Making Connections

 

Introduction

The main business of science is to uncover patterns.  Often we represent those patterns as algebraic expressions, graphs, or tables of numbers (data).  Being able to make connections among those various representations is one of the most vital skills to possess.

 

Task

In this project you attempt to make precise connections among these three ways of representing patterns.

 

Instructions

Work with at least one other person to complete the following exercises.  Solve each problem in order and save your work along the way.  You will create a presentation on one of the four parts to be given to your class.

 

·         First Problem – Connecting Algebraic Expressions and Graphs:  Factor each of the following expressions completely, and then compare the factored form with the “picture” of the expression that is shown in the graph on the right.  Describe any connections that you see, and then repeat for the next expression.  In the end, formulate a conjecture as to how an algebraic expression in factored form is related to its corresponding graph.  Keep in mind that we are not assuming that you have any knowledge whatsoever about graphs.  That is what makes this “detective work” so fun!

 


 

Algebraic Expression

Graph

Possible Relationship

Description: 1.jpg

 

Factored Form

Description: 2.jpg

 

Factored Form

Description: 3.jpg

 

Factored Form

Description: 4.jpg

 

Factored Form

Description: 5.jpg.jpg

 

Factored Form

Description: 6.jpg

 

Factored Form

Description: 7.jpg

 

Factored Form

Description: 8.jpg

 

Factored Form

Description: 9.jpg

 

Factored Form

Description: 12.jpg

 

Factored Form

Description: 13.jpg

 

Factored Form

The following problems are a special challenge.  You may have to adjust the relationship you

expressed above to accommodate these new examples.

Description: 10.jpg

 

Factored Form

Description: 11.jpg

 

Factored Form


·         Second Problem – Connecting Algebraic Expressions and Tables: For each of the same algebraic expressions that you examined above, compare the factored form with the table of values associated with the expression.  (For example, if the expression is , then the value associated with it when x=1 will be .In the end, formulate a conjecture that describes how an algebraic expression in factored form is related to its corresponding data table.

 

Algebraic Expression

Table

Possible Relationship

 

a

-8

-1024

-7

-686

-6

-432

-5

-250

-4

-128

-3

-54

-2

-16

-1

-2

0

0

1

2

2

16

3

54

4

128

5

250

6

432

7

686

8

1024

 

 

 

Factored Form

 

b

-8

-36

-7

-33

-6

-30

-5

-27

-4

-24

-3

-21

-2

-18

-1

-15

0

-12

1

-9

2

-6

3

-3

4

0

5

3

6

6

7

9

8

12

 

 

 

Factored Form

 

x

 

-8

330

-7

240

-6

162

-5

96

-4

42

-3

0

-2

-30

-1

-48

0

-54

1

-48

2

-30

3

0

4

42

5

96

6

162

7

240

8

330

 

 

 

Factored Form

 

x

 

-8

25

-7

16

-6

9

-5

4

-4

1

-3

0

-2

1

-1

4

0

9

1

16

2

25

3

36

4

49

5

64

6

81

7

100

8

121

 

 

 

Factored Form

 

y

 

-8

169

-7

144

-6

121

-5

100

-4

81

-3

64

-2

49

-1

36

0

25

1

16

2

9

3

4

4

1

5

0

6

1

7

4

8

9

 

 

 

Factored Form

 

c

 

-8

90

-7

64

-6

42

-5

24

-4

10

-3

0

-2

-6

-1

-8

0

-6

1

0

2

10

3

24

4

42

5

64

6

90

7

120

8

154

 

 

 

Factored Form

 

a

 

-8

-1560

-7

-1053

-6

-672

-5

-399

-4

-216

-3

-105

-2

-48

-1

-27

0

-24

1

-21

2

0

3

57

4

168

5

351

6

624

7

1005

8

1512

 

 

 

Factored Form

 

a

 

-8

-1512

-7

-1005

-6

-624

-5

-351

-4

-168

-3

-57

-2

0

-1

21

0

24

1

27

2

48

3

105

4

216

5

399

6

672

7

1053

8

1560

 

 

 

Factored Form

 

x

 

-8

130

-7

96

-6

66

-5

40

-4

18

-3

0

-2

-14

-1

-24

0

-30

1

-32

2

-30

3

-24

4

-14

5

0

6

18

7

40

8

66

 

 

 

Factored Form

 

d

 

-8

28160

-7

17150

-6

9720

-5

5000

-4

2240

-3

810

-2

200

-1

20

0

0

1

-10

2

-40

3

0

4

320

5

1250

6

3240

7

6860

8

12800

 

 

 

Factored Form

 

 

x

-8

-66560

-7

-32928

-6

-14256

-5

-5000

-4

-1152

-3

0

-2

112

-1

24

0

0

1

-32

2

-240

3

-648

4

-896

5

0

6

3888

7

13720

8

33792

 

 

 

Factored Form

 

x

 

-8

85

-7

60

-6

39

-5

22

-4

9

-3

0

-2

-5

-1

-6

0

-3

1

4

2

15

3

30

4

49

5

72

6

99

7

130

8

165

 

 

 

Factored Form

 

t

 

-8

390

-7

299

-6

220

-5

153

-4

98

-3

55

-2

24

-1

5

0

-2

1

3

2

20

3

49

4

90

5

143

6

208

7

285

8

374

 

 

 

Factored Form

 

 

·         Third Problem – Applying Your Findings:  For each expression, factor it completely and write the factored form beneath the expression.  Then match it to its corresponding table or graph by writing the letter corresponding to the expression on its matching table or graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Description: f.jpg

Description: b.jpg

 

Variable

Expression

-8

-92160

-7

-46305

-6

-20736

-5

-7875

-4

-2304

-3

-405

-2

0

-1

9

0

0

1

-9

2

0

3

405

4

2304

5

7875

6

20736

7

46305

8

92160

 

 

 

 

Description: d.jpg

Description: i.jpg

 

Variable

Expression

-8

-6291456

-7

-2470629

-6

-839808

-5

-234375

-4

-49152

-3

-6561

-2

-384

-1

-3

0

0

1

3

2

384

3

6561

4

49152

5

234375

6

839808

7

2470629

8

6291456

 

Variable

Expression

-8

-1040

-7

-702

-6

-448

-5

-266

-4

-144

-3

-70

-2

-32

-1

-18

0

-16

1

-14

2

0

3

38

4

112

5

234

6

416

7

670

8

1008

 

Variable

Expression

-8

-36

-7

-33

-6

-30

-5

-27

-4

-24

-3

-21

-2

-18

-1

-15

0

-12

1

-9

2

-6

3

-3

4

0

5

3

6

6

7

9

8

12

 

 

 

·         Fourth Problem – Predicting the Unknown: One of the primary reasons to make connections is to be able to explain or predict previously unobserved behavior.  Below we provide you with some tables and some graphs.  Based on these alone, determine whether the expression associated with them can be factored.  Explain the reasoning behind your decision.  [Hint:  You should make use of your observations from the problems above to determine what is means for an expression to not be factorable.]

 

 

Description: A.jpg

 

Variable

Expression

-8

-550

-7

-376

-6

-244

-5

-148

-4

-82

-3

-40

-2

-16

-1

-4

0

2

1

8

2

20

3

44

4

86

5

152

6

248

7

380

8

554

Description: AAA.jpg

Description: AA.jpg

 

Variable

Expression

-8

344

-7

193

-6

98

-5

44

-4

18

-3

8

-2

7

-1

8

0

9

1

6

2

2

3

1

4

8

5

32

6

84

7

176

8

325

 


 

Conclusions

 

With those from another group, compare your answers and your way of talking about the connections between the factored form of the expressions and the graphs and tables.  Work to make sure that your explanation is clear and concise. 

 

Prepare a presentation which:

1.      Explains the connection between the factored expression and the corresponding graphs and tables.

2.      Describes briefly how you determined this connection (you may want to discuss some of your original ideas and how you needed to refine them as you looked at more examples).

3.      Gives a test for determining whether a given expression can be factored if you are given a graph or table associated with the expression.

 

Finally, present your solution to your instructor.


Grade

Your project should correctly identify and justify mathematically: (1) the factored form of each expression, (2) the connection between the factored expression and the corresponding graphs and tables, (3) how you determined this connection, and (4) a test for determining whether a given expression can be factored if you are given a graph or table associated with the expression.

 

Score

Content

Presentation/Communication

4

•       The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution.

•       The solution completely addresses all mathematical components presented in the task.

•       The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results.

•       Mathematically relevant observations and/or connections are made.

•       There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made.

•       Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem.

•       There is precise and appropriate use of mathematical terminology and notation.

•       Your project is professional looking with graphics and effective use of color. 

3

•       The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution.

•       The solution addresses all of the mathematical components presented in the task.

•       The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem.

•       Most parts of the project are correct with only minor mathematical errors.

•       There is a clear explanation.

•       There is appropriate use of accurate mathematical representation.

•       There is effective use of mathematical terminology and notation.

•       Your project is neat with graphics and effective use of color. 

2

•       The solution is not complete indicating that parts of the problem are not understood.

•       The solution addresses some, but not all of the mathematical components presented in the task.

•       The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning.

•       Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures.

•       Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.

•       There is some use of appropriate mathematical representation.

•       There is some use of mathematical terminology and notation appropriate for the problem.

•       Your project contains low quality graphics and colors that do not add interest to the project.

1

•       There is no solution, or the solution has no relationship to the task.

•       No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem.

•       The solution addresses none of the mathematical components presented in the task.

•       There were so many errors in mathematical procedures that the problem could not be solved.

•       There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem.

•       There is no use, or inappropriate use, of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.).

•       There is no use, or mostly inappropriate use, of mathematical terminology and notation.

•       Your project is missing graphics and uses little to no color.


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