Number Sense Grades 3-5

1. Number SenseGrades 3-5 Day 3 Math & Science Collaborative at the Allegheny Intermediate Unit

2. Sharing • Share the mental math activity and game you did with your class. • What successes did your students have? • What challenges did they face? Math & Science Collaborative at the Allegheny Intermediate Unit

3. Read and Discuss • Discuss: “Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them.” from TCM, August 2006 • What are some reasons why students have difficulty mastering basic combinations? • What new ideas did you discuss in your group about the relationship between number sense and fluency after reading the article? Math & Science Collaborative at the Allegheny Intermediate Unit

4. Modules are built on the strands Math & Science Collaborative at the Allegheny Intermediate Unit

5. foundational development occurs PreK- Grade 2 25% 16% K-2=49% 30% Math & Science Collaborative at the Allegheny Intermediate Unit

6. Mental Math • Encourages students to build on number relationships to solve problems instead of memorized procedures • Using number relationships helps students develop efficient, flexible strategies with accuracy • Causes students to be efficient to avoid holding numerous quantities in their heads • Strengthens students’ understanding of place value Math & Science Collaborative at the Allegheny Intermediate Unit

7. Video: Doubling and Halving • The teacher chooses to use a context as students consider the relationship between 4 x 7 and 2 x 14. How does this context help support student thinking? • How is the commutative property addressed in the discussion? • How do the teacher’s questions help students begin to build an understanding of doubling and halving? • The teacher is continually assessing her students during the number talk. What student understandings and misconceptions will she use to guide her next instructional steps? Math & Science Collaborative at the Allegheny Intermediate Unit

8. Array Discussion • The class had shared strategies for solving 8 x 25 before the teacher introduced the array model. Why do you think this instructional decision was made? How did she link previous strategies to the array? • Noel incorrectly refers to columns as rows in the array. The teacher does not correct her. Why do you think she choose to ignore this error? • How does the teacher connect the students’ additive thinking to multiplication? • How does the array support student understanding of multiplication? The commutative property? The distributive property? Math & Science Collaborative at the Allegheny Intermediate Unit

9. 32 X 15 • What examples support the strong learning community that exists in the classroom? • What strategy closely resembles the standard algorithm? How are they similar? Different? • The distributive property is interwoven in most of the students’ strategies. What examples do you notice of this property being used? • The topic of efficiency surfaces in this discussion. Which strategies would you deem ot be most efficient and why? • Choose one of the strategies to model with an open array. Math & Science Collaborative at the Allegheny Intermediate Unit

10. 496 ÷ 8 • How does the succession of problems provide a scaffold for students to solve 496 ÷ 8? • How does the scaffold provide multiple ways for students to access the problem,? • In what other ways could 496 ÷ 8 be solved using the prior problems? • Jillian had a lovely strategy for solving this problem; however, she struggled to state what her answer was. Where is Jillian's answer in the recording on the board? What mathematical concepts are embraced in her strategy? • Students struggle to follow Jackson’s strategy. What components of division and multiplication does Jackson understand? • The teacher repeatedly asks students to explain where their answer is in their strategy. Why is that an important focal point throughout the discussion? Math & Science Collaborative at the Allegheny Intermediate Unit

11. Place Value Standards • PA Core Standards talk about “strategies based on place-value, properties of operations, and/or relationships between operations.” • Look at the standards. What strategies are they focusing on? What are “strategies based on place-value, properties of operations, and/or relationships between operations.? Math & Science Collaborative at the Allegheny Intermediate Unit

12. PA Core Math & Science Collaborative at the Allegheny Intermediate Unit

13. 2.OA.2 Math & Science Collaborative at the Allegheny Intermediate Unit

14. 1.OA.6 Math & Science Collaborative at the Allegheny Intermediate Unit

15. Precursors to Fluency • The sequence of number names, both starting at 1 and not starting at 1 • How to count a set, keeping track of the items they counted • Understanding relationships of more, less , and same • Skip counting starting from 1 and from other numbers • Cardinality • Conservation • 1-to-1 correspondence • Making tens – link to understanding place value • Subitizing • Decomposing and composing numbers • Understanding part-part-whole • Number sense Math & Science Collaborative at the Allegheny Intermediate Unit

16. Fluency demands… • An understanding of the meaning of operations and their relationship to each other. • The knowledge of a large repertoire of number relationships. (The “patterns” of our number system.) Math & Science Collaborative at the Allegheny Intermediate Unit

17. Fluency demands… • A thorough understanding of the base-ten number system, how numbers are structured in this system, and how this system behaves in different operations. • Knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems. Involves three components. Math & Science Collaborative at the Allegheny Intermediate Unit

18. Fluency • Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the problem. • Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number combinations and other important number relationships, and concern for double-checking results. • Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the problem at hand and also to use one method to solve a problem and another method to double-check the results. Math & Science Collaborative at the Allegheny Intermediate Unit

19. How Can We Help Students with Facts? • Ongoing practice and engagement with math facts tasks • Hands-on activities and thoughtful discussions • Conceptual understanding of operations • Strategic thinking Math & Science Collaborative at the Allegheny Intermediate Unit

20. Conceptual Understanding • Understanding operations • Symbolic representations • Relationship between parts and whole • Investigating the meaning of facts through hands-on activities and thoughtful discussions • Understanding is gained through: • Problem posing • Hands-on exploration • Classroom discussions • Real-world examples Math & Science Collaborative at the Allegheny Intermediate Unit

21. Meaningful Practice Builds on understanding of operations and using strategic reasoning to explore math facts • Practice 5 – 10 minutes daily throughout the school year • Vary the practice activities - ensures that students are motivated and engaged • Automaticity is achieved through brief, frequent, interactive activities Math & Science Collaborative at the Allegheny Intermediate Unit

22. Understanding Multiplication and Division • Address the big ideas • Guide the types of questions that are posed • Explore symbolic representations • Use models to represent multiplication and division • Number lines, manipulatives, area models, arrays • Explore concepts through problems and literature Math & Science Collaborative at the Allegheny Intermediate Unit

23. Understanding Multiplication and Division • What does it mean to understand multiplication? • Look at the standards for the operations. Read the standards dealing with multiplication and division. • Then talk with your group. Develop a chart, graphic, or model showing what students need to know to show an understanding of multiplication. Math & Science Collaborative at the Allegheny Intermediate Unit

24. Building Understanding While Focusing on Fluency • Use models to represent multiplication and division • Arrays, set models, area models, number lines • Use problem contexts/real-life situations • Make sure all four categories of problems are addressed • Productive talk • Classroom environment Math & Science Collaborative at the Allegheny Intermediate Unit

25. Big Ideas Central to Understanding Multiplication and Division • Our number system is a system of patterns. • Numbers can count objects or groups. • The order of factors does not change the product. • Addition and multiplication are related operations. • Multiplication and division are inverse operations. • Numbers are flexible. Math & Science Collaborative at the Allegheny Intermediate Unit

26. Questions Posed • What do the numbers in the equation represent? • What patterns do you notice in the factors and products? • Does the order of the factors affect the product? Give examples to justify your thinking? • Do you notice a connection between this multiplication equation and this division equation? Explain. • Can you break apart one of the factors to help you find the product? Math & Science Collaborative at the Allegheny Intermediate Unit

27. Classroom Environment • Discussion • Partner Work • Interactive bulletin boards • Word walls • Centers Math & Science Collaborative at the Allegheny Intermediate Unit

28. Introducing Concepts of Multiplication and Division • Compare methods for solving problems. • Provide multiple opportunities for students to represent and solve multiplication problems. • Set models • Arrays • Area Models • Number lines • Explore division Math & Science Collaborative at the Allegheny Intermediate Unit

29. Looking at One Example: x2 What are the big ideas around multiplying by 2? • Multiplication by 2 is same as doubling. • Numbers stand for a variety of things. Operation symbols help us determine what the numbers represent. • Our number system is a system of patterns. • Order of factors does not change the product. Math & Science Collaborative at the Allegheny Intermediate Unit

30. Possible Questions to Support the Big Ideas X2 • What does it mean to have twice as much? What does it mean to double a quantity? • What does it mean to have half as much? • What do the numbers in the equation mean? • What patterns do you notice in the products? • Does the order of the factors affect the products? Give examples to justify your thinking. • How are a sum and a product the same? How are they different? Math & Science Collaborative at the Allegheny Intermediate Unit

31. Literature Connection • Two of Everything– read and discuss. • After story, discuss what doubled. Ask: • Are doubling and twice as many the same? Explain. • Can you find twice as many by adding? How? • Can you find twice as many by multiplying? How? Math & Science Collaborative at the Allegheny Intermediate Unit

32. Word Problems • Students need to visualize the facts using a concrete model and move from concrete/visual experiences to symbolic representations. They need to use concrete items and draw pictures. Math & Science Collaborative at the Allegheny Intermediate Unit

33. Word Problems • Pose problem such as the following: • Mrs. Short baked some chocolate brownies. She placed 6 plates on the table and put 2 brownies on each plate. How many brownies did she put on plates? Math & Science Collaborative at the Allegheny Intermediate Unit

34. Observe Patterns with Twos • Have students think about a series of brownie problems, and write a multiplication equation to solve each one. • 1 plate with 2 brownies on each plate. • 2 plates with 2 brownies on each plate. ….. • 10 plates with 2 brownies on each plate. • What patterns do you notice? Math & Science Collaborative at the Allegheny Intermediate Unit

35. Commutative Property • Provide students with manipulatives and paper. The paper can represent the baskets. Have them determine the answer to the following: • Colin had 2 baskets with 3 apples in each basket, how many apples did he have? • Colin had 3 baskets with 2 apples in each basket. How many apples did he have? Math & Science Collaborative at the Allegheny Intermediate Unit

36. Building Automaticity • Short practice – daily routine • Games • Rolling for Doubles • Double Up • Fact Card Jumps • Doubles Match Up • Connect to Division Math & Science Collaborative at the Allegheny Intermediate Unit

37. Other Groups of Facts • For the group of facts assigned to your group, read the pages for your set of facts. • X10 • X5 • x1 • X0 • x3 • X4 • X6 • X9 • X8 • x7 Math & Science Collaborative at the Allegheny Intermediate Unit

38. Sharing • Give a brief overview of you group of facts. • Share: • An interesting activity for your fact Math & Science Collaborative at the Allegheny Intermediate Unit

39. Article Discussion • Read and discuss: “Fluency: Simply Fast and Accurate? I Think Not!” by NCTM Past President Linda M. Gojak • How do the ideas in the article resonate with your ideas? • What is the same? What is different? • What stood out for you? Math & Science Collaborative at the Allegheny Intermediate Unit

40. Fluency • Look back at your ideas about fluency. • What ideas, if any, do you want to add to your ideas? Math & Science Collaborative at the Allegheny Intermediate Unit