Number and Operations in Base Ten

1. Number and Operations in Base Ten CCSSM in the Fourth Grade Oliver F. Jenkins MathEd Constructs, LLC www.mathedconstructs.com

2. Grade 4 CCSSM Domains • Operations and Algebraic Thinking • Use the four operations with whole numbers to solve problems. • Gain familiarity with factors and multiples. • Generate and analyze patterns. • Number and Operations in Base Ten • Generalize place value understanding for multi-digit whole numbers. • Use place value understanding and properties of operations to perform multi-digit arithmetic. • Number and Operations – Fractions • Extend understanding of fraction equivalence and ordering. • Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. • Understand decimal notation for fractions, and compare decimal fractions. • Measurement and Data • Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. • Represent and interpret data. • Geometric measurement: understand concepts of angle and measure angles. • Geometry • Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

3. Algebraic Thinking Stream Number and Operations in Base Ten The Number System Algebra Number and Operations: Fractions Operations and Algebraic Thinking Expressions and Equations 9 – 12 K – 5 6 – 8 3 – 5

4. Domain: Number and Operations in Base Ten • Cluster: • Use place value understanding and properties of operations to perform multi-digit arithmetic. • Content Standard 4.NBT.5: • Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5. What must students know and and be able to do in order to master this standard? • Content Standard 4.NBT.5: • Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

6. Unwrapping Content Standards Instructional Targets Knowledge and understanding (Conceptual understandings) Reasoning (Mathematical practices) Performance skills (Procedural skill and fluency) Products (Applications)

7. What is the significance of . . . . . . using strategies based on place value and the properties of operations. Illustrate and explain . . . using equations, rectangular arrays, and/or area models. . . . in content standard 4.NBT.5?

8. Extending Our Analysis of Content Standard 4.NBT.5 Computation Strategies, Place Value, Properties of Operations, Array and Area Models

9. Computation Algorithms and Strategies • Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. • Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. • Special strategies. Either cannot be extended to all numbers represented in the base-ten system or require considerable modification in order to do so. • General methods. Extend to all numbers represented in the base-ten system. A general method is not necessarily efficient. However, general methods based on place value are more efficient and can be viewed as closely connected with standard algorithms.

10. Invented Strategies Invented strategies are flexible methods of computing that vary with the numbers and the situation. Successful use of the strategies requires that they be understood by the one who is using them – hence, the term invented. Strategies may be invented by a peer or the class as a whole; they may even be suggested by the teacher. However, they must be constructed by the student.

11. Strategies versus Algorithms Computation Strategies Computation Algorithms Digit oriented Right-handed “One right way” • Number oriented • Left-handed • Flexible

12. Benefits of Strategies • Students make fewer errors. • Less re-teaching is required. • Students develop number sense. • Strategies are the basis for mental computation and estimation. • Flexible methods are often faster that the traditional algorithms. • Algorithm invention is itself a significantly important process of “doing mathematics.”

13. Place Value • Base-ten numeration system • Based on the principles of grouping and place value • Objects are grouped by tens, then by tens of tens (hundreds), and so on • As you move to the left in base ten numbers, the value of the place is multiplied by 10 • Place value understandings underlie all computation strategies and algorithms

14. Computations Based on Place Value and Properties of Operations Standard algorithms for base-ten computations rely on decomposing numbers written in base-ten notation into base-ten units The properties of operations then allow any multi-digit computation to be reduced to a collection of single-digit computations which, in turn, sometimes require the composition or decomposition of a base-ten unit Example

15. Rectangular Array 20 + 4 4

16. Abbreviated Array Model 20 + 4 4

17. Another Abbreviated Array 20 + 3 10 + 2

18. Area Model

19. Thinking aboutStudent-Invented Strategies • Describe a strategy that students might invent to find: • Describe a strategy that students might invent to find:

20. TeachingMulti-digit Multiplication

21. Supporting Research • Findings • When students’ computation strategies reflect their understanding of numbers, understanding and fluency develop together. • Understanding is the basis for procedural fluency. • Children can and do devise or invent strategies for carrying out multi-digit computations. • Students learn well from a variety of instructional approaches. • Sustained experience with select physical models may be more effective than limited experience with a variety of different materials. • Conclusions • Building algorithms on the strategies that student invent promotes both understanding and fluency • A focus on array and area models is likely to be effective

22. Creating an Environment for Inventing Strategies • Expect and encourage student-to-student interactions, discussions, and conjectures • Celebrate when students clarify previous knowledge and attempt to construct new ideas • Encourage curiosity and an open mind to trying new things • Talk about both right and wrong ideas in a non-evaluative or non-threatening way • Move unsophisticated ideas to more sophisticated thinking through coaxing, coaching, and guided questioning • Use contexts and story problems to capture student interest • Consider carefully whether you should step in or step back when students are formulating new ideas (when in doubt – step back)

23. Bruner’s Stages of Representation • Enactive:Concrete stage. Learning begins with an action – touching, feeling, and manipulating. • Iconic:Pictorial stage. Students are drawing on paper what they already know how to do with the concrete manipulatives. • Symbolic:Abstract stage. The words and symbols representing information do not have any inherent connection to the information.

24. TeachingMulti-digit Multiplication An Instructional Progression

25. Allow understandings to develop through student invented or devised strategies for multi-digit multiplication; do not begin by teaching the standard algorithms • Students should be able to understand and explain the methods they invent • Encourage the use of visual representations such as area and array diagrams • These representations are known to further understandings and facilitate explanations • Some teacher modeling may be necessary to ensure productive application of arrays and area models

26. Build on third grade experiences with products of one-digit numbers and multiples of 10 by extending what was learned to products of one-digit numbers and multiples of 100 and 1,000 3.NBT.3.Multiply one-digit whole numbers by multiples of 10 in the range 10 – 90 (e.g., , ) using strategies based on place value and properties of operations. Student explanations should be based on place value and properties (commutative and associative) of multiplication and may use equations, arrays, or area models.

27. Prior to multiplying one-digit numbers by whole numbers up to four digits, have students develop and practice using the patterns in relationships among products such as, , , and . • Students should apply place value reasoning that they developed in the preceding step. • Encourage continued use of diagrams of arrays or areas to support students’ reasoning. • Student explanations should be based on place value and multiplication properties (commutative and associative).

28. Extend understandings of one-digit by multi-digit multiplication to two-digit by two-digit multiplication. • Students work individually or cooperatively to invent their own strategies. • Explanations are grounded in knowledge of place value and multiplication properties (commutative, associative, and distributive). • Continued use of rectangular arrays and area models is encouraged.

29. By reasoning repeatedly about the connection between math drawings and written numerical work (applications of invented computation strategies), students can come to see multiplication algorithms as abbreviations or summaries of their reasoning about quantities. This builds understandings requisite to content standard 5.NBT.5 – Fluently multiply multi-digit whole numbers using the standard algorithm.

30. A Problem-Solving Approach

31. Desirable Features ofProblem-Solving Tasks Genuine problems that reflect the goals of school mathematics Motivating situations that consider students’ interests and experiences, local contexts, puzzles, and applications Interesting tasks that have multiple solution strategies, multiple representations, and multiple solutions Rich opportunities for mathematical communication Appropriate content considering students’ ability levels and prior knowledge Reasonable difficulty levels that challenge yet not discourage

32. Problem Types Contextual Problems. Context problems are connected as closely as possible to children’s lives, rather than to “school mathematics.” They are designed to anticipate and to develop children’s mathematical modeling of the real world. Model Problems. The model is a thinking tool to help children both understand what is happening in the problem and a means of keeping track of the numbers and solving the problem.

33. Equal-Group Problems • Whole Unknown • Mark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? (repeated addition) • If apples cost 7 cents each, how much did Jill have to pay for 5 apples? (rate) • Peter walked for 3 hours at 4 miles per hour. How far did he walk? (rate) • Lucy needs 5 feet of material to make a scarf. She plans to make 8 scarfs. How many feet of material does she need? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

34. Combinations: Product Unknown • Sam bought 4 pairs of pants and 3 jackets, and they all can be worn together. How many different outfits consisting of a pair of pants and a jacket does Sam have? • An experiment involves tossing a coin and rolling a die. How many different possible results or outcomes can this experiment have? 1 2 3 4 6 5 H T

35. Area Problems • Sam’s garden is 19 feet long and 9 feet wide. What is the area of Sam’s garden in square feet? • Supporting standard: 3.MD.7b – Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems . . . 19 ft. 9 ft.

36. Problem-Solving Lesson Format Pose a problem Students’ problem solving Whole-class discussion Summing up Exercises or extensions (optional)

37. Design a Problem-Based Lesson • Identify lesson objectives aligned with standard 4.NBT.5 • Create contextual and/or model problems for teaching multi-digit multiplication • Construct a problem-solving lesson • Describe how the class and lesson materials will be organized • Write three questions that you will ask students during each of the first four lesson phases Solving problems is fun!