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Different Situations, Same Operation... Doubly True for Division!
Task
Review the different division situations and their relationship to multiplication situations.
For each type of division problem listed in the link above, draw a diagram to that represents both the semantic structure of the problem and its context. Use Section 3.2a in your digital handout to organize your drawings.
(TIP: If you're in groups, you might wish to assign a different person to each type and then share your representations.)
Discuss
How do your diagrams -- and the thinking behind your diagrams -- differ if you're representing a quotative/measurement situation (e.g., where the quota or size of group is known -- the unit) or a partitive/sharing one (where the number of parts are known -- the multiplier)? Which do you think students might find more challenging? Why?
Which of the five types of multiplication-division problems do you think students see most? Is one type more problematic than others?Which did you find most challenging to recognize as a multiplication or division situation?
Are students equally able to use multiplication or division to solve a problem or is one operation “dominant” in their thinking? Why might this be? How could we help?
Watch
Name the Operation
Discuss
Consider how you would solve the following situation:
Is this a multiplication or a division situation?
Have a quick conversation with a partner and share your reasoning.
Watch
One Situation, Different Number Sentences (continued)
Task
Categorizing and Analyzing Problem Types
While it isn't important that students -- or even educators! -- know the fancy terms to describe the different division types, it is important that we can recognize what is known and what is unknown in a given problem. This enables students to know when division or multiplication is the most efficient operation. And it helps us, as educators, to ensure that our students have plenty of opportunity to experience both types of division problems. (Warning: This task may hurt your head! :)
Analyze and categorize the following problems in Section 3.2b in your digital handout.
For each problem or situation, identify:
- the problem type
- what we know and what we're looking for
- the semantic and standard number sentence(s).
(TIP: You may wish to divide the problems among your group to avoid overload. Then share with each other your reasoning behind your answers.)
Curious to see how Craig and Moses filled out the chart? Have a look at their chart, but be skeptical; they've been known to make errors before!
Discuss
Of the problems you just looked at in the above task, which one(s) do you predict students would find most challenging? Why do you think so?
Before discussing this as a group, write down your ideas in Section 3.2c in your digital handout, and then discuss as a group.
Talk about what you might do to help them understand and gain access into this problem.
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