A Story of Units

1. A Story of Units Grade 2 – Module 6

2. Session Objectives • Examine the development of mathematical understanding across the module using a focus on Concept Development within the lessons. • Introduce mathematical models and instructional strategies to support implementation of A Story of Units.

3. Agenda Introduction to the Module Concept Development Module Review

4. Curriculum Overview of A Story of Units

5. Module Overview • Read the narrative. • *key concepts • models and tools • Important vocabulary • What is new and different about the way these concepts are presented?

6. Agenda Introduction to the Module Concept Development Module Review

7. Topic A • Formation of Equal Groups • How does addition provide a foundation for understanding multiplication? • How can equal groups be represented mathematically?

8. Lesson 1 Lesson Objective: Use manipulatives to create equal groups. Are these groups equal or unequal? How do you know?

9. Lesson 1 Problem Set Lesson 1, Student Debrief • For Problem 5, how did you go about making the three groups equal? • Make a prediction: How are these equal groups related to addition?

10. Lessons 2 and 3 Lesson Objective: Use math drawings to represent equal groups, and relate to repeated addition.

11. Lesson 3 Lesson Objective: Use math drawings to represent equal groups, and relate to repeated addition.

12. Lesson 4 Lesson Objective: Represent equal groups with tape diagrams, and relate to repeated addition.

13. Lesson 4 Application Problem

14. Topic B • Arrays and Equal Groups • Why is the array a useful model for representing equal groups?

15. Spatial Relationships and Structuring Geometry Progression, pages 10-11 “Another type of composition and decomposition is essential to students’ mathematical development – spatial structuring. Students need to conceptually structure an array to understand two-dimensional regions as truly two-dimensional. … Spatial structuring is thus the mental operation of constructing an organization or form for an object or set of objects in space, a form of abstraction. … As the mental structuring process helps them organize their counting, they become more systematic, using the array structure to guide the quantification. Eventually, they begin to use repeated addition of the number in each row or each column. Such spatial structuring precedes meaningful mathematical use of the structures, including multiplication and, later, area, volume, and the coordinated plane.”

16. Lesson 5 Lesson Objective: Compose arrays from rows and columns, and count to find the total using objects.

17. Lesson 6 Lesson Objective: Decompose arrays into rows and columns, and relate to repeated addition.

18. Lesson 6 Problem Set Lesson 6, Student Debrief • How is decomposing arrays similar to decomposing numbers? • For Problem 3(a), did you write the repeated addition number sentence in terms of rows or columns? Why didn’t it matter? • How does adding or removing rows or columns change the repeated addition number sentence?

19. Lesson 7 Concrete to Pictorial Lesson Objective: Represent arrays and distinguish rows and columns using math drawings.

20. Lesson 8 Lesson Objective: Create arrays using square tiles with gaps.

21. Lesson 8 Application Problem

22. Lesson 9 Problem Set • Draw a tape diagram and array. Then write an addition number sentence to match. • In a card game, 3 players get 4 cards each. One more player joins the game. How many total cards should be dealt now?

23. Topic C • Rectangular Arrays as a Foundation for Multiplication and Division • How does the composition of arrays build a foundation for multiplication? • How does the decomposition of arrays build a foundation for division?

24. Lessons 10 and 11 Lesson 10, Concept Development Lesson Objective: Use square tiles to compose a rectangle, and relate to the array model. Can we make a square array with 10 tiles?

25. Lesson 12 Lesson 12, Student Debrief • Lesson Objective: Use math drawings to compose a rectangle with square tiles. • Explain to your partner how to draw a rectangle with one square tile. Why was precision important today? How is this different from drawing an array with X’s? • How does drawing a rectangle support the idea of composing a larger unit from smaller units? Use the terms square, rows, and columns in your response.

26. Lesson 13 Lesson Objective: Use square tiles to decompose a rectangle.

27. Lesson 14 Lesson Objective: Use scissors to partition a rectangle into same-size squares, and compose arrays with the squares.

28. Lesson 15 Lesson 15, Student Debrief • Lesson Objective: Use math drawings to partition a rectangle with square tiles, and relate to repeated addition. • For Problem 4, if you were to continue adding columns of 2, would your array look more like a train or a tower? If you wanted to increase the total number of tiles quickly, would you suggest adding more rows or columns? Why? • Explain why you couldn’t draw another column of 2 in Problem 5

29. Lesson 16 Lesson 16, Student Debrief Lesson Objective: Use grid paper to create designs to develop spatial structuring. Why do you think we spent time creating designs and learning about tessellations today?

30. Topic D • The Meaning of Even and Odd Numbers • What strategies can be used to determine if a number is even or odd?

31. Lesson 17 Lesson 17, Student Debrief Lesson Objective: Relate doubles to even numbers, and write number sentences to express the sum. What connections do you seebetween even numbers and skip counting?

32. Lesson 18 Lesson 18, Student Debrief and UDL Box • Lesson Objective: Pair objects and skip-count to relate to even numbers. • What connections can you make between pairing objects and equal groups? If a number is even, can you make equal groups? If it is not even? • How can you prove whether 52 is even? How about 73?

33. Lesson 19 Lesson Objective: Investigate the pattern of even numbers: 0, 2, 4, 6, and 8 in the ones place, and relate to odd numbers.

34. Lesson 19 Homework

35. Lesson 20 Lesson Objective: Use rectangular arrays to investigate odd and even numbers.

36. Agenda Introduction to the Module Concept Development Module Review

37. Biggest Takeaway • Turn and Talk: • What insights have you gained about the importance of establishing a strong foundation in equal groups, repeated addition, and arrays in relation to multiplication and division? • How will this benefit student understanding of multiplication and division in Grade 3? • What connections do you see between the composition and decomposition in Module 6 and the composition and decomposition in previous modules?

38. Key Points • Module 6 focuses on using repeated addition to find the total number of objects arranged in rectangular arrays (2.OA.4), partitioning rectangles into rows and columns of same-size squares (2.G.2), and determining numbers as even and odd (2.OA.3). • This foundational understanding is crucial, since multiplication and division is the major work of Grade 3. • Module 6 also supports the required fluency of Grade 3: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations.