All the resources on this page come from the GADOE Framework 3rd Grade Unit 2 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 2 Overview

In this unit, students will:

• begin to understand the concepts of multiplication and division

• learn the basic facts of multiplication and their related division facts

• apply properties of operations (commutative, associative, and distributive) as strategies to multiply and divide

• understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

• fluently multiply and divide within 100, using strategies such as the patterns and relationships between multiplication and division

• understand multiplication and division as inverse operations

• solve problems and explain their processes of solving division problems that can also be represented as unknown factor multiplication problems.

• represent and interpret data

“Multiplication and division are commonly taught separately. However, it is very important to combine the two shortly after multiplication has been introduced. This will help the students to see the connection between the two.” (Van de Walle and Lovin, Teaching Student-Centered Mathematics 3-5, p. 60)

Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.

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 during this cluster are: operation, multiply, divide, factor, product, quotient, strategies, and properties-rules about how numbers work.

STANDARDS FOR MATHEMATICAL PRACTICE (SMP)

The Standards for Mathematical Practice describe varieties of expertise that mathematics

educators at all levels should seek to develop in their students. These practices rest on important

“processes and proficiencies” with longstanding importance in mathematics education.

This section provides examples of learning experiences for this unit that support the development

of the proficiencies described in the Standards for Mathematical Practice. 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 make sense of

problems involving multiplication and division.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning by

connecting arrays with multiplication problems.

3. Construct viable arguments and critique the reasoning of others. Students construct

and critique arguments regarding mental math strategies focusing on multiplication and

division.

4. Model with mathematics. Students are asked to use tiles to model various

understandings of multiplication by creating arrays or groups. They record their thinking

using words, pictures, and numbers to further explain their reasoning.

5. Use appropriate tools strategically. Students use graph paper to find all the possible

rectangles for a given product.

6. Attend to precision. Students will learn to use terms such as multiply, divide, factor, and

product with increasing precision.

7. Look for and make use of structure. Students use the distributive property of

multiplication as a strategy to multiply.

8. Look for and express regularity in repeated reasoning. Students use the distributive

property of multiplication to solve for products they do not know.

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

CONTENT STANDARDS

Represent and solve problems involving multiplication and division.

MGSE3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of

objects in 5 groups of 7 objects each. For example, describe a context in which a total number

of objects can be expressed as 5 x 7.

MGSE3.OA.2 Interpret whole number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the

number of objects in each share when 56 objects are partitioned equally into 8 shares (How many

in each group?), or as a number of shares when 56 objects are partitioned into equal shares of 8

objects each (How many groups can you make?). For example, describe a context in which a

number of shares or a number of groups can be expressed as 56 ÷ 8.

MGSE3.OA.3 Use multiplication and division within 100 to solve word problems in situations

involving equal groups, arrays, and measurement quantities, e.g., by using drawings and

equations with a symbol for the unknown number to represent the problem.3

MGSE3.OA.4 Determine the unknown whole number in a multiplication or division equation

relating three whole numbers using the inverse relationship of multiplication and division. For

example, determine the unknown number that makes the equation true in each of the equations, 8

× ? = 48, 5 = □ ÷ 3, 6 × 6 = ?.

Understand properties of multiplication and the relationship between multiplication and

division.

MGSE3.OA.5. Apply properties of operations as strategies to multiply and divide.4

Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of

multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 ×

10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one

can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Use arrays, area models, and manipulatives to develop understanding of properties.

MGSE3.OA.6. Understand division as an unknown-factor problem.

For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Conversations should also include connections between division and subtraction.

MGSE3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship

between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or

properties of operations. By the end of Grade 3, know from memory all products of two one-digit

numbers.

Use place value understanding and properties of operations to perform multi-digit

arithmetic.

MGSE3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9

x 80, 5 x 60) using strategies based on place value and properties of operations.

Represent and interpret data.

MGSE3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with

several categories. Solve one- and two-step “how many more” and “how many less” problems

using information presented in scaled bar graphs. For example, draw a bar graph in which each

square in the bar graph might represent 5 pets.

MGSE3.MD.4. Generate measurement data by measuring lengths using rulers marked with

halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is

marked off in appropriate units— whole numbers, halves, or quarters.

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.

BIG IDEAS

• Multiplication and division are inverse operations.

• Multiplication and division can be modeled with arrays.

• Multiplication is commutative, but division is not.

• There are two common situations where division may be used.

o Partition (or fair-sharing) - given the total amount and the number of equal

groups, determine how many/much in each group

o Measurement (or repeated subtraction) - given the total amount and the amount in

a group, determine how many groups of the same size can be created.

• As the divisor increases, the quotient decreases; as the divisor decreases, the quotient

increases.

• There is a relationship between the divisor, the dividend, the quotient, and any

remainder.

• The associative property of multiplication can be used to simplify computation.

• The distributive property of multiplication allows us to find partial products and then

find their sum.

ESSENTIAL QUESTIONS

• How are multiplication and division related?

• How can you write a mathematical sentence to represent a multiplication or division

model we have made?

• How do estimation, multiplication, and division help us solve problems in everyday life?

• How does understanding the properties of operations help us multiply large numbers?

CONCEPTS & SKILLS TO MAINTAIN

In Grade 2, instructional time focused on four critical areas:

• Furthering their understanding for the base-ten system. Students worked with counting in

fives, tens and multiples of hundreds, tens and ones. Students also recognize that the

digits in each place of a number represent the amounts of thousands, hundreds, tens, or

ones.

• Using their understanding of addition to develop fluency within 100. They solve

problems within 1,000 by using models of addition and subtraction.

• Recognizing the need for units of measure (centimeter and inch) and understand how to

use rulers and other measurement tools to get linear measurement.

• Developing an understanding of shapes by analyzing and describing them based on their

sides and angles.

Specifically, 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.

• odd and even numbers

• skip counting by twos, threes, fives, and tens

• determining reasonableness using estimation

• addition and subtraction as inverse operations

• commutative, associative, and identity properties of addition

• basic addition facts

• making tens in a variety of ways

• basic subtraction facts

• place value for ones, tens, hundreds, and thousands

• modeling numbers using base 10 blocks and on grid paper

• using addition to find the total number of objects in a rectangular array

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: http://joboaler.com/timed-testsand-the-development-of-math-anxiety/

STRATEGIES FOR TEACHING AND LEARNING
(Adapted from NC Dept. of Public Instruction)

Represent and solve problems involving multiplication and division.

In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 x 5,

and wrote equations to represent the sum. This strategy is a foundation for multiplication because

students should make a connection between repeated addition and multiplication.

Students need to experience problem-solving involving equal groups (whole unknown or size of

group is unknown) and multiplicative comparison (unknown product, group size unknown or

number of groups unknown) as shown in the table in the unit overview. No attempt should be

made to teach the abstract structure of these problems.

Encourage students to solve these problems in different ways to show the same idea and be able

to explain their thinking verbally and in written expression. Allowing students to present several

different strategies provides the opportunity for them to compare strategies.

Sets of counters, number lines to skip count and relate to multiplication and arrays/area models

will aid students in solving problems involving multiplication and division. Allow students to

model problems using these tools. They should represent the model used as a drawing or

equation to find the solution.

This shows multiplication using grouping with 3 groups of 5 objects and can be written as 3 × 5.

Provide a variety of contexts and tasks so that students will have more opportunity to develop

and use thinking strategies to support and reinforce learning of basic multiplication and division

facts.

Have students create multiplication problem situations in which they interpret the product of

whole numbers as the total number of objects in a group and write as an expression. Also, have

students create division-problem situations in which they interpret the quotient of whole numbers

as the number of shares.

Students can use known multiplication facts to determine the unknown fact in a multiplication or

division problem. Have them write a multiplication or division equation and the related

multiplication or division equation. For example, to determine the unknown whole number in 27

÷ = 3, students should use knowledge of the related multiplication fact of 3 × 9 = 27. They

should ask themselves questions such as, “How many 3s are in 27?” or “3 times what number is

27?” Have them justify their thinking with models or drawings.

Students need to apply properties of operations (commutative, associative and distributive) as

strategies to multiply and divide. Applying the concept involved is more important than students

knowing the name of the property. Understanding the commutative property of multiplication is

developed through the use of models as basic multiplication facts are learned. For example, the

result of multiplying 3 x 5 (15) is the same as the result of multiplying 5 x 3 (15).

To find the product of three numbers, students can use what they know about the product of two

of the factors and multiply this by the third factor. For example, to multiply 5 x 7 x 2, students

know that 5 x 2 is 10. Then, they can use mental math to find the product of 10 x 7 (70). Allow

students to use their own strategies and share with the class when applying the associative

property of multiplication.

Splitting arrays can help students understand the distributive property. They can use a known

fact to learn other facts that may cause difficulty. For example, students can split a 6 x 9 array

into 6 groups of 5 and 6 groups of 4; then, add the sums of the groups.

The 6 groups of 5 is 30 and the 6 groups of 4 is 24. Students can write 6 x 9 as 6 x 5 + 6 x 4.

Students’ understanding of the part/whole relationships is critical in understanding the

connection between multiplication and division.

Multiply and divide within 100

Students need to understand the part/whole relationships in order to understand the connection

between multiplication and division. They need to develop efficient strategies that lead to the big

ideas of multiplication and division. These big ideas include understanding the properties of

operations, such as the commutative and associative properties of multiplication and the

distributive property. The naming of the property is not necessary at this stage of learning.

In Grade 2, students found the total number of objects using rectangular arrays, such as a 5 x 5,

and wrote equations to represent the sum. This is called unitizing, and it requires students to

count groups, not just objects. They see the whole as a number of groups of a number of objects.

This strategy is a foundation for multiplication in that students should make a connection

between repeated addition and multiplication.

As students create arrays for multiplication using objects or drawing on graph paper, they may

discover that three groups of four and four groups of three yield the same results. They should

observe that the arrays stay the same, although how they are viewed changes. Provide numerous

situations for students to develop this understanding.

To develop an understanding of the distributive property, students need decompose the whole into groups. Arrays can be used to develop this understanding. To find the product of 3 × 9, students can decompose 9 into the sum of 4 and 5 and find 3 × (4 + 5).

The distributive property is the basis for the standard multiplication algorithm that students can use to fluently multiply multi-digit whole numbers in Grade 5.

Once students have an understanding of multiplication using efficient strategies, they should make the connection to division. Using various strategies to solve different contextual problems that use the same two one-digit whole numbers requiring multiplication allows for students to commit to memory all products of two one-digit numbers.

Represent and interpret data.

Representation of a data set is extended from picture graphs and bar graphs with single-unit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to multiplication and division with 100 (product is 100 or less and numbers used in division are 100 or less). In picture graphs, use values for the icons in which students are having difficulty with multiplication facts. For example,☺ represents 7 people. If there are three ☺, students should use known facts to determine that the three icons represents 21 people. The intervals on the vertical scale in bar graphs should not exceed 100.

Students are to draw picture graphs in which a symbol or picture represents more than one object. Bar graphs are drawn with intervals greater than one. Ask questions that require students to compare quantities and use mathematical concepts and skills. Use symbols on picture graphs that student can easily represent half of, or know how many half of the symbol represents. Students are to measure lengths using rulers marked with halves and fourths of an inch and record the data on a line plot. The horizontal scale of the line plot is marked off in whole numbers, halves or fourths. Students can create rulers with appropriate markings and use the ruler to create the line plots.

Although intervals on a bar graph are not in single units, students count each square as one. To avoid this error, have students include tick marks between each interval. Students should begin each scale with 0. They should think of skip- counting when determining the value of a bar since the scale is not in single units.

Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an example of a pictograph with symbols that represent multiple units. Graphs should include a title, categories, category label, key, and data. How many more books did Juan read than Nancy?

Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale, scale label, categories, category label, and data.

Analyze and Interpret data:

• How many more nonfiction books where read than fantasy books?

• Did more people read biography and mystery books or fiction and fantasy books?

• About how many books in all genres were read?

• Using the data from the graphs, what type of book was read more often than a mystery but less

often than a fairytale?

• What interval was used for this scale?

• What can we say about types of books read? What is a typical type of book read?

• If you were to purchase a book for the class library which would be the best genre? Why?

Students in second grade measured length in whole units using both metric and U.S. customary systems. It is important to review with students how to read and use a standard ruler including details about halves and quarter marks on the ruler. Students should connect their understanding of fractions to measuring to one-half and one-quarter inch. Third graders need many opportunities measuring the length of various objects in their environment. This standard provides a context for students to work with fractions by measuring objects to a quarter of an inch.

Example:

Measure objects in your desk to the nearest ½ or ¼ of an inch, display data collected on a line

plot. How many objects measured ¼? ½? etc. …

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. Mathematics Glossary • array • array • associative property of multiplication • bar graph • commutative property of multiplication • distributive property • dividend • division • divisor • equal groups • equations • factor • fourths • groups of • halves • identity property of multiplication • inch • line plot • measurement division (or repeated subtraction) • multiplicand • multiplication • multiplier • partial products • partition division • partitioned equally • picture graph • product • quotient • remainder • scale • strategy • unknown • whole numbers