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Patterns, Functions, and Algebra
 
Session 2 Part A Part B Part C Part D Part E Homework
 
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Session 6 Materials:
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Session 2:
Homework

Problem H1

Solution  

Look at this table:

Input

 

Output

1

 

6

2

 

10

3

 

14

4

 

18

5

 

22

6

 

26

?

 

?

?

 

?

 

a. 

Find and describe several patterns in the table.

b. 

If you used a variable in your description, explain its meaning.

c. 

What is the 100th entry in the table?

d. 

Is 102 ever an "output"? How do you know?


 

Problem H2

Solution  

Mr. Lewis looked at the table in Problem H1 and said, "Oh, I get it. Just do these steps."

1. 

First, you multiply the input number by itself and multiply that answer by 6.

2. 

Then you add 6 to the answer you just got, and call that number A.

3. 

Then, start over. Multiply the input number by itself and add 11, and call that B.

4. 

Multiply the input number by 5, and call that C.

5. 

Then, multiply B and C together, and divide by 6. Call that D.

6. 

Finally, subtract D from A, and that's all!

If you do all this, you'll get the numbers in the table." Does Mr. Lewis get it? Does Mr. Lewis' method agree with yours for the first three outputs? For the next three? Which method is correct?


 

Problem H3

Solution  

A frog climbs up the side of a well and slides back while resting. Every minute the frog leaps forward 5 meters (and it leaps forward precisely at the end of the minute). Then it rests for a minute. At the end of the rest, the frog slips back 3 meters. At the end of the next minute it leaps (5 meters), then a minute later it slides back (3 meters), and so on.

Here is a picture of a well 11 meters deep.

Well 11 meters deep

You may find it helpful to work with someone to model what's happening in the problem.

a. 

Fill in the table below. Why is there only one correct way to fill in the table?

Height of well

 

Time it takes to get out

1

 

2

 

3

 

4

 

5

 

6

 

7

 

8

 

9

 

10

 

11

 

12

 

13

 

14

 

15

 

16

 

17

 

 

show answers

 

Height of well

 

Time it takes to get out

1

 

1 minute

2

 

1 minute

3

 

1

4

 

1

5

 

1

6

 

3

7

 

3

8

 

5

9

 

5

10

 

7

11

 

7

12

 

9

13

 

9

14

 

11

15

 

11

16

 

13

17

 

13

 

hide answers

 

b. 

There are other ways to represent situations besides tables. Try drawing a visual image (a graph or a picture) that conveys information about the frog in the well.

c. 

How long would it take the frog to get out of a well that is 30 meters deep? One that is 100 meters? One that is 103 meters?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Try to predict using a rule rather than extending a table.   Close Tip

 

d. 

The (tired) frog took 13 minutes to get out of the well. How deep could the well have been?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Is there only one answer to this question?   Close Tip

Take it Further

Problem H4

Solution

a. 

Suppose our frog could climb 6 meters per minute and slid only 2 meters while resting. How long would it take to get out of a 100-meter well?

b. 

Suppose our frog could climb n meters per minute, and slid only 2 meters while resting. Describe, in terms of n, how long it would take to get out of a 100-meter well.

c. 

There are several numbers that contribute to the frog's predicament. There's the climbing rate (5 meters per minute in the original problem), the slippage rate (originally 3 meters per minute), and the height of the well. Figure out a rule that allows you to calculate the escape time if you are given the climbing rate, the slippage rate, and the height of the well. (Such a rule is called a function -- we'll look at functions in depth later on in the course.)


Similar reasoning to some of the Eric problems will be helpful here, but this
is a true challenge.   Close Tip


 

 

Suggested Readings:

Vance, James. "Number Operations from an Algebraic Perspective."
Reproduced with permission from Teaching Children Mathematics, © 1998 by the National Council of Teachers of Mathematics. All rights reserved.

Download PDF file

Solomon, Jesse, Carol Martignette-Boswell, et al. "Toward a Cooperative Model of Math Staff Development."
Reproduced with permission from Mathematics Teaching in the Middle School, © 1997 by the National Council of Teachers of Mathematics. All rights reserved.

Download PDF File:
Chapter 1


Chapter 2



Usiskin, Zalman. "Conceptions of School Algebra and Uses of Variables."
Reproduced with permission from The Ideas of Algebra, Grades K-12: 1988 NCTM Yearbook, © 1988 by the National Council of Teachers of Mathematics. All rights reserved.

Download PDF File:
Chapter 1


Chapter 2




Next > Session 3: Functions and Algorithms

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