Learning experiences and assessment opportunities

Extended Form of Multiplication (adapted from Mathematics K–6 Sample Units of Work, p 125)

Students multiply numbers by breaking the calculation into two parts

eg 32 × 21 = 32 × 20 + 32 × 1

Students are shown how these parts can be combined when using the extended form (without trading) of multiplication for two-digit numbers by two-digit numbers.

Part A
The teacher models the procedure as follows:

Step 1: The teacher poses the problem: ‘Jason swims 32 laps per day for 21 days. How many laps does he swim altogether?’

Step 2: The teacher poses the question: ‘What operation should I use to solve the problem?’

Step 3: The teacher says, ‘To solve this problem I can use the extended form of multiplication (“long multiplication”)’.

Step 4: The teacher says, ‘Before solving the problem using the extended form of multiplication, I estimate the answer. I know that 30 multiplied by 20 is 600, so I know that my answer will be more than 600.’

Step 5: The teacher writes the following on the board:

The teacher says, ‘I write the larger number first and the smaller number underneath, making sure that the ones in the bottom number are under the ones in the top number, and that the tens in the bottom number are under the tens in the top number. I write the multiplication sign to the left of the bottom number and provide space for working out and the answer.’

Step 6: The teacher models this step of the procedure by pointing to numbers as they are used, and recording the results. The teacher says, ‘The first line in our working is where I record the result of multiplying the top number by the number of ones in the bottom number. I multiply 32 by 1. To do this I first multiply 2 by 1, and record 2 in the ones column. Then I multiply 3 (which represents 3 tens or 30) by 1 and record 3 in the tens column.’

Step 7: The teacher models this step of the procedure by pointing to numbers as they are used, and recording the results. The teacher says, ‘The second line in our working is where I record the result of multiplying the top number by the number of tens in the bottom number. I multiply 32 by 2 tens (or 20). To do this, I first multiply 2 in the top number by 2 (tens) and record 4 in the tens column. Zero is recorded in the empty ones column because the 4 in the tens column represents 40. (It will always be the case that zero needs to be recorded in the ones column in this part of the procedure, so the zero can be recorded when starting the second line of working.) I then multiply 3 (which represents 3 tens or 30) by 2 (which represents 2 tens or 20), and record 6 (which represents 600, or 30 by 20) in the hundreds column.’

Step 8 : The teacher models this step of the procedure by pointing to numbers as they are used, and recording the results. The teacher says, ‘Now I add the two numbers in my lines of working together using a formal written algorithm for addition to get the answer to the problem, 672’.

Step 9: The teacher reiterates the procedure: ‘First I multiplied 32 by 1 and got 32. Then I multiplied 32 by 20 and got 640. Then I added the 32 and 640 together and got the answer 672. Jason swam 672 laps in 21 days.’

Part B

The teacher provides guided practice for the procedure, by asking questions about each step of the procedure. The teacher either rephrases or corrects student responses to the questions.

Step 1 : The teacher poses the problem: ‘There are 23 teams in a soccer competition. With 12 players on each team, what is the total number of players in the competition?’

Step 2: The teacher asks, ‘What operation should I use to solve the problem?’

Step 3: The teacher asks, ‘What procedure can I use to solve the problem?’

Step 4 : The teacher asks, ‘What should I do before using the procedure, so that I can check my answer?’

Step 5: The teacher says, ‘Now I am going to record the problem.
‘Which number should I write first?’
‘Where should I write the other number?’
‘Where do I write the multiplication sign?’

The teacher writes the following on the board:

• the teacher provides Robert with immediate feedback
• the teacher might need to repeat guided practice with Robert, with additional questions
  
  

Step 6: The teacher asks:
‘What do we do for the first line of working?’
‘Which numbers do I multiply first?’
‘What is 3 multiplied by 2?’
‘What does the 6 represent?’
‘Where do I record the result?’
The teacher records the result.

‘Which numbers do I multiply next?’
‘What is 2 multiplied by 2?’
‘What does the 4 represent?’
‘Where do I record the result?’
The teacher records the result.

‘What is 23 multiplied by 2?’

Step 7: The teacher asks:
‘What do we do for the second line of working?’
‘What do I record in the ones column of the second line of working?’
‘Why?’
‘What numbers do I multiply first?’
‘What is 3 multiplied by 1?’
‘What does the 3 represent?’
‘Where do I record the result?’
The teacher records the result.

‘What numbers do I multiply next?’
‘What is 2 multiplied by 1?’
‘What does the 2 represent?’
‘Where do I record the result?’
The teacher records the result.

‘What is 23 multiplied by 10?’

Step 8: The teacher asks:
‘What do I do next?’
‘What procedure do I use to add the results?’
The teacher guides the students in adding 46 and 230 using a formal written algorithm for addition.

‘How many players are there in the soccer competition?’

Step 9: The teacher reiterates the procedure.

• given that Robert has demonstrated that he can multiply two-digit numbers by one-digit numbers, the teacher could ask Robert questions about Step 6 of the procedure to increase his likelihood of success

Part C

Students undertake independent practice, completing worksheets of problems that can be solved using extended multiplication.

Extension: The teacher models and guides student practice for multiplying two-digit by two-digit numbers (with trading) and three-digit numbers by two-digit numbers, that can be solved using the extended form of multiplication. For example:

• Robert is provided with grid paper to assist him to align his work
• to assist Robert in his acquisition of the procedure, and so that he experiences a high level of success, the teacher provides him with examples that do not include the following:

• Robert is prompted to use his multiplication grid for multiplication facts that he does not recall readily
• Robert multiplies two-digit numbers by two-digit numbers using the extended form of multiplication fluently, before attempting word problems that require him to use the procedure
• the number of problems completed correctly by Robert is recorded on a graph