Module Focus: Grade 8 – Module 5 Sequence of Sessions

Overarching Objectives of this February 2014 Network Team Institute Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate

how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

High-Level Purpose of this Session● Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.● Standards alignment and focus: Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the

module addresses the major work of the grade.● Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 8 curriculum, A Story of Ratios.

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.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Participants will develop a deeper understanding of the sequence of Participants will be able to articulate the key points listed above.

mathematical concepts within the specified modules and will be able to articulate how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to the Module

15 minsEstablish the instructional focus of Grade 8 Module 5.

Grade 8 Module 5 Grade 8 Module 5 PPT

Review Grade 8 Module 5

Topic A Lessons 140 minsExamine lessons from Module 5 Topic A.

Grade 8 Module 5 Grade 8 Module 5 PPT

Review Topic A.

Topic B Lessons 60 minsExamine lessons from Module 5 Topic B.

Grade 8 Module 5 Grade 8 Module 5 PPT

Review Topic B.

End-of Module Assessment

60 minsComplete, score, and discuss portions of the assessment.

Grade 8 Module 5 Grade 8 Module 5 PPT

Complete assessment, review rubric and sample solutions.

Session Roadmap

Section: Time:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 min 1. Welcome! In this module focus session, we will examine Grade 8 – Module 5.

1 min 2. 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.

1 min 3. 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.

1 min 4. 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.

12 min

5. “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.”

6.

3 min 7. 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.”

1 min 8. “In this lesson students are introduced to functions and shown that functions serve the purpose of making predictions about the world around them."

3 min 9. Allow time for participants to write the equation on the white boards. Then select a participant to share their answer with the group. Remind participants that they must define their variables, as learned in Module 4.

3 min 10. 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.”

2 min 11. Read through Example 2. Provide time for participants to discuss whether or not we can assume constant speed.

2 min 12. 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.

3 min 13. Have participants make a new prediction about how many feet the stone will drop in 3.5 seconds. This data was not on the graph of the last slide so they have to use a proportion (or other method) to determine an answer. “We assume that most students will rely on proportions to come up with an answer. These are the answers that are produced when a proportion is used with a fraction that represents the data from the graph on the last slide.” Ask the question in the second bullet. Participants should state that these answers are not reasonable. Based on the graph in the last slide the stone dropped 144 ft in three seconds, making the first two answers completely wrong. The only reasonable answer is the third, but we cannot be sure it is correct because the first two were not. Ask the question in the third bullet. Participants should state that this is not a reliable method for making a prediction about the number of feet the stone drops in a given number of seconds.

3 min 14. 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.”

2 min 15. “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.”

3 min 16. 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.

1 min 17. “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.”

3 min 18. “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.”

1 min 19. “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.”

3 min 20. “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.”

3 min 21. “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.”

2 min 22. “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.”

1 min 23. “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.”

3 min 24. 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.

3 min 25. 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.

3 min 26. 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.

5 min 27. 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.

5 min 28. 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.

1 min 29. “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.”

10 min

30. 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.

2 min 31. Ask participants to provide answers to the questions then click to show what we expect students to write/think.

2 min 32. Ask participants to provide answers to the questions then click to show what we expect students to write/think.

2 min 33. Ask participants to provide answers to the questions then click to show what we expect students to write/think.

2 min 34. Ask participants to provide answers to the questions then click to show what we expect students to write/think.

2 min 35. 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.”

4 min 36. 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.”

2 min 37. “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.”

2 min 38. “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.”

2 min 39. “Is this the graph of a function? Explain.” Participants should respond, “Yes, because each input has exactly one output.”

2 min 40. “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.”

1 min 41. “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.”

3 min 42. “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.

2 min 43. “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.”

2 min 44. Read through the bullets on the slide. Once everyone is ready, continue with the fluency activity on the next three slides.

3 min 45. Give participants one minute per equation. After the three equations have been solved, click to show that the solution to each was x = -2.

3 min 46. 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).

2 min 47. Give participants one minute per equation. After the three equations have been solved, click to show that the solution to each was x = 5.

3 min 48. 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.”

3 min 49. 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!”

3 min 50. 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.”

1 min 51. “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.”

5 min 52. “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.

4 min 53. 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.

2 min 54. “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).”

1 min 55. “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.”

3 min 56. “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.”

3 min 57. “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.”

2 min 58. “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.”

2 min 59. “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.”

1 min 60. 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.”

1 min 61. “In this lesson, students explore patterns related to the area and volume of figures and write a function based on the patterns observed.”

3 min 62. “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.”

10 min

63. “Complete Exercises 7-10 in your handout.” Once most participants have finished, click to show the answers.”

3 min 64. “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.”

1 min 65. “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.”

10 min

66. “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.

2 min 67. “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.”

1 min 68. “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.”

10 min

69. “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.

2 min 70. “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.”

4 min 71. “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.

60 mins

72. (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.

2 min 73. “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.”

2 min 74. 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.

2 min 75. Let’s review some key points of this session.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

Grade 8 Module 5 Grade 8 Module 5 PPT Grade 8 Module 5 Handout

Additional Suggested Resources

A Story of Ratios Curriculum Overview