Multiplication as Scaling
Students will be able to interpret multiplication as scaling by comparing the size of a factor to the size of the product, without solving the equation.
- Write a multiplication problem on the board, like 3 x 6 = 18.
- Point out that the product is greater than either factor, and explain, "When we multiply whole numbers, the product is greater than the factors. In this problem, 18 is six times larger than three."
- Tell students that today we are going to think about multiplication as a way to scale, or compare the size of numbers.
Explicit Instruction/Teacher modeling(15 minutes)
- Write a multiplication problem on the board, like 8 x ⅖. Tell students that, in this problem, we are going to multiply a whole number and a fraction.
- Ask, "Do you think that the product of this multiplication problem will be less than or greater than the factors?" Give students a moment to talk with a partner, then have students show you their guess by pointing their finger up to answer "greater than" or pointing down to answer "less than."
- Solve the problem (8 x ⅖ = 3 ⅕) and explain, "When we multiply a whole number by a fraction that is less than one, the product will be less than the whole number factor. In this problem, 3 ⅕ is ⅖ as big as 8."
- Repeat with another problem, like 5 x 1 ½.
- After solving (5 x 1 ½ = 7 ½), explain, "When we multiply a whole number by a fraction that is greater than one, the product will be greater than the whole number factor. In this problem, 7 ½ is 1 ½ times greater than 5."
- Ask students what might happen if you multiplied a whole number by a fraction that is equal to 1, like 4/4.
- Discuss as a class (i.e., when we multiply by 1, the value stays the same).
- Display the anchor chart and review it.
Guided Practice(10 minutes)
- Hold up a series of prepared cards with multiplication problems (e.g., ⅓ x 9, 5/5 x 7, 1 ¼ x 20, and 8/4 x 2).
- As you hold up each card, ask students to show you (with a finger pointing up or a finger pointing down) whether the product will be less than or greater than the whole number factor.
Independent working time(10 minutes)
- Hand out the Scaling Practice worksheet.
- Do one problem with the class as an example.
- Have students complete the rest of the worksheet independently.
- Go over the worksheet with the class.
- Provide simplified practice problems with benchmark fractions (i.e., ½ x 4, ⅓ x 9, 1¼ x 20).
- Have students apply the strategies learned in the lesson to solve word problems.
- Have students apply scaling to area (see optional materials).
- As the class is engaged in independent practice, walk around to each student and hold up one of the prepared cards that was used during guided practice.
- Ask each student to tell you whether the product of the expression will be greater than, less than, or equal to the whole number factor.
- Take note of students who need additional support.
Review and closing(5 minutes)
- Ask, "How will this skill (scaling) help us think about numbers in our head?"
- Discuss as a class (i.e., we can understand whether a number will get bigger or smaller when we multiply it by a fraction; we can estimate what the product will be).