a story of ratios

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 7 Module 1 – First Half of Lessons


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

Objectives

• Articulate and model the instructional approaches to teaching the content of the first half of the lessons.

• Examine how the topics and lessons promote mastery of the focus standards and address the major work of the grade.

• Articulate connections from the content of previous grade levels to the content of this module.



Participant Poll• Classroom teacher• School leader• Principal• District leader• BOCES representative



Agenda• Review of Module Overview• In-Depth Examination of Module 1 Topics A and B: Lessons 1-10

• Analysis of Topic Openers• Features of Student and Teacher Materials• Modeling of Lesson Components • Coherence Across Grade Levels

• Closure and Reflections



A Story of Ratios Curriculum Overview



Review of Module Structure

Module Overview

Topic

L1 L2L3-

L4

L5-

L6

Topic

L7 L8-L9 L10

Topic L11-

L12L13 L14 L15

TopicL1

6

L1

7

L1

8

L1

9

L2

0

L2

1-

L2

2



Grade 7 Module 1 Overview



What’s in G7-M1? Topic A: Explore what it means to be proportional to and how to determine if

two types of quantities are in a proportional relationship, examining both tables and graphs

Topic B: Define the constant of proportionality and use it to represent proportional relationships with an equation, interpret the meaning of key points on the graph- (0,0) and (1,r) where r is the unit rate

Topic C: Compute unit rates involving fractions, find equivalent ratios of two partial quantities given a part-to-part ratio and the total of the quantities, solve multi-step ratio problems including markup, markdown, and commission

Topic D: Explore scale drawings and recognize the term scale factor as the constant of proportionality



Topic Openers

• Read the concept chart and the descriptive narrative.

• Make note of the important information that will help teachers implement these lessons.



Types of Lessons1. Problem Set

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

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 and teacher practice part of the modeling cycle with problems that are ill-defined and have a real world context.



Lesson 1: An Experience in Relationships as Measuring RateSample Exercise: Paper PassingDirections• One participant at each table, take stack of papers

out of large envelope labeled A. • On my command, take one and pass the remaining

stack to the left. Continue passing until all participants have one paper.

• Last person Stand Up when all participants have a paper (remain standing until further instructions).



Ratio and Rate from Grade 6Ratio: A pair of numbersValue of a Ratio: Students described the fraction A / B associated with the

ratio A : B as the value of the ratio A to B.

Rate: A ratio of two quantitiesUnit rate: The value of the ratioRate’s unit: The label, e.g. mph

Equivalent Ratios: Two ratios : and : are equivalent ratios if there is a 𝐴 𝐵 𝐶 𝐷positive number, , such that = and = . Students understood 𝑐 𝐶 𝑐𝐴 𝐷 𝑐𝐵equivalent ratios to have the same value.



Lesson 1: An Experience in Relationships as Measuring Rate

Example 1: Paper Passing • Why this problem?



Ratio and Rate from Grade 6Ratio: A pair of numbersValue of a Ratio: Students described the fraction A / B associated with the

ratio A : B as the value of the ratio A to B.

Rate: A ratio of two quantitiesUnit rate: The value of the ratioRate’s unit: The label, e.g. mph

Equivalent Ratios: Two ratios : and : are equivalent ratios if there is a 𝐴 𝐵 𝐶 𝐷positive number, , such that = and = . Students understood 𝑐 𝐶 𝑐𝐴 𝐷 𝑐𝐵equivalent ratios to have the same value.



Meet Your TableTalk to your table to determine who is currently teaching in a classroom.

Write a ratio of teachers to non-teachers to reflect the participants at your table.

Enter this information in the first line of your table.



Lesson 1: An Experience in Relationships as Measuring RateContentScan the lesson to identify what content is addressed in the problems for this lesson. What are kids being asked to do?

Turn and talk with others at your table about your observations.



Lesson OrganizationTeacher• Student Outcomes• Lesson Notes (in select lessons)• Classwork

• General directions and guidance, including timing guidance• Discussion points with expected student responses• Student classwork with solutions• Scaffolding Boxes

• Exit Ticket• Problem Set (with solutions)

Student

• Classwork

• Problem Set



Lesson 1: An Experience in Relationships as Measuring RateExit Ticket

Tillman the English BulldogInsert video linkhttp://www.youtube.com/watch?feature=player_embedded&v=tCKstDXMslQ

http://www.youtube.com/watch?feature=player_embedded&v=tCKstDXMslQ



Exit Ticket Reflection What do you notice about the exit ticket? How does it compare to what you

have used in the classroom or have seen teachers use in the past?

Take 3 minutes to talk with your table.



Lesson 2: Proportional Relationships• This lesson

introduces the notion of Proportional Relationship

• Serves as the connecting piece from Grade 6 to Grade 8

Grade 6 •Ratios•Equivalent Ratios•Rate and Unit Rate

Grade 7 •Proportional Relationship•Unit Rate as Constant of Proportionality

Grade 8

and up

•Linear Relationships•Study of Functions



Lesson 2: Proportional Relationships

Weight (ounces)

12.5 10 5 8

Cost ($) 5 4 2 3.20

Classwork

Example 1: Pay by the Ounce Frozen Yogurt!

A new self-serve frozen yogurt store opened this summer that sells its yogurt at a price based upon the

total weight of the yogurt and its toppings in a dish. Each member of Isabelle’s family weighed their dish

and this is what they found.

Cost ____ ______________________ ____ Weight. is proportional to



Lesson 2: Proportional Relationships

Classwork

Example 1: Pay by the Ounce Frozen Yogurt!

• 0.4



Lesson 2: Proportional Relationships Example 2: A Cooking Cheat Sheet!In the back of a recipe book, a diagram provides easy conversions to use while

cooking.

Ounces ____ ______________________ ____ Cups. Exercise 1 During Jose’s physical education class today, students visited activity stations.

Next to each station was a chart depicting how many Calories (on average) would be burned by completing the activity.

Calories burned while Jumping Rope

0 1 2 3 4Time (minutes)Calories Burned

0 11 22 33 44 Is the number of Calories burned proportional to time? How do you know?



Lesson 2: Proportional RelationshipsExercise 1 During Jose’s physical education class today, students visited activity stations.

Next to each station was a chart depicting how many Calories (on average) would be burned by completing the activity.

a. Is the number of Calories burned proportional to time? How do you know?b. If Jose jumped rope for 6.5 minutes, how many calories would he expect to

burn?



Lesson 2: Proportional RelationshipsClosingHow do we know if two quantities are proportional to each other? Two quantities are proportional to each other if there is one constant number

that is multiplied by each measure in the first quantity to give the corresponding measure in the second quantity.

How can we recognize a proportional relationship when looking at a table or a set of ratios?

If each of the measures in the second quantity is divided by its corresponding

measure in the first quantity and it produces the same number, called a

constant, then the two quantities are proportional to each other.



Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in TablesLesson 3 Opening Exercise: You have been hired by your neighbor to

babysit their children Friday night. You are paid $8 per hour. Complete the table relating your pay to the number of hours you worked.

Based on the table above, is pay proportional to hours worked? How do you know?



Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in Tables

Lesson 3: Compare and contrast Examples 1 and 2. Share your observations with a neighbor.



Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in TablesLesson 4 Example: Which team will win Randy’s race?

Take 10 minutes to complete all parts of the problem independently.

Take 5 minutes to share your solutions at your table.

• Be sure to explain your approach to the problem.

• Be prepared to share your approach with the whole group.



Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs

xChocolate Bars Sold

yMoney

Received ($)

2 34 58 9

12 12

Classwork

Opening Exercise:

Isaiah is selling candy bars to help raise money for his scouting troop. The table shows

the amount of candy he sold to the money he received.

Is the amount of chocolate bars sold proportional to the money Isaiah received?

How do you know?



Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs

x y2 3

Example 1: From a table to graph

Create another ratio table that contain two sets of quantities that are proportional to each other using the first set of ratios on the table.



Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in GraphsWhat is a common mistake a student might make when deciding whether a

graph of two quantities shows that they are proportional to each other?

How might we counteract this common misconception?



Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in GraphsTable Activity: Look in Envelope 6 and take 10 minutes to discuss the

given problem and have one table member record responses on the chart paper. At the end of the time, we will post the chart paper on the wall and have an opportunity to participate in gallery walk.



Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in GraphsGallery WalkUse Sticky Notes to…• Note differences found in groups who had the same

ratios. • Note any common mistakes and how it could be fixed.• Note any a poster that stood out that represented

their problems and findings exceptionally clear.



Conclusion of Topic ATake a few moments to reflect upon how you will be able to promote

successful implementation of these lessons in your classroom, school, district, and/or BOCES?

What do you think teachers would need to know?



Topic Opener

In Topic B, students learn to identify the constant of proportionality by finding

the unit rate in the collection of equivalent ratios. They represent this

relationship with equations of the form y = kx, where k is the constant of

proportionality (7.RP.2, 7.RP.2c). In Lessons 8 and 9, students derive the

constant of proportionality from the description of a real world context and

relate the equation representing the relationship to a corresponding ratio

table and/or graphical representation (7.RP.2b, 7.EE.4). Topic B concludes

with students consolidating their graphical understandings of proportional

relationships as they interpret the meanings of the points (0,0) and (1, r),

where r is the unit rate, in terms of the situation or context of a given

problem (7.RP.2d).



Lesson 7: Unit Rate as the Constant of Proportionality

Haven’t we already learned about the Constant of Proportionality?

Why is this lesson here?




Closing Questions What is another name for the number that relates the measures of two

quantities? (How do I get from x to y?)

How is the Constant of Proportionality related to the unit rate?



Lesson 7: Unit Rate as the Constant of ProportionalityExit Ticket

Susan and John are buying cold drinks for a neighborhood picnic, where each person is expected to drink 1 can of soda. Susan says that if you multiply the unit price for a can of soda by the number of people attending the picnic, you will be able to determine the total cost of the soda. John says that if you divide the cost of a 12-pack of soda by the number of sodas, you will be able to determine the total cost of the sodas. Who is right and why?




Exit Ticket Solution



Lesson 8-9: Representing Proportional Relationships with

EquationsTake 15 minutes to go through Lesson 8, completing Examples 1 and 2 and any 2 questions from the Problem Set. Feel free to compare your answers with what is provided in the teacher materials.




EquationsDiscuss at your table: What is the main idea of this lesson? How do the Examples prepare students to find success with the Problem Set?




EquationsLesson 9 Exit TicketOscar and Maria each wrote an equation that they felt represented the

proportional relationship between distance in km and distance in miles. One entry in the table paired 150 km with 93 miles. If k = number of kilometers and m= number of miles, who wrote the correct equation that would relate miles to kilometers and why?



Lesson 10: Interpreting Graphs of Proportional

RelationshipsSpend 3 minutes previewing Lesson 10 materials.

What are your initial observations about this lesson?

Take 15 minutes to complete Examples 1 and 2.

How do you anticipate students will respond to this lesson?



Biggest Takeaway• What is your biggest takeaway from this session? Think for just 1 minute.

• “Whip Around” the table to share.



Key Points • Lessons are designed for students to be active learners. Teacher

questioning guides students to explore ideas, investigate natural questions, and develop the big ideas that are the focus of the module. They are uncovering the story.

• Students use the study of ratios in Grade 6 as a foundation to examine the characteristics of a proportional relationship in Grade 7.

• Students make connections between multiple representations of proportional relationships (verbal, diagram, table, graph) using each one to develop a deeper understanding of what really make a proportional relationship.


NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosG7-M1 : Mid-Module Assessment



Objectives

• To articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment



Agenda• View and complete the Mid-Module Assessment.

• Discuss the rubric and use it to score each other’s assessment.

• Share notable examples of work and scoring.

• Round-table and whole group discussions at various times in the session related to the assessment questions and scoring.



G7-M1: Ratios and Proportional Relationships



G7-M1 Mid-Module Assessment



Standards Assessed

• Which Topics and Standards will be assessed?



Understanding the Assessment



A Progression Toward Mastery



Scoring Student Work



Sharing Student Work



Standards for Mathematical Practice



Evidence of the Mathematical Practices

• How and where do they appear in the assessment items?



Biggest Takeaway



Key Points • Assessment questions

• Are directly aligned to the CCSSM• May address multiple learning standards • Are formatted to invoke higher level thinking and address MPs

• Scoring Rubric• Progression Toward Mastery

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