Number and Operations in

Base TenCCSSM in the Fifth Grade

Oliver F. JenkinsMathEd 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

Number and Operations:

Fractions

Operations and Algebraic Thinking

The Number System

Expressions and

Equations

Algebra

K – 5

3 – 5

6 – 8 9 – 12

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 TargetsKnowledge and

understanding (Conceptual understandings)

Reasoning (Mathematical practices)

Performance skills (Procedural skill and fluency)

Products (Applications)

Extending Our Analysis of Content

Standard 5.NBT.6Computation 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 StrategiesInvented 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

Number oriented Left-handed Flexible

Computation Algorithms

Digit oriented Right-handed “One right way”

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

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:with 7 tens left over

(Note: Some students may need to break this into several steps.)

Array or Area Model

23

276

Using an Array to Find the Missing Side Length

10

+2

20 + 3

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 UnknownPartition 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

EqualSet

EqualSet

EqualSet

1

2

3

n

. .

.

Product

(Whole)

Number of sets

Number of Groups UnknownMeasurement 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

EqualSet

EqualSet

EqualSet

1

2

3

n

. .

.

Product

(Whole)

Number of sets

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 ones23 × 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

736

23 20 3

30

2

600 90

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 problemStudents’ problem solvingWhole-class discussionSumming upExercises 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!