show them that pressing the return key gets a response from Maddy.
• Ask the students to check their answer using estimation. Note that this problem could be checked by multiplying 70 x 20 (to get 1400) and allowing for the answer to be a larger number due to the numbers in the ones place. Is that answer reasonable? How do you know? Does that answer seem about right? How could you check?
• Using 73 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 73 leads to the easiest calculation? ( 73 = 70 + 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?
  
• 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.
• 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 73 x 30 and 73 x 6. By working these out to give 2190 and 438, the answer to 73 x 24 can be written as 2190 - 438 = 1752 (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?
• 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 24 becomes 80 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 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?
• Now let them work for a while by doing several examples using The Multiplier: generate your own hard multiplications. Remember to encourage them to check the reasonableness of their answers by estimation. After they have worked on several examples 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 43 carpet tiles in one direction and 26 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 43 vertically and the 26 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 40 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 43 into 40 and 3 is a helpful thing to do as it makes the multiplication easier. They might also see that changing 26 to 20 + 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?

  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?
  How did you know your answer was reasonable?

  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.