A Story of Ratios

1. A Story of Ratios Grade 6 – Module 2

2. Session Objectives • Examine the development of mathematical understanding across the module using a focus on concept development within the lessons. • Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.

3. Agenda Introduction to the Module Concept Development Module Review

4. Curriculum Overview of A Story of Ratios

5. Agenda Introduction to the Module Concept Development Module Review

6. CCSS Progressions Document- The Number System (6-8) • Take 3 minutes to read through the Progressions for Grade 6. • Take 2 minutes to discuss the overview at your table. • Share with whole group your findings.

7. Topic A: Dividing Fractions by Fractions • Take 2 minutes to read through the Topic Overview for Topic A. • Take 5 minutes to discuss the overview at your table. • Share with whole group your findings.

8. Topic A Lesson 1-2: Interpreting Division of a Whole Number by a Fraction—Visual Models • Lesson 3-4: Interpreting and Computing Division of a Fraction by a Fraction – More Models • Lesson 5: Creating Division Stories • Lesson 6: More Division Stories • Lesson 7: The Relationship Between Visual Fraction Models and Equations • Lesson 8: Dividing Fractions and Mixed Numbers

9. Topic A: Lesson Demonstration Let’s take a look at specific examples from the lessons in Topic A.

10. Lesson 1 - Interpreting Division of a Whole Number by a Fraction—Visual Models • Outcomes: • Students use visual models such as fraction bars, number lines, and area models to show the quotient of whole numbers and fractions. Students use the models to show the connection between those models and the multiplication of fractions. • Students divide a fraction by a whole number.

11. Lesson 2 - Interpreting Division of a Whole Number by a Fraction—Visual Models • Outcomes: • Students use visual models such as fraction bars, number lines, and area models to show the quotient of whole numbers and fractions. Students use the models to show the connection between those models and the multiplication of fractions. • Students understand the difference between a whole number being divided by a fraction and a fraction being divided by a whole number.

12. Lesson 3 - Interpreting and Computing Division of a Fraction by a Fraction—More Models • Outcomes: • Students use visual models such as fraction bars and area models to show the division of fractions by fractions with common denominators. • Students make connections to the multiplication of fractions. In addition, students understand that the division of fractions require students to ask, “How many groups of the divisor are in the dividend?” to get the quotient.

13. Lesson 4 - Interpreting and Computing Division of a Fraction by a Fraction—More Models • Outcomes: • Students use visual models such as fraction bars and area models to divide fractions by fractions with different denominators. • Students make connections between visual models and multiplication of fractions.

14. Lesson 5 – Creating Division Stories • Outcomes: • Students demonstrate further understanding of division of fractions when they create their own word problems. • Students choose a measurement division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they discover that they must try several situations and units before finding which are realistic with given numbers.

15. Lesson 6 – More Division Stories • Outcomes: • Students demonstrate further understanding of division of fractions when they create their own word problems. • Students choose a measurement division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they discover that they must try several situations and units before finding which are realistic with given numbers. • Take 2 minutes to look over the teacher notes for this lesson and note the exercises students will be completing.

16. Lesson 7 – The Relationship Between Visual Fraction Models and Equations • Outcome: • Students formally connect models of fractions to multiplication through the use of multiplicative inverses as they are represented in models.

17. Lesson 8 – Dividing Fractions and Mixed Numbers • Outcomes: • Students divide fractions by mixed numbers by first converting the mixed numbers into a fraction with a value larger than one. • Students use equations to find quotients. • Take 2 minutes to look over the teacher notes for this lesson and note the exercises students will be completing.

18. Topic A: Debrief • What visual representations are students utilizing in Topic A? • How will these representations help students in visualizing division of fractions and making connections to multiplication? • What is the importance of student created division stories? • What are your takeaways about partitive and measurement interpretations of division? • Why is it important that students know the difference • between them and are able to choose the appropriate • interpretation?

19. Topic B: Multi-Digit Decimal Operations—Adding, Subtracting, and Multiplying • Take 2 minutes to read through the Topic Overview for Topic B. • Take 2 minutes to discuss the overview at your table. • Share with whole group your findings.

20. Topic B Lesson 9: Sums and Differences of Decimals • Lesson 10: The Distributive Property and the Products of Decimals • Lesson 11: Fraction Multiplication and the Products of Decimals

21. Topic B: Lesson Demonstration Let’s take a look at specific examples from the lessons in Topic B.

22. Lesson 9 – Sums and Differences of Decimals • Outcomes: • Students relate decimals to mixed numbers and round addends, minuends, and subtrahends to whole numbers in order to predict reasonable answers. • Students use their knowledge of adding and subtracting multi-digit numbers to find the sums and differences of decimals. • Students understand the importance of place value and solve problems in real-world contexts.

23. Lesson 10 – The Distributive Property and the Product of Decimals • Outcome: • Through the use of arrays and partial products, students strategize and apply the distributive property to find the product of decimals.

24. Lesson 11 – Fraction Multiplication and the Products of Decimals • Outcomes: • Students use estimation and place value to determine the placement of the decimal point in products and to determine that the size of the product is relative to each factor. • Students discover and use connections between fraction multiplication and decimal multiplication. • Students recognize that the sum of the number of decimal digits in the factors yields the decimal digits in the product.

25. Topic B: Debrief • How is Lesson 9 related to Topic A? • What do you notice about the progression of the lessons from Topics A and B in terms of content building, specifically when it comes to place value?

26. Mid-Module Assessment Participant Task Look at the number card you were given. Move to the table labeled with your number. Work with your teammates to answer each portion of your question without looking at the student exemplars. You have 7 minutes to work the problem.

27. Mid-Module Assessment Participant Task TEAM SHARE

28. Topic C: Dividing Whole Numbers and Decimals • Take 2 minutes to read through the Topic Overview for Topic C. • Take 2 minutes to discuss the overview at your table. • Share with whole group your findings.

29. Topic C Lesson 12: Estimating Digits in a Quotient • Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm • Lesson 14: The Division Algorithm – Converting Decimal Division into Whole Number Division Using Fractions • Lesson 15: The Division Algorithm – Converting Decimal Division into Whole Number Division Using Mental Math

30. Topic C: Lesson Demonstration Let’s take a look at specific examples from the lessons in Topic C.

31. Lesson 12 – Estimating Digits in a Quotient • Outcome: • Students connect estimation with place value in order to determine the standard algorithm for division.

32. Lesson 13 – Dividing Multi-Digit Numbers Using the Algorithm • Outcome: • Students understand that the standard algorithm of division is simply a tally system arranged in place value columns.

33. Lesson 14 – The Division Algorithm—Converting Decimal Division into Whole Number Division Using Fractions • Outcomes: • Students use the algorithm to divide multi-digit numbers with and without remainders. Students compare their answer to estimates to justify reasonable quotients. • Students understand that when they “bring down” the next digit in the algorithm, they are distributing, recording, and shifting to the next place value.

34. Lesson 15 – The Division Algorithm—Converting Decimal Division to Whole Number Division Using Mental Math • Outcomes: • Students use their knowledge of dividing multi-digit numbers to solve for quotients of multi-digit decimals. • Students understand the mathematical concept of decimal placement in the divisor and the dividend and its connection to multiplying by powers of 10.

35. Topic C: Debrief • Discuss the importance of estimation in this Topic. • What connections do you see between Topics A, B and C?

36. Topic D: Number Theory – Thinking Logically About Multiplicative Arithmetic • Take 2 minutes to read through the Topic Overview for Topic D. • Take 2 minutes to discuss the overview at your table. • Share with whole group your findings.

37. Topic D Lesson 16: Odd and Even Numbers • Lesson 17: Divisibility Tests for 3 and 9 • Lesson 18: Least Common Multiple and Greatest Common Factor • Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm

38. Topic D: Lesson Demonstration Let’s take a look at specific examples from the lessons in Topic D.

39. Topic D: Participant Lesson Demonstration • Look at your number card. If you are a 16, go to your designated table. If you are a 17, go to your designated table, if you are an 18, go to your designated table. • Your group will have 7 minutes to read through the lesson you’ve been assigned. • You will discuss the representations and the flow of the lesson with your group. • After discussion, your group will present the outcomes of the • lesson and model one example from the lesson for the whole group.

40. Lesson 19 – The Euclidean Algorithm as an Application of the Long Division Algorithm • Outcome: • Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division.

41. Biggest Takeaway • Turn and Talk: • What questions were answered for you? • What new questions have surfaced?

42. Key Points • Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. • Students explore partitive and measurement interpretations of dividing fractions to make connections to the relationship between multiplication and division. • Estimation and place value enable students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. • Students learn to use connections between fraction multiplication and decimal multiplication. • Students connect estimation to place value and determine that the standard division algorithm is simply a tally system arranged in place value columns • Students make connections to division of fractions and rely on mental math strategies to implement the division algorithm when finding the quotients of decimals. • Students explore and discover that Euclid’s Algorithm is a more efficient way to find the greatest common factor of larger numbers and see that Euclid’s Algorithm is based on long division.