a docs/yr4_worddocs/gr8 al… · web viewa. mathematical processes and f. algebraic relationships...

A. Mathematical Processes and F. Algebraic Relationships(From the WKCE-CRT Mathematics Assessment Framework, Beginning of Grade 10)

A. Mathematical Processes

• Use reasoning and logic to: perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies and test reasonableness of results.• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g., using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models.• Connect mathematics to the real world as well as within mathematics.• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

F. Algebraic Relationships * F.a.Subskill: Patterns, Relations, and Functions• Describe, recognize, interpret and translate graphical representations of mathematical and real-world phenomena on coordinate grids, e.g., slope, intercepts, rate of change, linear and non-linear functions, and quadratic, exponential and constant functions.• Analyze, generalize and represent patterns of change, e.g., direct and inverse variations, including numerical sequences, patterns to a given term, algebraic expressions and equations.

F.b. Subskill: Expressions, Equations, and Inequalities• Solve linear equations.• Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations and inequalities, e.g., linear, exponential, quadratic.• Translate between different representations and describe the relationship among variable quantities in a problem, e.g., tables, graphs, functional notations, formulas.

F.c. Subskill: Properties• Demonstrate understanding of properties by evaluating and simplifying expressions.• Demonstrate understanding of properties by solving linear equations.

* Slightly modified to reflect “Sharing the 10th Grade Descriptors”

Note:While all three Grade 8 learning targets for algebraic relationships are given for each of the following CABS, it is the target in BOLD font which aligns with that particular CABS.

Grade 8 Classroom Assessments Based on State Standards

(CABS)

Examining the 10th Gr. Descriptors: What’s “New” for 8th Graders?

The 8th Grade Classroom Assessments Based on State Standards were created to reflect the 10th grade state descriptors and the MPS learning targets for 8th grade. The 10th grade state descriptors should be used by both 8th and 9th grade math teachers. See your MTL for the document, “Sharing the 10th Grade Descriptors”, or visit the Portal (math page, inside the teacher community), for the delineated shared responsibilities.

The 10th grade state descriptors contain both previously introduced and new concepts. It is the intent of the 8th - 9th Grades Math Assessment Pilot, after considerable feedback from Kevin McLeod, UWM Mathematics Professor and Co-Investigator, MMP, to emphasize the new concepts. Alerting teachers to what is new for 8th graders should have classroom implications.

The 8th grade math CABS are therefore split into two categories. First, a series of “Power CABS” which can be used to assess student understanding of concepts not found in earlier grade level state descriptors. Second, a series of CABS not denoted with “Power” which can be used to assess student understanding of concepts found in both earlier state descriptors and in the 10th grade descriptors.

The chart below identifies the descriptors that are new to 8th grade as determined by the 8th - 9th Grades Math Assessment Pilot, 2006-2007.

F. Algebraic Relationships: Objective/Subskill 10th Gr. Descriptor Piece that is “New” NotesF.a “Describe, recognize, interpret and translate

graphical representations of mathematical and real-world phenomena on coordinate grids, e.g. slope, intercepts, rate of change, non-linear functions, and quadratic, exponential and constant functions”

F.a “Analyze, generalize and represent patterns of change e.g. direct and inverse variations, algebraic relationships, and equations”

F.b “Model… a variety of mathematical and real-world problems by using algebraic expressions, equations, and inequalities e.g. linear, exponential, quadratic”

We revised this in “Sharing the 10th Grade Descriptors”: 8th is emphasizing modeling; 9th is emphasizing solving.

F.b “Translate between different representations and describe the relationship among variable quantities in a problem e.g. tables, graphs, functional notations, formulas”

Not noted in earlier grade descriptors, so all new for 8th.

F.c “Demonstrate understanding of the Distributive, Commutative, and Associative properties by evaluating and simplifying expressions”

All three of these properties have been introduced in earlier grades; however, students have not been asked to evaluate prior to this year.

F.c “Demonstrate understanding of properties by solving linear equations”

Name __________________________________________________________ Date _______________________

Mathematics Grade 8Classroom Assessment Based on Standards

Power CABS Identifier: “Slope”

MPS Learning Target: Algebraic Relationships• MPS Learning Target #6: Analyze, describe, and generalize mathematical and real-world patterns of change and functional relationships with emphasis on the role of variable quantities.• MPS Learning Target #7: Model, justify, and solve linear equations and relationships using translations among tables, graphs/grids, and symbolic forms.• MPS Learning Target #8: Explain use of properties (e.g., commutative, associative, distributive) to evaluate expressions and solve linear equations.

Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Patterns, Relations, and FunctionsDescriptor:Describe, recognize, interpret and translate graphical representations of mathematical and real-world phenomena on coordinate grids, e.g. slope, intercepts, rate of change, linear and non-linear functions, and quadratic, exponential and constant functions.

Objective: A. Mathematical ProcessesDescriptors:• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Which of the following equations best represents the relationship shown in the graph?

A. 3 6x− + B. 3 12x− + C. 12 3x − D. 3 122

x + E. 3 122

x− +

Page 1 of 2

Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.

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What is the slope of the line on this graph?

Explain 2 different ways of identifying the slope of this line:

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Similarities and Differences”



Objective: A. Mathematical ProcessesDescriptors:• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

Compare the two graphs below.

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Page 1 of 2


Page 2 of 2

List as many similarities and differences as you can between the two graphs. What information in each graph supports your answer? Fill in your answers in the T-chart below.

Similarities and Differences Support for your Answer

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “All About the Line”



Objective: A. Mathematical ProcessesDescriptors:• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Solve and analyze routine and non-routine problems.

A line is graphed on the grid below.

Page 1 of 2


Page 2 of 2

What is the slope of the line? ______How do you know?

Where would the line intersect the y axis? ____________Explain your reasoning.

Where would the line intersect the x axis? ___________Explain your reasoning.

Using the above information, write an equation for the line.

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Bacterial Growth”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptor:Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations and inequalities, e.g. linear, exponential, and quadratic.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

You are performing an experiment to determine the rate at which bacteria grows. The following graph represents the results you obtained.

-20020406080100120140160180200220240260

-2 -1 0 1 2 3 4 5 6Hours

Number of Bacteria

Page 1 of 2

Which of these expressions would represent the relationship between the number of hours and the number of bacteria shown in the graph?

A. x3 D. x3 + 10

B. 3x E. 3x + 10

C. 3(x-1) F. 3x + 10

Grade 8 Algebraic Relationships


Page 2 of 2

Explain the relationship between the number of hours passing and the number of bacteria. Use data from the graph to support your statement.

Estimate the y-intercept: _______How do you know that?

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Number Pattern”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Patterns, Relations, and FunctionsDescriptor:Analyze, generalize and represent patterns of change, e.g. direct and inverse variations, including numerical sequences,, patterns to a given term, algebraic expressions and equations.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Solve and analyze routine and non-routine problems.

Study this sequence:5, 7, 2, 4, -1, 1, -4 . . .

What is the 10th term in this sequence?Answer: ______


How did you determine what this number is?

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Triangle Pattern”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Patterns, Relations, and FunctionsDescriptor:Analyze, generalize and represent patterns of change, e.g. direct and inverse variations, including numerical sequences,, patterns to a given term, algebraic expressions and equations.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

A. The three figures below are divided into small congruent triangles. Complete the table on the next page.

1 3 2 4

1 5 7 2 6 8

Figure 1 Figure 2 Figure 3

Page 1 of 2

Figure Number of Small Triangles

1 22 8

34

B. The sequence of figures is extended to the 7th figure. How many small triangles would be needed for Figure 7?

Answer: _________

Reproduced with permission from the TIMSS and PIRLS International Study Center, Lynch Education, Boston College. 2003


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C. The sequence of figures is extended to the nth figure. Explain a way to find the number of small triangles in the nth figure that does not involve drawing it and counting the number of triangles.

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Sam’s Idea”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptor:Solve linear equations.


Sam wanted to find three consecutive even numbers that add up to 84. He wrote the equation:k + (k + 2) + (k + 4) = 84.

What does the letter k represent?

A. The least of the three even numbersB. The middle even numberC. The greatest of the three even numbersD. The average of the three even numbers

What are the three numbers? _________________________

Page 1 of 2

Adapted from the TIMSS and PIRLS International Study Center, Lynch Education, Boston College.


Page 2 of 2

How did you determine the numbers?

Name __________________________________________________________ Date _______________________


CABS Identifier: “Representing Hats”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptor:Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations and inequalities, e.g. linear, exponential, quadratic.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Solve and analyze routine and non-routine problems.

Juan has 5 fewer hats than Maria, and Clarissa has 3 times as many hats as Juan. If Maria has n hats, which of these represents the number of hats that Clarissa has?

A. 3n

B. n – 5

C. 5 – 3n

D. 3n – 5

E. 3(n – 5)

Reproduced with permission from the TIMSS and PIRLS International Study Center, Lynch Education, Boston College.


Name __________________________________________________________ Date _______________________


Power CABS Identifier: “How Many Books?”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptor:Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations and inequalities, e.g. linear, exponential, quadratic.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Solve and analyze routine and non-routine problems.

Graham has twice as many books as Bob. Chan has six more books than Bob. If Bob has x books, which of the following represents the total number of books the three boys have?

A. 2x + 6

B. 3x + 6

C. 3x +8

D. 4x + 6

E. 5x + 6

Adapted and reproduced with permission from the TIMSS and PIRLS International Study Center, Lynch Education, Boston College.


Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Equations and Table”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptor:Translate between different representations and describe the relationship among variable quantities in a problem, e.g. tables, graphs, functional notations, formulas.

Objective: A. Mathematical ProcessesDescriptors:• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

x y-3 -9

2 1

1 7

Which of the following equations is true for the three pairs of x and y values in the table above?

A) x – 3 = y

B) 2x – 3 = y

C) 3x – 2 = y

D) 3x + 2 = y

E) 2x + 3 = y

U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP)


Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Create Your Own”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: Expressions, Equations, and InequalitiesDescriptors:• Model and solve a variety of mathematical and real-world problems by using algebraic expressions, equations and inequalities, e.g. linear, exponential, and constant functions.• Translate between different representations and describe the relationship among variable quantities in a problem, e.g. tables, graphs, functional notations, formulas.

Objective: A. Mathematical ProcessesDescriptors:• Use reasoning and logic to perceive patterns, identify relationships, formulate questions, pose problems, make conjectures, justify strategies, and test reasonableness of results• Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways (e.g. using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models).• Connect mathematics to the real world as well as within mathematics.• Create and use representations to organize, record, and communicate mathematical ideas.• Solve and analyze routine and non-routine problems.

Create a situation that involves a linear relationship that is decreasing.A. Describe the situation in words.

B. Write an equation that could be used to represent your situation and identify what the variables represent.

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C. Create a table that could be used to represent the mathematical relationship for the variables.

D. Sketch a graph that could be used to represent the mathematical relationship for the situation. Identify the slope, y-intercept, and explain what each represents in your situation.


Page 2 of 2

Name __________________________________________________________ Date _______________________


Power CABS Identifier: “Distributive Property”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskill: PropertiesDescriptor:Demonstrate understanding of properties by solving equations.


Jill and Chris worked with the equation shown below:

3(5 2) 36x + =

Jill said she can use the Distributive Property this way:

3(5 ) 3(2) 36x + = 15 6 36x + =

15 (6 6) 36 6x + − = −

15 3015 15

x=

2x =

Chris said he can use the Distributive Property in this way:

Page 1 of 2


Page 2 of 2

Which person used the Distributive Property correctly and why?

Name __________________________________________________________ Date _______________________


CABS Identifier: “Equivalent or Not?”


Wisconsin Assessment Framework for MathematicsObjective: F. Algebraic RelationshipsSubskills: Patterns, Relations, and Functions; Expressions, Equations, and Inequalities; PropertiesDescriptors:• Analyze, generalize and represent patterns of change, e.g. direct and inverse variations, including numerical sequences,, patterns to a given term, algebraic expressions and equations.• Translate between different representations and describe the relationship among variable quantities in a problem, e.g. tables, graphs, functional notations, formulas.• Demonstrate understanding of properties by evaluating and simplifying expressions.


State whether each of the following expressions is equivalent or not to: 5 + 3(x – 4) and why.

Page 1 of 2

8(x – 4)

3x – 12 + 5

20


Page 2 of 2

3x – 4 + 5

5 + 3x – 3(4)

5 + 3(-4 + x)

Write your own equivalent expression and explain why it is equivalent:

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