Year Four - Illustrating Multiplication & Division
Home - Illustrating Multiplication & Division Map
About this Big Idea
In this Big Idea students expand and enhance their number sense as they investigate strategies to engage flexibly with numbers. Research shows that "the difference between low- and high-achieving children between the ages 7 and 13 was not that the high achievers knew more but that they had learned to be flexible with numbers." (Boaler, Munson, Williams. p. 177).
Understanding Goals:
Students will understand:
- the relationship between multiplication and division
- strategies of number sense
Background:
The emphasis in multiplication and division is on students developing mental strategies and using their own (informal) methods for recording their strategies. Comparing their own method of solution with the methods of other students will lead to the identification of efficient mental and written strategies. One problem may have several acceptable methods of solution. An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations.
Core Content from the Syllabus:
Working Mathematically
Multiplication & Division
- select and use a variety of mental and informal written strategies to solve multiplication and division problems
- develop efficient mental and written strategies, and use appropriate digital technologies, for multiplication and for division where there is no remainder
- model and apply the associative property of multiplication to aid mental computation
- use mental and informal written strategies to multiply a two-digit number by a one-digit number
- use mental strategies to divide a two-digit number by a one-digit number where there is no remainder
Language:
- multiply, multiplied by, product, multiplication, multiplication facts, tens, ones, double, multiple, factor, shared between, divide, divided by, division, halve, remainder, equals, is the same as, strategy, digit.
Connected to:
Mindset Mathematics Learning Activities
Visualise
In this activity, students will explore the connections between visual and numerical models for multiplication. Students create their own models as visual proofs. Students engage in a feedback and revision process of this work, just as mathematicians do. - See page 177
Questions for reflection:
- What makes a visual proof particularly clear and understandable?
- What features on the different posters did you appreciate? Why were they useful?
- What kinds of things did you revise? Why? What do you think your revisions did for your proof?
- What is going on in the proof? How did the group solve the problem?
- Does the solution work? Why or why not?
- If the solution doesn't work, what would need to happen to make it work? Is there something missing or something that needs to be revised?
- How would you explain this proof to others?
- How do the visual models of multiplication help you understand how to multiply?
Play
Students play with the idea of decomposing rectangles by trying to “cover the field”— a large array— with smaller rectangles they create by rolling dice. Students must think flexibly about what rectangles they make and how they fit them into their field. - See page 187
Questions for reflection:
- How did you decide how to use your dice to make a rectangle? What were you thinking about?
- How did your decisions change from the beginning of the game to the end of the game?
- If you wanted to make the biggest rectangle you could, how would you use your dice?
- If you wanted to make the smallest rectangle you could, how would you use your dice?
- What made this game challenging for you? How did you handle that challenge?
- How was this like our work with visualizing multiplication? How was it different?
- What advice would you give to someone who was going to play this game?
Investigate
Students extend their thinking about using area models for multiplication to consider how they can be used for division. - See Page 195
- Which models do you like the best? Why?
- Which ones help you see the missing side length? Why?
- Which models look efficient or easy to understand? Why?
- Does the order of the chunks matter? Why or why not?
- How can you use a visual model to divide?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
