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Section 1.7 More with Patterns and Variables: Rectangular Patterns in the 100s Chart

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Throughout this chapter, you have experienced how problems can be approached in a variety of ways. For example, to solve arithmetic problems such as 43 − 28, students found at least 8 ways to do this. Using four fours and four numbers, you found many ways to express most of the numbers 1 through 20. You found many patterns in the 100s chart. By now, we hope you are finding that mathematics is a way of thinking and communicating that can be very creative. One of the hallmarks of mathematics is sense-making in your own way, not in a way prescribed for you. In this lesson you will use your creativity based in sense making to find more ways to simplify an arithmetic expression and look for more patterns in the 100s Chart.

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Subsection 1.7.1 Number Talk: 93 − 27

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Solve the following problem in as many ways as you can.

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Mentally solve 93 − 27 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.

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One way a student solved 93 − 27 was by adding 3 to both 93 and 27 to get 96 − 30 = 66. The student also solved the problem by adding 4 to both numbers to get 97 − 31 = 66. Does this strategy always work? Why or why not? Is the student's process related to the 100s chart? Find the numbers on Table 1.6.3.1 and explain what's happening to the numbers. Use algebra to explain why this process works.

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Student Page 1.7.2 Rectangle Patterns in the 100s Chart
A4US

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In Analyzing the 100s Chart you found patterns in rows, columns, or diagonals. This time your pattern search will be confined to rectangles.

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1.

Consider the rectangle in Table 1.7.2.1. What patterns do you see in the numbers in this rectangle?

Table 1.7.2.1. Rectangle for Rectangle Patterns in the 100s Chart.
84 85 86
74 75 76
64 65 66
54 55 56

Let x=84. In the rectangle, write the other values in the rectangle in terms of x.

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(a)

Use your relabeling to justify any numeric patterns you found.

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(b)

Find another pattern in the relabeled rectangle.

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(c)

Does your pattern work for a 4 by 3 rectangle with upper left corner labeled 51?

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(d)

Does your pattern work for any other 4 by 3 rectangles? Why or why not?

Work on the the following exercises alone for at least 10 minutes. When each member of your group has found at least one rectangle pattern, share the patterns you found with your group.

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2.

Using a different color to outline each rectangle, draw three rectangles following the lines in Table 1.6.3.1 (reprinted below) with the following rules:

100 101 102 103 104 105 106 107 108 109
90 91 92 93 94 95 96 97 98 99
80 81 82 83 84 85 86 87 88 89
70 71 72 73 74 75 76 77 78 79
60 61 62 63 64 65 66 67 68 69
50 51 52 53 54 55 56 57 58 59
40 41 42 43 44 45 46 47 48 49
30 31 32 33 34 35 36 37 38 39
20 21 22 23 24 25 26 27 28 29
10 11 12 13 14 15 16 17 18 19
0 1 2 3 4 5 6 7 8 9
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(a)

Make sure all three rectangles have different dimensions.

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(b)

Make one rectangle large and one rectangle small.

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(c)

Make only one square.

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3.

What patterns do you notice among the numbers in one of the rectangles? Find at least three patterns. Each pattern you find must not extend beyond the rectangle.

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4.

Of the patterns you found, do any work for all three rectangles? Explain.

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5.

Choose one of the patterns you found. Explain why the pattern works.

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6.

Challenge yourself to use variables to show why the pattern works.

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As a group, choose your 3 favorite patterns, including at least one that you think other groups won't find. Share your patterns with the class. As a group, choose 2 patterns shared by other groups to justify. Use variables and algebra to justify that each pattern works.

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Homework 1.7.3 Homework

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2.

Work on Rectangle Patterns in the 100s Chart. For a pattern that has not been justified in class, explain why the pattern works. Challenge yourself to use variables to generalize the pattern so that it works regardless of where you draw the rectangle in the 100s chart.