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Section 1.8 Patterns and Variables: Justifying Patterns with Algebra

As in Section 1.7 some of our number talks are related to our work with the 100s Chart. Solve the next problem in as many ways as you can. Think about how it is related to the 100s Chart (Table 1.6.3.1).

Subsection 1.8.1 Number Talk: 96 + 29

Mentally solve 96 + 29 in as many ways as you can.

Raise a thumb when you have a solution. Raise a finger for each additional solution process you find.

  • How does this number talk relate to your work with patterns in the Table¬†1.6.3.1?

  • Will your process work for adding any two numbers? Why or why not?

Activity 3. Sharing Rectangle Patterns in the 100s Chart.

Share your justifications of patterns you found for homework with your group. As a group, decide which two patterns to justify in words and algebraically for the class. For your algebraic justification, label your rectangle as generally as you can, with \(x\) as the upper left corner of the rectangle and the other numbers in the rectangle based on \(x\text{.}\)

Homework 1.8.2 Homework

2.

(a)

Draw a 3 by 3 box around 9 numbers in the October 2020 calendar. An example is shown in Figure 1.8.2.1.

Figure 1.8.2.1. The October 2020 Calendar for Exercise 1.8.2.2

Use the example or draw your own 3 by 3 box.

(b)

Find two different patterns among the numbers in the 3 by 3 box. Describe each pattern carefully.

(c)

Use algebra to justify one of the patterns you found in Task 1.8.2.2.b. Indicate which pattern you are justifying.

3.

A famous pattern called Pascal's Triangle is shown in Figure 1.8.2.2.

Figure 1.8.2.2. Pascal's Triangle
(a)

Study the pattern. How are the numbers from one row related to the numbers in the next row?

(b)

Fill-in the missing numbers in the last row. Extend the pattern at least one additional row.

(c)

Describe 2 patterns that you see in this arrangement of numbers.

4.

(a)

Using only mental arithmetic, solve each problem in Table 1.8.2.3 below. Record how you thought about the numbers to help you add them mentally.

Table 1.8.2.3. Table for Exercise 1.8.2.4
Simplify Solution How did you solve the problem?
\(25 + 26\)    
\(15 + 16\)    
\(39 + 40\)    
\(18 + 19\)    
\(25 + 26\)    
\(163 + 164\)    
(b)

Write the sum \(25 + 26\) in terms of \(x\) with \(x = 25\text{.}\) Show the sum in terms of \(x\text{.}\)

(c)

What other problems did you solve using the same strategy you used to solve \(25 + 26\text{?}\) Color code the problems that you solve using the same strategy.

(d)

Did you use the same strategy to mentally solve all of the problems? If you used more than one strategy, color code problems you solved using the same strategy. Do this for each strategy you used. Describe each strategy.