© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios A Story of Ratios Grade 7 Module 1 – First Half of Lessons
Feb 24, 2016
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios
A Story of RatiosGrade 7 Module 1 – First Half of Lessons
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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.
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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
Participant Poll• Classroom teacher• School leader• Principal• District leader• BOCES representative
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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
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
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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
A Story of Ratios Curriculum Overview
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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
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
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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
Grade 7 Module 1 Overview
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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
Grade 7 Module 1 Overview
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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
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
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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
Topic Openers
• Read the concept chart and the descriptive narrative.
• Make note of the important information that will help teachers implement these lessons.
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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
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.
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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
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).
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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
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.
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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
Lesson 1: An Experience in Relationships as Measuring Rate
Example 1: Paper Passing • Why this problem?
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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
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.
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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
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.
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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
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.
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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
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
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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
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
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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
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.
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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
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
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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
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
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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
Lesson 2: Proportional Relationships
Classwork
Example 1: Pay by the Ounce Frozen Yogurt!
• 0.4
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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
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?
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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
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?
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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
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.
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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
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 $8 per hour. Complete the table relating your pay to the number of hours you worked.
Based on the table above, is pay proportional to hours worked? How do you know?
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N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios
Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in Tables
Lesson 3: Compare and contrast Examples 1 and 2. Share your observations with a neighbor.
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N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios
Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in TablesLesson 4 Example: Which team will win Randy’s race?
Take 10 minutes to complete all parts of the problem independently.
Take 5 minutes to share your solutions at your table.
• Be sure to explain your approach to the problem.
• Be prepared to share your approach with the whole group.
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N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios
Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs
xChocolate Bars Sold
yMoney
Received ($)
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?
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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
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.
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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
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?
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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
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.
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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
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.
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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
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?
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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
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).
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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
Lesson 7: Unit Rate as the Constant of Proportionality
Haven’t we already learned about the Constant of Proportionality?
Why is this lesson here?
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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
Lesson 7: Unit Rate as the Constant of Proportionality
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?
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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
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?
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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
Lesson 7: Unit Rate as the Constant of Proportionality
Exit Ticket Solution
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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
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.
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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
Lesson 8-9: Representing Proportional Relationships with
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?
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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
Lesson 8-9: Representing Proportional Relationships with
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?
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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
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?
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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
Biggest Takeaway• What is your biggest takeaway from this session? Think for just 1 minute.
• “Whip Around” the table to share.
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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
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.
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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios
A Story of RatiosG7-M1 : Mid-Module Assessment
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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
• To articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment
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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
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.
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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
G7-M1: Ratios and Proportional Relationships
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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
G7-M1 Mid-Module Assessment
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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
Standards Assessed
• Which Topics and Standards will be assessed?
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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
Understanding the Assessment
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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
A Progression Toward Mastery
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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
Scoring Student Work
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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
Sharing Student Work
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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
Standards for Mathematical Practice
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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
Evidence of the Mathematical Practices
• How and where do they appear in the assessment items?
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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
Biggest Takeaway
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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
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