### Student Page 1.6.3 Analyzing the 100s Chart

Generalization is an important concept in algebra. It is the basis of the use of variables. Think about what the word, variable, means when you are not thinking about mathematics. Share your ideas about this.

We will continue to refine our meaning of variable through today's activities. Work on Analyzing the 100s Chart. As with the Four Fours collection of activities, use your creativity. Work to give clear descriptions of any patterns you find. Color code each pattern and illustrate it in the same color on the 100s Chart.

#### 1.

Study Table 1.6.3.1.

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 |

##### (a)

What must be added to a number to get the number immediately to its right?

##### (b)

What must be added to a number to get the number directly below the number?

#### 2.

Replace 6 with the letter \(n\) so that \(n = 6\text{.}\) When \(n = 6\text{,}\) notice that \(7 = n + 1\text{.}\) Rewrite 8 and 5 in terms of n where \(n = 6\text{.}\) Also write the expressions in Table 1.6.3.1.

#### 3.

Replace the number 23 with the letter \(m\) so that \(m = 23\text{.}\) What are the next two odd numbers after 23? Write these odd numbers using the letter \(m\) instead of 23. Write the expressions in Table 1.6.3.1.

#### 4.

How can you represent the number just below 23 on the 100s chart in terms of \(m\) where \(m = 23\text{?}\) How can you represent the number just above 23 on the 100s chart using \(m\text{?}\) Write the expressions in Table 1.6.3.1.

#### 5.

Suppose Table 1.6.3.2 is located somewhere in Table 1.6.3.1. Write each value in the rectangle in terms of \(x\text{.}\)

\(x\) | ||

#### 6.

Notice that every rectangle in Table 1.6.3.1 with the same dimensions as Table 1.6.3.2 can be labeled exactly the same way. The initial value of \(x\) is irrelevant. Why is that?

#### 7.

What patterns do you see in Table 1.6.3.1? Find at least 3. Color code each pattern in the chart. Describe each pattern in writing. Include a description of how to move to successive numbers and how much is added or subtracted each time you make the move. Explain why the pattern works as you say it does.

For Analyzing the 100s Chart, share your solutions to each problem with your group. Share two of the more challenging patterns your group found with the class.

Of the patterns shared by the class, choose 2 to work on with your group. Together, describe the pattern carefully then figure out why each pattern works. Does the starting number matter? Be ready to share with the class.