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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Vertical Coherence in A Story of Ratios & A Story of Functions Grades 6-9

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Session Objectives 2 Review actions that NTI participants should have completed or be in the process of completing in preparation for the start of the 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 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grades 6 and 7 3 Ratios, Rates, and Unit Rate Proportional Relationships and the Constant of Proportionality Scale Drawings and Scale Factor

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 4 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 9 5 Sequences Leading to Functions Analyzing Tables of Data/Patterns in Rate of Change

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Key Areas of Focus in Mathematics 6 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grades 6 and 7 7 Ratios, Rates, and Unit Rate Proportional Relationships and the Constant of Proportionality Scale Drawings and Scale Factor

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM The transition into ratios and proportional relationships: 8 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships When do we really use ratios? 9 Consider: The ratio of Jerry’s allowance to Sam’s allowance is 2:1, the ratio of Sam’s allowance to Misti’s allowance is 1:3. If Misti gets $12 a month, how much is Jerry’s allowance? 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 real- world.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Transitioning into ratio problems 10 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Definitions and Descriptions 11 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 12 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 13 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 14 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Intervention 15 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 16 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 17 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 18 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 19 Example 6: The ratio of the number of Ingrid’s stamps to the number of Ray’s stamps is 3 : 7. If Ingrid gives one- sixth 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Double Number Line Diagrams 20 Example: Rate Problems A photocopier can print 12 copies in 36 seconds. At this rate, how many copies can it print in 1 minute?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 21 Ratios & Proportional Relationships Representations of Equivalent Ratios

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 22 Monique walks 3 miles in 25 minutes. vs. Sean spends 5 minutes watching television for every 2 minutes he spends on homework. Ratios & Proportional Relationships Equivalent Ratios-Tape Diagrams vs. Double Number Line Diagrams

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Rate and Unit Rate 23 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grades 6 and 7 24 Ratios, Rates, and Unit Rate Proportional Relationships and the Constant of Proportionality Scale Drawings and Scale Factor

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Ratio & Proportions Exercise 1 25 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 26 Cups of Grapes Cups of Peaches Cups of Peaches Cups of Grapes Cups of Peaches Cups of Grapes Cups of Peaches

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Ratio & Proportions Exercise 2 27 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 28 MilesHours HoursMiles Hours Miles Hours

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© 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 29 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).

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© 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 30

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© 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 31 The equation of the line depicted will be: 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. Cups of Grapes Cups of Peaches Cups of Peaches Cups of Grapes

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM 32 Where and how is the constant of proportionality represented? Ratios & Proportional Relationships Constant of Proportionality (unit rate of the ratio y:d ) Song Downloads cost $3 each.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Testing for Proportional Relationships 33 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Intervention 34 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grades 6 and 7 35 Ratios, Rates, and Unit Rate Proportional Relationships and the Constant of Proportionality Scale Drawings and Scale Factor

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Scale Factor 36 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 37 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Why move things around? 38 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Transformations of the Plane 39 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Three Rigid Transformations 40 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Three Rigid Transformations 41 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Congruence 42 Rigid transformations can be sequenced. Does order matter? Can sequences of rigid transformations be reversed? Work through: G8 M2 L8 G8 M2 L9

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Congruence 43 A congruence is a sequence of basic rigid motions (translations, reflections, or rotations) that maps one figure onto another. Work through:G8 M2 L10

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Congruence 44 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. Work through: G8 M2 L11

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry What does “same shape” mean? 45 Similarity is often referred to as “same shape” (but not necessarily same size). But what does “same shape” mean?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Dilations 46 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Dilations 47 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Dilations 48 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Similarity 49 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. Exercise leading to The Fundamental Theory of Similarity

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 50 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Transformations of Functions 51 G9-M3 Lesson 17 Example Dilations in Grade 8 vs. Scaling of the Graph of a Function in Grade 9

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 52 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Similarity – The AA Criterion 53 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Slope of a Line 54 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Deriving the Equation for a Line 55 For a line going through the origin:

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Deriving the Equation for a Line 56 For a line passing through point b on the y-axis:

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 57 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Ratios & Proportional Relationships Average Rate vs. Constant Rate 58 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Linear functions and rate of change 59 Grade 8 Module 4 defines slope as a number that describes “steepness” or “slant” of a line. It is the constant rate of change. Students then are asked to connect that terminology to meaningful context: “The information provided for each faucet’s rate of change, i.e. slope, allows us to answer a question… ‘Which faucet is leaking at a faster rate?’”

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Linear functions and rate of change 60

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 8 61 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Why study functions? 62 Functions allow us to make predictions, classify the data in our environment. G8-M5 Lesson 1 Examples 1 & 2

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions What is a function? 63

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Building on Concepts from EE 64 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Linear Equations and Functions Building on Concepts from EE 65 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 L2 Problem Set G8 M5 L5 Exercise 4

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 9 66 Sequences Leading to Functions Analyzing Tables of Data/Patterns in Rate of Change

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Integer Sequences: Should You Believe in Patterns? 67 G9-M3 Lesson 1

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Terms, Term Numbers, and “the term” 68 What do I mean by “the n th term”? Create a table of the terms of the sequence. Term NumberTerm or Value of the Term What would be an appropriate heading for each of our columns? G9-M3 Lesson 1 = 2 0 = 2 1 = 2 2 = 2 3 … 100 n = 2 99 = 2 n

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Introducing the f(n) Notation 69 G9-M3 Lesson 1

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Consolidating Understanding 70 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Arithmetic Sequences 71 Term 1: 5 Term 2: 8 = Term 3: 11 = Term 4: 14 = Term 5: 17 = Term n: G9-M3 Lesson 2 Example 1 = x ? = x 1 = x 2 = x 3 = x 4 = x ?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Recursive Formulas for Sequences 72 G9-M3 Lesson 2 Example 1 G9-M3 Lesson 2

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Geometric Sequences 73 G9-M3 Lesson 2 Example 1 G9-M3 Lesson 3 Example 1

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Arithmetic and Geometric Sequences 74

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Arithmetic and Geometric Sequences 75 G9-M3 Lesson 2 Example 1

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Exponential Functions 76 Which is better? Getting paid $33, 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 2 nd option turn out to be better? What if the experiment only went on for 15 days? G9-M3 Lesson 2 Example 1

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Exponential Growth vs. Linear Growth 77 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

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Why stay with whole numbers? 78 Why are square numbers called square numbers? If S(n) denotes the n th 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) cm 2 of a square of side length x cm. Review Exercises 9–12, taking time to do Problems 10 and 12. Do Exercises G9-M3 Lesson 8

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Agenda: Grade 9 79 Sequences Leading to Functions Analyzing Tables of Data/Patterns in Rate of Change

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Differences and Diagonals 80 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, …

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Differences and Diagonals 81 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, … n 2 : 1, 4, 9, 16, 25, 36, … n 3 : 1, 8, 27, 64, 125, 216, …

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Differences and Diagonals 82 What do you suppose the leading diagonal would look like for n 2 + n? If I gave you only the leading diagonal could you find an explicit formula for the sequence?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Objects in Motion 83 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?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Culminating Exercise from G9-M5 84 Complete the following exercise:

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Key Points 85 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.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Intervention 86 How might the material, as presented to you today, help in providing scaffolding and intervention, whether it be planned or spontaneous?

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© 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios and Functions NYS COMMON CORE MATHEMATICS CURRICULUM Biggest Takeaways 87 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?

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