Unit 3 Linear Equations and Functions Math 8

Unit 3 Linear Equations and Functions

Number of Days: 34 days 11/27/17 ? 1/26/18

Unit Goals ? Stage 1

Math 8

Unit Description: Students learn about linear equations in two variables. Students explore concepts of slope and intercepts as they write and graph linear equations in two variables. A study of linear equations will provide students with opportunities to model relationships between two quantities. Students are also introduced to functions. Students will use equations, tables, and/or graphs to compare properties of functions and distinguish between linear and nonlinear functions.

Materials: calculators, graph paper, Desmos, AngLegs

Standards for Mathematical

Transfer Goals

Practice

Students will be able to independently use their learning to...

SMP 1 Make sense of problems ? Make sense of never-before-seen problems and persevere in solving them.

SMP 2 SMP 3

SMP 4 SMP 5 SMP 6 SMP 7 SMP 8

and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated

? Construct viable arguments and critique the reasoning of others. Making Meaning

UNDERSTANDINGS Students will understand that... ? A linear equation of the form y = mx + b may represent either a

proportional ( b = 0 ) or non-proportional ( b 0 ) relationship depending on the value of b. ? The slope of a non-vertical line is the same between any two points along that line. ? In a proportional relationship between two quantities, the constant of proportionality, the slope of the graph, and the unit rate of the relationship are equal. ? Functions can be represented in different ways, such as algebraically, graphically, numerically in tables, or by verbal descriptions.

ESSENTIAL QUESTIONS Students will keep considering... ? Why is y = mx + b considered

a linear equation? ? How can similar triangles be

used to explain why the slope is the same between any two points on a non-vertical line? ? How can you represent a function in different ways?

reasoning.

Acquisition

KNOWLEDGE

SKILLS

Standards for Mathematical

Students will know...

Students will be skilled at and/or be able to...

Content Clusters Addressed

? The definition of academic vocabulary words, ? Use slope-intercept form to graph or write linear equations.

[m] 8.EE.B Understand the

such as linear equation, function, rate of

? Find the slope of a line by using slope triangles or the

connections between

change, slope, and slope-intercept form.

slope formula.

[m] 8.F.A [m] 8.F.B

proportional relationships, lines, and linear equations. Define, evaluate, and compare functions. Use functions to model relationships between quantities.

? A linear equation can be represented by y = mx + b and its graph is a straight line.

? The slope of a line is the change in y divided by the change in x.

? A function is a relationship that assigns exactly one output for each input. The graph of a function is the set of ordered pairs consisting of an input, x, and the corresponding output, y.

? Graph proportional relationships and interpret the unit rate as the slope of the graph.

? Compare proportional relationships represented in different ways.

? Identify relationships that are functions from mapping diagrams, tables, equations, and graphs.

? Construct a function to model a linear relationship between two quantities.

? Sketch a graph from a description of a function.

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Unit 3 Linear Equations and Functions

Assessed Grade Level Standards

Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning.

Math 8

Standards for Mathematical Content

[m] 8.EE.B Understand the connections between proportional relationships, lines, and linear equations.

8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships

represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving

objects has greater speed.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane;

derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

[m] 8.F.A Define, evaluate, and compare functions.

8.F.1

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs

consisting of an input and the corresponding output.

8.F.2

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic

expression, determine which function has the greater rate of change.

8.F.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not

linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains

the points (1, 1), (2, 4), and (3, 9), which are not on a straight line.

[m] 8.F.B Use functions to model relationships between quantities.

8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function

from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of

change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or

decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Key: [m] = major clusters; [s] = supporting clusters; [a] = additional clusters

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Unit 3 Linear Equations and Functions

Assessment Evidence Unit Assessment

Evidence of Learning ? Stage 2

Math 8

Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency. Concepts and skills that may be assessed in Claim1: 8.EE.B ? The student graphs proportional relationships. ? The student interprets the unit rate as the slope of the graph of a proportional relationship. ? The student compares two different proportional relationships represented in different formats. ? The student finds the equation y = mx or y = mx + b for a line.

8.F.A ? The student recognizes that a function is a rule that assigns to each input exactly one output. ? The student identifies or produces input and output pairs for given functions. ? The student recognizes the same function written in different forms (algebraic, graphic, tabular, or verbal). ? The student compares properties of two functions, each represented in a different way (algebraic, graphic, tabular, or verbal). ? The student recognizes and gives examples of functions that are not linear.

8.F.B ? The student constructs a function to model a linear relationship between two quantities. ? The student determines the rate of change and initial value of a function, either from a description of a relationship or from two (x,y) values, including

reading the rate of change and/or the value of the function from a table or graph. ? The student interprets features of a linear function, such as rate of change and initial value, in terms of the situation it models, its graph, or a table of

values. ? The student qualitatively describes the functional relationship between two quantities by analyzing a graph (e.g., whether the function is increasing or

decreasing, or whether the graph in linear or nonlinear). ? The student draws a graph that exhibits the qualitative features of a function that has been described in writing.

Claim 2: Students can solve a range of wellposed problems in pure and applied mathematics, making productive use of knowledge and problem-solving strategies. Standard clusters that may be assessed in Claim 2: ? 8.EE.B ? 8.F.A ? 8.F.B

Claim 3: The student can clearly and precisely construct viable arguments to support their own reasoning and critique the reasoning of others. Standard clusters that may be assessed in Claim 3: ? 8.EE.B ? 8.F.A

Claim 4: The student can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Standard clusters that may be assessed in Claim 4: ? 8.EE.B ? 8.F.B

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Unit 3 Linear Equations and Functions

Evidence of Learning ? Stage 2

Other Evidence

Formative Assessment Opportunities

? Informal teacher observations

? Modeling Lessons (SMP 4)

? Checking for understanding using active participation strategies

? Formative Assessment Lessons (FAL)

? Exit slips/summaries

? Quizzes / Chapter Tests

? Tasks

? SBAC Interim Assessment Blocks

Math 8

Access Using Formative Assessment for Differentiation for suggestions. Located on the LBUSD website ? "M" Mathematics ? Curriculum Documents

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Unit 3 Linear Equations and Functions

Days

Learning Target

I will explore linear equations and functions by participating in the Opening Task.

1 day

Learning Plan ? Stage 3

Suggested Sequence of Key Learning Events and Instruction

Expectations

Big Ideas Math Course 3 (Activities and Lessons)

OPENING TASK ? How Do I See The Pattern

Growing?

This Opening Task asks students to think about how

they see a pattern growing. Give students time to work

independently and then have them work with a partner.

Next, facilitate a class discussion utilizing Talk Moves

about how they see the pattern growing. This task is a

gateway into the entire unit on linear equations.

Math 8

Supplemental Resources Conceptual Understanding: ? How Do I See The Pattern

Growing?

I will investigate linear equations by...

3-4 days

? Identifying solutions. ? Using a table of values to graph the relationship. ? Exploring the equations and graphs of vertical and

horizontal lines. ? Using technology, such as graphing calculators or

Desmos, to explore characteristics of the graphs. (SMP 5) ? Answering questions such as... o How can you recognize a linear equation? o What does the graph of a linear equation look

like? o Explain why any solution point of a linear

equation will lie on the graph of its line. (SMP 3) o Describe the graphs of x = a and y = b. o Synergy Item Bank: Item ID 20087, 20089, 71224

? Section 4.1 (Activity 1; Examples 1, 2, and 3)

? STEM Video: Hurricanes!

Procedural Skills and Fluency: ? Graphing Linear Equations

Group Activity

Application: ? STEM Video Performance

Task: Anatomy of a Hurricane

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Unit 3 Linear Equations and Functions

Days

Learning Target

I will explore the slope of a linear equation by...

4-5 days

Learning Plan ? Stage 3

Suggested Sequence of Key Learning Events and Instruction

Expectations

Big Ideas Math Course 3 (Activities and Lessons)

? Describing the different types of slopes (positive,

? Section 4.2

negative, zero, and undefined).

(Activities 1, 2, and 3;

? Using slope triangles to find the slope from a graph.

Examples 1, 2, and 3)

? Explaining that the slope is the same between any

two distinct points on a non-vertical line in the

coordinate plane using similar triangles. (SMP 3)

? Using the slope formula to find the slope from two

points on the line.

? Answering questions such as...

o Explain why you can use any two points on the

line to find slope. (SMP 3)

o How can you find the slope of a linear equation

from a table? From a graph?

o Synergy Item Bank: Item ID 56648, 51255

I will represent proportional relationships by ...

4-5 days

? Writing linear equations in the form y = mx. ? Interpreting the unit rate as equal to the slope of the

graph and the constant of proportionality. ? Using tables, graphs, equations, and verbal

descriptions to represent real-world proportional relationships. (SMP 2) ? Using tables, graphs, equations, and verbal descriptions to compare two different proportional relationships. ? Answering questions such as... o How can you use the unit rate to write an

equation to represent a proportional relationship? o How is slope related to the unit rate and constant

of proportionality in proportional relationships? o Synergy Item Bank: Item ID 52081, 65748

? Section 4.3 (Activities 1, 2, and 3; Examples 1 and 3)

Math 8

Supplemental Resources

Conceptual Understanding: ? Desmos: Put the Point on

the Line ? Point, Point, Slope Task ? Rise-Run Triangles Task ? Similar Triangles and

Slope Interactive Activity ? Slope From Two Points

Task ? Understanding Slope with

Similar Triangles Video

Procedural Skills and Fluency: ? Finding Slope Solo-Team-

Teach ? Slope PowerPoint

Application: ? Modeling: Baby Beats

(SMP 4) Application: ? Desmos: Sugar, Sugar ? On Your Mark Activity ? Peaches and Plums Task ? Shelves Task ? Sore Throats Task ? Squares and Circles Task ? Stuffing Envelopes Task

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Unit 3 Linear Equations and Functions

Days

Learning Target

I will represent linear non-proportional relationships by ...

Learning Plan ? Stage 3

Suggested Sequence of Key Learning Events and Instruction

Expectations

Big Ideas Math Course 3 (Activities and Lessons)

? Using patterns to build conceptual understanding of ? Section 4.4

linear non-proportional relationships represented in

(Activity 1;

tables, graphs, equations, and verbal descriptions.

Examples 1, 2, and 3)

(SMP 2)

? Section 4.6

? Graphing linear equations in slope-intercept form

(Activities 1 and 3;

using the slope and y-intercept.

Examples 1, 2, and 3)

? Determining the slope and y-intercept from

analyzing an equation or graph.

? Writing an equation in slope-intercept form from a

graph, a verbal description, or a table. (SMP 2)

? Writing linear equations in the form y = mx + b and

4-5 days

explaining that when b 0 , the relationship between x and y is non-proportional. (SMP 3)

? Answering questions such as... o How can you determine the slope and the y-intercept of a linear equation from a graph? o How can you graph a line using the slope and y-intercept? o How do you write a linear equation in slopeintercept form to model a linear relationship given a graph, given a table, or a description? o Explain how you can identify a linear non-proportional relationship from a table, a graph, and an equation. (SMP 3) o Synergy Item Bank: Item ID 65933, 65934

Math 8

Supplemental Resources

Conceptual Understanding: ? myPD Course #3054:

Building Conceptual Understanding of Functions Using Desmos ? Linear Equation From a Table of Values Task ? Writing Linear Functions Task ? Linear Equations--Which One Doesn't Belong? ? Graphs of Linear Equations--Which One Doesn't Belong?

Procedural Skills and Fluency: ? Slope-Intercept Form

Frame ? Desmos Polygraph: Lines ? Desmos Polygraph: Lines,

Part 2

Application: ? Desmos: Picture Perfect ? Gym Membership Plans

Task ? Pig Pen Task ? School Lunch Task ? Modeling: Styrofoam Cups

(SMP 4)

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Unit 3 Linear Equations and Functions

Days

2-3 days

Learning Target

I will check my understanding of linear equations by participating in the FAL. I will explore functions by...

3-4 days

Learning Plan ? Stage 3

Suggested Sequence of Key Learning Events and Instruction

Expectations

Big Ideas Math Course 3 (Activities and Lessons)

OPTION #1: FORMATIVE ASSESSMENT LESSON

? Defining Lines by Points, Slopes, and Equations

(SMP 1, 2, 3, 5, 6, 7, 8)

? Producing input and output pairs. ? Explaining that a function is a rule that assigns to

each input exactly one output. (SMP 3) ? Identifying relationships that are functions from

mapping diagrams, tables, equations, and graphs. (SMP 2) ? Answering questions such as... o How can you tell if a relationship is a function? o Synergy Item Bank: Item ID 50426, 50652,

53912, 53943

? Section 6.1 (Activities 1 and 2; Examples 1, 2, and 3)

? Section 6.2 (Activities 2 and 3; Examples 1, 2, 3, and 4)

I will create and compare functions by ...

? Explaining that because non-vertical linear equations have exactly one output of y for each value of x, the relationship is a linear function. (SMP 3)

? Constructing a function to model a linear relationship between two quantities.

4-5 days

? Interpreting the rate of change and the initial value of a linear function in the terms of the situation it models. (SMP 2)

? Comparing properties of functions represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions). (SMP 2)

? Identifying whether a function is linear or nonlinear. ? Giving examples of nonlinear functions. ? Answering questions such as...

o How are linear equations and linear functions related?

o Is every linear equation also a linear function? o How can you use a table, equation, or a graph to

determine if a function is linear or nonlinear? o Synergy Item Bank: Item ID 50270, 51153,

51257, 52181, 53911, 53936, 53938

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? Section 6.3 (Activity 1; Examples 1, 2, 3, and 4)

? STEM Video: Apparent Temperature

? Section 6.4 (Activity 2; Examples 1, 2, 3, and 4)

Math 8

Supplemental Resources

Conceptual Understanding: ? Building Conceptual

Development of Functions ? The Certainty in Functions

Procedural Skills and Fluency: ? Identifying Functions

Connect 4 ? Is It a Function?

Procedural Skills and Fluency: ? Desmos: Match My Line ? Desmos Card Sort: Linear

Functions ? Desmos: Land the Plane ? Functions Frame ? Identifying Linear and

Nonlinear Functions Represented by Equations ? Identifying Linear and Nonlinear Functions Represented by Tables

Application: ? Cross Country Task ? Desmos: Lego Prices ? Patrick's Patterns Task ? STEM Video Performance

Task: Heat Index ? Modeling: In-N-Out Burger

(SMP 4) Posted 9/11/17

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