Grade 6 Module 1 Study Mid-Module Assessment Presentation

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosModule 1 Focus - Grade 6


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.

• Examine lesson components including Examples vs. Exercises, Application Problems, Concept Development, Problem Sets, and Exit Tickets.



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



Agenda• Review of Module Structure• Examination of the Module Overview and the Topic Openers• Lesson Study• Coherence Across the Module



A Story of Ratios



Review of Module Structure

Module Overview

Topic A Topic B Topic C Topic D

Lessons 1

- 8

Lessons 9

- 15

Lessons

16 -23

Lessons

24 - 29






Module Overview



Topic Openers – Topic A

• Read the descriptive narrative. • Make note of important information that will help educators implement these

lessons.



Topic Openers – Topic B

• Read the descriptive narrative. • Make note of important information that will help educators implement these

lessons.



Topic Openers – Topics A and B• How does each topic contribute

to the overall instructional goal of the module?

• How are the Topic Openers useful as a planning tool?

• What is the relationship between the Topic Opener and the other components of the module?






Lesson Study• Examine the development and function

of each lesson component.

• Lesson Type (Socratic, Modeling Cycle, Exploration, Problem Set)

• Lesson Title• Student Outcomes• Optional Lesson Notes• Classwork: Examples and Exercises • Application Problems• Concept Development• Practice Set*• Student Debrief/Exit Ticket



Lesson 1Student Outcomes• Students understand that a ratio is an ordered pair of non-negative

numbers, which are not both zero. • Students understand that a ratio is often used in lieu of describing the first

number as a multiple of the second. • Students use the precise language and notation of ratios (3: 2, 3 to 2).• Students understand that the order of the pair of numbers in a ratio

matters, and that the description of the ratio relationship determines the correct order of the numbers.

• Students conceive of real-world contextual situations to match a given ratio.



Lesson 1 - RatiosLesson Notes:• The first two lessons of this module will develop the students

understanding and definition of the term ratio. • A ratio is always a pair of numbers, like 2:3, and never a pair of quantities

like 2 cm : 3 sec. Keeping this straight for students will require consistently correct use of the term ratio. It will require keeping track of the units in a word problem separately.

• To help distinguish between ratios and statements about quantities that define ratios, we use the term ratio relationship to describe an English phrase in a word problem that indicates a ratio.

• Typical examples of ratio relationship descriptions include, “3 cups to 4 cups,” “5 miles in 4 hours,” etc. The ratios for these ratio relationships are 3:4 and 5:4, respectively.



Lesson 1- RatiosThe coed soccer team has four times as many boys on it as it has girls. We say

the ratio of the number of boys to the number of girls on the team is 4:1, we read this as “four to one.”

• Let’s create a table to show how many boys and how many girls are on the team.

# of Boys # of Girls Total # of Players

4 1 5

8 2 10

12 3 15

From the table, we can see that there are four boys for every one girl on the team.



Lesson 1- RatiosSuppose the ratio of boys to girls on a different soccer team is 3:2Create a table like the one shown below to show possibilities of the number of

boys and girls on the soccer team.

# of Boys # of Girls Total # of Players

3 2 5

6 4

10 9 6

15



Lesson 1- RatiosCan we make a tape diagram (or bar model) that shows that there are 3/2 as

many boys as girls? Which description makes the relationship easier to visualize? Saying the ratio

is 3 to 2 or saying there are 3 halves as many boys as girls? There is no right or wrong answer, have students explain why they picked their

choice.



Lesson 1 - RatiosRead through the rest of the lesson to see how it develops. Share your thoughts with your neighbor and write down key points you would

like to discuss and share with the whole group.

Five minute share.



Lesson 2 - Ratios• Students reinforce their understanding that a ratio is an ordered pair of

non-negative numbers, which are not both zero. • Students continue to learn and use the precise language and notation of

ratios (3: 2, 3 to 2).• Students recognize ratio language as such.• Students demonstrate their understanding that the order of the pair of

numbers in a ratio matters.• Students create multiple ratios from a context in which more than two

quantities are given.• Students conceive of real-world contextual situations to match a given

ratio.



Lesson 2 - Ratios



Lesson 3 – Equivalent Ratios• Students develop an intuitive understanding of equivalent ratios by using

tape diagrams to explore possible quantities of each part given the part to part ratio.

• Students use tape diagrams to solve problems where the part to part ratio is given and the value of one of the quantities is given.

• Students formalize a definition of equivalent ratios: Two ratios : and 𝐴 𝐵: are equivalent ratios if there is a positive number, , such that = 𝐶 𝐷 𝑐 𝐶 𝑐𝐴

and = .𝐷 𝑐𝐵



Lesson 3 – Equivalent RatiosShanni and Mel are using ribbon to decorate a project in their art class. The ratio of the length of

Shanni’s ribbon to Mel’s ribbon is 7:3. • Create a table.• Draw a tape diagram to represent this ratio.

Shanni

Mel

7 inches

3 inches

7:3

7 to 3

2 m2 m 2 m2 m 2 m2 m 2 m

2 m 2 m2 m

14 meters

6 meters

14:6

14 to 6

21 inches

9 inches

21:9

21 to 9

What ratios can we say are equivalent to 7:3?



Lesson 3 – Equivalent RatiosTake 5 minutes to work through Exercise 4 with your neighbor.

Share your thoughts!



Lesson 4 – Equivalent Ratios• Given a ratio, students will identify equivalent ratios.• Students use tape diagrams and the description of equivalent ratios to

determine if two ratios are equivalent. • Students relate the common factor, c, in the description of equivalent

ratios to the tape diagrams they’ve been using to find equivalent ratios.



Lesson 4 – Equivalent Ratios



Lesson 4- Equivalent RatiosExercise 3:In a bag of mixed walnuts and cashews, the ratio of walnuts to cashews is 5:6.

Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by determining equivalent ratios.

54

walnuts

cashews



Lesson 4- Equivalent RatiosExercise 3:In a bag of mixed walnuts and cashews, the ratio of walnuts to cashews is 5:6.

Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by determining equivalent ratios.



Correlations Between Lessons 1-2 and 3-4• Take 3 minutes to discuss with your table partners the correlations you see

between Lessons 1 and 2 to Lessons 3 and 4. • Share with whole group.



Lesson 5 – Solving Problems by Finding Equivalent Ratios

• Students use tape diagrams to find an equivalent ratio given the part to part ratio and the total of those quantities.

• Students use tape diagrams to find an equivalent ratio given the part to part ratio and the difference between those two quantities.

• Students make the connection between the common factor, c, in definition of equivalent ratios and the value of the unit in the tape diagram used to solve ratio problems.



Lesson 5 – Solving Problems by Finding Equivalent RatiosA County Superintendent of Highways is interested in the numbers of different types of

vehicles that regularly travel within his county. In the month of August a total of 192 registrations were purchased for passenger cars and pickup trucks at the local Department of Motor Vehicles. They reported that in the month of August, for every 5 passenger cars registered there were 7 pickup trucks registered. How many of each type of vehicle were registered there in the month of August?

5:7 part-to-part7:5 part-to-part5 to 12 part-to-whole7 to12 part-to-whole

Passenger Cars

Pickup Trucks

12 equal parts

192 vehicles

192 divided by 12 = 16

5 x 16 = 80 passenger cars

7 x 16 = 112 pickup trucks



Lesson 6 – Solving Problems by Finding Equivalent Ratios

Gallery Walk! Spend 2 minutes at each station to determine the answers to each question.

Share with the group the rigor you find in the problems provided in the examples.

• Students use tape diagrams to solve problems given a ratio between two quantities, and a change to those quantities that

changes the ratio.




between Lessons 1 -4 and Lessons 5-6. • Share with whole group.



Lesson 7 – Associated Ratios and the Value of a Ratio

• Students understand the relationship between ratio and fractions.• Students understand that given a ratio A : B, different ratios can be formed

from the numbers A and B, such as B : A, A : (A + B), and B : (A + B), that are associated with the same ratio relationship.

• Students describe the fraction A / B associated with the ratio A : B as the value of the ratio A to B.







Take five minutes to discuss the duration of the lesson with your neighbor.

Take note of how the lesson evolves, noting connections to the past lessons, as well as the rigor that is encompassed within the lesson.



Lesson 8 - Equivalent Ratios Defined Through Value of a Ratio

• Students understand the value of a ratio A:B is A/B. They understand that if two ratios are equivalent they have the same value.

• Students use the value of a ratio to solve ratio problems in a real-world context.

• Students use the value of a ratio in determining if two ratios are equivalent.




• Recall that given a ratio A : B, where B ≠ 0, we call the quotient, A / B, the value of the ratio.

• Circle any equivalent ratios from the list below. Ratio: 1 : 2 Value of the Ratio: Ratio: 5 : 10 Value of the Ratio:Ratio: 6 : 16 Value of the Ratio:Ratio: 12 : 32 Value of the Ratio: • What do you notice about the value of the equivalent ratios?• Note that 1:2 is not the same ratio as 5:10, we don’t say they are equal.

The ratios are not the same, but their values are equal. Would this always be the case? Would the values of equivalent fractions always be equal?

1/2

1/2

3/8

3/8



Exercise 2:Here is a theorem: If two ratios are equivalent, then they have the same value.Can you provide any counter-examples to the theorem above?• Allow students to try this in pairs. Observe the progress of students and question

student’s counter-examples. Ask for further clarification or proof that the two ratios are equivalent, but do not have the same value. If students still think they have discovered a counter-example, share the example with the class and discuss why it is not a counter-example.

• Ask entire class if anyone thought of a counter-example. If students share examples, have others explain why they are not counter-examples. Then discuss why there are not possible counter-examples to the given theorem. It is important for students to understand that the theorem is always true so it is not possible to come up with a counter-example.




• Read through the duration of the lesson with a table partner. Be prepared to discuss your findings with the group.

• SHARE!





between Lessons 1 an- 6 to Lessons 7 and 8. • Share with whole group.



Lesson 9 – Tables of Equivalent Ratios • Students understand that a ratio is often used to prescribe the relationship

between the amount of one quantity and the amount of another quantity as in the cases of mixtures or constant rates.

• Students understand that a ratio table is a table of equivalent ratios. • Students use ratio tables to solve problems.• The approach of this lesson and those that follow is for the teacher to

model the use of tables in problem solving. There is no need to engage in an explanation of why or how they are useful; simply modeling their use in this lesson, examining their structure in the next lesson, and repeated use for problem solving in the remaining lessons of the topic should sufficiently promote tables as a tool for problem solving with collections of equivalent ratios.



Lesson 9 – Tables of Equivalent Ratios • Read through Example 1 of Lesson 9 in the Teacher Materials.• Highlight and label with lesson numbers where previous skills from the

module prepare students for this lesson. • Notice Example 2 moves the students from the model to independent

creation of the table. • What skills are being addressed in this lesson?



Lesson 10 – The Structure of Ratio Tables• Students identify both the additive and multiplicative structure of a ratio

table and use the structure to make additional entries in the table.• Students use ratio tables to solve problems.

• Take 5 minutes to read through Lesson 10.• Talk with your table partners to make correlations to Lesson 9. • How does Lesson 10 extend Lesson 9?

• Share with whole group.



• Students solve problems by comparing different ratios using two or more ratio tables.

• Take 5 minutes to read through Lesson 11.• Talk with your table partners to make correlations to Lessons 9 and 10. • How does Lesson 11 extend Lessons 9 and 10?

• Share with whole group.

Lesson 11 – The Structure of Ratio Tables



Lesson 12- From Ratio Tables to Double Number Line Diagrams • Each participant is given a card with a ratio on it. • Move around the room in search of other participants who have ratios that

are equivalent to theirs. • Those with equivalent ratios will form a group and create a ratio table,

which contains all of the equivalent ratios. • Examine Exercise 3 and Example 1 in the Teacher Materials.• Share with the group your thoughts on the questioning strategies in

Exercise 3.• What major points do you walk away with from Example 1?



Lesson 13 - From Ratio Tables to Equations Using the Value of a Ratio

Jorge is mixing a special shade of orange. He has mixed 1 gallon of red paint with three gallons of yellow paint.

Based on the ratio of gallons of yellow paint to gallons of red paint, which of the following statements are true?

• 3/4 of a 4-gallon mix would be yellow. Every 1 gallon of yellow requires 13 gallon of red.

• Every 1 gallon of red requires 3 gallons of yellow.• There is 1 gallon of red in a 4-gallon mix of orange paint.• There are 2 gallons of yellow paint in an 8-gallon mix of orange paint.



• Take 5 minutes to work through the rest of the lesson with a partner or at your table.

• Share your thoughts!

Lesson 13 - From Ratio Tables to Equations Using the Value of a Ratio



Lesson 14 - From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate Plane • Students associate with each ratio A : B the ordered pair (A, B) and plot it in

the x-y coordinate plane.• Students represent ratios in ratio tables, equations and double number line

diagrams, then represent those ratios in the coordinate plane.



Lesson 14 - From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate Plane • Based on their previous knowledge from earlier lessons in this module, and

with predetermined groups, students complete tables to satisfy missing values, create double line diagrams to support the values, and develop an equation to support the values.

• From there, a Socratic approach is used in order to guide students to build a graph on the coordinate plane.

• Read through the questioning in Lesson 14 and discuss with your table partner(s). What do you notice about the questioning?



Lesson 15 - From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate Plane

Gallery Walk! Spend 2 minutes at each station to determine the answers to each question.

Share with the group the rigor you find in the problems provided in the examples.



Biggest Takeaway

• What is your biggest takeaway from this session?

• Be ready to share.



Key Points • Modules Overviews and Topic Openers provide essential information

about the instructional path of the module and are key tools in planning for successful implementation.

• Each of the lesson components are necessary in order to achieve balanced, rigorous instruction and to bring the Standards to life.

• The Exit Ticket is an essential piece of the Student Debrief and provides daily formative assessment.

• Opportunities to nurture the Standards for Mathematical Practice are embedded throughout the lesson.


NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosG6-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• Experience the Mid-Module Assessment from a student-

standpoint.

• Link the Assessment back to the Focus and Foundational standards.

• Complete and score the assessment using the rubric

• Identify the Mathematical Practices embodied in each assessment task.



G6-M1: Ratios and Unit Rates



G6-M1 Mid-Module Assessment



Standards Assessed

• Which Standards and Topics 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 CCSS• May address Multiple Focus Standards • Invoke Higher-Level Thinking and the Mathematical Practices

Scoring Rubric• Progression Toward Mastery• Standards Referenced for each Assessment Item

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Grade 6 Module 1 Study Mid-Module Assessment Presentation

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