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Students will explore the relationship between partitive and quotative division.
In this task, students will begin making generalizations about the relationship between quotative and partitive division. Using the same box of 2 400 donuts, students will be tasked with a partitive division problem today.
Some of the big ideas that will emerge in today’s task include:
Revisit the context from Day 2, the box of 2 400 donuts partitioned into layers of 800 donuts. Use the following information to practice similar quotative division problems. Students are encouraged to use models as tools and to represent their thinking. The question for each of the following scenarios is, how many layers of donuts are in the box?
400 donuts per layer, 1 200 donuts altogether
700 donuts per layers, 2 800 donuts altogether
450 donuts per layer, 2 250 donuts altogether
325 donuts per layer, 1 300 donuts altogether
If your school bought this box of doughnuts to split between 8 classes, how many would each class get?
Students are encouraged to work through this problem with partners or in small groups. Students should use models to defend their thinking and convince their community that their answer is correct.
Monitor student thinking by circulating around the room and listening to the mathematical discourse. Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:
The tools and representations you might see students using to convince their peers and/or the teacher include:
Have students share their strategies and reasoning for determining the total number of donuts that each class will receive.
Discuss their strategies and elicit student thinking during your consolidation to build off of their current prior knowledge and understanding rather than “fixing” or “funnelling” student thinking to a strategy and/or model that does not connect to their strategy and/or approach.
I know that I have to take all 2 400 donuts and divide them evenly amongst the 8 classes. I decided to break the 2 400 up into groups of 100 (because 100 is a friendly number) so that I could start sharing them. I drew the 8 different classes on the right, and started giving each class 100 donuts at a time. I crossed off a “100” every time I gave those donuts to one class. By the time I gave three groups of 100 donuts to each class, I ran out of donuts, and all classes had the same number of donuts, 300.
Approach:
This student accessed this partitive division problem by decomposing the dividend into groups of 100 and fair sharing into 8 groups (the divisor in this context) to reveal how many donuts per group (or class).
I know that right now there are 2 400 being shared by 8 classes. I thought to myself, if they were only being shared by 1 class, that would be all 2 400, and shared by 2 classes, that’s half of 2 400, so 1 200. But what about 4 classes? That’s just half again, so 600 donuts. But if I half one more time, that will show me how many donuts each class will get if 2 400 donuts are split between 8 classes. I just halved the quantity of donuts 3 times.
Approach:
This student partitioned a bar model by halving the quantity of donuts until the bar represented the quantity of donuts in one of 8 equal parts.
I know that dividing a quantity into 8 equal parts is the same as finding one-eight of that quantity. I could find half of the donuts, that would show how many donuts in 2 classes. Halving that would show me one-fourth of the total quantity, and halving again represents one-eight of the total quantity of donuts. I used the number line to represent my thinking and to justify my answer of 300 donuts.
Approach:
This student used a symbolic representation as a tool and a number line to represent their thinking. This student demonstrates an understanding of the relationship between partitive division and multiplying by a fraction. The student leveraged this understanding to determine the total quantity by halving until the total quantity of donuts was partitioned into eighths.
Consolidate learning using student generated solution strategies based on what you selected and how you sequenced while monitoring during the Sense Making portion of the lesson.
The purpose of the consolidation is to make connections between strategies and reveal and/or solidify big ideas.
In particular, consider the difference between the type of division explored through the task on Day 2 in comparison to the type of division explored today. A main purpose of today’s consolidation is to solidify an understanding of the relationship between quotative and partitive division.
In today’s scenario, the total number of donuts was known, as well as the number of parts (the number of classes). Students were tasked with revealing a quota (or rate), the number of donuts per class. Compare today’s scenario to yesterday’s using students generated models and strategies.
Provide students an opportunity to reflect on their learning by offering this consolidation prompt to be completed independently.
Consolidation Prompt:
Rayhan believes that 3 600 divided by 600 can represent two different contexts that require two different types of division. Use a box of donuts with a quantity of 3 600 donuts divided by 600 to write two different scenarios (one for quotative and one for partitive). Solve for both and use a model to justify your thinking.
We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.
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This Make Math Moments Task was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, and extensions to elicit and emerge mathematical models and strategies.
Click the links at the top of this task to head to the other related lessons created for this unit of study.
You might choose to simply open the document to display via your projector/TV, make a copy for editing, download as a PDF or upload to your LMS and/or print for students to have a physical copy.
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