a story of ratios

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 8 – Module 5


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios

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.



AgendaIntroduction to the ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Ratios



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



AgendaIntroduction to the ModuleConcept DevelopmentModule Review



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



L1: The Concept of a Function

Lesson 1, Concept Development

• Are functions just like linear equations?• What predictions do functions allow us to make?



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



Example 1

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



Example 2

• 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.

Discuss in groups, make notes on the handout.



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.



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?



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.





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).



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?





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:



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?



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.



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.



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.”



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.



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.



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



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.



Example 4

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



Exercise 3Use your handout to complete Exercise 3.



Problem Set 2-Just for Fun



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.



Exercise 1• Complete Exercise 1 independently or in pairs.



Exercise 1 (cont.)



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.



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



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



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



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



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.



Exercise 1

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



Exercise 1 (cont.)



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?



Fluency Activity



Fluency Activity

• What do you notice about this set of equations?



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.



Exercise 4Complete Exercise 4 using your handout.



Exercise 4 (cont.)



Discussion• 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.



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.



Exercise 2



Exercise 2 (cont.)



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?



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



L9: Examples of Functions from Geometry


• Students write equations to describe functions related to geometry.

• Students review some basic assumptions about volume.



Basic Assumptions



Exercises 7-10• Complete Exercises 7 – 10 using your handout.



Exercises 7-10 (cont.)• Connection between knowledge of functions and geometry.• Development of volume formula, V = Bh, where B is the base of the solid.



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.



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?



Development of Formula• Since it took 3 cones to fill the cylinder, then the volume for a cone is:



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.



Demonstration

• 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?



Development of Formula

• 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:



Problem Set #6



End-of-Module Assessment

• 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.



End-of-Module Assessment Scores

1a 1b 1c 1d 2a 2b 2c 3a 3b 3c

S11 4 2 3 2 2 3 2/3 1 2 3

S12 3 2 4 3 4 3 4 2 4 2

S13 4 4 4 4 4 4 4 4 4 4

S14 2 1 2 2 2 1 1 1 1 2



Biggest TakeawayTurn and Talk:• What questions were answered for you?• What new questions have surfaced?



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.

a story of ratios

Documents

module overview

content of module

module focus sessionmodule

story of ratiosgrade

network team institutegrade

minutes materials

mathematical models

instructional strategies