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

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.

How are students organizing the components of their solutions to keep track? This investigation asks students to juggle four fractions of four whole numbers with a manimum sum. With so many parts, students will need systems for organizing the pieces they are collecting and using toward a complete solution. IF you notice students with lots of parts getting somewhat lost in their own work, ask: What parts have you figured out? How can you record or organize those parts so you can see what you have or what you still have left to find? Students many want to use the Number Visuals Sheets, one per solution, as a way of organizing. Labels often help keep track of the values, such as recording ½ or 14 as 7.

Are students using one solution to help them find new ones? Rather than starting over each time, students can modify one solution repeatedly to generate new ones. Some students might like to put each component on a sticky note so that they can rearrange them to make new solutions, such as pulling out ½ or 14 = 7 and substituting ½ of 10 = 5 for a new solution. This kind of thinking focuses on patterning and the underlying structure of the problem.

How are students reasoning about finding a fraction of a whole number? Students may connect this work to multiplication or division of whole numbers, or to the work done in the Play activity. Students may benefit from using the Number Visuals Sheet again to better see and justify their thinking.

How are students translating their work into number sentences? At the end of the lesson, you will give students the opportunity to reframe their work as multiplication of fractions by a whole number. You may also introduce grouping symbols. Pay attention to how students are connecting the meaning of these symbols to their own work. The symbols should serve as a new way to label their existing thinking. Unconventional use of the symbols does not necessarily mean that students did not understand their work throughout the lesson.

Gather students together with their evidence to discuss the following questions. Chart students’ thinking and examples of their evidence:

· What solutions did you find?

· How did you use what you found to help you find new solutions?

Show students how to turn their findings into number sentences by naming what they have been doing as multiplication. For instance, tell them that “1/5 of 20 is 4” is the same thing as 1/5 x 20 = 4.. Ask students to help you rewrite some of the parts of a solution as multiplication number sentences. You may want to combine these individual number sentences into a single equation using parentheses to show the entire solution, such as (1/2 x 10) + (1/3 x 24) + (1/4 x 12) + (1/5 x 20) = 20.

Then ask students to look back over all the solutions they have come up with as a class. Ask students, Is there a way to organize these solutions to look for patterns? Give students a moment to turn and talk to a partner about ideas. Discuss ideas for how to organize solutions. Students may suggest some kind of table or rewriting all of the solutions using equations. Students might suggest writing each solution on a piece of paper or index card and organizing them on a bulletin board in some way.

Come to agreement about some way to organize and display all the solutions so that you can look for patterns , then implement your plan. Looking at your organized solutions, discuss the following questions:

· What do you notice about our solutions?

· Which numbers did we all use the most often? Why?

· What numbers did not get used? Why?

· How would you use these patterns to search for other solutions?

If students are interested, you might ask them to go back and search for additional solutions using their observations from the discussion.

Exit Ticket: When can you imagine needing to find a fraction of a whole number? Write your own fraction multiplication problem and solve it.

SEL Self-Assessments (English) and Teacher Rubric