Handy Facts

Est. Class Sessions: 2

Developing the Lesson

Part 2. Building a Multiplication Table

Use Invented Strategies. Display and direct students' attention to the My Multiplication Table page in the Student Activity Book. Review the terms columns and rows. Show students that columns make a vertical or up and down line on the table and that rows make a horizontal line that goes across the table. Tell students they will be filling in the columns under the starred numbers (0, 1, 2, 3, 5, 10) shortly. These columns were chosen because these multiplication facts are easy to remember and are frequently familiar to children.

Be sure students save their partially completed My Multiplication Table. They will complete them in Lessons 4 and 5.

Students benefit from using strategies and concrete ways to figure out their multiplication facts. Begin by having students look at the column on the multiplication table under 0. Point out the multiplication sign in the top left corner and show how multiplying 0 times 0 resulted in the recording of the 0 in the box on the table. Then move to the column under 2. Show how multiplying the 2 in that column by the 0 in that row results in recording a 0 in the proper box on the table.

Before assigning the task of filling in the table, have a discussion about strategies students could use to fill in the numbers. Ask a student to draw a number line, or use a blank number line on a display of the Handy Constant Hoppers page from the Student Activity Book to show how the moves of a constant hopper could help fill in the multiplication table.

Ask:

Show how you could use a constant hopper to help you fill in the 2s column of the table; e.g., 2 × 1, 2 × 2, and so on. (A +2 constant hopper would start at 0 and land on 2 after 1 hop, so 2 × 1 = 2. After two hops of 2 it would land on 4, so 2 × 2 = 4. After three hops of 2 it would land on 6, so 2 × 3 = 6 and so on.)

Have students say the number sentences as you slowly demonstrate how to fill in each box under the 2 column as they do the same on their tables. Make sure they are matching the numbers in the sentences to the numbers on the table. When complete, demonstrate how to use the table to find 2 × 3 as shown in Figure 3. Remind students that 2 and 3 are factors in this number sentence, and that 6 is the product. We say that six is a multiple of 2 because it is the product of 2 and another whole number, 3.

Tell students there are many ways to figure out multiplication facts.

Ask:

Does someone have a different way to fill in the 2s column? Show or tell us your strategy. (Encourage students to share their strategies. These may include skip counting, doubling, drawing a picture, using counters, drawing arrays, or using a 200 Chart. If students do not share some of these strategies use these discussion prompts to encourage this discussion.)
Show how you could use the 200 Chart to fill in the 2s column. (Use a display of the 200 Chart page from the Student Guide Reference section.)
Show how to draw a picture or use tally marks to figure out 2 × 7. (See Figure 4.)
Show how to use counters to figure out 2 × 9. (See Figure 5.)

It is important for students who have memorized the facts and are quickly filling in the multiplication table to be able to articulate and describe the strategies and patterns that could be used to solve the multiplication facts. Without this foundation, students will be unable to apply these patterns and properties when working with larger numbers and algebraic concepts.

Fill in Multiplication Table. Have students discuss strategies for filling in the 1, 3, 5, and 10 columns with a partner. Then assign the task of completing these columns on the multiplication table independently. When students have completed these parts of the table, have them work with a partner to discuss any patterns they see in the table and record them on the back of the page.

Share Strategies. Ask students to share any new strategies they used to complete the table. Allow students to share a variety of strategies.

Ask questions similar to the following to generate discussion:

What strategies did you use to complete the 3s column? (Skip counting, repeated addition, doubling and then adding. See Figure 6.)
Is there another way to do this?
What were you thinking in your head when you filled in the 5s column? (skip counting by 5s or multiplying the factor by 10 and then diving that product in half)
How did you fill in the table for a fact like 5 × 7? (Possible responses: I know 5 × 5 = 25. So 5 × 7 = 5 × 5 + 10 or 35; I skip counted by 5s on my number line counting 7 hops and landed on 35.)
Can you use a different strategy to check your answer?

Discuss Zero. The 0 column deserves a special discussion. To help students understand the zero facts, tell them two types of stories:

For problems such as 7 × 0:

I have 7 pockets. Each pocket contains 0 pennies. How many pennies do I have in my pockets?

For problems such as 0 × 7:

The queen had several boxes that each contained 7 diamonds. She gave me 0 of the boxes. How many of the diamonds did she give me?

Some students may confuse the property of zero in addition: when you add zero to any number the answer is that number, with the property of zero in multiplication: when you multiply any number by zero the answer will be zero. Use the following prompt to generate discussion to help clarify this possible misconception.

Ask:

We know 5 + 0 = 5. Why doesn't 5 × 0 equal 5? Show or tell how you know with a drawing. (See Figure 7 for a possible explanation with a drawing.)

Remind students of the 0 constant hopper in Question D. Ask students to make up and share some stories of their own about zero. Afterwards, have them fill in the 0 column and 0 row of their multiplication tables.

Describe Patterns on Table. To check the accuracy of the students' tables, ask students to skip count aloud by 1s, 2s, 3s, 5s, and 10s as you go down the columns for these numbers. Explain to students that the numbers being added to the table are called multiples. When skip counting by a certain number, the multiples of that number are said aloud. Have students correct any incorrect products on their tables.

The class will add to the Patterns For Remembering the Facts chart in Lessons 4 and 5.

They will use the chart to help them solve problems in Lessons 8 and 9.

After each column, pause and ask students to describe any patterns they found in that column. Use the terms factor, product, and multiple where appropriate. Record the patterns students found on the Patterns for Remembering the Facts chart prepared prior to the lesson. See Materials Preparation and Figure 8 for a sample class chart.

Use prompts similar to the following as necessary to discuss the patterns:

What patterns do you see in the zero column and the zero row? (All the numbers are zero.)
What does that tell you about the product of zero and any number? (Any number times zero is zero.)
What patterns do you see in the ones column? (It is just counting by ones from zero to ten.)
What does that tell you about the product of one and any number? (Any number times one is itself.)
What patterns do you see in the multiples in the twos column? (All the numbers end in 2, 4, 6, 8, or 0.)
What does that tell you about the multiples of two? (They are all even numbers.)
What patterns do you see in the multiples in the fives column? (All the numbers end in five or zero.)
Which multiples of five end in zero? Write the number sentences for the products that end in zero. What can you say about the factors in these number sentences? (0 × 5 = 0, 2 × 5 = 10, 4 × 5 = 20, ... 10 × 5 = 50; the other factor is an even number.)
Which multiples of five end in five? Write the number sentences for the products that end in five. What can you say about the factors in these number sentences? (1 × 5 = 5, 3 × 5 = 15, 5 × 5 = 25, ... 9 × 5 = 45; the other factor is an odd number.)
What patterns do you see in the tens column? Try to use the term multiple in your description of the pattern. (All the multiples of ten end in zero.)
What patterns do you see in the threes column? What can you say about the product of three and an even number? (Three times an even number is an even number.)
What can you say about the product of three and an odd number? (Three times an odd number is odd.)

Students might also observe that they get the same answer when they change the order of the factors. For example, 10 × 3 = 3 × 10. This will be more apparent later when they work with a completed multiplication table.

Student Guide:

Student Activity Book: 261

Answers