a story of ratios

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

NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

A Story of RatiosImplementation in a Transition YearPrincipal’s Session



Session Objectives– Closely examine a series of lessons in order to make

choices regarding implementation.– Understand how specific choices support successful

pacing while bridging gaps in prior knowledge and honoring the balance of rigor.

– Share strategies and successes in implementation of A Story of Ratios.

– Practice using the information in our daily administrative routines

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Agenda

Tuesday– Getting Focused– Grade 6– Grade 7– Grade 8

Wednesday– Implementation in a Transition Year Panel– Grade 9– Professional Development for SY 14-15

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A Story of RatiosImplementation in a Transition Year



Getting Started• Introductions• What are the issues I would love to have time

to discuss with colleagues?• Synthesize learning goals• Share out focus for SY 14-15 PD



A Story of RatiosModule Focus Grade 6



CCLS Checklist for A Story of Ratios

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What’s In a Module?• Teacher Materials• Module Overview• Topic Overviews• Daily Lessons• Assessments• Student Materials• Daily Lessons with Problem Sets• Copy Ready Materials• Exit Tickets • Fluency Worksheets / Sprints• Assessments



Types of Lessons1. Problem Set

Students and teachers work through examples and complete exercises to develop or reinforce a concept.

2. SocraticTeacher leads students in a conversation to develop a specific concept or proof.

3. ExplorationIndependent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge.

4. ModelingStudents practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined.



What’s In a Lesson?• Teacher Materials Lessons• Student Outcomes and Lesson Notes (in select lessons)• Classwork • General directions and guidance, including timing guidance• Bulleted discussion points with expected student responses• Student classwork with solutions (boxed)

• Exit Ticket with Solutions• Problem Set with Solutions• Student Materials• Classwork• Problem Set



Grade 6 Module 4



G6-M4: Module Overview



Lesson 1 – Introduction to Fluency Activities



Lesson 2 – Exploratory Challenge!• Work in pairs or small groups to determine

number sentences to show the relationship between multiplication and division. Use tape diagrams to provide support for your findings.

• Create two number sentences to show the relationship between multiplication and division. These number sentences should be identities and include variables. Use the squares to develop these number sentences.

• • Write your number sentences on large paper.

Show a series of tape diagrams to defend each of your number sentences.



Lesson 4Tape Diagram Demonstration



Lesson 4



Lesson 5



Lesson 6



Lesson 8: Replacing Numbers with Letters• Students understand that a letter in an expression can represent a number.

When that number is replaced with a letter, an expression is stated.• Students discover the commutative properties of addition and

multiplication, the additive identity property of 0, and the multiplicative identity property of 1. They determine that g÷ 1 = g, g ÷ g = 1, and 1 ÷ g = 1/g.



Lesson 8



Tape Diagram Demonstration

• Lesson 9



Lesson 11: Factoring Expressions

• Students model and write equivalent expressions using the distributive property and move from a factored form to an expanded form of an expression.



Lesson 11: Factoring Expressions




• Lesson 11



Lesson 12

y

4x 5+

4xy 5y



Lesson 23 – True and False Number Sentences• Students explain the meaning of equality and inequality symbols. They

determine if a number sentence is true or false based on the meaning of the equality/inequality symbol.




• Lesson 26

• This lesson serves as a means for students to solve one-step equations through the use of tape diagrams. Through the construction of tape diagrams, students create algebraic equations and solve for one variable. In this lesson, students continue their study of the properties of operations and identity and develop intuition of the properties of equality. This lesson continues the informal study of the properties of equality students have practiced since first grade and also serves as a springboard to the formal study, use, and application of the properties of equality seen in seventh grade. While students will intuitively use the properties of equality, understand that diagrams are driving the meaning behind the content of this lesson. This lesson purposefully omits focus on the actual properties of equality, which is reserved for Grade 7. Students will relate an equation directly to diagrams and verbalize what they do with diagrams to construct and solve algebraic equations.




• Lesson 27

• This lesson teaches students to solve one-step equations using tape diagrams. Through the construction of tape diagrams, students will create algebraic equations and solve for one variable. This lesson not only allows students to continue studying the properties of operations and identity, but also allows students to develop intuition of the properties of equality. This lesson continues the informal study of the properties of equality students have practiced since first grade, and also serves as a springboard to the formal study, use, and application of the properties of equality seen in Grade 7. Understand that, while students will intuitively use the properties of equality, diagrams are the focus of this lesson. This lesson purposefully omits focusing on the actual properties of equality, which will be covered in Grade 7. Students will relate an equation directly to diagrams and verbalize what they do with diagrams to construct and solve algebraic equations.



Your Most Important Takeaway from Grade 6 Module 4• Take a moment to reflect on this module and

make a note of the key takeaways.



Grade 6 Module 5



Module’s Foundation

• Standards:• Pages 18-20 in the Progressions Document (K-

6,Geometry,) serves as a foundation.



TerminologyNew or Recently Introduced Terms Triangular region Altitude and base of a triangle Pentagon Hexagon Line Perpendicular to a plane Parallel planes Right rectangular prism Cube Surface of a prism



Building upon prior work



Students discover through manipulation that the area of all triangles is exactly half the area of its corresponding rectangle.



Additional Scaffolding



Key Points• Students utilize and build upon their previous understanding of composition and

decomposition from Grades 1 – 5. • Area and volume are additive. • Students determine the formulas for area from their knowledge of the area of a

rectangle and how it can be composed and decomposed. • Students use their prior knowledge of Module 2 to calculate the volume of right

rectangular prisms with unit cubes with fractional lengths. • Students use their prior knowledge of Module 3 to determine area and surface

area of polygons through distance calculations. • Students use their prior knowledge of Module 4 to use equations to determine

missing angles, as well as to use formulas for area, volume, and surface area. • Students apply formulas to real-life contexts and distinguish between the need to

find surface area or volume within contextual situations.



Grade 7 Module 4



Percent

• Color in a 10 x 10 grid to represent each fraction:

• Lesson 1 / Opening Exercise



• What are equivalent representations of ?

Percent




• What are equivalent representations of Percent




Comparing Quantities with Percent

• Six club members decided to evenly split the total number of bracelets to be produced [300 bracelets]. Of the 54 bracelets produced over the weekend, Anna produced 32 bracelets. Compare the number of bracelets that Emily produced [22] as a percent of those that Anna produced.

• Compare the number of bracelets that Anna produced as a percent of the number that Emily produced.

• Lesson 3 / Example 1(a) and (b)



Find One Hundred Percent Given Another Percent• Nick currently has 7,200 points in his fantasy

baseball league which is 20% more points than Adam has. How many points does Adam have?

• Lesson 5 / Exercise 2



Lesson 7: Markup and Markdown Problems• Students understand the terms original price, selling price,

markup, markdown, markup rate, and markdown rate.

• Students identify the original price as the whole and use their knowledge of percent and proportional relationships to solve multi-step markup and markdown problems.

• Students understand equations for markup and markdown problems and use them to solve markup and markdown problems.• Lesson 7 / Student Outcomes



Markup and Markdown Problems•Exercise 4:a) Write an equation to determine the selling price, , on an item

that is originally price dollars after a markup of 25%.

b) Create a table (and label it) showing five possible solutions to your equation.

c) Create a graph (and label it) of your equation.

d) Interpret the points and .• Lesson 7 / Exercise (4)



Lesson 8: Percent Error Problems• Given the exact value, , of a quantity and an approximate value, ,

of the quantity, students use the absolute error, , to compute the percent error by using the formula .

• Students understand the meaning of percent error: the percent the absolute error is of the exact value.

• Students understand that when an exact value is not known, an estimate of the percent error can still be computed when given a range determined by two inclusive values; (e.g., if there are known to be between 6,000 and 7,000 black bears in New York, but the exact number is not known, the percent error can be estimated at most , which is .• Lesson 8 / Student Outcomes



Percent Error Problems

• The attendance at a musical event was counted several times. All counts were between and . If the actual attendance number is between and , inclusive, what is the most and least the percent error could be? Explain your answer.

• Lesson 8 / Example 3



Understanding Simple Interest• Larry invests $100 in a savings plan. The plan pays interest each year on

his $100 account balance. The following chart shows the balance on his account after each year for the next five years. He did not make any deposits or withdrawals during this time.


Time (in years) Balance (in dollars)

1 104.50

2 109.00

3 113.50

4 118.00

5 122.50



Lesson 12: The Scale Factor as a Percent for a Scale Drawing• Given a scale factor in percent, students make a scale drawing of a

picture or geometric figure using that scale, recognizing that the enlarged or reduced distances in a scale drawing are proportional to the corresponding distances in the original picture.

• Students understand scale factor to be the constant of proportionality.

• Student make scale drawings in which the horizontal and vertical scales are different.

• Lesson 12 / Student Outcomes



Changing Scales• A regular octagon is an eight-sided polygon with side lengths that are all equal. All

three octagons are scale drawings of each other. Use the chart and the side lengths to compute each scale factor as a percent. How can we check our answers?




Computing Actual Lengths from a Scale Drawing

•The distance around the entire small boat is 28.4 units. The larger figure is a scale drawing of the smaller sketch of the boat. State the scale factor as a percent, and then use the scale factor to find the distance

around the scale drawing.




Solving Area Problems Using Scale Drawings

• What percent of the area of the large disk lies outside the smaller disk?




Solving Area Problems Using Scale Drawings• Use Figure 1 below and the enlarged scale drawing to justify why the area

of the scale drawing is times the area of the original figure.




Counting Problems1. How many 4-letter passwords can be formed using the

letters “A” and “B”?

2. What percentage of the 4-letter passwords contain:a) No “A’s”?b) Exactly one “A”?c) Exactly 2 “A’s”?d) Exactly 3 “A’s”?e) 4 “A’s”?• Lesson 18 / Exercises 1-2



Key Points• Percent is a part to whole relationship.• Focus on identifying the whole quantity (or quantities) in percent

problems.• Greater fluency with percent improves problem solving abilities.• Percent problems can be represented in a variety of models including

equations, visual models, and numeric models.• Percent can compare a part of a quantity to the whole quantity, or can

compare two separate quantities.• A percent of a set of quantities represents a proportional relationship.• Percent error is not just a formula to memorize … it has meaning.



Key Points

• The scale factor is the unit rate (in percent form)• Scale drawings may have more than one scale

factor (horizontal and vertical scales).• Given a drawing , and scale drawing of drawing

with a scale factor , drawing is a scale drawing of drawing with scale factor .

• Work with percents in module 4 ushers in the topics of probability and statistics in module 5.



Grade 7 Module 5



Statistics and Probability Module

Topic ACalculating and

Interpreting Probabilities

Topic BEstimating

Probabilities

Topic CRandom Sampling and Estimated Population

Characteristics

Topic DComparing Populations

Mid-Module Assessment

End-of-Module Assessment

• Has 7 lessons• Covers Standards

7.SP.C.5 7.SP.C.67.SP.C.7 7.SP.C.8a-b


7.SP.C.6 7.SP.C.77.SP.C.8c


7.SP.A.1 7.SP.A.2


7.SP.B.3 7.SP.B.4



The Terminology of Probability•Jamal, a 7th grader, wants to design a game that involves tossing paper cups. Jamal tosses a paper cup five times and records the outcome of each toss. An outcome is the result of a single trial of an experiment. •Here are the results of each toss:

•Jamal noted that the paper cup could land in one of three ways: on its side, right side up, or upside down. The collection of these three outcomes is called the sample space of the experiment. The sample space of an experiment is the set of all possible outcomes of that experiment. • Lesson 3 – Example 1



Equally Likely Outcomes?•The sample space for the paper cup toss was on its side, right side up, and upside down. Do you think each of these outcomes has the same chance of occurring? If they do, then they are equally likely to occur.

•The outcomes of an experiment are equally likely to occur when the probability of each outcome is equal.

•You and your partner toss the paper cup 30 times and record in a table the results of each toss. • Lesson 3 – Example 2



Using Simulation to Estimate a Probability• 1. Using colored disks, describe how one at-bat could be

simulated for a baseball player who has a batting average of 0.300. Note that a batting average of 0.300 means the player gets a hit (on average) three times out of every ten times at bat. Be sure to state clearly what a color represents.

• Lesson 11 Exercise 1



Using Simulation to Estimate a Probability• 2. Using colored disks, describe how one at-bat could be simulated for a player who

has a batting average of 0.273. Note that a batting average of 0.273 means that on average, the player gets 273 hits out of 1000 at-bats.

•Discuss with your partner why it is NOT a good ideas to use colored disks to simulate this situation.•Use 000 – 272 to represent a hit• 273 – 999 represents a miss


25256 65205 72597 00562 12683 90674 78923 96568 32177 33855 76635 92290 88864 72794 14333 79019 05943 77510 74051 87238 07895 86481 94036 12749 24005 80718 13144 66934 54730 77140



•The owners of a gym have been keeping track of how long each person spends at the gym. Eight hundred of these times (in minutes) are shown in the population tables located at the end of the lesson. From this population you will take a random sample.

•What are the five observations in your sample?

•For the sample that you selected, calculate the sample mean.

Sampling Variability

• Lesson 17 Exercises 1 - 4

Do you think that the mean of these five observations is exactly correct for the population mean?

Could the population mean be greater than the number you

calculated? Smaller?

Suppose you were to take another random sample from the same population of times

at the gym. Could the new sample mean be closer to the population mean than the mean of these five observations? Further?



Sampling Variability •Use all the sample means to make a dot plot using the axis given below. (Remember, if you have repeated or close values, stack the dots one above the other.)

• Lesson 17 Exercises 8-11

What do you see in the dot plot that demonstrates sampling variability?

Suppose that a statistician plans to take a random sample of size 5 from the population of times spent at the gym and that he or she will use the sample mean

as an estimate of the population mean. Approximately how far can the statistician expect the sample mean to

be from the population mean?



Why Worry about Sampling Variability? •There are three bags, Bag , Bag , and Bag , with 100 numbers 𝐴 𝐵 𝐶in each bag. You and your classmates will investigate the population mean (the mean of all 100 numbers) in each bag. Each set of numbers has the same range. However, the population means of each set may or may not be the same. We will see who can uncover the mystery of the bags!

• Lesson 21

A B C



Why Worry about Sampling Variability? •Each bag has a population mean that is either 10.5 or 14.5. State what you think the population mean is for each bag. Explain your choice for each bag.


14.5

A B C

14.5 10.5



End-of-Module Assessment•Students in a random sample of 57 students were asked to measure their hand-spans (distance from outside of thumb to outside of little finger when the hand is stretched out as far as possible). The graphs below show the results for the males and females.

• Exercise 2



End-of-Module Assessment• a. Based on these data, do you think there is a difference

between the population mean hand-span for males and the population mean hand-span for females? Justify your answer.

• Exercise 2



End-of-Module Assessment• b. The same students were asked to measure their heights, with

the results shown below.

•Are these height data more or less convincing of a difference in the population mean height than the hand span data are of a difference in population mean handspan? Explain.• Exercise 2



Grade 8 Module 5



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



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?



Discussion• We can write a mathematical rule to describe the

movement of the falling stone.

• 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?



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.



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.



Example 4

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

in the above table?



Exercise 3



Problem Set 2-Just for Fun



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



Exercise 1



Fluency Activity



Exercise 4



Exercise 2



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.



L11: Volume of a Sphere

• Lesson 11, Concept Development

• Students know the volume formula for a sphere as it relates to the volume of a right cylinder with the same diameter and height.



Grade 8 Module 6



Topic A: Linear Functions



Topic B: Bivariate Numerical Data

• Lesson 6 Scatter Plots

• Lesson 7 Patterns in Scatter Plots

• Lesson 8 Informally Fitting a Line

• Lesson 9 Determining the Equation of a Line Fit to Data



Exit Ticket for Lesson 8



Mid- Module Assessment



Topic C: Linear and Nonlinear Models• Lesson 10: Linear Models

• Lesson 11: Using Linear Models in a Data Context

• Lesson 12: Nonlinear Models in a Data Context (Optional lesson)



Topic D: Bivariate Categorical Data• Lesson 13: Summarizing Bivariate Categorical Data in a Two-

Way Table

• Lesson 14: Association between Categorical Variables



End of Module Assessment



Module Summary: Key Ideas• Grade 8 Module 6 is a key. More use of functions and preparation

for Grade 9 Algebra and Functions Conceptual Categories.• Big theme is ASSOCIATION• Key ideas: • •Patterns in Scatter plots (Association between two

numerical variables)• • Equation of a line • • Fitting a line to data• • Analyzing categorical data from two-way frequency tables.• • Association between two categorical variables


NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions

A Story of FunctionsImplementation in a Transition Year



CCLS Checklist for A Story of Functions

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Key Points• The CCLS Checklist for A Story of Functions is a valuable tool

in keeping an eye on the big picture at each grade-level.• When making decisions regarding implementation, it is

important to honor the objective of each lesson.• The exit tickets of each lesson are helpful in this endeavor.• The lesson objectives are sequenced carefully toward

mastery of each topic.• Careful attention should be given to maintaining a balance of

rigor.

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What’s In a Module?• Teacher Materials• Module Overview• Topic Overviews• Daily Lessons• Assessments• Student Materials• Daily Lessons with Problem Sets• Copy Ready Materials• Exit Tickets • Fluency Worksheets / Sprints• Assessments



Types of Lessons1. Problem Set

Students and teachers work through examples and complete exercises to develop or reinforce a concept.

2. SocraticTeacher leads students in a conversation to develop a specific concept or proof.

3. ExplorationIndependent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge.

4. ModelingStudents practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined.



What’s In a Lesson?• Teacher Materials Lessons• Student Outcomes and Lesson Notes (in select lessons)• Classwork • General directions and guidance, including timing guidance• Bulleted discussion points with expected student responses• Student classwork with solutions (boxed)

• Exit Ticket with Solutions• Problem Set with Solutions• Student Materials• Classwork• Problem Set



Grade 9 Module 4



A Foundation for the Study of Quadratics: Part 3 – A closer look at u-shaped curves1. Tape the ends of the chain so that the lowest part of the chain

falls right at the origin.2. Identify several other points that the chain goes through. 3. Create a quadratic equation that goes through the points you

identified.



Flow of Module 4• Topic A: Quadratic Expressions, Equations, Functions, and their

Connection to Rectangles• Reversing multiplication yields factored expressions (recall

geometric models); practice factoring of quadratics.• When combined with the Zero Product Property we have a

new power to solve quadratic equations.• What does the graph of a quadratic equation look like? It’s

symmetric (and u-shaped).• Factored form + symmetry makes graphing simple.• Relating quadratic equations and their graphs to real-world

context, giving contextual interpretations of key features.



Flow of Module 4• Topic B: Using Different Forms for Quadratic Functions• Other ways to see structure in quadratics – solving by

completing the square; the quadratic formula.• Why does completing the square yield something we call

“vertex form”? The relationship between vertex form and transformations; the helpfulness of vertex form in graphing.

• Further examination of quadratic functions and their graphs in context.



Flow of Module 4• Topic C: Function Transformations and Modeling• The square root function and its relationship to the basic

quadratic function; the cube root and cubic functions.• Transformations of all of these types of functions.• Analyzing and comparing functions represented in different

forms, all done in context.



What are students coming in with?• Experience multiplying with polynomials using the distributive

property (G9-M1)• Experience relating the distributive property to an area model

or an modified area model (the tabular method) (G9-M1)• Experience writing a sum as a product of two factors (G7-M3)

and factoring out a greatest common factor (G6-M2)• Experience transforming graphs, transforming functions and

relating the transformed function to the transformed graph (G9-M3)



Key Points – Topic A• Consider having students come up with their own summaries for

how they approach factoring /solving / graphing a quadratic.• It’s better to study deeply a given application problem and the

analysis of its graph’s features than to do multiple problems.• Introduce concepts like domain, range, increasing, decreasing,

average rate of change, etc. by using words that feel natural in the context, and then repeat the statement or question using the more formal words.

• Scaffolds are a critical tool for successful implementation. In addition to those given in the module, consider the ones we explored in this session. (Take time now to reflect and take note of them.)



Key Points – Topic B• Completing the square has a geometric meaning.• A scaffold for completing the square when the leading coefficient is

not 1 involves multiplying the equation through first by the leading coefficient (if not already a perfect square) and then by the factor , if the coefficient of the term is not easily halved.

• This same scaffold used with the geometric model provides an alternative to the purely algebraic derivation of the quadratic formula.

• The final lesson should include a reflection on the student’s general strategy for graphing quadratic functions.

• Lessons 16-21 in Topics B and C provide a second opportunity for students to master transformations of functions.



Key Points – Topic C• Comparing features of functions provided in different forms

deepens and consolidates student understanding of the relationship between the structure of expressions and equations, the graphs of equations and functions, and the contexts they model.

• Students should walk away from quadratics understanding that a primary use of these functions is in modeling height over time of projectile objects, that they are naturally related to rectangular area problems, and that there are also used in an early study of business applications.



Key Points – End-of-Module Assessment• End of Module assessment are designed to assess all standards

of the module (at least at the cluster level) with an emphasis on assessing thoroughly those presented in the second half of the module.

• Recall, as much as possible, assessment items are designed to asses the standards while emulating PARCC Type 2 and Type 3 tasks.

• Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades.



• Take a moment to reflect on this module and make a note of the key takeaways.

Your Most Important Takeaway from Grade 9 Module 4

a story of ratios

Documents

successes of implementation

work session

module focus session

share strategies

transition year panelgrade

specific choices

minutes materials

balance of rigor