Completing the Table

Strategies for the Nines. The Patterns for the Nines pages in the Student Activity Book ask students to complete a table that will help them explore the patterns for the product of 9 and the numbers one through nine and to develop strategies for learning these facts.

Show a display of the page and begin by completing the first two rows of the table together. Show students how to follow the rule at the top of each column on the table and how to use their multiplication tables to check. Ask student pairs to finish completing the table and to discuss Questions 1–4 with their partners.

After students have had some time to describe patterns in the table to a partner, discuss Questions 1–4 as a class. Encourage students to use the terms product and factor throughout the discussion.

Question 1 asks students to look for patterns in the second column. Students might observe the following:

1. When the products of 9 and the numbers 1–9 are listed in a column, it is easy to see that the digits in the tens place count up by ones (0, 1, 2, 3, etc.) and that the digits in the ones place count down by ones (9, 8, 7, etc.).
2. The sums of the two digits in each of the products listed is nine. For example, 3 + 6 = 9 and 7 + 2 = 9. In fact, the sum of the digits of any multiple of 9 is also a multiple of 9.

See the Sample Dialog for a student's description of a pattern he saw while working with his father.

Question 2 asks students to look for patterns in the third column, which has the products of ten and the numbers one to nine. Students will be reminded that when you multiply ten times another number, the first digit of the product is the same as one of the factors and the last digit is zero.

Questions 3–4 develop the strategy for the nines using the familiar tens facts. For example, 10 × 4 is 40, so 9 × 4 is 4 less: 40 − 4 = 36. Allow students time to see the patterns in the columns and rows on the table to develop the strategy and describe it in their own words. Students may also find it helpful to use a rectangle to visualize it. See Figure 3.

Identify Multiples of Nines. Students will need calculators to complete Question 5. Students explore how the process of adding the digits in multiples of nine can be repeated until nine itself results. As illustrated in the example, demonstrate how to use a calculator to multiply 9 × 634. The product is 5706. Then add the digits in the product: 5 + 7 + 0 + 6 = 18. Add the new answer's digits and a nine results: 1 + 8 = 9. Students will work with other multiples of nine to discover that this pattern is consistent. See the TIMS Tip.

Assign Questions 5–6 as a challenge to students, or continue to complete the questions together as a class. See Meeting Individual Needs.

Add student descriptions of patterns for the nines to the Patterns for Remembering the Facts class display. See Figure 4. Students will use this chart to solve problems during the remainder of this unit.