Choosing Multiplication and Division: 3.25
Supporting materials
- Related Progression Points
- Developmental Overview of Proportional Reasoning and Multiplicative Thinking (PDF - 41Kb)
- Developmental Overview of Numbers and Operations (PDF - 2.8Mb)
Indicator of Progress
Students choose to use efficient multiplication and division strategies to solve problems.
Previously, they will have used repeated addition or subtraction, even when this was inefficient.
Illustration 1: Diagnostic task
Here are solutions provided by 7 upper primary children to the problem summarised below. A teacher would classify this as a division problem, whereas the students have solved it using addition, subtraction and multiplication as well. The solutions are ordered to show different levels of development from low to high. Amy and Ellen have not engaged with the main challenge of the problem. Daniel and Elisha are thinking additively and may not recognise that multiplication or division are applicable. Ben and Bob know multiplicative thinking is involved whereas Con is the only one who gives evidence that he recognises that division is applicable.
Problem: John has to take 20ml of medicine three times a day. How long will a 300ml bottle last? (reference)
Click on the students' names below to see how each of them worked through the problem.
| Amy is at the lowest developmental level. She uses her common sense and real world experience but she does not engage mathematically with the problem. |
| Ellen has identified only that 3 times 20 is required. The word times in the instruction may have prompted her to multiply. |
| Daniel uses repeated addition of 60, until he gets to 300. |
| Elisha uses repeated subtraction of 60, until all the medicine is gone. |
| Ben knows that multiplication is involved, and uses repeated addition to show that there are 15 doses in 300ml of medicine. Ben may not know division is useful here. |
| Bob reasons multiplicatively, building up from his known fact that there are five 20ml doses in 100ml to find that there are 15 doses in 300 ml. Bob may not know division is useful here. |
| Con identifies that division is appropriate and carries it out efficiently. |
Illustration 2: Why is it important to learn to recognise division?
John's medicine problem above is a quotition division problem (i.e. how many 20s in 300), but as we see, students can solve it in many different ways. Why should they change?
It is important that students recognise division situations because later they will not be able to apply the more elementary strategies (repeated addition, repeated subtraction, lucky guess multiplication) that are shown in the sample solutions. When the numbers are more awkward or there are decimals and fractions involved, only multiplication or division is useable. For example, you can repeatedly subtract 20 from 300 easily enough to see that 300÷20 = 15, but repeated subtraction is a very poor way of finding 1040 ÷ 1.2, (the average speed of an aeroplane which travels 1040 km in 1.2 hours). Also repeated subtraction cannot be used when the divisor is greater than the dividend (eg to find 0.1 divided by 0.25, or 1/6 divided by 1/2).
Illustration 3
Examples of the types of tasks that would be illustrative of multiplication and division concepts, aligned from the Mathematics Online Interview:
- Question 32 - Multiplication tasks
- Question 33 - Division tasks
- Question 34 - Off to the Circus – 97 people are going to the circus. How many buses are needed if 20 people can ride in each bus?
Teaching Strategies
Firstly, use a diagnostic task such as John's medicine (see above) to determine who uses division and multiplication and who does not. A short conversation with those still in the repeated addition/subtraction stage will determine whether the student:
- recognises but is avoiding division/multiplication, or
- does not recognise that it applies to a given problem situation.
Students in these two categories need different teaching strategies.
Activity 1: Seeing a reason to change, and building confidence and skills is for students avoiding division/multiplication, but who recognise that it is appropriate. The strategy here centres on building confidence and skills with multiplication and division operations, after showing the advantages of using the more sophisticated approach.
Activity 2: Strengthening recognition of operations is for students who do not recognise the applicability of division/multiplication so it centres around developing meaning for the operations.
Activity 3: Arrays and multiplication particularly highlights the array interpretation of multiplication because of its importance to the meaning of multiplication.
Activity 1: Seeing a reason to change, and building confidence and skills.
Some students will know division/multiplication is involved, but they still use other methods. Students avoid methods which they are not confident will give the right numerical answer. Others are reluctant to move on from methods that they have been using for several years.
Step 1. Develop a reason to change
In a group discussion, ask students to solve a few problems where repeated addition or repeated subtraction are extremely cumbersome. Discuss how using multiplication/division is much more efficient.
Sample problems: How many weeks in 364 days? How many months/days has your grandmother/the oldest person in the world been alive? Look up world records for some examples with large numbers.
Even with a calculator, repeated addition and repeated subtraction are tedious in these cases. Multiplication and division are just what you need. Have a race with calculators to see who gets answers first: those who multiply/divide or those who use repeated addition/subtraction.
Step 2. Building confidence and skills
These students now need practice in numerical multiplication/division, including:
- building fluent multiplication tables facts (6 × 7 = 42)
- extending to fact families e.g. 7 × 6 = 42, 42 ÷ 6 = 7, 42 ÷ 7 = 6
- combining with base ten knowledge e.g. 7 × 60 = 420, 4200 ÷ 6 = 700, 420 ÷ 60 = 7, etc.
- regular, well graded, immediately-checked practice of written multiplication and division algorithms.
Activity 2: Strengthening recognition of operations
To strengthen the ability to recognise multiplication and division in problem situations, students can work with a calculator. Using a calculator is good because the button press highlights the choice of operation. The intention at this point is to focus on choice of operation, not getting bogged down in actually calculating answers.
Recognising situations where multiplication applies.
At the level of these students, most situations for multiplication will have an equal groups structure: one quantity taken a whole number of times (e.g. 3 lots of $12.30) . Provide a variety of situations where students have to identify the equal groups, and link to multiplication.
Examples:
| 3 apples at 65c each | 65c + 65c + 65c | 3 groups of 65c | 3 × 65c |
| hours in a week | 24 hr + 24 hr +... + 24 hr | 7 groups of 24 hours | 7 × 24 hrs |
Recognising situations where division applies.
At this level, most situations for division will be either partition or quotition. Partition division problems (sharing problems) split a quantity into a given number of parts. Quotition division problems allocate a given quota to an unknown number of recipients.
Examples:
| I spent $1.95 on 3 apples. How much each? | 3 groups of ? = 195c | 3 × ? = 195 | partition situation |
| I spent $1.95 on some 65c apples. How many did I buy? | ? groups of 65c = 195c | ? × 65 = 195 | quotition situation |
For more information Early division ideas (Level 2.25) addresses concepts about partition and quotition situations.
Activity 3: Arrays and multiplication
A rectangular array of objects is a fundamental tool in teaching about multiplication, but some students in the middle levels do not have a thorough understanding of the link.
Place 13 counters in a row on a table and a second row underneath it. Ask students how they could work out the number of counters in total.
Discuss responses, especially highlighting 2 rows of 13 (2 × 13) and 13 columns of 2. Link these expressions to 2 groups of 13 and 13 groups of 2 and to 2 × 13 and 13 × 2. Ensure that students see the array from both of these points of view. Ask students to point out the 'equal groups'.
2 groups of 13
13 groups of 2
Add more rows asking similar questions. Then ask students to use calculators to find the number of counters in arrays with more rows (eg 8) both by repeated addition and by multiplication.
Repeat with a variety of 'array' situations from real life - seats in a stadium or hall, soldiers on parade, large delivery boxes of small packets of sweets, children at assembly, etc.
References
The problem John's medicine is abbreviated from ACER's Profiles of Problem Solving.
Stacey, K., Groves, S., Bourke, S. & Doig, B. (1993) Profiles of Problem Solving, Camberwell: Australian Council for Educational Research
Further Resources
The following resource contains sections that may be useful when designing learning experiences:
Digilearn object *
School Canteen: best buys level 1 – students buy supplies online for a school canteen. Students purchase given amounts of items such as bottles of orange juice and boxes of sultanas. Students check the prices for a range of packaging sizes. For example, choose how to order forty tubs of yoghurt that are available in boxes of 4, 8 or 10.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4559)
* Note that Digilearn is a secure site; DEECD login required.