 # Grade 8 – Module 5 Module Focus Session

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Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X A Story of Ratios Grade 8 – Module 5 (1 min) Welcome! In this module focus session, we will examine Grade 8 – Module 5.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Session Objectives Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios. (1 min) Our objectives for this session are to: Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review (1 min) We will begin by exploring the module overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the concept development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole. Let’s get started with the module overview.

Curriculum Overview of A Story of Ratios
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Curriculum Overview of A Story of Ratios (1 min) The fifth module in Grade 8 is called Examples of Functions From Geometry. The module is allotted 15 instructional days. It challenges students to build on understandings from previous modules by using what they know about constant rate, linear equations, non-linear expressions to understand the concept of a fraction and its graph. Students apply knowledge of volume from previous grade levels to determine the volume of cylinders, cones and spheres.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Module 5 Overview Table of Contents Overview Focus Standards Foundational Standards Focus Standards for Mathematical Practice Terminology Tools Assessment Summary (12 min) “I want to give you some time to familiarize yourself with the content of Module 5 by reading the Module Overview. Please take about 10 minutes to quietly read through the following sections (point to sections on slide).” Once participants are finished reading say “Notice that this module has only an End-of-Module Assessment. With just 11 lessons, there wasn’t a need for two summative assessments. This is the first time in this grade level that only one assessment is provided for the module. Certainly encourage your teachers to continue to use the Exit Tickets and other informal assessments throughout both of the topics of the module.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic A: Functions Function is introduced conceptually, then defined formally Functions are useful in making predictions Discrete and continuous rates The graph of a function is identical to the graph of the equation that describes it A constant rate of change implies a linear function and rates can be used for comparison of functions Graphs of non-linear functions (3 min) Read the bullet points on the slide, “These are the basic concepts in Topic A. Next we will look at the specific lessons within this topic.”

L1: The Concept of a Function
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L1: The Concept of a Function Are functions just like linear equations? What predictions do functions allow us to make? (1 min) “In this lesson students are introduced to functions and shown that functions serve the purpose of making predictions about the world around them." Lesson 1, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Work on the handout, write equation on white board. (3 min) Allow time for participants to write the equation on the handout (then on the whiteboard). Then select a participant to share their answer with the group. Remind participants that they must define their variables, as learned in Module 4.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 (3 min) Ask the first question. Possible response “In one minute the object travels 64 feet.” Instruct participants to complete the table in the handout. Ask the average speed question. Response “The average speed is 64 feet per second. We know that the object has a constant rate of change because it is traveling at a constant speed, therefore we expect the average speed to be the same over any time interval.” What predictions can we make? Complete the table. What is the average speed of the object from zero to three seconds?

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 Discuss in groups, make notes on the handout. (2 min) Read through Example 2. Provide time for participants to discuss whether or not we can assume constant speed. If this situation is linear, then the answer is no different than that of Example 1. The stone will drop 192 feet in either interval of 3 seconds.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 Shown is actual data about the distance traveled by the stone. How many feet did the stone drop in 3 seconds? How does your answer compare to that in Example 1? Complete the table. (2 min) Read through the first two bullets. After participants answer “144 ft” to the first question, ask them how that compares to the table they produced in example 1. They should state that the answers are not the same. In example 1, after three seconds the object traveled 192 ft. Then have them complete the table for example 2 using the graph on the slide.

Grade 8 – Module 5 Module Focus Session

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 There is an infinite amount of data that we could gather about the falling stone. Consider all of the possible time intervals from 0 to 4 seconds! Compare the average speed in each interval of 0.5 seconds (Exercise 5): The average speed is not equal to the same constant over each time interval. Therefore, the stone is not falling at a constant speed. How reasonable are these answers? Is this a reliable method for making a prediction about the number of feet the stone drops for a given number of seconds? (3 min) Read the first bullet. Say, “We could gather data about the stone in very small time intervals, tenths of a second, hundredths of a second, etc.” Click to show bullet 2. “What we want to do now is observe the average speed in intervals of a half second.” Click to show bullet 3. “Notice that the average speed is not the same for each interval of 0.5 seconds. Since the average speed is not equal to the same constant over each time interval, then we know that the stone is not falling at a constant speed.”

L2: Formal Definition of a Function
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L2: Formal Definition of a Function A function assigns to each input exactly one output. Students examine tables of values and decide if the data represents a function or not. A function can be described by a rule or formula, but not every rule will be mathematical. It may be a description. There are limitations to the predictions that can be made with functions (allusions to domain and range). (2 min) “In this lesson, students learn the formal definition of a function and know that most functions can be described by a mathematical rule or formula. We want to learn about functions to make predictions about the world around us and for that reason we need to consider each situation when discussing the reliability of those predictions based on the math that we do.” Lesson 2, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Opening TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Provide participants with 1 minute to answer the two questions at their tables. Say “Recall that we want to learn about functions so that we can make reliable predictions. What did you notice about the predictions about how many feet an object could travel in 1 second?” Participants should note that the table on the right produces two different distances for the same time of 1 second. Therefore the data in the table to the right would not be reliable for making predictions. Using the table on the left, how many feet did the object travel in 1 second? Using the table on the right, how many feet did the object travel in 1 second? How reasonable are these answers? Is this a reliable method for making a prediction about the number of feet the stone drops for a given number of seconds?

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X The table on the left allows for reliable predictions. It allows us to assign an exact distance for a given time. Therefore, the table on the left represents data from a function, where the table on the right does not. A function is like a machine: (1 min) “We show students that a function is like a machine. A certain time, t, is put into the machine and the machine uses information about rate to determine the distance traveled in t seconds. The machine should produce exactly one answer for each value of t used in the machine.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X We can write a mathematical rule to describe the movement of the falling stone. Not all functions can be described this way. Consider a function that allows you to predict the correct answers on a test. It would not be a mathematical rule. Functions have limitations. Consider the stone example again. Using the above rule, can we find a value for distance when t = -2? t = 5? Would it make sense in the context of the problem? (3 min) “In most cases we can write a mathematical rule to describe a function, in this case it is t = 16t^2.” Click to show second bullet and read aloud, then click to show the third bullet to reinforce this idea. Click to show the fourth bullet. Participants should answer that yes, we can certainly find a value for the distance at those given values of t. Click to show the last bullet. Give participants a moment to think about the question. They should state that t = -2 doesn’t make sense because it would mean that two seconds before the stone is dropped it has traveled 64 feet. They should state that t = 5 doesn’t make sense either because it shows that the stone has traveled 400 feet in 5 seconds, but the stone hit the ground after 4 seconds! Say “This discussion alludes to an understanding about the domain of a function, without actually calling it domain.”

L3: Linear Functions and Proportionality
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L3: Linear Functions and Proportionality Linear functions are related to constant speed and proportional relationships. Students use the language related to a function: Distance traveled is a function of the time spent traveling. (1 min) “In this lesson, students relate constant speed and proportional relationships to linear functions. They begin using the language of the topic, i.e., the distance traveled is a function of the time spent traveling.” Lesson 3, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Example 1 (PS #7 from L2) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Do you think this a linear function? Explain. The rate of change is the same for any number of bags purchased. This relationship can be described by y = 1.25 x. (3 min) “This example is a problem that was completed in the problem set of the previous lesson. It is used as a spring board for understanding that proportional relationships can be represented by linear function.” Ask participants how students would answer the question in the first bullet. They should state that students would examine the rate of change over various time periods and look to see if the rate was the same. Click to show the last bullet. Say “Students use the rate of change to write the rule that describes the function.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Example 1 (PS #7 from L2) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Consider the graph of the data from the table. Can x be a negative number? No; allusion to domain. Does the table/graph represent all possible inputs and outputs? No; 10 bags, for example, is not represented. “The function described by y = 1.25x has these values.” (3 min) “Now we consider the graph of the data from the table.” Click to show the second bullet. Participants should say “No because you can’t purchase -2 bags of candy.” Click to show the next bullet. Participants should say “No because you can purchase more than 8 bags of candy and that data is not shown on the table or the graph.” Then say “For this reason, we say that a function has the data in the table and graph and can be represented completely by y = 1.25 x for positive integer values of x.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Lesson 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Constant rates and proportional relationships can be described by a function, specifically a linear function where the rule is a linear equation. Functions are described in terms of their inputs and outputs. For example, the total at the store is a function of how many bags of candy are purchased. (2 min) “Students complete problems about constant rate and reach the same conclusion as they did with the bags of candy example, e.g., that a linear function can be used to represent a constant rate problem. Throughout the lesson we focus on having students verbally describe the function, as in the second bullet.”

L4: More Examples of Functions
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L4: More Examples of Functions Discrete and continuous rates. Examples of functions include books purchased and cost, volume of water flow over time, temperature change in soup over time; all of which can be described mathematically. Examples of functions that cannot be described mathematically. (1 min) “In this lesson, students are exposed to discrete and continuous rates. Students examine functions in a variety of contexts, including those that cannot be described by a mathematical rule.” Lesson 4, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Opening Discussion What are the differences between these two situations? (3 min) Read the question aloud and give participants time to discuss at their tables. Participants should state that Table A is about purchasing bags of candy and Table B is about distance traveled. Acknowledge any other differences that participants note (fractional inputs in Table B for example) then continue with the next slide.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Opening Discussion What restrictions are there to the x values of each situation? Allusion to domain. Discrete rates are those where the inputs must be separate or distinct, i.e., positive integers. Continuous rates are those where there are no gaps in the values of the input. (3 min) Read the first question. Click to show “allusion to domain”. Say “In Table A we are restricted to positive integer values of x because we are talking about the purchase of a bag of candy. It is not likely that a shop owner would allow you to open up a bag and buy just part of it! In Table B there are no restrictions to x other than it must be positive because we are talking about time in seconds.” Click and read through the next two bullets.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Example 4 TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (3 min) Show the example and let participants think for a moment about the first question. They should state that it is a function because each input has exactly one output. Click to show the second bullet. Again, give them a moment to think. They should respond that there is no mathematical rule that can describe the function that assigns heights to players. Is this a function? What mathematical rule can describe the data in the above table?

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Use your handout to complete Exercise 3. (5 min) Instruct participants to respond to Exercise 3 on their handouts. When most have finished, have participants offer their answers to each question and click to show what we expect students to write/think.

Problem Set 2-Just for Fun
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute Problem Set 2-Just for Fun TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (5 min) Give participant 2 minutes to figure out what the function is. Then ask them to give their answer to part b on the white boards. Then select a participant to describe the function.

L5: Graphs of Functions and Equations
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L5: Graphs of Functions and Equations Students understand that the inputs and outputs of a function correspond to ordered pairs on the coordinate plane. Students know that the graph of a function is identical to the graph of the equation that describes it. Students can determine if a graph represents a function by examining the inputs and corresponding outputs. (1 min) “In this lesson, students compare the graph of a function to the graph of an equation. They learn that the graph of a function is identical to the graph of the equation that describes it. Students examine points on a graph to determine if the graph is a graph of a function.” Lesson 5, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 1 independently or in pairs. (10 minutes total to work on Exercise 1, 10 minutes to debrief with this and the next 5 slides) Instruct participants to complete exercise 1 in the handout. Once most have finished, ask participants to provide answers to the questions then click to show what we expect students to write/think.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Ask participants to provide answers to the questions then click to show what we expect students to write/think.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Ask participants to provide answers to the questions then click to show what we expect students to write/think.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Ask participants to provide answers to the questions then click to show what we expect students to write/think.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Ask participants to provide answers to the questions then click to show what we expect students to write/think.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Ask participants to provide answers to the questions then click to show what we expect students to write/think. Say “The answer to this part of the exercise is the mathematical goal of the lesson. A discussion about this exercise follows to solidify this understanding.”

Discussion of Exercise 1
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute Discussion of Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Given an input, how did you determine the output that the function would assign? We use the rule. In place of x, we put the input. The number that was computed was the output. When you wrote your inputs and corresponding outputs as ordered pairs, what you were doing can be described generally by the ordered pair because How did the ordered pairs of the function compare to the ordered pairs of the equation? They were exactly the same. What does that mean about the graph of a function compared to the graph of the equation that describes it? The graph of the function is identical to the graph of the equation that describes it. (4 min) Read/click through the bullets on the slide. Ask participants for their responses if time. Once all bullets are shown say “Again, the goal of the lesson is for students to understand that the graph of a function is identical to the graph of the equation that describes it. Seeing the inputs and outputs as ordered pairs, then comparing the graphs leads students to this understanding.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 4: Graph 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Use your handout to complete Exercise 4. Is this the graph of a function? Explain. (2 min) “Once students know that graphs of functions are identical to the equations that describe them we ask them to examine a graph to determine if it is a function or not. Is this the graph of a function? Explain.” Participants should respond, “Yes, because each input has exactly one output.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 4: Graph 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Is this the graph of a function? Explain. (2 min) “Is this the graph of a function? Explain.” Participants should respond, “No because the input of 6 has two outputs, 4 and 6. For that reason, this graph cannot be a function.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 4: Graph 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Is this the graph of a function? Explain. (2 min) “Is this the graph of a function? Explain.” Participants should respond, “Yes, because each input has exactly one output.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Discussion: Graph 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Is this the graph of a function? Explain. (2 min) “Is this the graph of a function? Explain.” Participants should respond, “No because the input of -3, for example, has two outputs, 0 and 4. For that reason, this graph cannot be a function.”

L6: Graphs of Linear Functions and Rate of Change
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L6: Graphs of Linear Functions and Rate of Change Students use inputs and corresponding outputs from a table to determine if a function is a linear function by computing the rate of change. Students know that when the rate of change is constant, then the function is a linear function. (1 min) “In this lesson, students use inputs and outputs of functions, much like they did with points on a graph to determine the slope, to determine if a function is a linear function.” Lesson 6, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X How do you expect students to determine if the table has values of a linear function? (3 min) “Students are shown a table and asked to determine if the function that assigns those values is a linear function. How do you expect students to figure out the answer to this question?” Participants should say that they would examine the rate of change for different inputs and outputs. Click to show part a and its solution.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 1 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “Once students know that function is a linear function then they know that the equation that describes it must be a linear equation. Part b requires students to determine that equation.” Click to show part c. “Next, students are asked to describe the graph. From the last lesson they know that the graph of a function is identical to the equation that describes it. Therefore, the linear function must graph as a line.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Grab a white board and marker. You may need to share erasers. I will show you one equation at a time. You will have one minute to solve the equation. When I say “Show me” you will hold up your white board whether you have finished solving the equation or not. Ready? (2 min) Read through the bullets on the slide. Once everyone is ready, continue with the fluency activity on the next three slides.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Give participants one minute per equation. After the three equations have been solved, click to show that the solution to each was x = -2.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Give participants one minute per equation. After the three equations have been solved, click to show that the solution to each was x = -(3/14).

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) Give participants one minute per equation. After the three equations have been solved, click to show that the solution to each was x = 5.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X What do you notice about this set of equations? (3 min) Debrief the fluency by asking participants what they noticed about this set of equations. They should respond that the second and third equations are exactly like the first except a constant is multiplying both sides or dividing both sides. That is why the answer is the same for each equation. “We want students to look for and make use of the structure of the equations (MP 7) while developing fluency in solving multi-step equations.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X What do you notice about this set of equations? (3 min) Ask participants what they noticed about this set of equations. They should respond that the second and third equations are exactly like the first except a constant is multiplying both sides or dividing both sides. It is just like the last set of three equations except that the terms in the second equation were in a different order and the expressions in the third equation are on opposite sides of the equal sign. That is why the answer is the same for each equation. Say “We purposely included problems that had a negative fractional solution. Many students think they must be wrong when they get a fractional answer and we are trying to change that!”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Fluency Activity TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X What do you notice about this set of equations? (3 min) Ask participants what they noticed about this set of equations. They should respond that the second and third equations are exactly like the first except that you have to combine like terms to see it in the second equation. The third equation is just like the first except you have to combine like terms and then notice that the same constant is dividing both sides. That is why the answer is the same for each equation. “A debrief is not part of the actual lesson, but recommended if time permits. Ideally teachers would see the structure in these equations and be able to impart that knowledge and way of seeing equations onto the students.”

L7: Comparing Linear Functions and Graphs
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L7: Comparing Linear Functions and Graphs Similar to students comparing the graphs of linear equations represented in different ways, students now compare functions. Students work in small groups, discussing the various methods they can use (graphing, comparing the rates of change, using algebraic skills), before they begin solving and answering the questions in the exercises. (1 min) “In this lesson, students compare functions like they did with linear equation in Module 4. Students work in small groups and must discuss the problem before they actually begin solving it.” Lesson 7, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 4 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 4 using your handout. (5 min) “Complete Exercise 4 using your handout. Be sure to discuss with a partner before answering the questions.” When most participants have finished the exercise you can ask one to use the document camera to show their solution. Then show the next slide which is a sample of what we expect students to do/write.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 4 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (4 min) Show solution to exercise. Debrief the skills utilized in order to respond to the questions of the exercise: Must know how to find the slope of a line. Must know how to use a table to determine the rate of change and compare different values to determine if the rate of change is constant or not. Must interpret a slope of a graph and rate of change shown in a table that are equal to the same constant means that both Adam and Bianca are saving money at the same rate. Must be able to write the equation from a table using rate of change and y = mx + b. Must be able to identify the y-intercept from the graph. Must interpret the y-intercept of a graph and the value of b in an equation as the amount of money that each person started with.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Students describe their methods of solving each exercise. Was one method more efficient than the other? Does everyone agree? Why or why not? (MP 3) Was every problem completed the same way? Explain. (2 min) “Following the exercises, the teacher has a discussion with students about their methods of solving each using the questions above. We want students “talking” math, constructing viable arguments, and critiquing the reasoning of others (MP 3).”

L8: Graphs of Simple Non-Linear Functions
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L8: Graphs of Simple Non-Linear Functions Students examine the rate of change for non-linear function and conclude that non-linear functions do not have a constant rate of change. Students identify functions as linear or non-linear by examining the rate of change. (1 min) “In this lesson, students examine the rate of change of non-linear functions and conclude that non-linear functions are the result of non-linear equations that do not have a constant rate of change.” Lesson 8, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “We will look at Exercise 2 as an example of the work students do in this lesson. They begin by examining the rule, using what they learned in Module 4, to determine if the function would be linear or non-linear. “ Click to show part b. “Then students use given values of x, the input, and use the rule of the function to determine the output.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 2 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Next students graph the inputs and outputs as ordered pairs on a coordinate plane and describe the shape of the graph. Students are instructed to find the rate of change for a specific row in the table they completed in part b.” Click to show that answer to e, then click to show part f. “Students compare another set of rows from the table, noting that the rate of change is not the same.” Click to show part g. “Students are instructed to compare another set of rows, challenging them to think deeply about the definition of linear and whether or not the function could be linear because two pairs of rows did yield the same result.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercise 2 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “The last part of the problem has students compare their initial claim (part a shown here below) to what they think of the function now that they have examined the graph and the rate of change. After several exercises like this one students should conclude that if the rule that describes the function is non-linear then the function is non-linear.”

Discussion and More Exercises
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion and More Exercises What did you notice about the rate of change? What does this mean about the function? What do you think made the functions non-linear? (2 min) “The exercises are debriefed with the above three questions. Then students use equations that describe functions to make conclusions about whether the function is linear or not.” Click to show exercises 4-10 directions. “Students are given the option of graphing the equation to verify their thinking.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic B: Volume Writing functions to describe area and volume of familiar figures (rectangles and rectangular prisms) Building from knowledge of V = Bh to develop volume formula for cylinder. Using the volume formula for cylinder to determine the volume of cones and spheres (1 min) Read the bullet points on the slide, “These are the basic concepts in Topic B. Next we will look at the lessons of the topic.”

L9: Examples of Functions from Geometry
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L9: Examples of Functions from Geometry Students write equations to describe functions related to geometry. Students review some basic assumptions about volume. (1 min) “In this lesson, students explore patterns related to the area and volume of figures and write a function based on the patterns observed.” Lesson 9, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Basic Assumptions TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Here are the basic assumptions that are reviewed about volume. We suggest that the teacher read each one then ask students to paraphrase the assumption in their own words so the teacher can be sure they are understood.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercises 7-10 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercises 7 – 10 using your handout. (10 min) “Complete Exercises 7-10 in your handout.” Once most participants have finished, click to show the answers.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Exercises 7-10 (cont.) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Connection between knowledge of functions and geometry. Development of volume formula, V = Bh, where B is the base of the solid. (3 min) “The goal of this set of exercises is twofold; we want students to see how functions can be used in a geometric setting, but we also wanted to get at the general formula for the volume of a solid. We work with volume in this module and it reappears in Module 7. In Exercise 10 you can see how the parts lead the students to writing a function in the form of the general volume formula.”

L10: Volumes of Familiar Solids–Cones and Cylinders
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L10: Volumes of Familiar Solids–Cones and Cylinders Students use V = Bh to determine the volume of right cylinders. Students learn connection between volume formulas of cylinders and cones. Students determine the volumes of cylinders and cones. (1 min) “In this lesson, students lee volume formula for right cylinders and then use that formula to determine the formula for the volume of a cone. There is a demonstration in this lesson that I will do next.” Lesson 10, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Demonstration If we were to fill a cone of height, h, and radius, r, with rice/water/sand, how many cones do you think it would take to fill up a cylinder of the same height, h, and radius, r? (10 min) “Right now I’d like you to take on the role of a student. Discuss at your tables what the answer to the question stated here is.” If needed, show the cylinder and cone and point out the information regarding their dimensions. Once participants have discussed with their partners, ask them to share aloud their thoughts before doing the demonstration. Demonstration: Fill the cone completely with rice/water/sand. Then pour its contents into the cylinder. Repeat until the cylinder is full. It should take 3 cones.

Development of Formula
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Development of Formula Since it took 3 cones to fill the cylinder, then the volume for a cone is: (2 min) “Here is where we use that general formula. We know that the right cylinder has volume, V=Bh. We replace B with (pi)r^2. Given the information from the demonstration we can see that the volume of a cone is 1/3 (pi)r^2h.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L11: Volume of a Sphere Students know the volume formula for a sphere as it relates to the volume of a right cylinder with the same diameter and height. (1 min) “In this lesson, students learn that the volume of a sphere is connected to what they know about the volume of a cylinder. Another demonstration helps relate the two volume formulas.” Lesson 11, Concept Development

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Demonstration TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Given a cylinder and sphere with the same diameter, and a cylinder whose height is equal to the diameter, how much of the volume of the cylinder is taken up by the sphere? (10 min) “Again, I’d like you to take on the role of a student. Discuss at your tables what the answer to the question stated here is.” If needed, show the cylinder and sphere and point out the information regarding their dimensions. Once participants have discussed with their partners, ask them to share aloud their thoughts before doing the demonstration. Demonstration: Fill the sphere completely with rice/water/sand. Then pour its contents into the cylinder. It should fill 2/3 of the cylinder.

Development of Formula
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute Development of Formula TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X We saw that the volume of the sphere is exactly two-thirds the volume of the cylinder with the same diameter and height. Then the volume formula for a sphere is: (2 min) “We relate the volume of the sphere to the volume of the cylinder, based on what we observed in the demonstration. Then derive the volume formula for a sphere through computation.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute Problem Set #6 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (4 min) “This is an example of the kind of work we expect students to do after learning about the volumes of cylinders, cones, and spheres. We purposely wrote problems that combined all three formulas so that students would have to think about which formula to apply and when.” Provide time for participants to review/discuss the problem.

End-of-Module Assessment
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute End-of-Module Assessment TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Take about 20 minutes of quiet time to complete the assessment. After 20 minutes, review rubric and score sample assessments. Compare with table partners. Discuss any discrepancies in scoring and discuss any problematic language or skills gaps that may need to be addressed prior to use of assessment with students. (60 min total to take, discuss, and review the assessment) Ask participants to observe 20 minutes of quiet time while everyone works on the assessment. At the end of 20 minutes, instruct them to review the rubric and score the sample student assessments. Then discuss the assessment and rubric scores with their partners.

End-of-Module Assessment Scores
Grade 8 – Module 5 Module Focus Session February 2014 Network Team Institute End-of-Module Assessment Scores TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X 1a 1b 1c 1d 2a 2b 2c 3a 3b 3c S11 4 2 3 2/3 1 S12 S13 S14 (2 min) “Here is how we would score the student work. In your classrooms you know the students best so you may decide to score them differently. However, if there are any huge discrepancies we should probably have a discussion about it.”

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced? Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have? Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

Grade 8 – Module 5 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Key Points Functions are defined as an assignment where each input has exactly one output. Linear functions and their graphs relies on an understanding of linear equations and their graphs. Volume formulas for cylinders, cones and spheres are all related and stem from the general volume formula V = Bh. (2 min) Let’s review some key points of this session.