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A Story of Ratios

A Story of Ratios. Module 1 Focus - Grade 6. 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.

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A Story of Ratios

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  1. A Story of Ratios Module 1 Focus - Grade 6

  2. 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.

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

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

  5. A Story of Ratios

  6. 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

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

  8. Module Overview

  9. Topic Openers – Topic A • Read the descriptive narrative. • Make note of important information that will help educators implement these lessons.

  10. Topic Openers – Topic B • Read the descriptive narrative. • Make note of important information that will help educators implement these lessons.

  11. 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?

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

  13. 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

  14. Lesson 1 • Student 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.

  15. Lesson 1 - Ratios • Lesson 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.

  16. Lesson 1- Ratios 10 8 2 12 3 15 • From the table, we can see that there are four boys for every one girl on the team. • The 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.

  17. Lesson 1- Ratios 6 4 10 9 6 15 • Suppose the ratio of boys to girls on a different soccer team is 3:2 • Create a table like the one shown below to show possibilities of the number of boys and girls on the soccer team.

  18. Lesson 1- Ratios • Can 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.

  19. Lesson 1 - Ratios • Read 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.

  20. 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.

  21. Lesson 2 - Ratios

  22. 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 𝐴:𝐵 and 𝐶:𝐷 are equivalent ratios if there is a positive number, 𝑐, such that 𝐶=𝑐𝐴 and 𝐷=𝑐𝐵.

  23. Lesson 3 – Equivalent Ratios 7 inches 3 inches 7:3 7 to 3 21 inches 9 inches 21:9 21 to 9 2 m 2 m 2 m 2 m 2 m 2 m 14 meters 6 meters 14:6 14 to 6 2 m 2 m 2 m 2 m What ratios can we say are equivalent to 7:3? • Shanni 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

  24. Lesson 3 – Equivalent Ratios Take 5 minutes to work through Exercise 4 with your neighbor. Share your thoughts!

  25. 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.

  26. Lesson 4 – Equivalent Ratios

  27. walnuts 54 cashews Lesson 4- Equivalent Ratios • Exercise 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.

  28. Lesson 4- Equivalent Ratios • Exercise 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.

  29. 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.

  30. 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.

  31. Passenger Cars Pickup Trucks Lesson 5 – Solving Problems by Finding Equivalent Ratios 12 equal parts 192 divided by 12 = 16 7 x 16 = 112 pickup trucks 192 vehicles 5 x 16 = 80 passenger cars A 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-part 7:5 part-to-part 5 to 12 part-to-whole 7 to12 part-to-whole

  32. Lesson 6 – Solving Problems by Finding Equivalent Ratios • Students use tape diagrams to solve problems given a ratio between two quantities, and a change to those quantities that changes the ratio. 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.

  33. Correlations Between Lessons 1-4 and 5-6 • Take 3 minutes to discuss with your table partners the correlations you see between Lessons 1 -4 and Lessons 5-6. • Share with whole group.

  34. 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.

  35. Lesson 7 – Associated Ratios and the Value of a Ratio

  36. Lesson 7 – Associated Ratios and the Value of a Ratio 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.

  37. 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.

  38. Lesson 8 - Equivalent Ratios Defined Through Value of a Ratio 1/2 1/2 3/8 3/8 • 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?

  39. Lesson 8 - Equivalent Ratios Defined Through Value of a Ratio • 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.

  40. Lesson 8 - Equivalent Ratios Defined Through Value of a Ratio • Read through the duration of the lesson with a table partner. Be prepared to discuss your findings with the group. • SHARE!

  41. Correlations Between Lessons 1-6 and 7-8 • Take 3 minutes to discuss with your table partners the correlations you see between Lessons 1 an- 6 to Lessons 7 and 8. • Share with whole group.

  42. 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 tableis 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.

  43. 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?

  44. 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.

  45. Lesson 11 – The Structure of Ratio Tables • 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.

  46. 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?

  47. 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? • 3/4of 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.

  48. Lesson 13 - From Ratio Tables to Equations Using the Value of a Ratio • Take 5 minutes to work through the rest of the lesson with a partner or at your table. • Share your thoughts!

  49. 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.

  50. 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?

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