Multiplicative Thinking

Adrian Berenger3 February 2010

Teaching & Learning CoachMoreland Network

WARM-UP ACTIVITY

WITH WHOLE

NUMBER

267149385

673125948

712594836

Non-slip surface at railways stations

Count the Dots

• How many blue dots do you see?

• How do you know?

• How do children know how many blue dots there are?

Count the Dots

• How many blue dots are there now?

• How many blue dots are under the sign?

• How would students work out the number of blue dots under the sign?

GLENROYWEST

PRIMARYSCHOOL

What else do I know?

• ‘ I know that 6 X 4 = 24. What else do I know?’– Make a list using multiplication facts.

– Use a combination of multiplication and addition facts.

Using a Thinkboard

• Draw a version of the following thinkboard and use it to document 6 X 4 = 24

Structure of the Workshop

• Preliminary Activities using number facts (20 min)• What is multiplicative thinking and how could it

work in GWPS? (15 min)• ‘Quilt Designs’ (20 min)• Conclusion and Questions (5 min)

Background

• Research into problems at senior levels in mathematics can be traced back to students not being able to apply multiplication facts.

• Students with high levels of multiplicative thinking are better equipped to learn complex algebra, probability theory, statistics and to apply thinking and reasoning to unfamiliar problems.

What is ‘Multiplicative Thinking’?

• Multiplicative Thinking IS NOT …knowing your times tables

• Multiplicative Thinking IS …knowing when to apply multiplication to solve problems

How can we assess multiplicative thinking?

• Pre-screening all middle years students using a paper test to identify zones from 1 to 8.

• ZONE 1: Can solve simple multiplication and division problems involving relatively small whole numbers, but tends to rely on drawing, models and count-all strategies. May use skip counting (repeated addition) for groups less than 5. Can make simple observations from data given in a task and extend a simple pattern number pattern. Multiplicative thinking (MT) not really apparent as no indication that groups are perceived as composite units, dealt with systematically, or that the number of groups can be manipulated to support a more efficient calculation

• ZONE 8: Can use appropriate representations, language and symbols to solve and justify a wide range of problems involving unfamiliar multiplicative situations including fractions and decimals. Can justify partitioning. Can use and formally describe patterns in terms of general rules. Beginning to work more systematically with complex, open-ended problems

How do we target students?

• School-based decision on how multiplicative thinking becomes part of the regular mathematics program

• Structured activities that target student learning at their thinking level (scaffolding)

• Post-screen to look for growth and for opportunities of improvement

Quilt Designs

Problem Statement: Some children are making a quilt out of material in an art class. Each block is symmetrical and made up of 9 squares. Each block needs 6 black squares, 2 grey squares and 1 white square in the middle. How many black squares would you need if you had 6 grey squares?

POSSIBLE ANSWERS•Model a quilt block using the tiles and butcher paper•List all possible quilts blocks

•How many white, grey and black squares are needed for 99 quilt squares?

Clarifying Questions

• How many black squares are needed for each white square?

• How many grey squares are needed for each white square?

• How many white squares are needed for each grey square?

• What fraction of each block is made up of black squares?

• What is the ratio of black to grey to white squares?

Summarise Using a Table

Q1. How many grey and black squares are needed for 5 quilt blocks?Q2. How many grey squares are needed for 10 quilt blocks?Q3. How many black squares are needed for 20 quilt blocks?Q4. How many quilt blocks can be made from 50 grey squares?Q5. How many white, grey and black squares are needed for 99 quilt squares?

Conclusion• COMMON MISUNDERSTANDINGS

– Trusting the Count: Developing flexible mental objects for the numbers 0 to 10

– Place Value: The importance of moving beyond counting ones, the structure of the base ten numeration system

– Multiplicative Thinking: The key to understanding rational number and developing efficient mental and written computation strategies in later years

– Partitioning: The missing link in building common fraction and decimal knowledge and confidence

– Proportional Reasoning: Extending what is know about multiplication and division beyond rule based procedures to solve problems involving fractions, decimals, percent, ratio, rate and proportion

– Generalising: Skills and knowledge to support equivalence of number properties and patterns, and the use of algebraic test without which it is impossible to engage with broader curricula expectations at this level

• QUESTIONS?