## Finding Products

Students examine the role of commutativity and the multiplicative identity, play a multiplication game, and explore products where one of the factors is 6. They also create a "My Personal Multiplication Chart" to record products.

To assess prior knowledge, students will review the multiplication facts for which one factor is 5 or less.

Provide students with paper and crayons and ask them to draw six blue vertical lines on the paper. Now ask them to draw four red horizontal lines intersecting the vertical lines. Ask them to circle in purple each place there is an intersection and count the number of intersections. Challenge them to identify what multiplication fact they have just demonstrated. Tell them that in this model, the number of rows is given first. [4 ×6 = 24.] Ask them to turn their papers a quarter turn and name the multiplication fact now modeled. [6 ×4 = 24.]

Encourage them to generate other facts where one factor is 6, including 6 × 0 and 6 × 1.

Repeat with 7 as a factor.

It may be helpful for students to visualize the vertical lines as city streets, the horizontal lines as roads, and the intersections as marking where a stoplight is needed.

Distribute index cards to each pair and ask each student to make a set of 10 cards numbered 0 to 9, one to a card.

When they have finished, ask them to shuffle the two decks together and stack them face down. Tell them to take turns turning over the top card, multiplying the number drawn by 6 and then saying the product. As each card is used, it should be returned to the bottom of the deck. Give students time to play, and then ask the class to skip count in unison by 6. Encourage them to do so without looking at the game board. Repeat for 7 as a factor.

Ask students to save the numbered cards for later use.

Next, ask students to make a deck of triangle fact cards for the 6 and 7 tables by putting 2 factors, one of them a 6 or a 7, on 2 of the corners and the product in the third corner. They may wish to use red and blue for the factors and purple for the product. Remind the students that they can check the multiplication chart for products they are unsure of.

When they have made triangle fact cards for the facts 0 × 6 to 9 × 6 and 0 × 7 to 9 × 7, ask each student to cover the product on one card with his or her thumb, show the card to the other student, and ask him or her to tell the product. Encourage the students to separate the cars with the facts they know from those they are less sure of.

### Reference

Wirtz, Robert. Drill and Practice at the Problem Solving Level. Washington DC: CDA, 1976.

- Crayons
- Paper
- File Cards
- My Multiplication Chart Activity Sheet

**Assessment Option**

Distribute a copy of the My Multiplication Chart activity sheet to each student. Students should complete the chart for the facts they currently know. Collect the charts to assess student progress.

**Extensions**

- What products can you get when you multiply by 7? How many are even? Will you get an even product when you multiply by 3? By 6? By 8? By 9? How can you tell if the product will be even?
- How many ways can you have a product of 6? Of 12? Of 25? Of 42? Of 1?
- Move on to the next lesson,
*Multiplication Stories*. - Students who need additional practice may use the Times Table Tool.

Times Table Tool

**Questions for Students**

1. If you spin a 5, what would the product of 6 and that number be? What multiplication fact would show that?

[30; 5 × 6 = 30.]

2. If you spin a 1, what would the product of that number and 6 be? Why are you sure of that?

[6; The multiplicative identity.]

3. How could you model with streets and roads the multiplication facts 3 × 6 = 18 and 6 × 3 = 18? What is alike between these multiplication sentences? What is different?

[Student responses may vary.]

4. What numbers do you say when you skip count by 7's to 70? Which of these are even numbers?

[7, 14, 21, 28, 35, 42, 49, 56, 63, 70; When the other factor is even, the product is even.]

**Teacher Reflection**

- Are some students reluctant to participate? How can I encourage their participation?
- Which students are able to identify with accuracy the facts they know by heart? How can the other students be helped to do this?
- Which students were able to stay on task while they played the game? Should some pairs be changed in the next lesson?
- What extension activities are appropriate for students who have memorized all their multiplication facts?

### Multiplication Stories

### Learning Objectives

Students will:

- Find products where one factor is 6 or 7.
- Use the commutative property.
- Explore the results of multiplying by 1 and by 0.
- Practice the multiplication facts with factors of 6 and 7.
- Create a learning tool for mastering products.

### NCTM Standards and Expectations

- Understand the effects of multiplying and dividing whole numbers.

- Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30x50.

- Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.

### Common Core State Standards – Mathematics

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.A.3

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.B.5

Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.C.7

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Grade 3, Algebraic Thinking

- CCSS.Math.Content.3.OA.D.8

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.A.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.A.3

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.B.4

Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Grade 4, Algebraic Thinking

- CCSS.Math.Content.4.OA.A.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP6

Attend to precision.