Stage 3 multiplication and division - multi-step problem solving
- Find common multiples of two numbers
- Find relationships between numbers
- Use division to solve problems
Students need to use their knowledge of multiples and the inverse relationship between multiplication and division to solve problems that involve ratios. Students need to be able to find a common divisor for both units that are being compared e.g. 400 grams and 24 cupcakes can both be divided by 4. This assist students in finding a base unit to then find the solution to the problem. Students can also see the pattern of multiples to solve problems involving ratios. The concepts of the relationship between division and multiplication and pattern relationships develops into Stage 4 in rates and ratios where proportional reasoning is involved.
Activities to support the strategies
Activity - 'real world' problems
Using real problems is a way to develop students’ working mathematically skills in communicating, problem solving and reasoning in the classroom. This following problem involves students using their knowledge of multiples and their relationship to each other to solve problems. This leads to understanding ratios in Stage 4.
Tony is paid the same amount for each car he washes. He gets paid $15 for washing 3 cars.
- How many cars does he need to wash to get paid $45?
Using Newman’s prompt questions (PDF 41.18KB) while problem solving is a good guiding structure for problems
1. Please read the question to me.
2. Tell me what the question is asking you to do.
3. Tell me how you are going to find the answer.
4. Show me what to do to get the answer. “Talk aloud” as you do it, so that I can understand how you are thinking.
5. Now, write down your answer to the question.
Work through the problem as a whole class. Using toy cars and plastic money recreate the problem and have students act it out. Allow students to use these resources and others to work on solving the problem. Allow students time to work in pairs or small groups to trial and number of different strategies for solving the problem. It is important that students can also visualise the problem.
Do students see that 15 can be multiplied 3 times to get 45?
Do students see that finding out how much it costs to wash one car could help in solving the problem?
Australian curriculum reference: ACMNA101
Solve problems involving division by a one – digit number, including those that result in a remainder; ACMNA173: Recognise and solve problems involving simple ratios; ACMNA188: Solve a range of problems involving ratios and rates, with and without the use of digital technologies
NSW syllabus reference: MA3-6NA
Use mental and written strategies to divide a number with three or more digits by a one – digit divisor where there is no remainder; show the connection between division and multiplication; apply appropriate mental and written strategies, and digital technologies, to solve division word problems; recognise when division is required to solve word problems (Problem Solving); MA4-7NA; use ratios to compare quantities measured in the same units; solve a variety of real – life problems involving ratios; convert given information into a simplified rate
NSW numeracy continuum reference:
Aspect 5: Multiplication and Division as operations.
- Number sequences and relationships - lessons from Talking about patterns and algebra Stage 3 section
- NRICH Enriching Mathematics website – In proportion includes a range of problems that show how proportional reasoning is used.
- Numeracy Wrap – Painting in Proportion – interactive activities that explore ratios in a practical context.