Part 2
From Multiplication to Division
From Multiplication to Division
WATCH: Different Situations, Same Operation... Doubly True for Division!
WATCH: Different Situations, Same Operation... Doubly True for Division!
3.03_Doubly True (#94).movLooking for a Summary?
Looking for a Summary?
To review the different division situations and their relationship to multiplication situations, click here.
To review the different division situations and their relationship to multiplication situations, click here.
TASK: Representing Division Situations
TASK: Representing Division Situations
For each type of DIVISION problem listed in the handout above, draw a diagram to that represents both the semantic structure of the problem and its context. Download and print this handout and use it to organize your drawings.
For each type of DIVISION problem listed in the handout above, draw a diagram to that represents both the semantic structure of the problem and its context. Download and print this handout and use it 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.)
(TIP: If you're in groups, you might wish to assign a different person to each type and then share your representations.)
Discuss:
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?
- 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?
- 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?
- 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
WATCH: Name the Operation
3.04_Name the Operation (#97).movTASK: What do you think?
TASK: What do you think?
Consider how you would solve the following situation:
Consider how you would solve the following situation:
Discuss:
Discuss:
- Is this a multiplication or a division situation?
- Is this a multiplication or a division situation?
- Have a quick conversation with a partner and share your reasoning.
- Have a quick conversation with a partner and share your reasoning.
WATCH: One Situation, Different Number Sentences (continued)
WATCH: One Situation, Different Number Sentences (continued)
3.05_OneSituation (#98)*.mp4TASK: Categorizing and Analyzing Problem Types
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.
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!
Warning: This task may hurt your head!
Analyze and categorize the following problems. You can download and print the handout by clicking here. For each problem or situation, identify:
Analyze and categorize the following problems. You can download and print the handout by clicking here. For each problem or situation, identify:
- the problem type
- the problem type
- what we know and what we're looking for
- what we know and what we're looking for
- the semantic and standard number sentence(s).
- 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.)
(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?
Curious?
Click here to download the thinking of Moses and Craig on these problems and compare your answers to theirs. But be skeptical. They've been known to make errors before! And fairly regularly.
Click here to download the thinking of Moses and Craig on these problems and compare your answers to theirs. But be skeptical. They've been known to make errors before! And fairly regularly.
