Big Idea: Multiplication and Division: use knowledge of numbers and operations to solve mathematical problems encountered in everyday life

Expectations:

• B2.10 divide whole numbers by proper fractions, using appropriate tools and strategies

teacher background

This activity centres on making cards and using paper as a model for dividing a whole number by a unit fraction. Students might have different traditions for making cards, or you may be teaching this lesson as a time of year when cards are often exchanged, such as Valentine’s Day.

A critical component to this lesson is developing intuition about fraction division, which for so many people defies reason. Students are likely to have learned that dividing gives a smaller answer or is about making things smaller, which does not always work with fractions. Algorithms for whole-number division also cannot be generalized to fraction division. Students often end up learning tricks that they do not understand, making errors they will not notice.

KEY CONCEPTS:

• Multiplication and division are related. The same situation or problem can be represented with a division or a multiplication sentence. For example, the division question 6 ÷ 34 = ? can also be thought of as a multiplication question, 34 × ? = 6.

• The strategies used to divide a whole number by a proper fraction may depend on the context of the problem.

  • If the situation involves scaling, 24 ÷ 34 may be interpreted as “some scale factor of three fourths gave a result of 24”.

    • Therefore, 34 × ? = 24

    • 3 × 14 × ? = 24 or 14 × ? = 8

    • Therefore, the quotient is 32 because 32 one fourths is 8.

• If the situation involves equal groups, 24 ÷ 34 may be interpreted as “How many three fourths are in 24?” Either three fourths is repeatedly added until it has a sum of 24 or it is repeatedly subtracted until the result is zero.

• If the situation involves area, 24 ÷ 34 may be interpreted as “What is the length of a rectangle that has an area of 24 square units, if its width is three fourths of a unit?” Therefore, 34 × ? = 24 may be determined by physically manipulating 20 square units so that a rectangle is formed such that one dimension is three fourths of one whole.

Note

• In choosing division situations that divide a whole number by a fraction, consider whether the problem results in a full group or a partial group (remainder). In Grade 6, students should solve problems that result in full groups.

Launch the lesson by showing students a piece of decorative paper. It could be a piece of construction paper that you have in class or something more elaborate. Tell them you are getting ready to make cards to give to friends and family. You have 6 sheets of this paper that you can use, and you know you can make a card with 1/3 of a piece of paper. Ask, How many cards can I make with my paper? How do you know? What shape could the cards be?

How are students labeling their work with number sentences? Sometimes students’ visual models are accurate and demonstrate their reasoning, but when they return to label their work with a number sentence, they might do so unconventionally, labeling their work as 1/3 divided by 6 = 18 or even 6 x 3 = 18. Honor the sense that students have made in their work before turning toward the conventions. If, as in 1/3 divided by 6 = 18, students have simply inverted the number sentence, ask them what it means when we say that one number is being “divided by” another, and then ask whether this matches what they have written. Students will often catch the misconception when they are asked to think aloud. In the case of 6 x 3 = 18, this is very likely what students were thinking and it makes sense, but it is not precisely what is happening in this situation. Point out that 3 isn’t in the situation and ask where it came from. You might ask what happened to 1/3 and whether there is a way to write a number sentence that uses 1/3. The similarity between 6 x 3 and 6 divided by 1/3 is an important one to discuss as a class and then to complicate later by looking at non unit fractions.

Are students attempting to use an algorithm? Some students may have had exposure to fraction algorithms without the opportunity to make sense of them, and these students might try to solve this problem algorithmically. A common error in this case is 6 divided by 1/3 = 1/18. Challenge students to draw a picture to show what is happening in the situation and why their solution makes sense. You might ask, Does it make sense that I can make 1/18 of a card with my 6 sheets of paper? Push students to reason about the numbers using the context of the situation, and then be sure to raise this as something for the class to think about in the discussion.

Are students overgeneralizing the division pattern and ignoring the numerator? The potential challenge of exploring only unit fractions is that students will come to see 6 divided by 1/3 as equal to 6 x 3 and learn to simply multiply the denominator by the whole number. The most straightforward way to avoid overgeneralizing this important pattern (for instance, students thinking that 6 divided by ¾ = 6 x 4) is simply to make sure that students have the chance to grapple with non unit fractions and see what happens. If students do make this error, support them in using reasoning and models to determine whether it makes sense. That is, if each card gets bigger and is now ¾ of a piece of paper, does it make sense that we would be able to make more cards? What would it look like to cut the paper into cards?

Exit Ticket: What is it that division with fractions can lead to a larger answer than you started with?

SEL Self-Assessments (English) and Teacher Rubric

Ask students to generate other situations in which they might divide a whole number by a fraction. Encourage students to brainstorm as many possibilities as they can. Students might create general situations or write specific problems. In either case, the goal is for students to move beyond the specific case of making cards to considering how this kind of fraction division occurs throughout their world. Be sure to have students share their examples with the class to support all students in making connections.