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Mental Strategies for Multiplication
Did You Catch It?
At around 3:35 of the video, Craig tries to represent doubling and halving by referencing the associative property. Unfortunately, he doesn’t quite nail the number sentence that represents the diagram (and Moses was apparently of no help! :)
What number sentence should he have written to more accurately reflect the associative property that sits underneath this strategy? He appreciates your help! You can offer your advice in Section 4.2a in your digital handout.
Task
The work you did in deriving basic facts -- and the work students do -- forms a fertile foundation for solving multi-digit computations mentally.
On your own, take some time to work out this problem in your head:
Use an open array to communicate your strategy with the group.
Identify the property of multiplication that you accessed.
In groups of two or three:
- use your open array to describe your strategy;
- identify the property you accessed;
- notice and name the additive or multiplicative number relationships you used;
- record your strategy with a number sentence.
Try some of these other problems.
- Note the properties you are using.
- Discuss what you see in these numbers that helps you choose a strategy.
- Talk about instructional moves that might help students strengthen their flexibility with computations.
Note: If you have trouble holding the numbers in your head, free free to jot them down, but try to perform the strategy mentally.
Share and Discuss
Post strategies from these questions to this Section 4.2b of your digital handout. Observe and analyze the strategies of others. Were there any that surprised you or that?
Discuss the value of having students develop flexible computation strategies. Augment your thinking with that in A Guide to Effective Instruction: Volume 5 (page 38).
Learn more about the surprising challenge of the array
An array is one of the most powerful models for representing multiplication and division. However, the row and column structure that we see so automatically isn't automatically apparent to everyone.
In a fascinating piece of research, learn what Michael Battista uncovers about the challenges students have with conceptualizing the array and the strategies he suggests to help build spatial structuring. Perhaps you will choose to use some of his tasks with your students.
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Mental Strategies for Division
Task
Representing our Mental Division Strategies
It's tempting to think that we can do multiplication mentally, but not division. But that's not the case. Take a moment to consider how you might solve this division question mentally.
In your head, solve:
How do you visualize what's happening in this problem?
Discuss
Describe how you "see" this question.
- Do you form groups of seven (Quotative Division)?
- Or do you share among seven parts (Partitive Division)?
How would you record your thinking with a number sentence? What operation(s) do you use?
Task
Try some of these other problems.
Talk about what you see in these numbers that helps you choose a strategy. Consider whether you think of these as "quotative" or "partitive" division situations. What makes the difference?
Talk about instructional moves that might help students strengthen their flexibility with division. How might these sorts of activities help build students' understanding of the operation?
Flashback: To refresh your memory on the types of division situations we talked about in Session 3, refer to this summary document.
Share and Discuss
Post strategies for the questions above to Section 4.2c in your digital handout.
Observe and analyze the strategies of others. Were there any that surprised you or that?
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