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Session 10, Grades 6-8, Part D:
More Problems That Illustrate Algebraic Thinking (25 minutes)
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At the 6-8 grade level, there are many problems that prepare students for their later work in algebra. There is some evidence that more students are, in fact, taking a formal course in algebra in the eighth grade. The problems included in this session, however, may be used in courses both before and during students' first formal course in algebra. Note 7
For each of the problems below, answer the following questions:
a. | What algebraic content is in the problems? |
b. | What content does it prepare students for later? |
c. | How does this content relate to the mathematical ideas in this course? |
d. | How would your students approach this problem? |
e. | What are other questions that might extend students' thinking about the problem? |
f. | Does your current program in mathematics at your school include problems of this type? |
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Problem D1 |  |
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a. | Which block, cylinder, sphere, or cube will balance Scale C? |
b. | List or draw the steps you followed to identify the block. |
c. | Which block, cylinder, sphere, or cube will balance Scale F? |
d. | List or draw the steps you followed to identify the block. |
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Problem D2 | |
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a. Find the weight of each block.
cylinder = ________ pounds
sphere = ________ pounds
cube = ________ pounds
b. Write or draw the steps you followed to solve the problem.
c. Find the weight of each block.
cylinder = ________ pounds
sphere = ________ pounds
cube = ________ pounds
d. Write or draw the steps you followed to solve the problem.
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Problem D3 | |
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These signs tell about some items for sale. Same items have same prices. Different items have different prices. How much is:
a. | A helmet? |
b. | A bell? |
c. | A bicycle lock? |
d. | Explain how you figured out the prices. |
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Problem D4 | |
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This is a staircase Deion built. He used 3 blocks for the 1st stair, 6 blocks for the 2nd stair, 9 blocks for the 3rd stair, and so on, using 3 more blocks for each higher stair.
a. | Make a three-column table. In Column 1 show the Stair Number. In Column 2 show Number of Blocks used. In Column 3 show the Total Number of Blocks used to build the stairs from Stair 1 up to and including Stair 5.
For example, for stairs 1-3, Deion used a total of 3 + 6 + 9, or 18, blocks. |
b. | Let S represent the stair number, and let N represent the number of blocks. Complete this function rule to show the number of blocks needed to build Stair S. N = ? |
c. | What is the number of blocks Deion would need to build Stair 10? |
d. | Explain your answer to the above question. |
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Problem D5 | |
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This is a series of growing shapes. Imagine that the building pattern continues. How many blocks will it take to build:
a. | Shape 8? |
b. | Shape 12? |
c. | Tell in words how you would build Shape 5. |
d. | Write a function rule to relate the number of blocks in a shape to the shape number. Let N represent the shape number. Let B represent the number of blocks in a shape. B = ? |
This is a series of growing cubes, with one small cube on top. Imagine the building pattern continues. How many blocks will it take to build:
e. | Cube 7? |
f. | Cube 20? |
g. | Tell in words how you would build Cube 5. |
h. | Write a function rule to relate the number of blocks in a cube to the cube number. Let N represent the cube number. Let B represent the number of blocks in a shape. B = ? |
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