Number and Operations in Base Ten

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

Grade 5 CCSSM Domains • Operations and Algebraic Thinking • Write and interpret numerical expressions. • Analyze patterns and relationships. • Number and Operations in Base Ten • Understand the place value system. • Perform operations with multi-digit whole numbers and with decimals to hundredths. • Number and Operations – Fractions • Use equivalent fractions as a strategy to add and subtract fractions. • Apply and extend previous understandings of multiplication and division to multiply and divide fractions. • Measurement and Data • Convert like measurement units within a given measurement system. • Represent and interpret data. • Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. • Geometry • Graph points on the coordinate plane to solve real-world and mathematical problems. • Classify two-dimensional figures into categories based on their properties.

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

Domain: Number and Operations in Base Ten • Cluster: • Perform operations with multi-digit whole numbers and with decimals to hundredths. • Content Standard 5.NBT.6: • Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

What must students know and and be able to do in order to master this standard? • Content Standard 5.NBT.6: • Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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

Extending Our Analysis of Content Standard 5.NBT.6 Computation Strategies, Place Value, Properties of Operations, Relationship between Multiplication and Division, Array and Area Models

What is the significance of . . . . . . using strategies based on place value,the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain . . . using equations, rectangular arrays, and/or area models. . . . in content standard 5.NBT.6?

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.

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.

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

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.”

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

Computations Based on Place Value and Properties of Operations with 7 tens left over (Note: Some students may need to break this into several steps.) 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:

Array or Area Model 23 276

Using an Array to Find the Missing Side Length + 3 20 10 + 2

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

TeachingMulti-digit Division

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

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)

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.

Allow understandings to develop through student invented or devised strategies for multi-digit division; do not begin by teaching the standard algorithms • Students should be able to understand and explain the methods they invent • Prompt a variety of invented strategies by periodically posing a division problem and having students solve the problem using two different strategies • Example: Solve 514 ÷ 8 in two different ways. Your ways may converge in similar places but begin with different first steps – or they may be completely different. • 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

Build on fourth grade experiences with one-digit divisors and up to four digit dividends. 4.NBT.6.Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. • 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.

Size of Groups Unknown Partition or Fair-Sharing Division • Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? (fair sharing) • Jill paid 35 cents for 5 apples. What was the cost of 1 apple? (rate) • Peter walked 12 miles in 3 hours. How many miles per hour (how fast) did he walk? (rate) • Lucy has 40 feet of material for making scarfs. She plans to make 8 scarfs. How many feet of material will she use for each scarf? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

Number of Groups Unknown Measurement or Repeated-Subtraction Division • Mark has 24 apples. He put then into bags containing 6 apples each. How many bags did Mark use? (repeated subtraction) • Jill bought apples at 7 cents apiece. The total cost of her apples was 35 cents. How many apples did Jill buy? (rate) • Peter walked 12 miles at a rate of 4 miles per hour. How many hours did it take Peter to walk the 12 miles? (rate) • Lucy has 40 feet of material to make scarfs. Five feet of material is needed to make a scarf? How many scarfs can she make? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

Side Length Unknown The surface area of Bob’s tablet is 88 square inches. If the tablet is 8 inches wide, then how long is it? 8 in. 88 sq. in.

Missing-Factor Strategy Cluster Problems

Fair-Sharing Strategy 736 ÷ 23 736 = 73 tens + 6 ones 23 × 3 tens = 69 tens with 4 tens left over 4 tens + 6 ones = 46 ones 23 × 2 ones = 46 ones So, 736 is shared among 23 groups, each consisting of 3 tens and 2 ones Therefore, 736 ÷ 23 = 3 tens + 2 ones = 32

Using an Array in Conjunction with a Missing-Factor Strategy 23 20 3 30 600 90 736 2 40 6

By reasoning repeatedly about the connection between math drawings and written numerical work (applications of invented computation strategies), students can come to see division algorithms as abbreviations or summaries of their reasoning about quantities. This builds understandings requisite to content standard 6.NS.2 – Fluently divide multi-digit numbers using the standard algorithm.

A Problem-Solving Approach

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

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.

Sample Problems • Size of Group Unknown • The bag has 783 jellybeans. Aidan and her 28 classmates want to share them equally. How many jellybeans will Aidan and each of her classmates get? • Number of Groups Unknown • Jumbo the elephant loves peanuts. His trainer has 736 peanuts. If he gives Jumbo 23 peanuts each day, how many days will the peanuts last? • Side Length Unknown • Monique’s porch is rectangular in shape with an area of 228 square feet. If the porch is 12 feet wide, how long is it?

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

Design a Problem-Based Lesson • Identify lesson objectives aligned with standard 5.NBT.6 • Create contextual and/or model problems for teaching multi-digit whole number division • 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!

Number and Operations in Base Ten