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A Story of Ratios

A Story of Ratios. Grade 8 – Module 5. Session Objectives. Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

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A Story of Ratios

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  1. A Story of Ratios Grade 8 – Module 5

  2. 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.

  3. Agenda Introduction to the Module Concept Development Module Review

  4. Curriculum Overview of A Story of Ratios

  5. Module 5 Overview • Table of Contents • Overview • Focus Standards • Foundational Standards • Focus Standards for Mathematical Practice • Terminology • Tools • Assessment Summary

  6. Agenda Introduction to the Module Concept Development Module Review

  7. 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

  8. L1: The Concept of a Function Lesson 1, Concept Development • Are functions just like linear equations? • What predictions do functions allow us to make?

  9. Example 1 • Work on the handout, write equation on white board.

  10. Example 1 • What predictions can we make? • Complete the table. • What is the average speed of the object from zero to three seconds?

  11. Example 2 • Discuss in groups, make notes on the handout. • 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.

  12. 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.

  13. Example 2 • Use the space in your handout to make a new prediction. How many feet will the stone drop in 3.5 seconds? • 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?

  14. 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?

  15. L2: Formal Definition of a Function Lesson 2, Concept Development • 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).

  16. Opening • 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?

  17. Discussion • 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:

  18. Discussion • 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?

  19. L3: Linear Functions and Proportionality Lesson 3, Concept Development • 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.

  20. Example 1 (PS #7 from L2) • 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.

  21. Example 1 (PS #7 from L2) • 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.”

  22. Lesson 3 • 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.

  23. L4: More Examples of Functions Lesson 4, Concept Development • 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.

  24. Opening Discussion • What are the differences between these two situations?

  25. 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.

  26. Example 4 • Is this a function? • What mathematical rule can describe the data in the above table?

  27. Exercise 3 • Use your handout to complete Exercise 3.

  28. Problem Set 2-Just for Fun

  29. L5: Graphs of Functions and Equations Lesson 5, Concept Development • 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.

  30. Exercise 1 • Complete Exercise 1 independently or in pairs.

  31. Exercise 1 (cont.)

  32. Exercise 1 (cont.)

  33. Exercise 1 (cont.)

  34. Exercise 1 (cont.)

  35. Exercise 1 (cont.)

  36. Discussion of Exercise 1 • 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.

  37. Exercise 4: Graph 1 • Is this the graph of a function? Explain. Use your handout to complete Exercise 4.

  38. Exercise 4: Graph 2 • Is this the graph of a function? Explain.

  39. Exercise 4: Graph 3 • Is this the graph of a function? Explain.

  40. Discussion: Graph 3 • Is this the graph of a function? Explain.

  41. L6: Graphs of Linear Functions and Rate of Change Lesson 6, Concept Development • 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.

  42. Exercise 1 How do you expect students to determine if the table has values of a linear function?

  43. Exercise 1 (cont.)

  44. Fluency Activity • 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?

  45. Fluency Activity

  46. Fluency Activity

  47. Fluency Activity

  48. Fluency Activity • What do you notice about this set of equations?

  49. Fluency Activity • What do you notice about this set of equations?

  50. Fluency Activity • What do you notice about this set of equations?

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