Mathematics 2016-17—Grade 3
Week 24—February
enVisionmath2.0—Topic 11
Critical Area(s): Multiplication and Division
Beyond the Critical Area(s): Solving multi-step problems
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:
Number and Operations in Base Ten
Clusters outlined in bold should drive the learning for this period of instruction.
3.OA.D Solve problems involving the four operations, and identify and
explain patterns in arithmetic.
3.OA.C Multiply and divide within
100.
Standards:
3.OA.D.8 Solve two-step word problems using the four operations. Represent 3.OA.C.7 Fluently multiply and divide
within 100, using strategies such as
these problems using equations with a letter standing for the unknown
the relationship between
quantity. Assess the reasonableness of answers using mental computation
multiplication and division (e.g.,
and estimation strategies including rounding.
knowing that 8 × 5 = 40, one knows
*This standard is limited to problems posed with whole numbers and having
40 ÷ 5 = 8) or properties of
whole number answers; students should know how to perform operations in
operations. By the end of Grade 3,
the conventional order when there are no parentheses to specify a particular
know from memory all products of
order (Order of Operations).
two one-digit numbers.
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3.NBT.A Use place value
understanding and properties of
operations to perform multi-digit
arithmetic.
3.NBT.A.1 Use place value
understanding to round whole
numbers to the nearest 10 or 100.
3.NBT.A.2 Fluently add and subtract
within 1000 using strategies and
algorithms based on place value,
properties of operations, and/or the
relationship between addition and
subtraction.
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Mathematics 2016-17—Grade 3
Week 24—February
enVisionmath2.0—Topic 11
Foundational Learning
2.OA.A.1
2.MD.B.5
3.OA.A.3
3.NBT.A.1-2
Future Learning
3.NF.A
3.MD.A.1-2
4.OA.A.2-3
4.NF.B
Key Student Understandings
 Students will focus on solving two-step word problems involving whole numbers.
Assessments


Students will use models/diagrams to make sense of and represent problems.

Formative Assessment Strategies
Students will understand that another way to represent word problem is by using
equations.

Evidence for Standards-Based Grading

Students will begin to use formal algebraic language, using a letter to represent a specific
unknown quantity in a problem.
Common Misconceptions/Challenges
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 need carefully constructed questions to help guide them in determining what to do, not 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.
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.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
 The use of terms like “round up” and “round down” confuses many students. For example, in rounding the number 37 to 40, some might say it “rounds
up” because the digit in the tens place changes from 3 to 4 (rounds up). This thinking causes a misconception when applied to rounding down. The
number 32 should be rounded (down) to 30, but using the logic mentioned for rounding up, some students may look at the digit in the tens place and
take it to the previous number, resulting in the incorrect value of 20. To remedy this misconception, students need to use a number line to visualize the
placement of the number and/or ask questions such as: “What are the two closest tens/decade numbers to 32, and which one is it closer to?” Developing
the understanding of what the answer choices are before rounding can alleviate much of the misconception and confusion related to rounding.
 Students may misalign digits when adding or subtracting, losing the meaning of place value. Emphasize the importance of attending to place value;
provide grid paper to help students align the numbers precisely.
 Students may not decompose numbers (based on place value) or use landmark/friendly numbers to develop flexible computation methods.
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Mathematics 2016-17—Grade 3
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enVisionmath2.0—Topic 11

Students do not consider the quantities or the magnitude of the numbers when subtracting or adding; they only see numbers as individual digits; or they
misuse the operation and flip the digits around in order to subtract:

Students misunderstand the value of a number or the number’s relationship to other numbers in order to use estimation skills flexibly. When multiplying
by multiples of 10, students may be confused if the product of a basic fact ends with a zero. Have students underline the product of the basic fact and
then write the other zero: 5 x 60 = 300.
Instructional Practices
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 two-step 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 singledigit 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
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8
8
?
Property of MPS
(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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enVisionmath2.0—Topic 11
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
Week 24—February
enVisionmath2.0—Topic 11

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 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 facts. Memorization of multiplication facts comes
as a by-product of developing number sense after having worked with representations and strategies for multiplication.
Domain: 3.NBT
Cluster: 3.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100.

This standard refers to place value understanding, which extends beyond an algorithm or memorized procedure for rounding. The expectation is that
students have a deep understanding of place value and number sense and can explain and reason about the answers they get when they round.
Students should have numerous experiences using a number line (horizontal and vertical) and a hundreds chart as tools to support work with rounding.

Students learn when and why to round numbers. They identify possible answers and halfway points. Then they narrow where the given number falls
between the possible answers and halfway points. They also understand that by convention if a number is exactly at the halfway point of the two
possible answers, the number is rounded up.
o Example: Round 178 to nearest 10.
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Mathematics 2016-17—Grade 3
Week 24—February
enVisionmath2.0—Topic 11
o
This beaker has 73 milliliters of water in it. Show the amount on a vertical number line. What is 73 milliliters rounded to the nearest ten?
Think: 73 is between what two tens number? (70/7 tens and 80/8 tens). What is halfway between 70 and 80? (75/7 tens and 5 ones) Is 73
milliliters more than halfway or less than halfway between 70 milliliters and 80 milliliters? (less than halfway) So… 73 mL is closest to 70 mL.

Prior to implementing rules for rounding students need to have opportunities to investigate place value. A strong understanding of place value is
essential for the developed number sense and the subsequent work that involves rounding numbers.

Building on previous understandings of the place value of digits in multi-digit numbers, place value is used to round whole numbers. Dependence on
learning rules can be eliminated with strategies such as the use of a number line to determine which multiple of 10 or of100, a number is nearest (5 or
more rounds up, less than 5 rounds down). As students’ understanding of place value increases, the strategies for rounding are valuable for estimating,
justifying, and predicting the reasonableness of solutions in problem-solving.
3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship
between addition and subtraction.

At Grade 3, the major focus is multiplication, so students’ work with addition and subtraction is focused on maintenance of fluency within 1000 for some
students and building fluency to within 1000 for others. Strategies used in Grade 2 to add and subtract two-digit numbers are developed to fluently add
and subtract whole numbers within 1000. These strategies should be discussed so that students can make comparisons and move toward efficient
methods. (Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, March 2015, page 12)

Number sense and computational understanding is built on a firm understanding of place value.

This standard refers to fluency, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as
the distributive property).
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Mathematics 2016-17—Grade 3
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enVisionmath2.0—Topic 11

The word algorithm refers to a procedure or a series of steps. There are algorithms other than the standard algorithm. Third grade students should have
experiences beyond the standard algorithm.
o 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.
o 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. (Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, April 2011, page 2)

Adding and subtracting fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them
flexibly, accurately, and efficiently. Students explain their thinking and show their work by using strategies and algorithms, and verify that their answer is
reasonable.

Problems should include both vertical and horizontal forms, including opportunities for students to apply the commutative and associative properties.
Students explain their thinking and show their work by using strategies and algorithms, and verify that their answer is reasonable.
o Example: There are 178 fourth graders and 225 fifth graders on the playground. What is the total number of students on the playground?
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Mathematics 2016-17—Grade 3
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enVisionmath2.0—Topic 11

Students should also be encouraged to use flexible thinking, number sense, and opportunistic mental strategies to add and subtract within 1000 when
possible.
o Example: Mary read 573 pages during her summer reading challenge. She was only required to read 399 pages. How many extra pages did Mary
read beyond the challenge requirements?
Students may use several approaches to solve the problem that are more efficient than the traditional algorithm. Examples of other methods
students may use are listed below:
Adding up
Compensation
Subtracting to count down
Adding by tens or hundreds
399 + 1 = 400
400 + 100 = 500
500 + 73 = 573
Therefore 1+ 100 + 73 = 174
pages
400 + 100 is 500
500 + 73 is 573
100 + 73 is 173
173 + 1 (399 to 400) is 174
Take away 73 from 573 to get to
500.
Take away 100 to get to 400.
Take away 1 to get to 399.
Then 73 + 100 + 1 = 174.
Differentiation
3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 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 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 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 concrete supports to figure out problems.
 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.
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.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
 Provide students with a number line 1 – 100. Have students circle the number “10” on the number line. With the
same color, draw a box around the numbers that round to 10. Continue with other multiples of 10. For advanced
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Property of MPS
399 + 1 is 400
500 is 100 more
510, 520, 530, 540, 550, 560,
570, (that’s 70 more), 571, 572,
573 (that’s 3 more)
so the total is 1 + 100 + 70 + 3 =
174
Literacy Connections

Academic Vocabulary Terms

Vocabulary Strategies

Literacy Strategies
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Mathematics 2016-17—Grade 3
Week 24—February
enVisionmath2.0—Topic 11


students, change the size of the numbers and the place value to which students will round the number.
Adjust the size of quantities students are working with when problem solving. Practice should focus on place value
and the quantity of the number. Use front-end estimation as well as rounding to help students estimate the
reasonableness of their answer.
Practice counting quantities in several different ways both forward and backward. Use multiple representations to
support student understanding of double digit addition, subtraction and multiplication strategies (e.g., open
number line, partial product, partial sum, base ten blocks, base ten drawings, arrays). Example: Starting at 1000,
count backwards by 100s, 25s or 10s.
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 Fact Thinking Strategies
Topic 11 Pacing Guide
Grade 3 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
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