A Story of Ratios

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## A Story of Ratios

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**A Story of Ratios**Grade 7 Module 1 – First Half of Lessons**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**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 Lessons**• Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept or procedure. • Socratic Teacher leads students in a conversation to develop a specific concept or proof. • Exploration Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge • Modeling Students 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 Rate • Sample Exercise: Paper Passing • Directions • 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 6**• Ratio: A pair of numbers • Value 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 quantities • Unit rate: The value of the ratio • Rate’s unit: The label, e.g. mph • 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. Students understood equivalent 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 6**• Ratio: A pair of numbers • Value 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 quantities • Unit rate: The value of the ratio • Rate’s unit: The label, e.g. mph • 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. Students understood equivalent ratios to have the same value.**Meet Your Table**Talk 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 Rate Content Scan 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 Organization**• Student • Classwork • Problem Set • Teacher • 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)**Lesson 1: An Experience in Relationships as**Measuring Rate Exit Ticket Tillman the English Bulldog Insert video link 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**Lesson 2: Proportional Relationships**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. is proportional to • Cost ____ ______________________ ____ Weight.**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 4 Time (minutes) Calories Burned 0 11 22 33 44 Is the number of Calories burned proportional to time? How do you know?**Lesson 2: Proportional Relationships**• 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. • 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 Relationships**• 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. • Closing • How 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?**Lessons 3-4: Identifying Proportional and Non-Proportional**Relationships in Tables Lesson 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 Tables 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. Lesson 4 Example: Which team will win Randy’s race?**Lessons 5-6: Identifying Proportional and Non-Proportional**Relationships in Graphs 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 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. Important Note: Characteristics of graphs of proportional relationships:**Lessons 5-6: Identifying Proportional and Non-Proportional**Relationships in Graphs • What 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 Graphs Table 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 Graphs • Gallery Walk • Use 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 A**Take 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?**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?**Lesson 7: Unit Rate as the Constant of**Proportionality Exit 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?**Lesson 7: Unit Rate as the Constant of**Proportionality Exit Ticket Solution**Lesson 8-9: Representing Proportional**Relationships with Equations • Take 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.**Lesson 8-9: Representing Proportional**Relationships with Equations Discuss at your table: What is the main idea of this lesson? How do the Examples prepare students to find success with the Problem Set?**Lesson 8-9: Representing Proportional**Relationships with Equations Lesson 9 Exit Ticket Oscar 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 Relationships Take 15 minutes to complete Examples 1 and 2. How do you anticipate students will respond to this lesson? Spend 3 minutes previewing Lesson 10 materials. What are your initial observations about 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.**A Story of Ratios**G7-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.