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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Vertical Coherence in A Story of Ratios & A Story of Functions Grades 6-9 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Session Objectives • Review actions that NTI participants should have completed or be in the process of completing in preparation for the start of the 2013-14 school year • Frame the overall learning objectives for this NTI’s mathematics sessions, giving the participants a clear understanding of the theme and focus of the sessions to come • Explore the vertical coherence of the curriculum Grades 6 - 9 • Engage in a deeper study of the Ratio and Proportion Progression for Grades 6 – 9, as well as the Expressions and Equations Progressions for Grades 6 - 9 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grades 6 and 7 • Ratios, Rates, and Unit Rate • Proportional Relationships and the Constant of Proportionality • Scale Drawings and Scale Factor 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 9 • Sequences Leading to Functions • Analyzing Tables of Data/Patterns in Rate of Change 5 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Key Areas of Focus in Mathematics K–2 Addition and subtraction - concepts, skills, and problem solving and place value 3–5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra and linear functions 9 Create, understand, solve, and represent equations and inequalities; interpret functions and linear models 6 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grades 6 and 7 • Ratios, Rates, and Unit Rate • Proportional Relationships and the Constant of Proportionality • Scale Drawings and Scale Factor 7 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions The transition into ratios and proportional relationships: • Why study ratios? • Make sense of proportional relationships • • • Definition and study of ratio, value of the ratio, equivalent ratios, unit rate of A:B, unit rate of B:A, constant of proportionality Percent Geometry • • Scale Factor and Scale Drawings Similarity • • Properties of Similar Figures Corresponding Side Lengths Equal in Ratio 8 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships When do we really use ratios? • Consider: Jimmy’s allowance is twice as much as Johnny’s and Jeremy’s is 4/5 of Johnny’s. How much allowance does Jimmy get? • List as many real-world applications of ratios and proportional relationships as you can think of. • Choose one of them to make up your own ratio application problem that seems especially realworld. 9 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Transitioning into ratio problems • Students have solved problems of multiplicative comparison. •There were 4 times as many boys as girls at the party. If there were 30 kids at the party, how many were boys? • What would motivate a student to want to use ratios to describe this type of situation? • Read through G6-M1, Lesson 1, Examples 1-2 10 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Definitions and Descriptions Ratio A pair of non-negative numbers, A:B, which are not both zero. They are used to indicate that there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity. Value of a Ratio For the ratio A:B, the value of the ratio is the quotient A/B as long as B is not zero. Likewise, for the ratio B:A, the value of the ratio is the quotient B/A as long as A is not zero. Equivalent Ratios Two ratios A:B and C:D are equivalent if there is a positive number, k, such that C=kA and D=kB. They are ratios that have the same value. 11 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 1: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. 12 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 2: Lena finds two boxes of printer paper in the teacher supply room. The ratio of the packs of paper in Box A to the packs of paper in Box B is 4:3. If half of the paper in Box A is moved to Box B, what is the new ratio of packs of paper in Box A to Box B? 13 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 3: Sana and Amy collect bottle caps. The ratio of the number of bottle caps Sana has to the number Amy has is 2 : 3. The ratio became 5 : 6 when Sana added 8 more bottle caps to her collection. How many bottle caps does Amy have? 14 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Intervention • What misconception is evident in the student work example? • Which foundational skills may the student be lacking? • How would you address the intervention for this student? 15 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 3: Sana and Amy collect bottle caps. The ratio of the number of bottle caps Sana has to the number Amy has is 2 : 3. The ratio became 5 : 6 when Sana added 8 more bottle caps to her collection. How many bottle caps does Amy have? 16 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 4: The ratio of songs on Jessa’s phone to songs on Tessie’s phone is 2 to 3. Tessie deletes half of her songs and now has 60 fewer songs than Jessa. How many songs does Jessa have? 17 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 5: Jack and Matteo had an equal amount of money each. After Jack spent $38 and Matteo spent $32, the ratio of Jack’s money to Matteo’s money was 3:5. How much did each boy have at first? 18 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Example 6: The ratio of the number of Ingrid’s stamps to the number of Ray’s stamps is 3 : 7. If Ingrid gives onesixth of her stamps to Ray, what will be the new ratio of the number of Ingrid’s stamps to the number of Ray’s stamps? 19 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Double Number Line Diagrams Example: Rate Problems A photocopier can print 12 copies in 36 seconds. At this rate, how many copies can it print in 1 minute? 20 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Representations of Equivalent Ratios 21 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Equivalent Ratios-Tape Diagrams vs. Double Number Line Diagrams Monique walks 3 miles in 25 minutes. vs. Sean spends 5 minutes watching television for every 2 minutes he spends on homework. 22 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Rate and Unit Rate Rate: If I traveled 180 miles in 3 hours; my average speed is 60 mph. The quantity, 60 mph, is an example of a rate. Unit Rate: The numeric value of the rate, e.g. in the rate 60 mph, the unit rate is 60. Rate’s Unit: The unit of measurement for the rate, e.g. mph. 23 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grades 6 and 7 • Ratios, Rates, and Unit Rate • Proportional Relationships and the Constant of Proportionality • Scale Drawings and Scale Factor 24 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Ratio & Proportions Exercise 1 In Jasmine’s favorite fruit salad, the ratio of the number of cups of grapes to number of cups of peaches is 5 to 2. a. How many cups of peaches will be used if 25 cups of grapes are used? b. Create both a ratio table and a graph of the proportional relationship to depict the relationship and find your answer. c. What is the constant of proportionality? d. Write the equation of the line depicted in the graph. 25 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 5 2 10 4 15 6 20 8 25 10 Cups of Peaches Cups of Grapes 2 5 4 10 6 15 8 20 10 25 Jasmine’s Fruit Salad Cups of Peaches Cups of Peaches 25 20 15 10 5 0 0 5 Cups of Grapes 10 15 20 25 Jasmine’s Fruit Salad Cups of Grapes Cups of Grapes 25 20 15 10 5 0 0 5 Cups of Peaches 10 15 20 25 26 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Ratio & Proportions Exercise 2 Jack is taking a hike through a forested park. He moves at a constant rate, covering 5 miles every 2 hours. a. How much time will have passed when he has hiked 9 miles? b. Create both a ratio table and a graph of the proportional relationship to depict the relationship and find your answer. c. What is the constant of proportionality? d. Write the equation of the line depicted in the graph. 27 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM 5 2 6 2.4 7 2.8 8 3.2 9 3.6 Hours Miles 2 5 2.4 6 2.8 7 3.2 8 3.6 9 Jack’s Hike Hours Hours 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Miles Jack’s Hike Miles Miles A Story of Ratios and Functions 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Hours 28 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Constant of Proportionality Given a ratio A : B, and given that one places the quantity associated with A on the x-axis and the quantity associated with B on the y-axis, then: The constant of proportionality for the ratio A : B is the unit rate of the ratio B : A (the value of the ratio B:A). 29 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Constant of Proportionality 30 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Constant of Proportionality The equation of the line depicted will be: Cups of Grapes Cups of Peaches 5 2 10 4 15 6 20 8 25 10 Cups of Peaches If a proportional relationship is described by the set of ordered pairs that satisfies the equation y = kx, where k is a positive constant, then k is called the constant of proportionality.; e.g., If the ratio of x to y is 5 to 2, then the constant of proportionality represents the ratio of y to x and is 2/5, and y = 2/5 x. Jasmine’s Fruit Salad 25 20 15 10 5 0 0 5 10 15 20 25 Cups of Grapes 31 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Constant of Proportionality Where and how is the constant of proportionality represented? Song Downloads cost $3 each. (unit rate of the ratio y:d) 32 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Testing for Proportional Relationships • G7-M1 Lesson 3 Exercises 1-3 • Now create graphs of the data in each table. What do you notice? • Recall the lesson from G8-M4 Lesson 10: • Calculating an average rate does not dictate that there was a constant rate on that interval, nor does it guarantee that the rate will continue at that same average rate. 33 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Intervention • Locate the student work for Grade 7 Module 1, Lesson 10. Determine first which student response(s) is correct. Determine misconceptions in student work. Create a course of action to address the misconceptions. • • • • What material/document(s) is available to you? • What scaffolds can you implement? • How will you implement the scaffold/intervention? • Whole Group? Small Group? Individual? Pre-Assessing? 34 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grades 6 and 7 • Ratios, Rates, and Unit Rate • Proportional Relationships and the Constant of Proportionality • Scale Drawings and Scale Factor 35 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Scale Factor • G7-M1 End of Module Assessment, Item 2 • Scale factor is a unit rate, and is therefore unit-less. To calculate the true scale factor, one must compare using the same unit of measure in each quantity. • Students work informally to know that the area of the scaled drawing is altered by a factor of (scale factor)2 • Does creating a scale drawing ever effect the measure of the angles in the drawing? 36 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 37 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Why move things around? • To avoid direct measurement… … while answering questions, making conjectures. • Work through: G8-M2 Lesson 1 Exploratory Challenge • Students describe motions intuitively, working towards comfort with formal concepts and language. 38 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Transformations of the Plane • Think of a plane as a sheet of overhead projector transparency, or a sheet of paper. • Consider an aerial photo of a portion of a city. Map each point on the street to a point on your paper (the map) in a way that intuitively “preserves the shape” (foreshadowing similarity). • Another way of mapping, we can project (using a light source) from one sheet to another. • A transformation of the plane, to be denoted by , is a rule that associates (or assigns) to each point of the plane to a unique point which will be denoted by . 39 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Three Rigid Transformations Understand and perform: • Translation: a transformation along a vector. • Reflection: a transformation across a line. • Rotation: a transformation about a point for a given angle measure 40 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Three Rigid Transformations • Do translations, reflections and rotations: o Map lines to lines, rays to rays, segments to segments, and angles to angles? o Preserve the lengths of segments? o Preserve the measures of angles? • Students verify these properties informally. Work through: G8 M2 Lesson 2; G8 M2 Lesson 4; G8 M2 Lesson 5 41 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Congruence • Rigid transformations can be sequenced. Does order matter? • Can sequences of rigid transformations be reversed? Work through: • G8 M2 L8 • G8 M2 L9 42 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Congruence • A congruence is a sequence of basic rigid motions (translations, reflections, or rotations) that maps one figure onto another. Work through: G8 M2 L10 • A two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Work through: G8 M2 L11 43 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Congruence • Are these properties true for a congruence? • Map lines to lines, rays to rays, segments to segments, and angles to angles. • Preserve the lengths of segments. • Preserve the measures of angles. • A two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. 44 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry What does “same shape” mean? • Similarity is often referred to as “same shape” (but not necessarily same size). • But what does “same shape” mean? Which if any of these properties would still apply to a ‘Similarity Transformation’? • Map lines to lines, rays to rays, segments to segments, and angles to angles? • Preserve the lengths of segments? • Preserve the measures of angles? 45 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Dilations • A dilation is a transformation of the plane with center O and scale factor r, that assigns to each point P of the plane a point Dilation(P) so that: • Dilation(O) = O and • If P ≠ O, then Dilation(P), denoted as P’, is the point on the ray OP so that |OP’| = r |OP|. • When r = 1, then the figures are congruent. • When 0 < r < 1, then the dilated figure is smaller than the original. • When r > 1, then the dilated figure is larger than the original. 46 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Dilations • Do dilations: o Map lines to lines, rays to rays, segments to segments, and angles to angles? o Preserve the lengths of segments? o Preserve the measures of angles? • G8-M3 Lesson 2: Examples 1 and 2 • Complete the exercises 47 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Dilations • Dilations: DO map lines to lines, rays to rays, segments to segments, and angles to angles. DO NOT preserve the lengths of segments. DO preserve the measures of angles. 48 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Similarity • • Two figures are said to be similar if you can map one onto another by a dilation followed by a congruence. Show figures are similar by describing the sequence of the dilation and congruence. 49 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 50 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Transformations of Functions • • G9-M3 Lesson 17 Example Dilations in Grade 8 vs. Scaling of the Graph of a Function in Grade 9 51 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 52 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometry Similarity – The AA Criterion • • G8-M3 Lesson 10 Two triangles are said to be similar if they have two pairs of corresponding angles that are equal. G8-M3 Lesson 11 If two triangles have one pair of equal corresponding angles and the ratio of corresponding sides (along each side of the given angle) are equal, then the triangles are similar. 53 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Slope of a Line • • • • Choose any 2 pairs of points on the line. Use the points to create right triangles. The AA criterion say these triangles are similar. Thus, the proportion of the vertical leg to the horizontal leg is the same for each triangle. Thus for any two points on a non-vertical line the ratio the vertical distance: horizontal distance between the two points is equal. 54 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Deriving the Equation for a Line For a line going through the origin: 55 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Deriving the Equation for a Line For a line passing through point b on the y-axis: 56 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 57 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Ratios & Proportional Relationships Average Rate vs. Constant Rate AVERAGE RATE. Let a time interval of hours be given. Suppose that an object travels a total distance of miles during this time interval. The object’s average rate in the given time interval is miles per hour. CONSTANT RATE. For any positive real number , an object travels at a constant rate of mph over a fixed time interval if the average rate is always equal to mph for any interval of time during the fixed time interval. 58 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Linear functions and rate of change • Grade 8 Module 4 defines slope as a number that describes “steepness” or “slant” of a line. It is the constant rate of change. • Review G8 M5 Lesson 7 Exercise 4 59 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Linear functions and rate of change • G8-M7 Lesson 22 introduces the concept of average rate of change. • Grade 9 Module 3 defines average rate of change: Given a function whose domain includes the interval and whose range is real numbers, the average rate of change on the interval is 60 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 8 • Transformations of the Plane • Rigid Motions, Congruence • Dilations, Similarity • • • • Transformations of Functions (Grade 9) Slope of a Line and Deriving the Equation of a Line Constant Speed vs. Average Speed and Rate of Change Early Study of Functions 61 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Why study functions? • Functions allow us to • • make predictions, classify the data in our environment. • G8-M5 Lesson 1 Examples 1 & 2 62 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions What is a function? • Grade 8: A function is a rule that assigns to each input exactly one output. • Grade 9: A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched to one and only one element of 𝑌. The set 𝑋 is called the domain; the set 𝑌 is called the range. • The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 63 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Building on Concepts from EE • • An expression in 1 variable defines a general calculation in which the variable can represent / can be replaced with a single number (chosen from a set of acceptable inputs). It’s useful to relate a function to an input-output machine with a variable representing the input, and an expression representing the output. • Review G8 M5 L2 Problem Set 64 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Linear Equations and Functions Building on Concepts from EE • • If we then choose a different variable to represent the output, we have an equation in two variables. Plotting points gives a visual representation of the relationship between the two variables. • • Work through: G8 M5 L5 Exercise 4 65 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 9 Sequences Leading to Functions Analyzing Tables of Data/Patterns in Rate of Change 66 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Integer Sequences: Should You Believe in Patterns? • What is the next number in the sequence? 2, 4, 6, 8, … • Is it 17? G9-M3 Lesson 1 67 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Terms, Term Numbers, and “the term” • What do I mean by “the nth term”? • Create a table of the terms of the sequence. G9-M3 Lesson 1 68 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Introducing the f(n) Notation • I’d like to have a formula that works like this: I pick any term number I want and plug it into the formula, and it will give me the value of that term. • In this case: A formula for the nth term = 2n-1 • Would it be okay if I wrote f(n) to stand for “a formula for the nth term?” f (n) = 2n-1 G9-M3 Lesson 1 69 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Consolidating Understanding Closing: • Why is it important to have a formula to represent a sequence? • Can one sequence have two different formulas? • What does f(n) represent? How is it read aloud? Lesson Summary: • A sequence can be thought of as an ordered list of elements. To define the pattern of the sequence, an explicit formula is often given, and unless specified otherwise, the first term is found by substituting 1 into the formula. G9-M3 Lesson 1 70 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Arithmetic Sequences Term 1: Term 2: Term 3: Term 4: Term 5: ... Term n: 5 8 = 5+3 11 = 5 + 3 + 3 14 = 5 + 3 + 3 + 3 17 = 5 + 3 + 3 + 3 + 3 G9-M3 Lesson 2 Example 1 71 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Recursive Formulas for Sequences • When Johnny saw Akeila’s sequence he wrote the following: 𝐴(𝑛 + 1) = 𝐴(𝑛) + 3 for 𝑛 ≥ 1 and 𝐴 1 = 5 • Why do you suppose he would write that? Can you make sense of what he is trying to convey? • What does the 𝐴(𝑛 + 1) part mean? G9-M3 Lesson 2 Example 1 G9-M3 Lesson 2 72 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Geometric Sequences • Look at the sequence: 1, 3, 9, 27 … • How is this sequence different from the one we just analyzed? • Write a recursive formula for the sequence. • Write an explicit formula for the sequence. • Plot the terms of the sequence on a coordinate plane. G9-M3 Lesson 2 Example 1 G9-M3 Lesson 3 Example 1 73 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Arithmetic and Geometric Sequences 1) −2, 2, 6, 10, … 2) 2, 4, 8, 16, … 3) 4) 1 3 5 , 1, , 2, , … 2 2 2 1 1 1 1, , , , … 3 9 27 5) 10, 1, 0.1, 0.01, 0.001, … 6) 4, −1, −6, −11, … 74 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Arithmetic and Geometric Sequences Describe as precisely as possible what an arithmetic sequence is. ARITHMETIC SEQUENCE - described as follows: A sequence is called arithmetic if there is a real number 𝑑 such that each term in the sequence is the sum of the previous term and 𝑑. GEOMETRIC SEQUENCE - described as follows: A sequence is called geometric if there is a real number 𝑟 such that each term in the sequence is a product of the previous term and 𝑟. G9-M3 Lesson 2 Example 1 75 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Exponential Functions Which is better? • Getting paid $33,333.34 every day for 30 days (for a total of just over $ 1 million dollars), OR • Getting paid $0.01 today and getting paid double the previous day’s pay for the 29 days that follow? • Why does the 2nd option turn out to be better? • What if the experiment only went on for 15 days? G9-M3 Lesson 2 Example 1 76 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Exponential Growth vs. Linear Growth • Is it fair to say that the values of the geometric sequence grow faster than the values of the arithmetic sequence? • Review G9-M3 Lesson 5 Opening Exercise and Examples 1 and 2 G9-M3 Lesson 2 Example 1 77 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Why stay with whole numbers? • • • • • Why are square numbers called square numbers? If S(n) denotes the nth square number, what is a formula for S(n) ? In this context what would be the meaning of S(0), S(π), S(-1) ? Exercises 5–8: Suppose we extend our thinking to consider squares of side length x cm… Create a formula for the area, A(x) cm2 of a square of side length x cm. Review Exercises 9–12, taking time to do Problems 10 and 12. Do Exercises 13 - 14 G9-M3 Lesson 8 78 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Agenda: Grade 9 Sequences Leading to Functions Analyzing Tables of Data/Patterns in Rate of Change 79 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Differences and Diagonals What is the next number in the sequence? • 4, 7, 10, 13, 16, … • 4, 5, 8, 13, 20, 29, … • 0, 2, 20, 72, 176, 350, 612, … • 1, 2, 4, 8, 16, 32, 64, 128, 256, … • 1, 4, 9, 16, 25, 36, 49, 64, 81, … • 1, 8, 27, 64, 125, 216, 343, … 80 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Differences and Diagonals • • If I only gave you the leading diagonal (and it eventually led to a row of zero’s) could you find every number in the sequence? What does the leading diagonal look like for each of the following: 1: 1, 1, 1, 1, 1, …. n: 1, 2, 3, 4, 5, 6, … n2: 1, 4, 9, 16, 25, 36, … n3: 1, 8, 27, 64, 125, 216, … 81 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Differences and Diagonals • • What do you suppose the leading diagonal would look like for n2 + n? If I gave you only the leading diagonal could you find an explicit formula for the sequence? 82 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Objects in Motion • Suppose I wish to test the speed and acceleration of a golf ball dropped from the top of a building that is 100 meters tall. • How would I set up my experiment? • What data would I need to collect? • How could I estimate the speed of the ball? • How could I estimate the acceleration of the ball? 83 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Culminating Exercise from G9-M5 Complete the following exercise: 84 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Key Points • • • • From a ratio relationship, students rely on multiplicative comparisons to determine a ratio of A:B. From a ratio, students are able to determine associated ratios and calculate rates and the unit rate. From associated ratios and unit rates, students discover that the unit rate of the ratio B:A represents the constant of proportionality, scale factor, rate of change and slope. They are able to derive an equation to a line and understand functions. From scale factor, students understand, rigid motions, congruence, dilation, and similarity. 85 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Intervention • How might the material, as presented to you today, help in providing scaffolding and intervention, whether it be planned or spontaneous? 86 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios and Functions Biggest Takeaways • • • Turn and talk with a partner at your table about your biggest takeaways from this session. Have you changed your feelings about our work together? Is this different than what you felt at the onset of today? What learning have you gained from our work together? What do you know now that you did not understand before we began? 87 © 2012 Common Core, Inc. All rights reserved. commoncore.org