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

Expectations:

• B2.9 multiply whole numbers by proper fractions, using appropriate tools and strategies

teacher background

The goal of this activity is to give students a visual experience of multiplying a fraction by a whole number before we name this as fraction multiplication. Here students just think about finding a fraction of a whole number. Later, we can substitute a multiplication symbol for the word of, and students will then have two equivalent and useful ways of thinking about multiplication: ⅓ x 18 is the same as ⅓ of 18. We encourage you to take advantage of the many connections students are likely to make as they use the Number Visuals Sheet to explore the puzzle. Students may implicitly move between thinking of finding, say,1/4 of 20 and dividing 20 by 4 to get 5. Students may think about finding numbers that work with 1/3 by counting by 3s or naming the multiples of 3 (that is, 3,6,9,12,15…). Students might say that all the even numbers work with 1/2, In each of these cases, students are making connections between finding a fraction of a whole number, factors, multiples, properties of numbers, multiplication, and division. These are useful connections that can make students’ thinking more flexible and grounded in sense making, rather than procedures. We encourage you to notice these aloud for students as they occur and ask whether these connections are always, or only sometimes, true. For instance, you might ask: Are you saying that you can find ⅓ of all the multiples of 3? Why?

KEY CONCEPTS:

• A proper fraction can be decomposed as a product of the count and its unit fraction (e.g., 34 = 3 × 14 or 14 × 3).

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

  • If the situation involves scaling, 5 × 34 may be interpreted as "the total number of unit fractions is five times greater". Thus, 5 × 34 = 5 × 3 × 14 = 15 × 14 = 154 (15 fourths).

  • If the situation involves equal groups, 5 × 34 may be interpreted as "five groups of three fourths". Thus, 5 × 34 = 34 + 34 + 34 + 34 + 34 = or 334.

  • If the situation involves area, 5 × 34 may be interpreted as "the area of a rectangle with a length of five units is multiplied by its width of three fourths of a unit". The area could be determined by finding the area of a rectangle with dimension 5 by 1 and then subtracting the extra area, which is 5 one fourths. Therefore:

5 × 34

= (5 × 1) − 5 × (14)

= 5 − 54

= 5 − 114

= 334

Note

• How tools are used to multiply a whole number by a proper fraction can be influenced by the contexts of a problem. For example:

  • A double number line may be used to show multiplication as scaling.

  • Hops on a number line may be used to show multiplication as repeat addition.

  • A grid may be used to show multiplication as area of a rectangle.

The strategies that are used to multiply a whole number by a proper fraction may depend on the type of numbers given. For example, 8 × 34 = 8 × 3 × 14. Using the associative property, the product of 8 × 14 may be multiplied first and then multiplied by 3. This results in 2 × 3 = 6. Another approach is to multiply 8 × 3 first, which results in 24, which is then multiplied by 14, resulting in 6.

Students work in partners with copies of the Number Visuals Sheet to mark up with colours, if they choose. Partners explore the following puzzles:

· Can you find a number represented on this page where ½, ⅓ , and ¼ of the number results in a whole number? Which numbers can you find that satisfy these conditions?

· What numbers pictured on this page satisfy these conditions? How might you draw them? How could you use your visuals to prove that they satisfy the conditions?

Students may want to play with each fraction separately at first, using a Number Visuals Sheet to identify numbers that they can find ½ of, then a separate sheet to identify numbers they can find 1/3 of, and so forth. Students could then look across their different sheets to see what numbers the sheets have in common. Alternatively, students might decide to search number by number. For instance, they might start with the number 1 and ask themselves: Can I find ½ of it? Can I find ⅓ of it? Can I find ¼ of it? This strategy enables students to eliminate numbers or find solutions, one by one. These are just two approaches students might use, and there are certainly others. Encourage students to develop their own process and to think about how to be systematic.

Are students connecting fractions and factors? The numbers students can find ½ of are also numbers that have 2 as a factor, and are even numbers. Connecting these mathematical ideas is a key outcome. It also makes sense to think of multiplying ½ as the same as dividing by 2. As students play with the number visuals, highlight places where students are moving across these different ideas and making connections between fractions, multiplication, division, factors, multiples, and number properties. You might want to bring these observations to the class discussion and have students debate whether they are always true. For instance, you might say that one partnership had the idea that all the numbers that you could find ½ of were even numbers, and ask whether students agree or disagree with this and why. You might chart these different connections and observations and ask students to see whether they can find any counterexamples.

Are students thinking systematically? Some students might start with a hunch and explore it first. For instance, a partnership might think that 12 is a flexible number and that maybe it will be a solution. This is fine as a starting place. But as students move into playing with the different fractions and number visuals, they will need a system if they hope to find all the numbers that satisfy the conditions of the problem. Ask students how they will know which numbers work and which ones don’t. You also might probe the hunces that they started with, as they might be built on some useful assumptions. For instance, the notion that 12 is flexible is useful, and it is flexible because of its many factors. IF students can articulate this, then that could lead to quickly testing and eliminating many other numbers, such as the primes.

How are students organizing their work to see useful patterns? Juggling three fractions and 35 number visuals on multiple pieces of paper can be an organizational challenge. Support students in thinking about how they will keep track of what they notice so that they can use those observations to hunt for solutions. You could ask: Will colour help? What labels could be useful? What is the job of each piece of paper? How are you recording what works and what doesn’t?

Gather students together with their Number Visuals Sheets, and discuss the following questions:

· What strategies did you use to find a number where ½, ⅓ , and ¼ of it is a whole number?

· How could we use the visuals to prove that these numbers satisfy these conditions?

· What patterns do you notice in the numbers that satisfy the conditions?

· What numbers not pictured could also satisfy these conditions? How do you know? How do the patterns in our solutions help us predict larger number solutions? How could they be drawn?

Use blank copies of the Number Visuals Sheet and chart paper to record students’ strategies for identifying solutions and eliminating numbers that do not work. As students are sharing their thinking, ask students to say not just which numbers have a whole-number solution, but what that solution is. For instance, students might say that you can find ½ of 14. Follow up by asking, What is ½ of 14? Record this on your chart or Number visuals Sheet as part of the evidence ½ of 14 is 7. Recording in this way will support students in seeing number patterns and connecting those to the visual patterns on the sheet.

Exit Ticket: What does it mean to find ⅕ (or ⅓ or ¼) of a number? Use examples or drawings to help you explain.

SEL Self-Assessments (English) and Teacher Rubric

Choose a different set of two or three unit fractions and see what numbers students can find that satisfy these conditions. You could have students select the two or three unit fractions with a partner, have the class choose them together, or you might choose them yourself. Keep in mind that some combinations of unit fractions, such as 1/10, ¼, and ⅛ , will lead to no solutions on the Number Visuals Sheet. This could be interesting itself, leading to the question, Does this mean there are no numbers that satisfy these conditions? For some students, however, this might be quite frustrating. Some interesting combinations could be: ⅓ , ⅕ , and 1/10 , or ½ and 1/7.