Mathematics 2016-17—Grade 3
Weeks 11-13—/November/December
enVisionmath2.0—Topic 5
Critical Area(s): Multiplication and Division
FOCUS for Grade 3
Major Work
Supporting Work
Additional Work
70% of time
20% of time
10% of time
3.OA.A.1-2-3-4
3.MD.B.3-4
3.NBT.A.1-2-3
3.OA.B.5-6
3.G.A.1-2
3.MD.D.8
3.OA.C.7
3.OA.D.8-9
3.NF.A.1-2-3
3.MD.A.1-2
3.MD.C.5-6-7
Fluency standards: 3.OA.C.7 and 3.NBT.A.2
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Standards in bold are specifically targeted within instructional materials.
Domains:
Operations and Algebraic Thinking
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
3.OA.A Represent and solve problems involving
multiplication and division.
3.OA.C Multiply and divide within 100.
3.OA.D Solve problems involving the four
operations, and identify and explain patterns in
arithmetic.
Standards:
3.OA.A.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.OA.C.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.
3.OA.A.4 Determine the unknown whole number
in a multiplication or division equation relating
three whole numbers. For example, determine the
unknown number that makes the equation true in
each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
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3.OA.D.8 Solve two-step word problems using the
four operations. Represent these problems using
equations with a letter standing for the unknown
quantity. Assess the reasonableness of answers
using mental computation and estimation
strategies including rounding.
3.OA.D.9 Identify arithmetic patterns (including
patterns in the addition table or multiplication
table), and explain them using properties of
operations. For example, observe that 4 times a
number is always even, and explain why 4 times a
number can be decomposed into two equal
addends.
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Mathematics 2016-17—Grade 3
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enVisionmath2.0—Topic 5
Foundational Learning
2.OA.C.4
3.OA.A
3.OA.B
Future Learning
3.MD.C.7
3.MD.A.2
3.MD.B.3
4.OA.A.1-2
4.NBT.B.5-6
4.NF.B.4
Key Student Understandings
 Students understand how to use patterns to see relationships among the multiplication
facts.


Students understand how to use known facts to multiply more difficult facts.
Students understand how to apply the Distributive Property to find new multiplication facts
from known facts.
Assessments

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
3.OA.C Develop strategies to multiply and divide within 100.
 Students guess products or quotients for given equations, rather than using a viable strategy to determine the answer to an equation. Allow students to
share strategies used (i.e., arrays, using a known fact, distributive property, relationship between multiplication and division) with peers, and spend time
exploring students’ invented strategies to determine their validity and discuss their efficiency.
3.OA.A Represent and solve problems involving multiplication and division.
 Students think in terms of individual things rather than in groups. A helpful approach to nurture the understanding of grouping is by the use of
rectangular arrays, equal groups drawings, jumps on a number line and tape diagrams as models.
 Students have difficulty identifying information in a problem situation (which number represents the total, the number of groups and/or the number of
items in a group). Students need more experience making explicit connections between their representations (concrete models or drawings) and
determining the number of groups or the number of items in a group.
3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 Students disregard quantities and their relationships when solving multi-step word problems. Model and encourage the use of a “Think Aloud” strategy
to truly make sense of problems before jumping into computation. Have students restate the problem in their own words. Students can also identify and
underline important information. Students may need carefully constructed questions to help guide them in determining what to do, but should NEVER be
told what to do.
 Students misuse estimation strategies when applying them to solve multi-step problems. Students solve problems first and then adjust their answer.
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Mathematics 2016-17—Grade 3
Weeks 11-13—/November/December
enVisionmath2.0—Topic 5
Instructional Practices
Domain: 3.OA
Cluster: 3.OA.C Develop strategies to multiply and divide within 100.
3.OA.C.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.
**Refer to: RESOURCES > DEVELOPING FLUENCY > MULTIPLICATION THINKING STRATEGIES on last page of this document.**

This standard uses the word fluently, which means accurately, efficiently (using a reasonable amount of steps and time), and flexibly (using strategies
such as the distributive property). “Know from memory” does not mean focusing only on timed tests and repetitive practice, but ample experiences
working with manipulatives, pictures, arrays, word problems, and numbers to internalize the basic facts (up to 9 x 9).

By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with
multiplication facts through 10, and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when
and how to use procedures appropriately, and skill in performing them flexibly, accurately, and efficiently.

Strategies students use to attain fluency may include:
o multiplication by zeros and ones
o doubles (2s facts), doubling twice (4s), doubling three times (8s)
o tens facts (relating to place value, 5 x 10 is 5 tens or 50)
o five facts (half of tens)
o skip counting (counting groups of __ and knowing how many groups have been counted)
o square numbers (ex: 3 x 3)
o nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3)
o decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6)
o turn-around facts (Commutative Property)
o fact families (ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24)
o missing factors

Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms. (Problems presented horizontally
encourage solving mentally).

Note that mastering this material, and reaching fluency in single-digit multiplications and related divisions with understanding, may be quite time
consuming because there are not general strategies for multiplying or dividing all single-digit numbers, as there are for addition and subtraction. Instead,
there are many patterns and strategies dependent upon specific numbers. So it is imperative that extra time and support be provided if needed.
(Progressions for the CCSSM; Operations and Algebraic Thinking, CCSS Writing Team, May 2011, page 22)
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Mathematics 2016-17—Grade 3
Weeks 11-13—/November/December
enVisionmath2.0—Topic 5

All of the understandings of multiplication and division situations (See Glossary, Table 2, page 89), of the levels of representation and solving, and of
patterns need to culminate by the end of Grade 3 in fluent multiplying and dividing of all single-digit numbers and 10. Such fluency may be reached by
becoming fluent for each number (e.g., the 2s, the 5s, etc.) and then extending the fluency to several, then all numbers mixed together. Organizing
practice so that it focuses most heavily on understood but not yet fluent products and unknown factors can speed learning. To achieve this by the end of
Grade 3, students must begin working toward fluency for the easy numbers as early as possible.

Because an unknown factor (a division) can be found from the related multiplication, the emphasis at the end of the year is on knowing from memory all
products of two one-digit numbers. As should be clear from the foregoing, this isn’t a matter of instilling facts divorced from their meanings, but rather
the outcome of a carefully designed learning process that heavily involves the interplay of practice and reasoning. All of the work on how different
numbers fit with the base-ten numbers culminates in these “just know” products and is necessary for learning products. Fluent dividing for all single-digit
numbers, which will combine just knows, knowing from a multiplication, patterns, and best strategy, is also part of this vital standard. (Progressions for
the CCSSM; Operations and Algebraic Thinking, CCSS Writing Team, May 2011, page 27)

NOTE: Memorizing multiplication facts from flash cards should not be sole or primary mode of learning the facts. Memorization of most of the
multiplication facts comes as a by-product of developing number sense after having worked with representations of multiplication.
Domain: 3.OA
Cluster: 3.OA.A Represent and solve problems involving multiplication and division.
3.OA.A.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.

The table below gives examples of a variety of problem solving contexts, in which students need to find the product, the group size, or the number of
groups. Students should be given ample experiences to explore all of the different problem structures. The easiest problem structure includes Unknown
Product (3 x 6 = ? or 18 ÷ 3 = 6). The more difficult problem structures include Group Size Unknown (3 x ? = 18 or 18 ÷ 3 = 6) or Number of Groups
Unknown (? x 6 = 18, 18 ÷ 6 = 3).
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enVisionmath2.0—Topic 5

Explore word problems using multiple representations (use real-world situations such as candy boxes, egg in cartons , crayons, and cars in a parking lot)
to help develop student conceptualize quantities. Discuss representations using the following models:
o Equations: 3 x 4 = ?, 4 x 3 = ?, 12 ÷ 4 = ? and 12 ÷ 3 = ?
o
Arrays:
o
Equal groups:
o
Bar diagrams (tape diagrams):
20
20 in all
4 4 4 4 4
5 groups
4 in each group
o

Number lines
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.
o This shows multiplication using grouping with 3 groups of 5 objects and can be written as 3 × 5.
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
Examples of multiplication:
o There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there?
 This task can be solved by drawing an array by putting 6 desks in each row. This is an array model:

This task can also be solved by drawing pictures of equal groups. 4 groups of 6 equals 24 objects:

A student can also reason through the problem mentally or verbally, “I know 6 and 6 are 12. 12 and 12 are 24. Therefore, there are 4
groups of 6 giving a total of 24 desks in the classroom.”

Students in Grade 3 should use a variety of pictures, such as stars, boxes, flowers to represent unknown numbers (variables). Letters are also introduced
to represent unknowns in third grade.

Examples of Division:
o There are some students at recess. The teacher divides the class into 4 lines with 6 students in each line. Write a division equation for this story
and determine how many students are in the class. (� ÷ 4 = 6. There are 24 students in the class).
o Determining the number of objects in each share (partition model of division, where the size of
the groups is unknown—see model at right):
The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many
hair clips will each person receive?
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Mathematics 2016-17—Grade 3
Weeks 11-13—/November/December
enVisionmath2.0—Topic 5
o
Measurement model of division (the number of shares/groups is unknown):
Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many days will the bananas last?
Starting Day 1
24
24-4=
20
Solution: The bananas will last for 6 days.
Day 2
20-4=
16
Day 3
16-4=
12
Day 4
12-4=
8
Day 5
8-4=
4
Day 6
4-4=
0
3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the
unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

This standard refers to Glossary page 89, Table 2 (table also included at the end of this document for your convenience) and equations for the different
types of multiplication and division problem structures. The easiest problem structure includes Unknown Product (3 x 6 = ? or 18 ÷ 3 = 6). The more
difficult problem structures include Group Size Unknown (3 x ? = 18 or 18 ÷ 3 = 6) or Number of Groups Unknown (? x 6 = 18, 18 ÷ 6 = 3).

The focus of 3.OA.A.4 extends beyond the traditional notion of fact families, by having students explore the inverse relationship of multiplication and
division.

Students extend work from earlier grades with their understanding of the meaning of the equal sign as “the same amount as” to interpret an equation
with an unknown. When given 4 x ? = 40, they might think:
o 4 groups of some number is the same as 40
o 4 times some number is the same as 40
o I know that 4 groups of 10 is 40 so the unknown number is 10
o The missing factor is 10 because 4 times 10 equals 40.

Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.
o Examples:
 Solve the equations below:
24 = ? x 6
72 ÷ = 9
Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel have altogether? 3 x 4 = m
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Domain: 3.OA
Cluster: 3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown
quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Students in Grade 3 begin the step to formal algebraic language by using a letter for the unknown quantity in expressions or equations for one and twostep problems. The symbols of arithmetic, x or • or * for multiplication and ÷ or / for division, continue to be used in Grades 3, 4, and 5. (Progressions for
the CCSSM; Operations and Algebraic Thinking, CCSS Writing Team, May 2011, page 27)

This standard refers to two-step word problems using the four operations. The size of the numbers should be limited to related Grade 3 standards. Adding and
subtracting numbers should include numbers within 1,000 (3.NBT.A.2), and multiplying and dividing numbers should include single-digit factors and products less
than 100 (3.OA.C.7).
o Example: Mike runs 2 miles a day. His goal is to run 25 miles. After 5 days, how many miles does Mike have left to run in order to meet his goal?
Write an equation and find the solution (2 x 5 + m = 25).

In the diagram at right, Carla’s bands are shown using 4 equal-sized bars that represent 4 x
8 or 32 bands. Agustin’s bands are directly below showing that the number that Agustin
has plus 15 equals 32. The diagram can also be drawn like this:
8
8
15
8
8
?
(Progressions for the CCSSM; Operations and Algebraic Thinking,
CCSS Writing Team, May 2011, page 28)
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Mathematics 2016-17—Grade 3
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3.OA.D.9 Identify arithmetic patterns, and explain the patterns using properties of operations.

This standard calls for students to examine arithmetic patterns involving both addition and multiplication. Arithmetic patterns are patterns that change
by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term.
o Doubles (multiples of 2) in a multiplication table fall on horizontal and vertical lines.
o All the multiples of 5 end in a 0 or 5, while all the multiples of 10 end with 0. Every other multiple of 5 is a multiple of 10.
o Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends (14=7+7).
o Multiples of even numbers (2, 4, 6, and 8) are always even numbers.
o On a multiplication chart, the products in each row and column increase by the same amount (skip counting).
o On an addition chart, the sums in each row and column increase by the same amount.
o When applying the Zero Property of Multiplication, the answer is always 0.
o When applying the Commutative Property of Multiplication, the answer is always the same.
o When applying the Identity Property of Multiplication, the answer is always the factor paired with the 1 factor.

Connect patterns to a 100s chart, a number line, or other visual model to help clarify the structure of the pattern.
o Example: What do you notice about the numbers highlighted in pink in the multiplication table? Explain a pattern using properties of operations.
Student: “When you change the order of the factors you will still get the same product. For example 6 x 5 = 30 and 5 x 6 = 30.” (commutative
property)
x
0
1
2
3
4
5
6
7
8
9
10
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
10
2
0
2
4
6
8
10
12
14
16
18
20
3
0
3
6
9
12
15
18
21
24
27
30
4
0
4
8
12
16
20
24
28
32
36
40
5
0
5
10
15
20
25
30
35
40
45
50
6
0
6
12
18
24
30
36
42
48
54
60
7
0
7
14
21
28
35
42
49
56
63
70
8
0
8
16
24
32
40
48
56
64
72
80
9
0
0
9
10
18
20
27
30
36
40
45
50
54
60
63
70
72
80
81
90
90
100
10
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Mathematics 2016-17—Grade 3
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enVisionmath2.0—Topic 5
Differentiation
3.OA.C Develop strategies to multiply and divide within 100.
 Teach thinking strategies explicitly, and allow students to share thinking aloud and compare strategies with peers.
 Allow struggling students to use concrete models/manipulatives to explore and model properties and the inverse
relationship of multiplication and division.
3.OA.A Represent and solve problems involving multiplication and division.
 When working with different operations, students should engage in applying quantitative reasoning strategy (using
the structure of the story to identify the operation needed in order to solve the problem.)
 Ask students to represent their thinking using different models – concrete, pictorial, verbal.
 Modify numbers in the problem to match student need.
 When working with story problem situations, students need to work with a variety of contexts and then compare
their solution strategies. Recording strategies on charts and discussing approaches is a form of differentiation.
Provide various hands on supports to figure out problems.
3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 Ask students to write down the verbal description of solving the problem or to represent their thinking using
different models—concrete, pictorial, verbal—to demonstrate how to record a solution in different ways.
 Modify numbers in problems to match student need.
 Give students a one-step problem and let them solve it, then have them create another problem using the answer
as a starting point. Combine both parts into one problem.
 When working with different operations, students should engage in applying quantitative reasoning strategies
(using the structure of the situation to identify the operation needed) in order to solve the problem.
Literacy Connections

Academic vocabulary terms

Vocabulary Strategies

Literacy Strategies
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students, and
Above Level Students.
Resources
enVisionmath2.0
Developing Fluency
Multiplication Thinking Strategies
Topic 5 Pacing Guide
Grade 3 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
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