Students will determine how to partition the cheese in order to fair share it based on the number of crackers. 

In this task, students will represent the partitioning of cheese slices in order to fair share them amongst the crackers without leaving any cheese leftover. At this point, students may not have a lot of experience dividing when the dividend is less than the divisor. This new experience will likely emerge practical tools for partitioning such as concrete materials, a bar model and/or a linear model.

Through the consolidation, students are prompted to make conjectures about the behaviour of this structure of division. For example, “I already know the whole amount of cheese (the dividend) and the quantity of crackers (the divisor). What I am trying to reveal is a rate, or how much per one cracker (part).”

You might also explore equivalent representations of the quotient. This could reveal that the quotient is the dividend (numerator) over the divisor (denominator).

Some of the big ideas that may emerge through this task include:

• Partitive division is one of the two structures of division;
• In partitive division, the dividend is the quota and the divisor is the number of parts or groups;
• When dividing partitively, the quotient reveals a rate, how many per part (or the quota per group);
• The quotient can be expressed as any fraction equivalent to the dividend over the divisor;
• The number of partitions determines the fraction unit and gives the quantity its name (i.e.: partitioning a whole into 5 parts creates “5 fifth” parts);
• Different denominators can be used to represent equivalent quantities.

Show students this 30 second video.

Then, ask students:

What do you notice?

What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other means to ensure students know that their voice is acknowledged and appreciated.

Some of the noticing and wondering that may come up include:

• I see a block of cheese.
• I wonder if it is cheddar.
• I see crackers.
• I wonder how many crackers there are.
• I saw three slices being cut.
• I wonder if he is going to cut anymore.

At this point, you can answer any wonders that you can cross off the list right away.

After students share out with the class, tell a story about the situation from the video and how annoying it is when you just randomly break off pieces of cheese for your crackers and then end up with too many crackers left with not enough cheese.

So, the question we begin with is:

How could the cheese be cut to split evenly with the crackers?
Make an estimate.

Encourage students to begin crafting an estimate. If some students quickly arrive at an estimate, encourage students to explain their thinking.

How many crackers did they estimate and why does their final estimate make sense?

Emphasize that you will be looking to see their thinking represented either concretely or pictorially (this representation might be in addition to their abstract representation if students decide to divide using an algorithm). You might consider allowing each student time to think through their plan for 3-5 minutes before asking them to collaborate with a partner or a small-group. This dedicated time for personal reflection allows all students to enter the conversation with the advantage of time to think through their current understanding before being potentially overwhelmed or bulldozed by the ideas of others.

Keep in mind, the intent here is to encourage all students to enter into this task. Since all students are estimating the number of crackers, this opens the door for a student who may not feel as confident to make an estimate that is easier for partitively dividing (or fair sharing) the 3 slices of cheese across the number of crackers. Estimating a number of crackers that is a multiple of 3 will make this process easier for students, so keep an eye for students who select 9, 12, or 15 as their number of crackers.

Some students might quickly comment that it is difficult to know for sure without confirming the number of crackers. When this point is expressed by a student, share the following video.

Now that students are aware of exactly how many crackers there are (12) and how many slices of cheese (3), we can refine our prompt for students:

How much of one slice of cheese can we put on each cracker? 

Refining this question to become more specific will encourage students to relate the quantity of cheese relative to a whole slice per cracker in order to more specifically encourage the use of fractional language or notation when describing the quotient.

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Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.

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Share this video to confirm student thinking.

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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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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