Section 1.6 Generalizations and Variables: Patterns in the 100s Chart
Subsection 1.6.1 Video: Brains Grow and Change
The video explains that making sense of mathematics is necessary to grow your brain. It shares brain science related to how to strengthen or create new pathways in your brain. If somewhere in your past, you have gotten the idea that you are not a math person, this video might change your point of view! βBrains Grow and Changeβ is from YouCubed.org, Week 4 of Inspirational Math, Day 5β12β.Subsection 1.6.2 Number Talk: 43 β 28
Number talks help you realize how many different, correct, ways there are to solve the same problem. They help you develop flexibility with numbers. Eventually, we will look for generalizations, extending our number talks into algebra. Consider the following problem:I had $43 with me when I went shoping last week.
I spent $28 on fruit. How much do I have left?
Solve the problem mentally. Find at least one other way to solve the problem mentally. When asked to do so, share your solution and one of the ways you solved the problem.
- Answer
Which solution are you defending?
- Ask
Does anyone have a question for me?
- Ask
Does anyone have another way to solve the problem?
Student Page 1.6.3 Analyzing 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
3.
Replace the number 23 with the letter
4.
How can you represent the number just below 23 on the 100s chart in terms of
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
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
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.
Homework 1.6.4 Homework
1.
Go back to How To Learn Math For Students Directions from Section 1.1, How to Learn Math for Students. Complete How to Learn Math for Students Exercise 1.1.1.6. You will hand in your written reflections for How to Learn Math for Students Exercise 1.1.1.6 in class.
2.
Revisit Student Page Exercise 1.6.3.6 in Analyzing the 100s Chart. Because the initial value of
(a)
(b)
(c)
(d)
3.
You probably have learned to combine like terms in previous mathematics classes. As with the Order of Operations, there are mathematical reasons underlying which terms can be combined.
When children learn to multiply, they use arrays such as those in Figure 1.6.4.1 to make sense of repeated addition. The product of the dimensions of an array is the number of squares in the array; the number of rows shows how many times the number of squares in the first row is repeated. Combining Like Terms builds on this experience in learning multiplication to help you think about the meaning of the terms 1,
4.
There are free apps online to help you make sense of combining like terms and adding polynomials using algebra tiles. Use the apps on this websiteβ13β. Read the introductory pages carefully. These pages tell you how to use the apps. Navigate through the introduction using the arrow in the upper right corner. Watch the videos and play with the apps for Simplifying Expressions and Adding Polynomials. Solve at least 2 problems at each difficulty level, until you conceptually understand the link between the algebra tiles and the expressions they represent. This work should take at most 20 minutes.
Student Page 1.6.5 Combining Like Terms
1.
Find the area of each of the algebra tiles shown. The side lengths are labeled. The variable,
2.
Translate each of the figures into algebra. Explain your translation. Simplify each expression as much as possible and explain why you cannot simplify further.
(a)
(b)
(c)
(d)
3.
Draw algebra tiles to illustrate each expression. Use your illustration to simplify each of the algebraic expressions. Explain how you know you are correct.
(a)
(b)
(c)
(d)
4.
Summarize your work with combining like terms. Which terms can you combine? How do you know?
5.
Simplify the following expressions. Show and explain your work.
(a)
(b)
(c)
(d)
bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/2015/06/Brains-Grow-Change.mp4
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