Students will divide whole numbers through partitive division to reveal a rate that is a fractional amount.

In today’s math talk and purposeful practice, students will continue solving problems involving the partitive division of two whole numbers that results in a fraction. Students are encouraged to use models to reveal how many per one part (or the rate).
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.

The following number talk consists of a string of related problems that can be explored with an animated approach leveraging the visual math talk prompt videos below or in a low-tech fashion by orally sharing with students and writing on a vertical surface like a chalkboard or whiteboard. Regardless of which approach you choose, it is highly encouraged that the facilitator annotate student thinking on a vertical surface such as a chalkboard, whiteboard or projector so students feel their thinking in honoured and worth sharing.

The purpose of this string of problems is to leverage partitive division of whole numbers to reveal a rate. The first division sentence in each pair of sentences will result in a whole number quotient, while the second will reveal a quotient that is a fraction of a whole.

If you choose to go with a low-tech version of this number talk, consider writing each of the following division sentences on the board, one at a time. Ask students to develop a partitive context (where the total quantity and number of parts is known) and to use a model to represent each equation. Alternatively, you could leverage the cheese and crackers context or another partitive context of your choosing. Honour student thinking by sharing a variety of models. If it does not emerge authentically, find an opportunity to model one or more of the scenarios below using an array model and/or a double bar model. These models might be a helpful tool as students navigate the upcoming tasks.

10 ÷  5

5 / 10

12 / 4

4 ÷ 12

9 ÷ 3

3 / 9

The string intentionally represents division using the division symbol and the fraction notation. The purpose of using both representations is to encourage students to see the relationship between division and fractions.

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If you choose to leverage the visual math talk prompts below, your students will be presented with the familiar context of cheese and crackers as they explored in the previous lesson.

Begin playing the following visual number talk prompt video and be ready to pause the video to allow for think time.

Pause the video where indicated, and share the following prompt:

How could the 10 cheese slices be cut to share evenly with the 5 crackers? Convince your neighbour.

With this first prompt, it is likely that many students can spatially partition the 10 slices of cheese into 5 groups or to fair share amongst the 5 crackers.

Despite this feeling obvious to some (possibly many) students, be sure to ask students what operation is being used and to ask them to model this both visually and symbolically. One such visual model that can be helpful here is the double bar model or double number line which clearly shows the 2 to 1 ratio relationship that these quantities share.

Therefore, 10 cheese slices ÷ 5 crackers results in a rate of 2 cheese slices per cracker.

Begin playing the next visual number talk prompt video and be ready to pause the video to allow for think time.

Pause the video where indicated and now, the dividend and divisor will switch as compared to the first visual math talk prompt:

How could the 5 cheese slices be cut to share evenly with 10 crackers? Convince your neighbour.

While not as obvious as the result of the first visual number talk prompt, it is likely that students will realize that they can only fair share 1 full slice of cheese to half of the crackers and therefore must partition the cheese slices into halves (hence partitive division) to share an equivalent amount of cheese to each cracker.

While some students may model using a concrete or visual fair sharing approach as we see above, be sure to ask students to show what this might look like on a double bar model or double number line as well as symbolically.

Be sure to draw the attention of students to the 1 to 2 ratio of cheese slices to crackers and compare and contrast to the division sentence from the first visual number talk prompt where the ratio was the inverse.

Continue on to the next visual math talk prompt video:

Pause the video where indicated, and share the following prompt:

How could the 12 cheese slices be cut to share evenly with 4 crackers? Convince your neighbour.

We now come to another prompt where the number of cheese slices is greater than the number of crackers which will result in a rate that is greater than 1, however some students may not recognize this immediately.

Be sure to monitor students as they work to assess who is leveraging an unstructured fair share scenario and who is leveraging another model like the double bar model, double number line or ratio table that we have emerged in the previous two prompts.

Be sure to draw the attention of the students to the 3 to 1 ratio relationship that exists here and how this idea in conjunction with the mathematical models we are leveraging can be helpful to solve other problems. For example, we can quickly see that 9 cheese slices and 3 crackers also results in a rate of 3 cheese slices per cracker and so on.

Show students the silent solution visual math talk prompt video:

Pause the video where indicated, and share the following prompt:

How could the 4 cheese slices be cut to share evenly with 12 crackers? Convince your neighbour.

We have another situation where we are switching our dividend with our divisor. More and more students may begin to realize that they can confidently take the reciprocal of the previous visual math talk prompt for the answer to this problem.

Since 12 cheese slices divided amongst 4 crackers was 3 cheese slices per cracker, we can say that 4 cheese slices divided amongst 12 crackers results in ⅓ cheese slices per cracker.

It is worth modelling this again to ensure that students see the value in this tool for thinking and also as a means to represent their thinking.

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Students will have an opportunity to independently complete the following practice questions. As students are working, you might consider using this opportunity to pose purposeful questions and document student thinking for the purpose of formative assessment.
Pay close attention to the strategies that students are using.
Are students:

• Using concrete materials such as fraction strips?
• Paper folding?
• Partitioning an area or bar model?
• Drawing an array?
• Partitioning a number line?
• Reasoning using relationships such as doubling and/or halving?

While students are working independently, you might also consider strategically pulling students individually or in small groups to offer guided instruction in order to move them along their developmental continuum.

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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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