All the resources on this page come from the GADOE Framework 5th Grade Unit 1 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 1 Overview

In this unit students will:

• Solve problems by representing mathematical relationships between quantities using mathematical expressions and equations.

• Use the four whole number operations efficiently, including the application of order of operations.

• Write, evaluate, and interpret mathematical expressions with and without using symbols.

• Apply strategies for multiplying a 2- or 3-digit number by a 2-digit number.

• Develop paper-and-pencil multiplication algorithms (the U.S. traditional algorithm is not an expectation) for 3- or 4-digit number multiplied by a 2- or 3-digit number.

• Apply paper-and-pencil strategies for division (the strategies should be based on place-value reasoning - the U.S. traditional algorithm is not an expectation)

• Solve problems involving multiplication and division.

• Investigate the effects of multiplying whole numbers by powers of 10.

Note: Fluent use of standard algorithm for long division is a grade 6 standard (MGSE6.NS.2).

WRITE AND INTERPRET NUMERICAL EXPRESSIONS

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: parentheses, brackets, braces and numerical expressions.

UNDERSTAND THE PLACE VALUE SYSTEM

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: numerical patterns, rules, power of ten.

PERFORM OPERATIONS WITH MULTI-DIGIT WHOLE NUMBERS
(Decimals are addressed in a later unit)

Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for whole numbers and number notation, and properties of operations to add and subtract whole numbers. They develop fluency in these computations and make reasonable estimates of their results. Students use the relationship between whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: multiplication/multiply, division/division, products, quotients, dividends, rectangular arrays, area models, addition/add, subtraction/subtract, (properties)-rules about how numbers work, reasoning.

Combining multiplication and division within lessons is very important to allow students to understand the relationship between the two operations. Students need guidance and multiple experiences to develop an understanding that groups of things can be a single entity while at the same time contain a given number of objects. These experiences are especially useful in contextual situations such as the tasks in this unit.

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed throughout the year. Ideas related to the eight standards of mathematical practices should be addressed continually as well. The first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the standards, essential questions, and formative assessment questions be reviewed early in the planning process. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources. The amount of time that it will take to complete each task will vary. Some tasks may be completed in one class period, and others may take several days to complete. There is no expectation that every student will complete all of the tasks presented in this unit.

*For more detailed information about unpacking the content standards, unpacking a task, math routings and rituals, maintenance activities and more, please refer to the Grade Level Overview.

STANDARDS FOR MATHEMATICAL PRACTICE

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students solve problems by applying their understanding of operations with whole numbers, including the order of operations. Students seek the meaning of a problem and look for efficient ways to solve it.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning to connect quantities to written symbols and create a logical representation of the problem at hand. Students write simple expressions that record calculations with numbers and represent numbers using place value concepts.

3. Construct viable arguments and critique the reasoning of others. Students construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics. Students use base ten blocks, drawings, and equations to represent place value and powers of ten. They interpret expressions and connect representations of them.

5. Use appropriate tools strategically. Students select and use tools such as estimation, graph paper, and place value charts to solve problems with whole number operations.

6. Attend to precision. Students use clear and precise language, (math talk) in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to expressions, place value, and powers of ten

7. Look for and make use of structure. Students use properties of operations as strategies to add, subtract, multiply, and divide with whole numbers. They explore and use patterns to evaluate expressions. Students utilize patterns in place value and powers of ten and relate them to graphical representations of them.

8. Look for and express regularity in repeated reasoning. Students use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and properties of operations to fluently perform operations.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

STANDARDS FOR MATHEMATICAL CONTENT

MGSE5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

MGSE5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

MGSE5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

MGSE5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

MGSE5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm (or other strategies demonstrating understanding of multiplication) up to a 3-digit by 2-digit factor.

MGSE5.NBT.6 Fluently divide up to 4-digit dividends and 2-digit divisors by using at least one of the following methods: 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 or concrete models. (e.g., rectangular arrays, area models)

BIG IDEAS

• Multiplication may be used to find the total number of objects when objects are arranged in equal groups, rectangular arrays/area models.

• One of the factors in multiplication indicates the number of objects in a group and the other factor indicates the number of groups.

• Unfamiliar multiplication problems may be solved by using, invented strategies or known multiplication facts and properties of multiplication and division. For example, 8 × 7 = (8 × 2) + (8 × 5) and 18 × 7 = (10 × 7) + (8 × 7).

• There are two common situations where division may be used: fair sharing (given the total amount and the number of equal groups, determine how many/much in each group) and measurement (given the total amount and the amount in a group, determine how many groups of the same size can be created).

• The dividend, divisor, quotient, and remainder are related in the following manner: dividend = divisor x quotient + remainder.

• Some division situations will produce a remainder, but the remainder will always be less than the divisor. If the remainder is greater than the divisor, that means at least one more can be given to each group (fair sharing) or at least one more group of the given size (the dividend) may be created.

ESSENTIAL QUESTIONS

• Why is it important to follow an order of operations?

• How can I effectively critique the reasoning of others?

• How can I write an expression that demonstrates a situation or context?

• How can an expression be written given a set value?

• What is the difference between an equation and an expression?

• In what kinds of real-world situations might we use equations and expressions?

• How can we evaluate expressions?

• How can an expression be written?

• How does multiplying a whole number by a power of ten affect the product?

• How can estimating help us when solving multiplication problems?

• What strategies can we use to efficiently solve multiplication problems?

• How can I use what I know about multiplying multiples of ten to multiply two whole numbers?

• How can I apply my understanding of area of a rectangle and square to determine the best buy for a football field?

• How can we compare the cost of materials? • How can estimating help us when solving division problems?

• What strategies can we use to efficiently solve division problems?

• How can I use the situation in a story problem to determine the best operation to use?

• How can I effectively explain my mathematical thinking and reasoning to others?

• How can identifying patterns help determine multiple solutions?

• How can you determine the most cost-efficient arrangement?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• Solve multi-step word problems using four operations

• Fluently multiply and divide within 100 using strategies

• Multiply one-digit whole numbers by multiples of 10

• Multiply a whole number of up to four digits by a one-digit whole number

• Multiply two two-digit numbers

• Divide up to four-digit dividends by one-digit divisors

• Use number talks to reinforce properties of operations and mental computation

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18- wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-tests.pdf

STRATEGIES FOR TEACHING AND LEARNING

Write and Interpret Numerical Expressions

MGSE5.OA.1
MGSE5.OA.2

Students should be given ample opportunities to explore and evaluate numerical expressions with mixed operations. Eventually this should include real-world contexts that would require the use of grouping symbols in order to describe the context as a single expression. This is the foundation for evaluating algebraic expressions that will include whole-number exponents in Grade 6.

There are conventions (rules) determined by mathematicians that must be learned with no conceptual basis. For example, multiplication and division are always done before addition and subtraction. Begin with expressions that have two operations without any grouping symbols (multiplication or division combined with addition or subtraction) before introducing expressions with multiple operations. Using the same digits, with the operations in a different order, have students evaluate the expressions and discuss why the value of the expression is different. For example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rules that must be followed. Have students insert parentheses around the multiplication or division part in an expression. A discussion should focus on the similarities and differences in the problems and the results. This leads to students being able to solve problem situations which require that they know the order in which operations should take place.

After students have evaluated expressions without grouping symbols, present problems with one grouping symbol, beginning with parentheses, then adding expressions that have brackets and/or braces.

Have students write numerical expressions in words without calculating the value. This is the foundation for writing algebraic expressions. Then, have students write numerical expressions from phrases without calculating them. Using both brackets and braces (nesting symbols) isn’t a fifth-grade expectation but it can be taught given an explanation. However, the main emphasis in fifth grade is the use of the parenthesis.

Understand the Place Value System

MGSE5.NBT.1

This standard calls for students to reason about the magnitude of numbers. Students should work with the idea that the tens place is ten times as much as the ones place, and the ones place is 1/10th the size of the tens place. In 4th grade, students examined the relationships of the digits in numbers for whole numbers only. This standard extends this understanding to the relationship of decimal fractions, however, that will be addressed in a later unit. Refer to the grade level overview for more information. Students use base ten blocks, pictures of base ten blocks, and interactive images of base ten blocks to manipulate and investigate the place value relationships. They use their understanding of unit fractions to compare decimal places and fractional language to describe those comparisons.Before considering the relationship of decimal fractions, students express their understanding that in multi-digit whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left.

When converting in the metric system, have students extend their prior knowledge of the base ten system as they multiply or divide by powers of ten (as referenced in Units 1 and 2). Teaching conversions should focus on the relationship of the measurements, not merely rote memorization. The questions ask the student to find out the size of each of the subsets. Students are not expected to know e.g. that there are 5280 feet in a mile. If this is to be used as an assessment task, the conversion factors should be given to the students. However, in a teaching situation it is worth having them realize that they need that information rather than giving it to them upfront; having students identify what information they need to have to solve the problem and knowing where to go to find it allows them to engage in Standard for Mathematical Practice 5, Use appropriate tools strategically. Retrieved from Illustrative Mathematics http://www.illustrativemathematics.org/standards/k8

MGSE5.NBT.2

This standard includes multiplying by multiples of 10 and powers of 10, including 102 which is 10 10=100, and 103 which is 10 10 10 =1,000. Students should have experiences working with connecting the pattern of the number of zeros in the product when you multiply by powers of 10. Students should notice the shift of the digits when multiplying by a power of 10.

Examples: 2.5 x 10^3 = 2.5 (10 x10 x10) = 2.5 x 1,000 = 2,500

Students should reason that the exponent above the 10 indicates how many places the digits are shifting (not just that the digits are shifting but that you are multiplying or making the number 10 times greater three times) when you multiply by a power of 10. When we multiply by a positive power of 10, the digits shift to the left because the number is becoming larger.

350 ÷ 10 = 35 350/10 = 35 (350 x 1 /10) = 35

This will relate well to subsequent work with operating with fractions. This example shows that when we divide by powers of 10, the exponent above the 10 indicates how many places the digits are shifting (how many times we are dividing by 10, the number becomes ten times smaller). When we divide by a positive power of 10, the digits shift to the right because the number is becoming smaller.

Students need to be provided with opportunities to explore this concept and come to this understanding; this should not just be taught procedurally.

Perform operations with multi-digit whole numbers and with decimals to hundredths

MGSE5.NBT.5

In previous grade levels, students have used various models and strategies to solve problems involving multiplication with whole numbers, so they should be able to transition to using standard algorithms effectively. With guidance from the teacher, they should understand the connection between the standard algorithm and their strategies.

Connections between the algorithm for multiplying multi-digit whole numbers and strategies such as partial products or lattice multiplication are necessary for students’ understanding. The multiplication can also be done without listing the partial products by multiplying the value of each digit from one factor by the value of each digit from the other factor. Understanding of place value is vital in using the standard algorithm. In using the standard algorithm for multiplication, when multiplying the ones, 32 ones is 3 tens and 2 ones. The 2 is written in the ones place. When multiplying the tens, the 24 tens is 2 hundreds and 4 tens. But, the 3 tens from the 32 ones need to be added to these 4 tens, for 7 tens. Multiplying the hundreds, the 16 hundreds is 1 thousand and 6 hundreds. But, the 2 hundreds from the 24 tens need to be added to these 6 hundreds, for 8 hundreds.

MGSE5.NBT.6

By fifth grade, students should understand that division can mean equal sharing or partitioning of equal groups or arrays. They should also understand that it is the same as repeated subtraction, and since it’s the inverse of multiplication, the quotient can be thought of as a missing factor. In fourth grade, students divided 4-digit dividends by 1-digit divisors. They also used contexts to interpret the meaning of remainders. Division is extended to 2-digit divisors in fifth grade, but fluency of the traditional algorithm is not expected until sixth grade. Division models and strategies that have been used in previous grade levels, such as arrays, number lines, and partial quotients, should continue to be used in fifth grade as students deepen their conceptual understanding of this division.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The terms below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

• Algorithm

• Distributive Property

• Dividend

• Divisor

• Equation

• Exponents

• Expression

• Measurement Division (or repeated subtraction)

• Multiplicand

• Multiplier

• Order of Operations

• Partition Division (or fair-sharing)

• Partial Product

• Partial Quotient

• Product

• Properties of Operations

• Quotient

• Remainder