Grade 8 + DIGITAL. EL Strategies. DOK 1-4 RTI Tiers 1-3. Flexible Supplemental K-8 ELA & Math Online & Print

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1 Standards PLUS Flexible Supplemental K-8 ELA & Math Online & Print Grade 8 SAMPLER Mathematics EL Strategies DOK 1-4 RTI Tiers Minute Lessons Assessments Consistent with CA Testing Technology Standards PLUS Targeted Intervention Ready to Teach RTI Tier Materials PRINT + DIGITAL Writing Program EL Strategies Performance Lessons Integrated Projects Written directly to the CA Standards by CA Educators

2 Close the Achievement Gap EL STRATEGIES All Standards Plus lessons explicitly teach communication skills, strategies, and conventions that meet the goal of EL Instruction. Standards PLUS Includes: Standards PLUS is so much more READY TO TEACH RTI / TIER Standards Plus Lessons provide: Whole Class Instruction Targeted Intervention Intense Intervention Standards PLUS is Seven Programs in One: MINUTE LESSONS DOK 1-2 / RTI Tiers 1-2 Research-based, Direct Instruction, K-8, ELA and Math lessons. Written to the state standards. PERFORMANCE LESSONS DOK 3 Students deepen and apply their knowledge into new applications. ASSESSMENTS DOK 1-2 Weekly formative assessments monitor student progress. Online assessments help students master digital item types. INTEGRATED PROJECTS DOK 4 Students apply knowledge to real-world situations. STANDARDS PLUS DIGITAL DOK 1-3 / RTI Tiers 1-3 Lessons and assessments match the the digital format of the state test. Students transfer their knowledge into a digital learning environment. TARGETED INTERVENTION LESSONS DOK 1-2 / RTI Tiers 2-3 Scaffolded lessons assigned based on assessment results. Digital program automates this process. WRITING PROGRAM (ELA Only) DOK 1-4 / RTI Tiers 1-2 Includes lessons on every writing genre. Writing performance lessons include skills trace, prompts, and rubrics. HOMEWORK/ PARENT CONNECTION (COMING SOON)

3 Sample Lessons Included in this Booklet Domain Lesson Focus Standard(s) Functions (Functions Standards: 8.F.1 5) 17 Comparing the Properties of Two Functions 8.F E5 Construct/Interpret a Function to Model a Linear Relationship Construct/Interpret a Function to Model a Linear Relationship Construct/Interpret a Function to Solve Problems Evaluation Constructing and Interpreting Functions 21 Sketch a Function Graph 22 Describe Functional Relationships 23 Describe Functional Relationships 24 Describe Functional Relationships E6 Evaluation Use Functions to Model Relationships 8.F.4 8.F.2, 8.F.4 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. 8.F.5 P7 Performance Lesson #7 Functional Relationships (8.F.2, 8.F.4, 8.F.5) See the lesson index for the entire program on pages

4 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Comparing the Properties of Two Functions Lesson: #17 Standard: 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Lesson Objective: Students will compare the properties of two functions. Introduction: Today we will continue to compare the properties of two functions in different forms. We have been comparing functions but we did not cover verbal descriptions. Today we will incorporate verbal descriptions in the comparisons. Instruction: We will read the verbal description and translate the verbal description to an equation. If the equation does not directly transfer to slope-intercept form, then we will re-write the equation into slope-intercept form. We already know how to write equations from graphs and tables and how to re-write standard form equations to slope-intercept form equations. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: Let s complete the example together. A number y is three more than half a number x. What is the translation to an equation? Yes, it s 1 y x 3. To be able to compare it 2 to the standard form equation, let s re-write function 2 to slope-intercept form. We first subtract 3x then multiply by 1. We end up with 3 y x 2. Now we are ready to write a comparison. 5 5 Function 2 has a greater rate of change since 3 1. Function 1 has a greater 5 2 y-intercept/initial value/vertical shift. Independent Practice: Follow the same process we did with the example problem to complete the practice problems. Review: When the students are finished, go over the problems. If the students do not have enough time to finish, then they can write down the answers as you go over the problems. Closure: Today you compared the properties of functions represented in two forms. One of the forms was a verbal description and the other form was written graphically, algebraically as an equation, or numerically in a table form. Answers: 1. Verbal description equation: y 3x 2; Graph equation: 3 y x 3 2 Comparison: The verbal description function has a greater rate of change, since 3, and a greater y-intercept/initial value. The graph has a greater 3 2 vertical shift. 2. Verbal description equation: 1 y x 3 ; Table equation: 1 y x Comparison: The table equation has a greater rate of change since, 1 1 and a greater y-intercept/initial value., 2 3 The verbal description has a greater vertical shift from the origin. 3. Verbal description equation: y 3x 5; Equation in slope-intercept form: y 2x 5. Comparison: The verbal description has a greater rate of change since 3 > 2. Both functions have the same y-intercept/initial value/vertical shift

5 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Comparing the Properties of Two Functions Lesson: #17 Standard: 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: Compare the properties of the two functions represented in different ways. Function 1 Function 2 Comparison A number y is three more than half a number x. 3x 5y 10 Directions: Compare the properties of the two functions represented in different ways. Function 1 Function 2 Comparison A number y is two less than three times a number x. 1. Eighteen is six times a number y less than two times a number x. x y Sample Daily Lesson - Student Response Page Toni releases a helium balloon from her hand that is 5 feet above the ground. The balloon ascends at a rate of 3 feet/second. Let y represent the height of the balloon from the ground in feet and x represent the time in seconds. 2x y

6 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Model a Linear Relationship Lesson: #18 Standard: 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. Lesson Objective: Students will construct and interpret a function to model a linear relationship as well adjust the function given additional information. Introduction: Today you will write a function that models a linear relationship given to you in words and in a table. You will also adjust the function when given new information. Sample Daily Lesson- Teacher Lesson Plan Instruction: We have had practice translating verbal descriptions to linear equations. Today we will be given a situation with two variables. You will need to determine the two variables, how to represent them algebraically, and which variable is the dependent variable, and which is the independent variable. You will then need to determine the rate of change and the initial value. Finally, you will write an equation in slope-intercept form. Once you have determined the equation, you will answer questions related to the function. Guided Practice: Let s complete the example together. An empty egg carton weighs 0.6 ounce. Each egg placed in the carton weighs on average 2.3 ounces. Represent the total weight of the carton, w, with x eggs. Our two variables are defined for us already. Which variable is the dependent variable? Yes, the weight of the carton. So the independent variable is the number of eggs, x. What is the initial value? Yes, it s the weight of the empty carton, which is 0.6 ounces. What is the rate of change? Yes, it s the ounces per egg, which is 2.3 ounces per egg. Since we identified all the parts we need to write an equation in slopeintercept form, we write w 2.3x 0.6. Independent Practice: It s your turn to apply the same process to the practice problems. Review: When the students are finished, go over the problems. Closure: Today you wrote a function that modeled a linear relationship given to you in words and in a table. You also adjusted your function when given new information. Answers: 1. Rate of change: 29 feet/hour; Initial value: -200 feet 2. y 29x x 200; x 6.9hours 4. The initial value is -250 feet and the rate of change is doubled to 58 feet per hour; y 58x Completed table. Time (hours) Depth (feet)

7 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Model a Linear Relationship Lesson: #18 Standard: 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. Example: An empty egg carton weighs 0.6 ounce. Each egg placed in the carton weighs on average 2.3 ounces. Represent the total weight of the carton, w, with x eggs. Directions: Complete the following problems using the information and table below. A submarine located 200 feet below sea level ascends at the rate shown below in the table. Submarine Depth Time (hours) Depth (feet) Identify the rate of change and the initial value. Include units. 2. Write a function to model the relationship between time and depth of the submarine. 3. How many hours will it take the submarine to reach the surface of the water? 4. If the submarine is located 250 feet below the surface of the water and its rate of speed is doubled, how will this change the equation? Write the new equation. 5. Using the new equation, complete a new table of data. Sample Daily Lesson - Student Response Page Submarine Depth with New Information Time (hours) Depth (feet)

8 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Model a Linear Relationship Lesson: #19 Standard: 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. Lesson Objective: Students will construct and interpret functions to model linear relationships and compare the properties of the functions. Introduction: Today you will continue to construct and interpret functions to model linear relationships. You will also compare the properties of the functions you construct. Instruction: You will be given a scenario and a graph. From those different representations, you will determine the variables, initial value, and rate of change. You will then write equations in slope-intercept form to model the relationships. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: Let s complete the example problem together. The cost of gym membership includes the sign-up fee of $75 and a monthly fee of $30. The gym will lower its sign-up fee to $25 after the holidays to attract more members. We need to write a function to model this relationship. First we need to define the variables. The dependent variable is the total cost of gym membership. We ll use y. The independent variable is the number of months. We ll use x. What is the initial value for the gym membership before the holidays? Yes, $75. What is the rate of change? Yes, $30 per month. We can write the function in slope-intercept form as y 30x 75. After the holidays the sign-up fee is reduced to $25. How does this change our function equation? Yes, we change the constant only. Our new function is y 30x 25. How does this change affect the graph of the original equation? Only the constant is changed. The constant is the initial value. The graph is shifted down from our original equation. Independent Practice: It s your turn to apply the same process to the practice problems. You must use the information in both the scenario and the graph to be able to answer the problems. Review: When the students are finished, go over the problems. Closure: Today you continued to construct and interpret functions to model linear relationships. You also compared the properties of the functions you constructed. Answers: 1. Plan A - Initial value: $ 84; Rate of change: - 12/mo y 12x Plan B - Initial value: $84; Rate of change: - $21/mo. 4. y 21x Plan B will pay back Teresa faster. The slope of the line representing Plan B is steeper than plan A. The steeper the line, the faster the balance due is decreasing

9 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Model a Linear Relationship Lesson: #19 Standard: 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. Example: The cost of gym membership includes the sign-up fee of $75 and a monthly fee of $30. The gym will lower its sign-up fee to $25 after the holidays to attract more members. Write a function to model each relationship. How does this change affect the graph of the original equation? Directions: Stacey owes Teresa money. Stacey sets up two different payment plans for paying Teresa back. Plan A is shown in the graph below. For Plan B, Stacey would pay Teresa $21 per month. Payment Plans Sample Daily Lesson - Student Response Page 1. What is the initial value and rate of change for Plan A? 2. Write the function that represents Plan A. 3. What is the initial value and rate of change for Plan B? 4. Write the function that represents Plan B. 5. Graph Plan B on the graph above and label. 6. Which plan would pay back the amount Stacey owes Teresa faster? How is that reflected in the slopes of the graphs?

10 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Solve Problems Lesson: #20 Standard: 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. Lesson Objective: Students will construct and interpret functions to model linear relationships and solve problems. Introduction: Today you will continue to construct and interpret functions to model linear relationships. You will use your model to graph and to solve problems. Instruction: You will be given a scenario and a model. From the information, you will determine the variables, initial value, and rate of change. You will then write equations in slope-intercept form to represent the relationships and solve problems. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: Let s complete the example together. We need to write a function to model the relationship between the number of ounces in the glass at any given time if it is topped off at a rate of 0.5 ounce per second. By looking at the picture, we see the capacity of the glass and the amount of water already in the glass. We are given the rate of change and the initial value. The rate of change is 0.5 ounce per second. The initial value is 7 ounces. We can write the function from these two pieces of information. What is the equation? Yes, it s y 0.5x 7, where y is the amount of ounces in the glass and x is time in seconds. At what point will the glass be full? Since our glass is 10 ounces, we set y equal to 10 and solve for x. We get x = 6 seconds. Our point is (6, 10). Independent Practice: It s your turn to apply the same process to the practice problems. You are given two other glasses and information that you will write functions for. You will graph the three functions on the same graph. You will use the functions and graph to answer questions. Review: When the students are finished, go over the problems. Closure: Today you continued to construct and interpret functions to model linear relationships. You used your model to graph and solve problems. Answers: 1. y x (Graph on right.) 2. y 2x Answers will vary, the justification is most important. If the student predicts that the 24-oz glass will be filled first, he or she could say because it is being filled at a faster rate. If the student says that the 10-oz glass will be filled first, it could be because it s the smallest size glass. The student could have solved all the equations to find the total seconds for each glass to be topped off. 5. The 24-ounce glass filled first. Student should respond why they expected this or not and why. The justification/ reasoning is very important for this activity. 6. The 24-ounce glass filled at a faster rate. By looking at the graph, the line is steeper. By looking at the equations, the coefficient of x is the greatest for the 24-ounce glass. 7. No. The glass filling at a faster rate will not always fill first. It will depend on both the initial amount of liquid in the glass, and the size of the glass

11 Student Page 1 of 2 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Solve Problems Lesson: #20 Standard: 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. Example: Write a function to model the relationship between the number of ounces in the glass at any given time if it is topped off at a rate of 0.5 ounce per second. Function: Point at which the glass is full: (, ) Directions: Complete the problems below. 1. Write a function to model the relationship between the number of ounces in the glass at any given time if it is topped off at a rate of 1 ounce per second. Sample Daily Lesson - Student Response Page 2. Write a function to model the relationship between the number of ounces in the glass at any given time if it is topped off at a rate of 2 ounces per second

12 Student Page 2 of 2 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Construct/Interpret a Function to Solve Problems Lesson: #20 Standard: 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. 3. Of the three glasses, the 10-ounce, 16-ounce, and 24-ounce, predict which glass will be topped off first. Justify your answer. 4. Graph the relationship between the number of ounces in each glass at any given time. You only need to draw the portion of the graph that makes sense. Mark and label the point on the graph at which each glass is full. Ounces in Glasses Sample Daily Lesson- Student Response Page 5. Which glass filled first? Are you surprised? Why or why not? 6. Which glass filled at a faster rate? How do you know by looking at the graph? At the equations? 7. Will a glass filling at a faster rate always fill first? Why or why not?

13 Sample lessons continue on the next page

14 Domain: Functions Common Core Standards Plus Mathematics Grade 8 Focus: Constructing and Interpreting Functions Evaluation: #5 Sample Daily Lesson- Teacher Lesson Plan The weekly evaluation may be used in the following ways: As a formative assessment of the students progress. As an additional opportunity to reinforce the vocabulary, concepts, and knowledge presented during the week of instruction. Standard: 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.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. Procedure: Read the directions aloud and ensure that students understand how to respond to each item. If you are using the weekly evaluation as a formative assessment, have the students complete the evaluation independently. If you are using it to reinforce the week s instruction, determine the items that will be completed as guided practice, and those that will be completed as independent practice. Review: Review the correct answers with students as soon as they are finished. Answers: 1. (8.F.2) y 3x 5 y = 4x + 10 Time (in seconds) Number of Ounces Time (in seconds) Number of Ounces (8.F.4) The 26-ounce glass is filling at a faster rate. It is filling at 4 ounces per second and the 14-ounce glass is filling at 3 ounces per second. 3. (8.F.4) The 14-ounce glass fills first. By looking at the table, it takes 3 second for the 14-ounce glass to fill and 4 seconds for the 26-ounce glass to fill. 4. (8.F.4) Various answers possible. One example is a 20-ounce glass with an initial amount of 4 ounces filling at a rate of 2 ounces per second represented by the equation y 2x 4. This glass would take 8 seconds to be topped off. The picture should be labeled accordingly

15 Domain: Functions Common Core Standards Plus Mathematics Grade 8 Directions: Complete the following problems independently. Focus: Constructing and Interpreting Functions Evaluation: #5 1. Shade and label each glass as described by the functions. Then complete the tables. y 3x 5 y 4x 10 Time (in seconds) Number of Ounces 2. Which glass is filling at a faster rate? How do you know? Time (in seconds) Number of Ounces Sample Daily Lesson - Student Response Page 3. Which glass fills first? Explain. 4. Label the glass below that will be topped off in 8 seconds. Write a function that represents your glass being topped off

16 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Sketch a Function Graph Lesson: #21 Standard: 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. Lesson Objective: Students will sketch a function graph given a scenario and describe qualitatively the relationship between quantities of a given graph. Introduction: This week we will conclude our explorations of functions. We will sketch graphs from a verbal description or from a model. We will also describe the relationship between two quantities given a graph. Instruction: To sketch a graph from a scenario, we must identify the variables. We must understand if the relationship between the variables is increasing or decreasing and determine if the relationship is linear or non-linear or a combination of the two. If it is linear, we need to see if there is a change in the rate for the same relationship between the variables. If we are sketching more than one relationship on a graph, we need to represent the different relationships relative to each other. To write a scenario from a sketch, we need to choose the variables and label the variables on the axes. Then we can write the scenario to match the line or curve of the graph. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: We have three containers of different sizes and shapes shown. Water is poured into each container at the same constant rate. The graph below shows how the height of water varies as the volume of the water increases for container A. On the same graph sketch the height-volume relationship for containers B and C. We need to label each line. Container B is half the height of container A and container B is also narrow, the height of the water will increase at a greater rate and the total volume will be less when filled. Since the sides of the container are vertical, we also have a line. We draw the line on the graph accordingly. You need to write this justification next to the graph in the space provided. Let s sketch the height-volume relationship for container C. The bottom of container C is wider than container A so the rate of the water s height increase will be less at first. Once the height of the water reaches the upper portion of the container, the height will increase at the same rate as it did for container B. This change is captured in the sketch by changing the slope of the line. The heights of containers A and C are equal but the volume of container C is greater. So we stop our line at the same height but reflect a greater volume by sketching the line further to the right. Select students to explain why the graph looks the way it does. Independent Practice: In these practice problems, you have to think in the other direction. You are given the graph, and you need to write a scenario. It is very important that you define your variables. Label each axis. There is no one correct answer. There are infinitely many correct answers, but there are incorrect answers. Your scenario must match your graph and use the variables you chose. Review: Have the students exchange their scenarios and have other students read, and if needed, correct each other s scenarios. When the students are finished, go over the problems. Select students to share their scenarios. Closure: Today we sketched graphs from a verbal description and model. We also described the relationship between two quantities given a graph. We had to determine and label the variables on the graph. Answers: Answers will vary for all problems. 1. Mike is one mile from home. He rides his bike further from home at a constant speed. The y- axis of graph should be labeled as distance and x-axis should be labeled time. 2. The height of a baby increases slowly at first. The baby has a growth spurt, and its height increases rapidly. The baby s increase in height slows down again. The y-axis should be labeled height and the x-axis should be labeled time. 3. Yolanda increases her speed while riding her bike. Once she reaches a certain speed, she travels at a constant speed. She then peddles up to a hill, and her speed decreases until she is barely moving. She reaches the top of the hill where the road is flat, and she travels at a much slower constant speed since she is tired from climbing the hill. Once she reaches her destination, she stops. The y-axis is labeled speed and x-axis is labeled as time

17 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Sketch a Function Graph Lesson: #21 Standard: 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. Example: Below are three containers. Water is poured into each container at the same constant rate. The graph below shows how the height of water varies as the volume of the water increases for container A. On the same graph sketch the height-volume relationship for containers B and C. Label each. Explain your graphs for containers B and C. Directions: Create scenarios for each of the graphs below. Label the meaning of the axes for each scenario. Sample Daily Lesson - Student Response Page

18 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #22 Standard: 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. Lesson Objective: Students will analyze graphs and interpret their meaning by answering questions. Introduction: Today we will continue to analyze function graphs and interpret their meanings to answer questions and justify our answers. Instruction: To be able to interpret function graphs, we must pay attention to the variables as defined by the axes labels, the characteristics of the function graph such as increasing or decreasing, linear or non-linear, and the behavior of the function in relation to another function that is sketched on the same graph. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: Let s complete the example together. The graph below shows how the speed of a racing car varies during the second lap of a raceway. What do you notice about the speed as the car travels around the second lap? The car slows down three times as it travels around the lap of the raceway. Does it slow down the same amount each time? No, it slows down the most during the second time and it slows down the least during the third time. When would a race car slow down on a raceway? As it s going around a curve or corner. How many curves do you think this raceway has? (3) How do you know? The graph shows the car slowing down three times. Now think about how the sharpness of a curve must look relative to the other curves by looking at the speed graph. You have five potential raceway circuits that the race car could be traveling. Point s represents where the race car would be at the start of the second lap. Which of these circuits was the racing car going around? Carefully examine each circuit. You should be able to justify your selection. Which circuit did you choose and why? You should have selected circuit B. Circuit B has three curves. But so does circuit C. Why is B a better choice than C? Notice that the second curve is the sharpest and would require the greatest reduction in speed and the third curve is the least sharp as compared to the other curves, and would require the least amount of speed reduction. This matches the speed-distance graph. Independent Practice: It s your turn to apply the same process as we did together in the example. You will analyze a graph and answer questions and justify your thinking. Review: When the students are finished, go over the problems. Closure: Today we analyzed function graphs and interpreted their meaning to answer questions and justify answers. Answers: 1. Car 2 travels a greater distance. Both cars start and stop at the same time. Car 2 travels faster than car 1 the entire time so therefore it travels farther. 2. Car 1 is traveling at a constant velocity between points E and F. 3. Car 1 accelerates two times. The first time is between points A and B. The second time is between points C and D. We know the car is accelerating because the velocity is increasing between these two sets of points. 4. Car 2 is accelerating between points A and I. It is accelerating greater between points A and H than between points H and I because the slope of the line segment between is A and H is steeper than the line segment between H and I

19 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #22 Standard: 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. Example: The graph below shows how the speed of a racing car varies during the second lap of a race. Point s shows the location of the racing car at the beginning of the car s second lap. Which of these circuits was the racing car going around? Explain. Source: Janvier, Girardon, and Morand 1993 Directions: Two cars begin and end their trip at the same time. Use the velocity vs. time graph to answer the questions. Sample Daily Lesson - Student Response Page 1. Which car travels a greater distance? How do you know? 2. Explain what is happening between points E and F for car How many times does car 1 accelerate? Explain. 4. Explain what is happening between points A and I for car

20 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #23 Standard: 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. Sample Daily Lesson- Teacher Lesson Plan Lesson Objective: Students will analyze graphs and answer questions about the scenario the function represents. Introduction: Today we will continue to analyze function graphs and interpret their meaning to answer questions and justify our answers. Instruction: To be able to interpret function graphs, we must pay attention to the variables as defined by the axes labels, the characteristics of the function graph such as increasing or decreasing, linear or non-linear, and the behavior of the function in relation to another function that is sketched on the same graph. Today we will have only one function sketched on the graph. Since our functions are graphed on axes with a scale, we must pay close attention to the scale and read the numbers correctly. Guided Practice: A farmer is building a rectangular animal pen. She purchases 50 feet of fencing, and she wants to use all 50 feet of fence. The graph on your page represents the relationship between the possible widths and corresponding areas of the pen assuming the farmer uses all 50 feet of fencing. The labeled point represents the maximum point on the graph. The first question asks what are the possible widths of the pen if it is 100 square feet? The possible widths for a 100-square foot area pen are 5 feet and 20 feet. We can determine this by drawing a horizontal line through the graph at 100 square feet and seeing where the horizontal line, represented by the equation y = 100, intersects with the curve. It intersects in two locations. It intersects at x = 5 and at x = 20. At what range of widths does the area increase? The area increases until it reaches the maximum value on the graph. The point marked and labeled on the graph represents the maximum area and at what width that occurs at. Therefore, as the widths increase from 0 to 12.5 feet, the area also increases. What is the maximum area of the pen possible? The maximum area is square feet. What are the dimensions of the pen with the maximum area? We know the width. It s12.5 feet. We have to calculate the length. Since the pen is rectangular, we divide the area by the width to find the length divided by 12.5 is This tells us the rectangle which the largest area is a square. We can use the graph to find other possible widths by drawing a horizontal line through any other area and finding the two intersections points with the curve. The intersection points will represent the widths. Once we identify the widths, we can calculate the length. We can only approximate the width since there are no other points that fall on the graph that are integer points. I will choose the area to be 140 square feet. If I draw a horizontal line through 140 square feet, the two points it intersects through are approximately x = 8.5 and x = So if the width is 8.5 feet then by calculation, the length is about 16.5 feet since 140 square feet divided by 8.5 is about If the width is 16.5 feet, then the length is 8.5 feet. Independent Practice: It is your turn to complete the practice problem. Notice the labels and scale on the axes. You will need to interpret these correctly to accurately answer the questions. Review: When the students are finished, go over the problems. Closure: Today we analyzed function graphs and interpreted their meaning. Answers: meters meters 3. 7 meters = is the diam. Half the diam. is the radius, so half of 14 is seconds. You can see the time it takes the Ferris wheel to go from its max. height to the next time it reaches its max. height (or any combination of time to complete 1 revolution). 5. Two times. 32 seconds has passed between points A and B. Since it takes 16 seconds for one complete revolution, it will have completed 2 revolutions in 32 seconds

21 Student Page 1 of 2 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #23 Standard: 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. Example: A farmer is building a rectangular animal pen. She purchases 50 feet of fencing to construct the pen. She wants to use all 50 feet of fence. The graph below represents the relationship between the possible widths and corresponding areas of the pen assuming the farmer uses all 50 feet of fencing. The labeled point represents the maximum point on the graph. 1. What are the possible widths of the pen if it is 100 square feet? Explain. Sample Daily Lesson - Student Response Page 2. Over what range of widths does the area of the pen increase? 3. What is the maximum area of the pen possible? What are the dimensions of the pen with the maximum area? 4. How can you use the graph to find other possible widths of the pen? Determine one other set of possible dimensions for the animal pen

22 Student Page 2 of 2 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #23 Standard: 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. Directions: Complete the following problems using the information and graph below. The graph below shows the varying height in meters of a Ferris wheel over time once the ride has begun. Sample Daily Lesson- Student Response Page 1. At what height does someone get on the Ferris wheel? 2. What is the maximum height the Ferris wheel? 3. What is the radius of the Ferris wheel? Explain. 4. How long does it take for the Ferris wheel to go all the way around one time? How do you know? 5. How many times has the Ferris wheel gone around from point A to point B? Explain

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24 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #24 Standard: 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. Lesson Objective: Students will analyze graphs and answer questions about the scenario the function represents. Introduction: Today we will continue to analyze function graphs and interpret their meaning to answer questions and justify our answers. Instruction: We have been interpreting function graphs by noting several things: the variables defined by the axes labels; the characteristics of the function graph as increasing, decreasing, linear, or non-linear; and the behavior of the function in relation to another function that is sketched on the same graph. Today only one function will be sketched on the graph. Since our functions are graphed on axes with a scale, we will note the scale and read the numbers correctly. Sample Daily Lesson- Teacher Lesson Plan Guided Practice: Let s complete the example together. You have a graph that represents the relationship between the height of a ball in feet and time in seconds. The x-scale is by ones and the y-scale is by tens. The first question asks what time does the ball reach its maximum height? Where do we look on the graph to find the answer? Yes, we look at the maximum height which is the maximum point on the graph. The maximum height occurs at 3 seconds. What is the speed of the ball at the maximum height? Be careful. We are not given speed on the graph. Think about this. Between zero and three seconds, the height of the ball is increasing. Between three and six seconds, the height of the ball is decreasing. What is the speed at the moment the ball reaches it maximum height? Yes, it s zero. The ball comes to a stop before it comes back down. We know this because it changes from the increasing height to decreasing height. What is the height of the ball at 4 seconds? It is close to 128 feet. At 5 seconds? The ball is at 80 feet. At 6 seconds? It s at zero height. So between 4 and 5 seconds the ball travels 48 feet. Between 5 and 6 seconds, the ball travels 80 feet. So is the speed of the ball increasing or decreasing between 3 and 6 seconds? Yes, it s increasing. We know this because the change in height over the same change in time increases. Independent Practice: It s your turn to apply the same process and complete the practice problem. Be sure to interpret the scale correctly. Review: When the students are finished, go over the problems. Closure: Today we analyzed non-linear function graphs and interpreted their meaning to answer questions and justify our answers. Answers: feet. 2. Between 0.5 second and 1 second. About 0.75 second bounces. 4. About 9 seconds. At that time it appears the line is flat. 5. After each bounce the ball decreases in height and the time between bounces becomes shorter. 6. To determine the total vertical distance the ball travels, add up the maximum height of each bounce and multiply it by two, then add in the initial height. 7. The maximum height changes non-linearly. You would draw a curve, not a line, connecting the maximum heights of each bounce

25 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #24 Standard: 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. Example: Use the graph below to answer the questions. Below is a graph of the height of a ball with respect to time. Student Page 1 of 2 1. At what time does the ball reach its maximum height? Sample Daily Lesson - Student Response Page 2. What is the speed of the ball at 3 seconds? How do you know? 3. What is the height of the ball at 4 seconds? At 5 seconds? At 6 seconds? 4. Is the speed of the ball increasing or decreasing from 3 to 6 seconds? Explain

26 Student Page 2 of 2 Common Core Standards Plus Mathematics Grade 8 Domain: Functions Focus: Describe Functional Relationships Lesson: #24 Standard: 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. Directions: Complete the following problems using the information and graph provided. A basketball is dropped from a certain height and bounces. The graph below represents the relationship between the height of the bouncing ball and time. Sample Daily Lesson- Teacher Lesson Plan 1. At what height is the ball initially dropped from? 2. How long does it take for the ball to hit the ground for the first time? 3. How many times does it bounce before it comes to rest? 4. How long does it take for the ball to come to rest? Explain. 5. What do you observe happens to the ball after each bounce? 6. How could you determine the total vertical distance the ball travels? 7. Does the maximum height of the ball change linearly or non-linearly after each bounce? Explain

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28 Domain: Functions Common Core Standards Plus Mathematics Grade 8 Focus: Define, Evaluate, and Compare Functions and Use Functions to Model Relationships Evaluation: #6 Sample Daily Lesson- Teacher Lesson Plan The weekly evaluation may be used in the following ways: As a formative assessment of the students progress. As an additional opportunity to reinforce the vocabulary, concepts, and knowledge presented during the week of instruction. Standard: 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. Procedure: Read the directions aloud and ensure that students understand how to respond to each item. If you are using the weekly evaluation as a formative assessment, have the students complete the evaluation independently. If you are using it to reinforce the week s instruction, determine the items that will be completed as guided practice, and those that will be completed as independent practice. Review: Review the correct answers with students as soon as they are finished. Answers: 1. (8.F.5) Function 2 since (8.F.5) A. Initial value: 0; Rate of change: Function: y 25 3 B. Initial value: 1000; Rate of change: x gallons per minute gallons per minute 20 Function: y x C. For Pool A solve the equation x. Pool A fills in minutes. For Pool B solve the equation x Pool B fills in 780 minutes. Pool A fills in the least time. 3. (8.F.5)Completed graph

29 Domain: Functions Common Core Standards Plus Mathematics Grade 8 Focus: Define, Evaluate, and Compare Functions and Use Functions to Model Relationships Evaluation: #6 Directions: Complete the following problems independently. Circle the functions. 1. Which linear function has a greater rate of change? Function 1 Function 2: x y 5 x y Write a function to model each relationship. Two pools are being filled with water. A. Pool A is being filled at a constant rate of 125 gallons per 15 minutes. Initial value: Rate of change: Function: B. Pool B is being filled at a constant rate of 200 gallons per half an hour. Pool B already contains 1000 gallons of water. Initial value: Rate of change: Function: C. Pool A holds 5500 gallons. Pool B holds 6200 gallons. Which pool will be filled in the least time? Justify your answer. 3. The cross sections of Pool C and Pool D are shown below. Sample Daily Lesson - Student Response Page The pools begin empty and are being filled starting at the same time and at the same constant rate. Sketch a graph to show how the height of water in each pool varies with time. The height is defined from the deepest section of the pool. Label the axes and each graph

30 Teacher Lesson Plan Common Core Standards Plus Mathematics Grade 8 Performance Task #7 Domain: Functions Sample Performance Lesson - Teacher Lesson Plan Standard Reference: 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.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. Required Student Materials: Student Pages: WB Pg. 167 (Student Worksheet) Lined paper Lesson Objective: The students will analyze and describe a functional relationship. Overview: Students will use their knowledge of functional relationships as addressed in Common Core Standards Plus Functions Daily Lessons 17-24, E5-E6. Students will: Analyze a given graph to write a scenario that would result in the same graph. Analyze and describe the functional relationship displayed in the graph. Guided Practice: (Required Student Materials: WB Pg. 167) Review vocabulary. Review functional relationships. Independent Practice: (Required Student Materials: WB Pg. 167) Have the students: Analyze a given graph. Write a scenario that would result in the given graph. Analyze and describe the functional relationship displayed in the graph. Review & Evaluation: Have students share their scenarios in small groups. Share a few examples of the students scenarios with the class. Discuss how there could be different scenarios that result in the same graph

31 Student Page Common Core Standards Plus Mathematics Grade 8 Performance Task #7 Domain: Functions Vocabulary: Relation: A set of ordered pairs. Ordered pairs: Corresponding numbers in a table that are used to locate a point on a coordinate plane. Function: A subset or special type of relation; a function is a relation where the value of one variable depends on the value of the other variable. The y- coordinate value is dependent on the x- coordinate value. Any x- value can be substituted into the equation and yield exactly one y- value out of the equation. Relation curve: The graph of a function. Vertical line test: A relation is a function if no vertical lines intersect the graph at more than one point. Linear function: The graph of a linear function results in a straight line; all the points described by the function lie on the same line. Slope (Rate of Change): The measure of how steep a line is, or vertical change divided by horizontal change; slope may be positive, moving up from left to right; negative, moving downward from left to right; or zero, moving neither up nor down from left to right. Coefficient: The number that is multiplied by the variable or an algebraic expression in an algebraic term. Constant: A number without a variable; a value that does not change. Directions: Write a scenario that could be shown using the graph below. Describe the functional relationship between the two quantities Sample Performance Lesoon - Student Repsone Page

32 Common Core Standards Plus - Math Grade 8 Lesson Index Domain Lesson Focus Standard(s) The Number System (The Number System Standards: 8.NS.) Expressions and Equations (Expressions and Equations Standards: 8.EE.1 7, 8.EE.7a b, 8.EE.8, 8.EE.8a c) 1 Types of Numbers 8.NS.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Student Page 2 Decimal Expansion Converting Repeating Decimals to Fractions Converting Repeating Decimals to Fractions 8 E1 Evaluation Irrational Numbers 9 5 Approximating Square Roots 8.NS.2: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 6 Compare Real Numbers 13 7 Order Real Numbers 14 8 Locate Real Numbers on the Real Number Line 15 E2 Evaluation Real Numbers P1 Performance Lesson #1 Rational and Irrational Numbers (8.NS.1, 8.NS.2) Square Numbers and Roots 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irra onal. 2 Using Square Roots 19 3 Evaluate Cube Roots 20 4 Using Cube Roots 21 E1 Square and Cube Roots 22 5 Properties of Exponents 6 Properties of Exponents 8.EE.1: Know and apply the properties of 24 7 Properties of Exponents integer exponents to generate equivalent numerical expressions. For example, Properties of Exponents = 3 3 = 1/3 3 = 1/ E2 Evaluation Properties of Exponents 27 P2 Performance Lesson #2 Square Roots, Cube Roots, and Exponents (8.EE.1, 8.EE.2) Scientific Notation 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. 10 Scientific Notation Scientific Notation Scientific Notation 35 E3 Evaluation Scientific Notation Operations Using Scientific Notation 14 Operations Using Scientific Notation Operations Using Scientific Notation 8.EE Using Technology w/ Scientific Notation 40 E4 Evaluation Scientific Notation 41 P3 Performance Lesson #3 Using Scientific Notation (8.EE.3, 8.EE.4) DOK Level Learning Plus Associates

33 Common Core Standards Plus - Math Grade 8 Lesson Index Domain Lesson Focus Standard(s) Expressions and Equations (Expressions and Equations Standards: 8.EE.1 7, 8.EE.7a b, 8.EE.8, 8.EE.8a c) Graph Proportional Relationships & Determine Unit Rate Graph Proportional Relationships & Determine Unit Rate 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. Student Page Comparing Proportional Relationships Comparing Proportional Relationships E5 Evaluation Graphing and Comparing Proportional Relationships 21 Simple Triangles and Slope 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. 22 Simple Triangles and Slope Derive the Equation y = mx Derive the Equation y = mx 60 E6 Evaluation Proportional Relationships, Lines, and Linear Equations ,63 P4 Performance Lesson #4 What is Slope? (8.EE.5, 8.EE.6) Types of Solutions to a Linear Equation 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 26 Linear Equations Solving 1 Step and 2 Step Equations 8.EE.7a, 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions E7 Evaluation Finding Solutions to 1 and 2 Step using the distributive property and collecting like terms. Linear Equations 28 Solving 1 Step and 2 Step Equations Distributive Property 30 Simplifying Expressions Multi Step Linear Equations 8.EE.7b Multi Step Linear Equations 76 E8 Solving Multi Step Linear Equations Multi Step Linear Equations 79 8.EE.7a 34 Multi Step Linear Equations Multi Step Linear Equations 81 8.EE.7b 36 Multi Step Linear Equations 82 E9 Solve Multi Step Linear Equations 8.EE.7a, 8.EE.7b Systems of Equations 38 Systems of Equations 86 8.EE.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 39 System of Equations Systems of Equations E10 Evaluation Systems of Equations 91, DOK Level Learning Plus Associates 33

34 Common Core Standards Plus - Math Grade 8 Lesson Index Domain Lesson Focus Standard(s) Expressions and Equations (Expressions and Equations Standards: 8.EE.1 7a b, 8.EE.8a c) 41 Systems of Equations 8.EE.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and Systems of Equations 96 Student Page 43 Systems of Equations Systems of Equations E11 Evaluation Solving Systems of Equations Algebraically 45 Systems of Equations , EE.8c: Solve real world and mathematical 46 Systems of Equations 107 problems leading to two linear equations in two variables. For example, given coordinates for two 47 Solving Systems of Equations pairs of points, determine whether the line through the first pair of points intersects the line 48 Systems of Equations through the second pair. E12 Evaluation Systems of Equations P5 Performance Lesson #5 Systems of Equations (8.EE.7a, 8.EE.7b, 8.EE.8a, 8.EE.8b, 8.EE.8c) Integrated Project #1 It s Your Future (8.NS.1, 8.NS.2, 8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4, 8.EE.5, 8.EE.6, 8.EE.7, 8.EE.7a, 8.EE.7b, 8.EE.8, 8.EE.8a, 8.EE.8b, 8.EE.8c) Prerequisite Common Core Standards Plus Domains: The Number System and Expressions and Equations Product: The students will research two possible careers and show the earning potential from the first year of employment to the 30 th year. Overview: In this project, the students will research two career choices that they each find appealing. They will discover the education requirements for the careers and the expected education costs to prepare to enter the career. They will determine the expected starting salary and annual growth expectations for each of the two careers. They will calculate and graph the annual salary for thirty years for each of the two careers. Then they will estimate possible savings for retirement based on multiple contingencies and analyze the results. They will share their findings in peer groups and provide a written self reflection of the process and how it may impact their futures. Since this is a learning activity, all components will be completed in class. DOK Level Learning Plus Associates

35 Common Core Standards Plus - Math Grade 8 Lesson Index Domain Lesson Focus Standard(s) Functions (Functions Standards: 8.F.1 5) Student Page 1 Defining Functions Defining Functions Defining Functions Defining 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 E1 Evaluation Defining Functions 123, Identifying Linear and Non Linear Functions 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 Identifying Linear and Non Linear Functions Identifying Linear and Non Linear Functions Linear Parent Function 8.F.3, 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) E2 Evaluation Comparing Functions Linear Functions in y = k Form 8.F.2, 8.F Rate of Change 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. 11 Rate of Change Rate of Change E3 Evaluation Comparing Functions 8.F.3, 8.F Rewrite Linear Equation into Slope Intercept Form 14 Comparing Properties of Two Functions Comparing Properties of Two Functions Comparing Properties of Two Functions 148 E4 Evaluation Comparing Functions F.2 P6 Performance Lesson #6 Linear Functions and Relationships (8.F.1, 8.F.2, 8.F.3, 8.F.4) Comparing the Properties of Two Functions 8.F E5 Construct/Interpret a Function to Model a Linear Relationship Construct/Interpret a Function to Model a Linear Relationship Construct/Interpret a Function to Solve Problems Evaluation Constructing and Interpreting Functions 21 Sketch a Function Graph 8.F F.2, 8.F 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. 22 Describe Functional Relationships Describe Functional Relationships Describe Functional Relationships 8.F Evaluation Use Functions to Model E6 165 Relationships P7 Performance Lesson #7 Functional Relationships (8.F.2, 8.F.4, 8.F.5) DOK Level Learning Plus Associates 35

36 Common Core Standards Plus - Math Grade 8 Lesson Index Domain Lesson Focus Statistics and Probability (Statistics and Probability Standards: 8.SP.1 4) 1 Associations of Bivariate Data Standard(s) 8.SP.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.1, 8.SP.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Student Page Construct Scatter Plots Create and Interpret Scatter Plots Line of Best Fit E1 Evaluation Scatter Plots and Line of Best Fit Evaluate and Write Linear Models 6 Find and Use Linear Models to Solve Problems 8.SP.2, 8.SP.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 7 Evaluate Goodness of Fit 8.SP Find and Use a Linear Model to Solve Problems 181 E2 Evaluation Linear Models of Scatter Plots 8.SP.2, 8.SP P8 Performance Lesson #8 Scatter Plots (8.SP.1, 8.SP.2, 8.SP.3) E3 Construct Two Way Frequency Tables Construct Two Way Frequency Tables Construct Two Way Relative Frequency Tables Two Way Relative Frequency Tables Evaluation Scatter Plots and Two Way Tables 8.SP.4: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two way table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables P9 Performance Lesson #9 Two Way Tables (8.SP.4) Integrated Project #2 Home Improvement Plans (8.F.1, 8.F.2, 8.F.3, 8.F.4, 8.F.5, 8.SP.1, 8.SP.2, 8.SP.3, 8.SP.4) Prerequisite Standards Plus Domains: Functions and Statistics & Probability DOK Level Project Objective: The students will work in teams of three of four to make plans to paint a house. They will analyze time and cost to do the work using a variety of tools. Each group will present their findings to the class. Overview: In this project, the students will work in groups to create a plan to paint given interior walls of a house. They will analyze the challenge, the room dimensions, the cost of paint, the coverage of paint, and the time to paint using different tools. They will work as a team to write a report that includes accurate calculations, graphs of related functions, a sketch of the floor plan of the house based on the dimensions, and expected outcomes. They will present the plan to the class. Since this is a learning activity, all components will be completed in class Learning Plus Associates

37 Common Core Standards Plus - Math Grade 8 Lesson Index Common Core Standards Plus Mathematics Grade 8 Lesson Index Domain Lesson Focus Standard(s) Geometry (Geometry Standards: 8.G.1 8.G.9) Student Page 8.G.1: Verify experimentally the properties of rotations, 1 Verifying Properties reflections, and translations. 8.G.2: See Below Showing Congruency 8.G.2: Understand that a two dimensional figure is 199 congruent to another if the second can be obtained 3 Mapping Figures from the first by a sequence of rotations, reflections, Mapping Figures and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. 201 Evaluation Using Rotations, Reflections, and Translations 8.G.1, 8.G G.3: Describe the effect of dilations, translations, 5 Dilating Figures rotations, and reflections on two dimensional figures Transforming Figures using coordinates G.4: Understand that a two dimensional figure is 7 Transforming Figures similar to another if the second can be obtained from Transforming Figures the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional 207 figures, describe a sequence that exhibits the similarity between them. E1 E2 Transforming Figures Describe a Sequence of Transformations 8.G Angle Sum and Exterior Angle Theorems 8.G.5: Use informal arguments to establish facts Applying the Angle Sum of a Triangle about the angle sum and exterior angle of triangles, about the angles created when parallel 211 Apply the Angle Sum and Exterior Angle of lines are cut by a transversal, and the angle angle 12 Triangles criterion for similarity of triangles. 212 Evaluation The Angle Sum and Exterior Angle E3 8.G.4, 8.G.5 of Triangles Defining Angles Made by a Transversal Measuring the Angles Formed by a Transversal Measuring Angles Formed by a Transversal 8.G Measuring Angles Formed by a Transversal 218 E4 Evaluation Parallel Lines Cut by a Transversal Parallel Lines Cut by a Transversal 221 Use Transversals to Find the Angle Sum of a 18 Triangle Properties and Criteria for Similar Triangles 8.G Criteria for Similar Triangles 224 Evaluation Transformations, Triangles, and E5 Parallel Lines Cut by Transversals 225 P10 Performance Lesson #10 2 D Figures & Transformations (8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5) Proof of the Pythagorean Theorem Proof of the Pythagorean Theorem Proof of the Pythagorean Theorem 8.G.6: Explain a proof of the Pythagorean Theorem and its converse. 24 Converse of the Pythagorean Theorem Evaluation Proofs of the Pythagorean E6 Theorem and It s Converse Applying the Pythagorean Theorem Applying the Pythagorean Theorem 8.G.7: Apply the Pythagorean Theorem to Applying the Pythagorean Theorem determine unknown side lengths in right triangles in real world and mathematical problems in two Applying the Pythagorean Theorem and three dimensions. 242 E7 Evaluation Apply the Pythagorean Theorem 243 DOK Level Learning Plus Associates 37

38 Common Core Standards Plus - Math Grade 8 Lesson Index Student Domain Lesson Focus Standard(s) Page Applying the Pythagorean Theorem and Its 29 Converse Applying the Pythagorean Theorem 246 Pythagorean Theorem and Special Right 31 Triangles 8.G Applying the Pythagorean Theorem to 3 32 Dimensional Problems 248 Evaluation Applying the Pythagorean E8 Theorem 249 Finding the Distance Between Points on a 33 Coordinate Plane Distance Formula 8.G.8: Apply the Pythagorean Theorem to find Geometry (Geometry Standards: 8.G.1 8.G.9) 35 Applying the Distance Formula 254 the distance between two points in a coordinate system. Distance Formula and the Converse of the Pythagorean Theorem E9 Evaluation Pythagorean Theorem P11 Performance Lesson #11 Pythagorean Theorem (8.G.6, 8.G.7, 8.G.8) Use the Volume Formula of Cylinders to Solve 37 Problems 260 Use the Volume Formula of Cylinders to Solve Problems 8.G.9: Know the formulas for the volumes of Use the Volume Formula of Cylinders to Solve cones, cylinders, and spheres and use them to Problems solve real world and mathematical problems. Use the Volume Formula of Cylinders to Solve 40 Problems 263 E10 Evaluation Volume of Cylinders and Cones 264 Use the Volume Formula of Spheres to Solve 41 Problems 265 Use the Volume Formula of Spheres and 42 Cylinders to Solve Problems 266 Use the Volume Formula of Three 43 Dimensional Shapes to Solve Problems 8.G Use the Volume Formula of Three 44 Dimensional Shapes to Solve Problems 268 E11 Evaluation 11 Use the Volume Formula 269 P12 Performance Lesson #12 Volume (8.G.9) Integrated Project #3 Pythagoras Who? (8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5, 8.G.6, 8.G.7, 8.G.8) Prerequisite Standards Plus Domain: Geometry Project Objective: The students will research and report on one proof of the Pythagorean Theorem and create a visual display that exhibits the proof. They will provide an oral presentation of the proof and the visual display. Overview: In this project, the students will explore proofs of the Pythagorean Theorem from throughout history. They will discuss the meaning and importance of the Pythagorean Theorem. The class will develop a rubric to use for creating the visual displays and for reporting aloud. Each student will select one proof, create a visual display that exhibits the proof, and provide an oral presentation of the proof and the visual display. Since this is a learning activity, all components will be completed in class. DOK Level Learning Plus Associates

39 Standards Plus is a perfect fit for California Schools Standards Plus has a proven record of closing achievement gaps in districts throughout California. Over 190+ Schools in California implemented Standards Plus in 2016 and exceeded the State Test average in one or more grade levels. Standards Plus Materials Benefit English Learners: Using Standards Plus instruction across grade levels ensures all students are given equal access to grade level, standards-based instruction. By explicitly targeting the standards Emphasizing academic vocabulary Accelerating language development Providing immediate feedback to students Improving student confidence Standards Plus Supplemental Materials have been independently reviewed and verified for alignment to the California Standards by learninglist.com

40 Standards Plus is Proven Effective in California Schools CALIFORNIA SBAC GROWTH RATE STANDARDS PLUS SCHOOLS SBAC GROWTH RATE* more than doubled OVER 83% of Schools that implemented Standards Plus in more than doubled the California SBAC growth rate in one or more grade level. Standards Plus Closes the Achievement Gap with 7 Different Programs in One Standards Plus includes: Today s Lesson Performance Lessons Integrated Projects Minute Direct Instruction Lessons in Print and Online Increase EL Performance with Equity ELA & Math in Grades K-8 Transfer of Knowledge to a Digital Learning Environment Intervention Materials Built-In Students Experience SBAC-Like Technology Fits into Every Budget starting at $ a Student