Third Grade MATHEMATICS - Volusia · 2019-10-10 · Third Grade MATHEMATICS Curriculum Map 2019 – 2020 Volusia County Schools Mathematics Florida Standards . Grade 3 Math Instructional

Third Grade MATHEMATICS

Curriculum Map

2019 – 2020

Volusia County Schools

Mathematics Florida Standards

Grade 3 Math Instructional Calendar Units Topics Standards Suggested Dates

U

nit 1

1 Understanding Multiplication and Division of Whole Numbers (12 days)

3.OA.1.1 3.OA.1.2 3.OA.1.3

Aug. 12-27 Optional Topic Assessment

2 Multiplication Facts: Use Patterns (9 days) 3.OA.3.7

3.OA.4.9 Aug. 28- Sept. 10 Sept. 2 (Labor Day) Optional Topic Assessment

3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8 (11 days)

3.OA.2.5 3.OA.3.7

Sept. 11- 26 Sept. 16 (PL Day) Optional Topic Assessment

4 Use Multiplication to Divide: Division Facts (11 days)

3.OA.1.4 3.OA.2.6 3.OA.3.7

Sept. 27- Oct. 11 Optional Topic Assessment enVision Unit 1

U

nit 2

5 Fluently Multiply and Divide within 100 (8 days) 3.OA.3.7

3.OA.4.9 Oct. 15- 24 Oct. 14 (TDD) Optional Topic Assessment

6 Connect Area to Multiplication and Addition (12 days)

3.MD.3.5 3.MD.3.6 3.MD.3.7

Oct. 25- Nov. 8 Optional Topic Assessment

7

Represent and Interpret Data (9 days) 3.MD.2.3 Nov. 12- Nov. 22 Nov. 11 (Veterans Day) Nov 25-29 (Thanksgiving) Optional Topic Assessment enVision Unit 2

Uni

t 3

8/9

Use Strategies and Properties to Add and Subtract and Fluently Add and Subtract within 1,000 (13 days)

3.OA.4.9 3.NBT.1.1 3.NBT.1.2

Dec. 2-18 Optional Topic Assessment Dec. 19 (TDD) Dec. 20-Jan. 3 (Winter Break)

10 Multiply by Multiples of 10 (4 days) 3.NBT.1.3 Jan. 6-10

11 Use Operations with Whole Numbers to Solve Problems (9 days)

3.OA.4.8

Jan. 13-24 Jan. 20 MLK Day Optional Topic Assessment

12

Understand Fractions as Numbers (21 days) 3.NF.1.1 3.NF.1.2 3.NF.1.3.c 3.MD.2.4 3.G.1.2

Jan. 27- Feb. 25 Feb. 17 (President’s Day) Optional Topic Assessment enVision Unit 3

Uni

t 4

13

Fraction Equivalence and Comparison (19 days)

3.NF.1.3 Feb 26- Mar 31 Mar. 13 (TDD) Mar 16-20 (Spring Break) Optional Topic Assessment

14 Solve Time, Capacity, and Mass Problems (10 days)

3.MD.1.1 3.MD.1.2

Apr. 1-14 Optional Topic Assessment

15 Attributes of Two-Dimensional Shapes (5 days)

3.G.1.1 Apr. 15-21

16 Solve Perimeter Problems (8 days) 3.MD.4.8 Apr. 22- May 1

17 Demonstrating Computational Fluency in Problem Solving (19 days)

3.OA.3.7 3.OA.4.8 3.NBT.1.2

May 4-29 FSA May 25 (Memorial Day)

Volusia County Schools Grade 3 Instructional Calendar Elementary Mathematics Department 2019-2020 Revised 8/9/19

2 Volusia County Schools Grade 3 Math Curriculum Map Mathematics Department June 2019

Unit 1 PACING: Aug. 12 – Oct. 11

Topic 1: Understanding Multiplication and Division of Whole Numbers Pacing: Aug. 12 - 27 Standards Academic

Language Interpret products of whole numbers, e.g., interpret 5 x 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. MAFS.3.OA.1.1 array

column dividend divisor equal groups/shares equation expression factors groups of product quotient row symbol

Students will: • interpret a situation requiring multiplication using objects, pictures, words, and expressions. • describe a context that could be represented by a given multiplication expression.

E.g., Jim purchased 4 packages of muffins. Each package contained 3 muffins. Describe another situation where there would be 4 groups of 3 or 4 x 3. NOTE: In a multiplication expression, the first factor describes the number of groups and the second factor describes the number in each group (e.g., 5 elephants with 4 legs each would be represented with the expression 5 x 4 since there are 5 groups, the elephants, and each group has 4 legs, rather than 4 x 5 which would represent 4 elephants with 5 legs each).

Assessment Limits Whole number factors may not exceed 10 x 10. Students may not be required to write an equation to represent a product of whole numbers.

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, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

MAFS.3.OA.1.2

Students will: • interpret a situation requiring division using objects, pictures, words, and expressions. • describe a context that could be represented by a given division expression.

E.g., 50 ÷ 10; can be 50 items divided into 10 equal groups, or 50 items divided into equal groups of 10. Represent a context that could be described as the quotient of two whole numbers (e.g., 8 ÷ 2 is a way to show the equal sharing of 8 cookies between 2 boys).

NOTE: In a division expression, the divisor can be the number of groups e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or the divisor can be the number in each group e.g., interpret 56 ÷ 8 as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Assessment Limits Whole number quotients and divisors may not exceed 10. Items may not require students to write an equation to represent a quotient of whole numbers.


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 a problem.

MAFS.3.OA.1.3

Students will: • represent multiplication and division word problems involving numbers within 100 using objects, drawings, and equations using a

symbol for the unknown number. • use the following situations to solve real-world and mathematical problems related to multiplication and division within 100.

o equal groups:

Stan has 4 bags of cookies with 5 cookies in each bag. How many cookies does he have? Hector has 12 wrenches. He puts 4 in each compartment in his tool box. How many compartments does it take to

hold all of his wrenches? o array model:

Mrs. Smith arranges the desks in her classroom. She has 4 rows with 3 desks in each row. How many desks are in her classroom?

A marching band has 100 members. The director puts the members into equal columns of 10. How many columns does it take to arrange all of the band members?

o measurement quantities: You need 3 lengths of string, each 6 inches long. How much string will you need altogether? A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

NOTE: See Common Multiplication and Division Situations Table on page 49. NOTE: Use the definition of multiplication (e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each) and the definition of division (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are portioned into equal shares of 8 objects each) implied by the grade 3 standards. Allow students to use repeated addition as a strategy to find products, but do not use it as a definition of multiplication. Similarly, allow students to use repeated subtraction as a strategy to find quotients, but do not use it as a definition of division.

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items may not contain more than one unknown per equation. Items may not contain the words “times as much/many.”

Aspects of Rigor targeted by the standards in this topic: • Conceptual understanding: The Standards call for conceptual understanding of key concepts. Students must be able to

access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures.

• Application: The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.


Applicable information from the progression documents: In Equal Groups, the roles of the factors differ. One factor is the number of objects in a group (like any quantity in addition and subtraction situations), and the other is a multiplier that indicates the number of groups. So, for example, 4 groups of 3 objects is arranged differently than 3 groups of 4 objects. Thus there are two kinds of division situations depending on which factor is the unknown (the number of objects in each group or the number of groups). In the Array situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the situation. But rows and columns depend on the orientation of the array. If an array is rotated 90º, the rows become columns and the columns become rows. This is useful for seeing the commutative property for multiplication in rectangular arrays and areas. Multiplication and division problem representations and solution methods can be considered as falling within three levels... Level 1 is making and counting all of the quantities involved in a multiplication or division. As before, the quantities can be represented by objects or with a diagram, but a diagram affords reflection and sharing when it is drawn on the board and explained by a student. The Grade 2 standards 2.OA.3 and 2.OA.4 are at this level but set the stage for Level 2….Level 2 is repeated counting on by a given number, such as for 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern. For 8 3s, you know the number of 3s and count by 3 until you reach 8 of them. For 24÷3, you count by 3 until you hear 24, then look at your tracking method to see how many 3s you have. Because listening for 24 is easier than monitoring the tracking method for 8 3s to stop at 8, dividing can be easier than multiplying. (See pp. 24-25 in the OA Progressions.)


Topic 2: Multiplication Facts: Use Patterns Pacing: Aug. 28- Sept. 10 Standards Academic

Language Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 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.

MAFS.3.OA.3.7 array column difference digit dividend divisor equal groups/shares equation expression factor groups of horizontal pattern product quotient row sum vertical

Students will:

• multiply within 100 choosing and using strategies such as the relationship between multiplication and division or properties of operations.

• divide within 100 by choosing and using strategies such as the relationship between multiplication and division or properties of operations.

• multiply and divide fluently within 100 by the end of Grade 3. NOTE: Computational fluency is defined as accuracy, efficiency, and flexibility. The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense (Boaler, 2015). The brain researchers concluded that automaticity should be reached through understanding of numerical relations, achieved through thinking about number strategies (Delazer et al, 2005).

• know from memory all products of two one-digit numbers by the end of Grade 3.

NOTE: By the end of Grade 3, students are to know “from memory” all products of two one-digit numbers. Accurate, efficient, and flexible strategies come from memory, not from mere memorization. Memorization via rote drills and repetitive timed tests is void of strategy while learning facts “from memory” relies on strategy. Automaticity comes through learning, repetition, and practicing strategies throughout the course of the year, until it becomes a natural response.

NOTE: All strategies for multiplying and dividing should be connected to conceptual understanding rather than teaching tricks.

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts.


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.

MAFS.3.OA.4.9

Students will: • discover and identify patterns that occur in multiplication tables. • explain multiplication patterns, including patterns that are not explicit, using properties of operations.

E.g.,

All even numbers can be divided by 2. A skip counting pattern occurs in each row and column. Changing the order of the factors does not change the product. The product of two even numbers is always even. The product of two odd numbers is always odd. The product of one even and one odd number is always even.

NOTE: Connect the work of looking for patterns in multiplication facts to grade 2 work with skip counting.

Assessment Limits All values in multiplication or division situations may not exceed whole number multiplication facts of 10 x 10 or the related division facts.



• Procedural skill and fluency: The Standards call for speed and accuracy in calculation. Students are given opportunities to practice core functions so that they have access to more complex concepts and procedures.


Applicable information from the progression document: In the Array situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the situation. But rows and columns depend on the orientation of the array. If an array is rotated 90º, the rows become columns and the columns become rows. This is useful for seeing the commutative property for multiplication in rectangular arrays and areas.

Multiplication and division problem representations and solution methods can be considered as falling within three levels related to the levels for addition and subtraction (see Appendix). Level 1 is making and counting all of the quantities involved in a multiplication or division. Level 2 is repeated counting on by a given number, such as for 3 3; 6; 9; 12; 15; 18; 21; 24; 27; 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern. For 8 x 3, you know the number of 3s and count by 3 until you reach 8 of them. For 24÷3, you count by 3 until you hear 24, then look at your tracking method to see how many 3s you have. Because listening for 24 is easier than monitoring the tracking method for 8 x 3 is to stop at 8, dividing can be easier than multiplying.

The difficulty of saying and remembering the count-by for a given number depends on how closely related it is to 10, the base for our written and spoken numbers. For example, the count-by sequence for 5 is easy, but the count-by sequence for 7 is difficult. Decomposing with respect to a ten can be useful in going over a decade within a count-by. For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14 + 7 =14 + 6 + 1= 20 + 1 =21. The count-by sequence can also be said with the factors, such as “one times three is three, two times three is six, three times three is nine, etc.” Seeing as well as hearing the count-bys and the equations for the multiplications or divisions can be helpful. Level 3 methods use the associative property or the distributive property to compose and decompose. These compositions and decompositions may be additive (as for addition and subtraction) or multiplicative. For example, students multiplicatively compose or decompose: 4 x 6 is easier to count by 3 eight times: 4 x 6 = 4 x (2 x 3) = (4 x 2) x 3 = 8 x 3: Students may know a product 1 or 2 ahead of or behind a given product and say: I know 6 x 5 is 30, so 7 x 5 is 30 + 5 more, which is 35. This implicitly uses the distributive property: 7 x 5 = (6 +1) x 5 = 6 x 5 + 1 x 5= 30 + 5 = 35.

Patterns make multiplication by some numbers easier to learn than multiplication by others, so approaches may teach multiplications and divisions in various orders depending on what numbers are seen as or are supported to be easiest.

All of the understandings of multiplication and division situations, 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…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.

(See pp.24-27 in the OA Progressions.)


Topic 3: Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8 Pacing: Sept. 11- 26 Standards Academic

Language Apply properties of operations as strategies to multiply and divide. E.g., If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative Property of multiplication). 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 – 30. (Associate Property of multiplication.) Knowing that 8 x 5 = 40, and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 (Distributive Property).

MAFS.3.OA.2.5

array decompose dividend divisor equation factor product quotient

Students will: • represent expressions using various objects, pictures, and words in order to develop understanding of the commutative,

distributive and associative properties of multiplication. • apply the Commutative Property as a strategy to multiply.

o Commutative Property of multiplication: E.g., 3 × 2 is the same value as 2 × 3 o Distributive Property of multiplication: E.g., 5 × 7 is the same value as (5 × 5) + (5 × 2) o Associative Property: E.g., 7 × 6 = 7 x (3 x 2), which is the same value as (7 x 3) x 2

NOTE: Highlight student thinking based on the application of properties of operations. Include a variety of strategies so that students can see how to determine unknown facts from facts that they personally know. NOTE: Students are not expected to identify the properties by name.

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items may contain no more than two properties in an equation (e.g., a x (b + c) = (a x b) + (c x a)).

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 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.

MAFS.3.OA.3.7

Students will:






NOTE: By the end of Grade 3, students are to know “from memory” all products of two one-digit numbers. Accurate, efficient, and

flexible strategies come from memory, not from mere memorization. Memorization via rote drills and repetitive timed tests is void of strategy while learning facts “from memory” relies on strategy. Automaticity comes through learning, repetition, and practicing strategies throughout the course of the year, until it becomes a natural response.





Applicable information from the progression document: Multiplication and division problem representations and solution methods can be considered as falling within three levels related to the levels for addition and subtraction (see Appendix). Level 1 is making and counting all of the quantities involved in a multiplication or division. Level 2 is repeated counting on by a given number, such as for 3 3; 6; 9; 12; 15; 18; 21; 24; 27; 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern. For 8 x 3, you know the number of 3s and count by 3 until you reach 8 of them. For 24÷3, you count by 3 until you hear 24, then look at your tracking method to see how many 3’s you have. Because listening for 24 is easier than monitoring the tracking method for 8 x 3s to stop at 8, dividing can be easier than multiplying. The difficulty of saying and remembering the count-by for a given number depends on how closely related it is to 10, the base for our written and spoken numbers. For example, the count-by sequence for 5 is easy, but the count-by sequence for 7 is difficult. Decomposing with respect to a ten can be useful in going over a decade within a count-by. For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14 + 7 =14 + 6 + 1= 20 + 1 =21. The count-by sequence can also be said with the factors, such as “one times three is three, two times three is six, three times three is nine, etc.” Seeing as well as hearing the count-bys and the equations for the multiplications or divisions can be helpful. Level 3 methods use the associative property or the distributive property to compose and decompose. These compositions and decompositions may be additive (as for addition and subtraction) or multiplicative. For example, students multiplicatively compose or decompose: 4x 6 is easier to count by 3 eight times: 4 x 6 = 4 x (2 x 3) = (4 x 2) x 3 = 8 x 3: Students may know a product 1 or 2 ahead of or behind a given product and say: I know 6 x 5 is 30, so 7 x 5 is 30 + 5 more, which is 35. This implicitly uses the distributive property: 7 x 5 = (6 + 1) x 5 = 6 x 5 +1 x 5= 30 + 5 = 35.



All of the understandings of multiplication and division situations, 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.

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.

(See pp. 24-27 in the OA Progressions.)


Topic 4: Use Multiplication to Divide: Division Facts Pacing: Sept. 27- Oct. 11 Standards Academic

Language 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 = ?.

MAFS.3.OA.1.4 dividend divisor factor inverse operations product quotient

Students will: • determine the unknown whole number in multiplication and division equations relating three whole numbers.

Assessment Limits All values in items may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Items must provide the equation. Students may not be required to create the equation.

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. MAFS.3.OA.2.6 Students will:

• understand division as an unknown-factor problem. • demonstrate and explain how a division equation can be rewritten as a related multiplication equation with an unknown factor. • solve division problems by finding the unknown factor in a related multiplication problem.



MAFS.3.OA.3.7

Students will:





• know from memory all products of two one-digit numbers (end-of-school year expectation).

NOTE: By the end of Grade 3, students are to know “from memory” all products of two one-digit numbers. Accurate, efficient, and

flexible strategies come from memory, not from mere memorization. Memorization via rote drills and repetitive timed tests is void of strategy while learning facts “from memory” relies on strategy. Automaticity comes through learning, repetition, and practicing strategies throughout the course of the year, until it becomes a natural response.





Applicable information from the progression document:

In the Array situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the situation. But rows and columns depend on the orientation of the array. If an array is rotated 90º, the rows become columns and the columns become rows. This is useful for seeing the commutative property for multiplication in rectangular arrays and areas.

Relating Equal Group situations to Arrays, and indicating rows or columns within arrays, can help students see that a corner object in an array (or a corner square in an area model) is not double counted: at a given time, it is counted as part of a row or as a part of a column but not both.

Problems in terms of “rows” and “columns,” e.g., “The apples in the grocery window are in 3 rows and 6 columns,” are difficult because of the distinction between the number of things in a row and the number of rows. There are 3 rows but the number of columns (6) tells how many are in each row. There are 6 columns but the number of rows (3) tells how many are in each column. Students do need to be able to use and understand these words, but this understanding can grow over time while students also learn and use the language in the other multiplication and division situations.

Multiplication and division problem representations and solution methods can be considered as falling within three levels related to the levels for addition and subtraction (see Appendix). Level 1 is making and counting all of the quantities involved in a multiplication or division. Level 2 is repeated counting on by a given number, such as for 3: 3; 6; 9; 12; 15; 18; 21; 24; 27; 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern. For 8 * 3, you know the number of 3s and count by 3 until you reach 8 of them. For 24÷3, you count by 3 until you hear 24, then look at your tracking method to see how many 3s you have. Because listening for 24 is easier than monitoring the tracking method for 8 x 3s to stop at 8, dividing can be easier than multiplying. The difficulty of saying and remembering the count-by for a given number depends on how closely related it is to 10, the base for our written and spoken numbers. For example, the count-by sequence for 5 is easy, but the count-by sequence for 7 is difficult. Decomposing with respect to a ten can be useful in going over a decade within a count-by. For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14+7 =14+6+1= 20+1 =21. The count-by sequence can also be said with the factors, such as “one times three is three, two times three is six, three times three is nine, etc.” Seeing as well as hearing the count-bys and the equations for the multiplications or divisions can be helpful.


Level 3 methods use the associative property or the distributive property to compose and decompose. These compositions and decompositions may be additive (as for addition and subtraction) or multiplicative. For example, students multiplicatively compose or decompose: 4 x 6 is easier to count by 3 eight times: 4 x 6 = 4 x (2 x 3) = (4 x 2) x 3 = 8 x 3: Students may know a product 1 or 2 ahead of or behind a given product and say: “I know 6 x 5 is 30, so 7 x 5 is 30 + 5 more, which is 35. This implicitly uses the distributive property: 7 x 5 = (6 +1) x 5 = 6 x 5 + 1 x 5 = 30 + 5 = 35.

Multiplications and divisions can be learned at the same time and can reinforce each other. Level 2 methods can be particularly easy for division, as discussed above. Level 3 methods may be more difficult for division than for multiplication.

Throughout multiplication and division learning, students gain fluency and begin to know certain products and unknown factors.

Use of two-step problems involving easy or middle difficulty adding and subtracting within 1,000 or one such adding or subtracting with one step of multiplication or division can help to maintain fluency with addition and subtraction while giving the needed time to the major Grade 3 multiplication and division standards.

All of the understandings of multiplication and division situations, 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.

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.


(See pp.24-27 in the OA Progressions.)


Unit 2 PACING: Oct. 15 - Nov. 22 Topic 5: Fluently Multiply and Divide within 100 Pacing: Oct. 15 - 24

Standards Academic Language


MAFS.3.OA.3.7 array column difference digit dividend divisor equal groups/shares equation expression factor groups of horizontal pattern product quotient row sum vertical

Students will:





NOTE: By the end of Grade 3, students are to know “from memory” all products of two one-digit numbers. Accurate, efficient,

and flexible strategies come from memory, not from mere memorization. Memorization via rote drills and repetitive timed tests is void of strategy while learning facts “from memory” relies on strategy. Automaticity comes through learning, repetition, and practicing strategies throughout the course of the year, until it becomes a natural response.

Students are NOT expected to identify the properties of operations by name.




MAFS.3.OA.4.9

Students will: • discover and identify patterns that occur in multiplication tables. • explain multiplication patterns, including patterns that are not explicit, using properties of operations.

E.g.,

All even numbers can be divided by 2. A skip counting pattern occurs in each row and column. Changing the order of the factors does not change the product. The product of two even numbers is always even. The product of two odd numbers is always odd. The product of one even and one odd number is always even.

Assessment Limits All values in multiplication or division situations may not exceed whole number multiplication facts of 10 x 10 or the related division facts.





Topic 6: Connect Area to Multiplication and Addition Pacing: Oct. 25- Nov. 8 Standards Academic

Language Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

MAFS.3.MD.3.5

attribute column equation equivalent expression factor length measure plane figure product row square unit unit square width

Students will: • define a unit square (i.e., a square with side lengths of 1 unit) and understand that it is used to measure the attribute of area.

NOTE: The side length of the square could be one customary unit (e.g., inch, foot), one metric unit (e.g., centimeter, meter) or one non-standard unit.

• explain that to measure the area of plane figures, the total number of square units can be determined by covering the figure,

without gaps or overlaps, with unit squares. NOTE: To avoid confusion in later grades, use precise language when talking about the definition of area. Do not define area as length times width; instead, students need to learn to conceptualize area as the amount of two-dimensional space in a bounded region and to measure it by choosing a unit of area, often a square.

Assessment Limits Items may include plane figures that can be covered by unit squares. Items may not include exponential notation for unit abbreviations (e.g., “cm2 ”).

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). MAFS.3.MD.3.6 Students will:

• measure the area of a plane figure covered with unit squares by counting the number of unit squares used to cover the figure.

NOTE: Nonrectangular plane figures used must be able to be decomposed and recomposed into rectangles by rearranging partial unit squares into whole unit squares that can be counted in order to find exact areas.

E.g., In order to find the area of the figure below, students can move the two triangles on the right so they match up with the two triangles on the left to form squares that are 1 inch on each side. Students can then find the area of the original figure by counting the whole square units on the newly composed rectangle.


Assessment Limits Items may include plane figures that can be covered by unit squares. Items may not include exponential notation for unit abbreviations (e.g., “cm2 ”).

Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it and show that the area is the same

as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving

real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems.

MAFS.3.MD.3.7

Students will: • use square tiles to find the area of rectangles with whole number side lengths. • discover the relationship between tiling and multiplying side lengths to find the area of rectangles. • explain the relationship between tiling and multiplying side lengths to find the area of rectangles.

NOTE: Students will not use the formula a = l × w until Grade 4. Students need to discover that the length of one dimension of a rectangle tells how many unit squares are in each row and the length of the other dimension of the rectangle tells how many unit squares are in each column and these lengths can be multiplied to find the area of the rectangle by using their prior understanding of arrays.

• solve real-world and mathematical problems involving finding the area of rectangles by multiplying side lengths. • use square tiles to connect the area of rectangles to the distributive property. • explain the relationship between tiling and multiplying decomposed side lengths to find the area of rectangles. • create area models to represent the distributive property for the area of a rectangle.


E.g.,

NOTE: As students employ the distributive property to find area, do not hold students responsible for writing equations using parentheses. Grouping symbols are not required until Grade 5 (OA.1.1).

• decompose a rectilinear figure into non-overlapping rectangles to find its area by recognizing area as additive. • solve real-world area problems involving rectilinear figures that can be decomposed into non-overlapping rectangles.

E.g.,

Assessment Limits Figures are limited to rectangles and shapes that can be decomposed into rectangles. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10.

Aspects of Rigor targeted by the standards in this topic:

• Conceptual understanding: The Standards call for conceptual understanding of key concepts. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures.


Applicable information from the progression document: Students need to learn to conceptualize area as the amount of two-dimensional space in a bounded region and to measure it by choosing a unit of area, often a square. A two-dimensional geometric figure that is covered by a certain number of squares without gaps or overlaps can be said to have an area of that number of square units. …students can be taught to multiply length measurements to find the area of a rectangular region. But, in order that they make sense of these quantities, they first learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the number of rows. They also learn to understand and explain that the area of a rectangular region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 x 5 or by adding two products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property. (See pp. 16-18 in the MD Progressions.)


Topic 7: Represent and Interpret Data Pacing: Nov. 12- Nov. 22 Standards Academic

Language 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.

MAFS.3.MD.2.3 bar graph category category labels column data horizontal interpret key picture graph represent row scale scale labels symbol title vertical

Students will: • create a bar graph representing up to 6 categories of data using the parts of a bar graph (title, scale, scale label, categories,

category label, and data and bar graph). NOTE: There should be gaps between each of the bars on the bar graph. Histograms will be taught in grade 6.

• solve one- and two-step “how many more” and “how many less” problems using information presented in bar graphs. • create a picture graph using the parts of a scaled picture graph (title, categories, category label, key, and data).

NOTE: The symbol representing data in a picture graph should be the same for all categories in that picture graph.

NOTE: Students need to also create horizontal graphs.


Assessment Limits The number of data categories are six or fewer. Items must provide appropriate scale and/or key unless item is assessing that feature. Only whole number marks may be labeled on number lines.

Aspects of Rigor targeted by the standards in this topic: • Application: The standards call for students to use math in situations that require mathematical knowledge. Correctly applying

mathematical knowledge depends on students having a solid conceptual understanding and procedural fluency.

Applicable information from the progression document: In Grade 3, the most important development in data representation for categorical data is that students now draw picture graphs in which each picture represents more than one object, and they draw bar graphs in which the height of a given bar in tick marks must be multiplied by the scale factor in order to yield the number of objects in the given category. These developments connect with the emphasis on multiplication in this grade. At the end of Grade 3, students can draw a scaled picture graph or a scaled bar graph to represent a data set with several categories (six or fewer categories)…Students can gather categorical data in authentic contexts, including contexts arising in their study of science, history, health, and so on. Of course, students do not have to generate the data every time they work on making bar graphs and picture graphs. That would be too time-consuming. After some experiences in generating the data, most work in producing bar graphs and picture graphs can be done by providing students with data sets. The Standards in Grades 1–3 do not require students to gather categorical data. (See p. 7 in the MD Progressions.) Use of two-step problems involving easy or middle difficulty adding and subtracting within 1,000 or one such adding or subtracting with one step of multiplication or division can help to maintain fluency with addition and subtraction while giving the needed time to the major Grade 3 multiplication and division standards. (See p. 28 of the OA Progressions.)


Unit 3 PACING: Dec. 2 – Feb. 25

Topic 8 & 9: Use Strategies and Properties to Add and Subtract & Fluently Add and Subtract within 1,000 Pacing: Dec. 2-18



MAFS.3.OA.4.9 column compose decompose decrease difference distance estimate horizontal increase length pattern reasonable row sum vertical

Students will: • discover and identify patterns that occur in addition tables. • explain addition patterns, including patterns that are not explicit, using properties of operations.

E.g.,

Any sum of two even numbers is even. Any sum of two odd numbers is even. Any sum of an even number and an odd number is odd. Changing the order of the addends does not change the sum.

Assessment Limits Adding and subtracting is limited to whole numbers within 1,000.

Use place value understanding to round whole numbers to the nearest 10 or 100. MAFS.3.NBT.1.1 Students will:

• understand that the purpose of rounding to estimate before calculation is to make mental math easier and to check the reasonableness of an answer.

• round whole numbers to the nearest 10 through the use of a number line, hundred chart, place value chart, etc. • round whole numbers to the nearest 100 through the use of a number line, hundred chart, place value chart, etc.

E.g.,


Assessment Limits Items may contain whole numbers up to 1,000.

•

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. MAFS.3.NBT.1.2 Students will:

• add and subtract within 1000, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

• fluently add (up to 3 addends) and subtract within 1000, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Strategy Clarification Example

Place Value Strategy

• student breaks each addend into its place value (expanded form) and like place value amounts are combined

Subtract by Adding Up

• student adds up from the number being subtracted (subtrahend) to the whole (minuend)

• the larger the chunks added, the more efficient the strategy

380 – 241 241 + 9 = 250 250 + 50 = 300 300 + 80 = 380 80 + 50 + 9 = 139 therefore 380 – 241 = 139

NOTE: The Standards distinguish strategies from algorithms. From the Standards glossary:

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. 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. Work with computation begins with use of strategies and “efficient, accurate, and generalizable methods.” For addition and subtraction, the culmination of this work is signaled in the Grade 4 Standards by use of the term “standard algorithm.”

Assessment Limits Addends and sums are less than or equal to 1,000. Minuends, subtrahends, and differences are less than or equal to 1,000. Items may not require students to name specific properties.





Applicable information from the progression document: At Grade 3, the major focus is multiplication, so students' work with addition and subtraction is limited to maintenance of fluency within 1000 for some students and building fluency to within 1000 for others...They focus on methods that generalize readily to larger numbers so that these methods can be extended to 1,000,000 in Grade 4 and fluency can be reached with such larger numbers. Fluency within 1000 implies that students use written methods without concrete models or drawings, though concrete models or drawings can be used with explanations to overcome errors and to continue to build understanding as needed. [Students] need to understand that when moving to the right across the places in a number (e.g., 456), the digits represent smaller units. When rounding to the nearest 10 or 100, the goal is to approximate the number by the closest number with no ones or no tens and ones (e.g., so 456 to the nearest ten is 460; and to the nearest hundred is 500). Rounding to the unit represented by the leftmost place is typically the sort of estimate that is easiest for students and often is sufficient for practical purposes. Rounding to the unit represented by a place in the middle of a number may be more difficult for students (the surrounding digits are sometimes distracting). Rounding two numbers before computing can take as long as just computing their sum or difference. (See p. 12 in the NBT Progressions.)


Topic 10: Multiply by Multiples of 10 Pacing: Jan. 6-10 Standards Academic

Language 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. MAFS.3.NBT.1.3

factors products Students will:

• use concrete tools or drawings to multiply one-digit numbers by multiples of 10. • apply place value understanding to multiply by multiples of 10.

E.g., 4 x 50 is 4 groups of 5 tens or 20 tens. 20 tens has the same value as 200.

• multiply one-digit numbers by multiples of 10 in the range of 10-90 using strategies based on place value and properties of operation.

E.g., 9 x 80 = 9 x (8 x 10) or (9 x 8) x 10

NOTE: This standard expects that students reason about their products rather than “just adding zeroes”.

• recognize patterns in multiplying by multiples of 10.

E.g., “When you multiply a number by 10, the digits shift and the value becomes 10 times as great.”

Assessment Limits Items may not require students to name specific properties.



Applicable information from the progression document: [T]he product 3 x 50 can be represented as 3 groups of 5 tens, which is 15 tens, which is 150. This reasoning relies on the associative property of multiplication: 3 x 50 = 3 x (5 x 10) = (3 x 5) x 10 = 15 x 10 = 150. It is an example of how to explain an instance of a calculation pattern for these products: calculate the product of the non-zero digits, then shift the product one place to the left to make the result ten times as large. (See p. 12 in the NBT Progressions.)


Topic 11: Use Operations with Whole Numbers to Solve Problems Pacing: Jan. 13-24


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.

MAFS.3.OA.4.8 difference dividend division divisor equation estimation factor multiply product quotient reasonableness sum unknown quantity

Students will: • use the four operations to solve two-step word problems representing the problems using equations with a letter standing for

the unknown quantity. • use mental computation and estimation strategies, including rounding (prior to solving), to determine the reasonableness of

answers to two-step word problems.

NOTE: See Common Addition and Subtractions Table on page 48 and the Common Multiplication and Division Situations Table on page 49.

NOTE: Present word problems to students and encourage them to use the models and strategies that are most helpful to them. Avoid recurring a specific model, strategy, or “first step” for solving. Record student work with equations using variables and ask students to do the same. Prompt students to assess the reasonableness of their answers and their classmates’ answers.

Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in multiplication or division situations may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Students may not be required to perform rounding in isolation. Equations may be provided in items.



Applicable information from the progression document: Use of two-step problems involving easy or middle difficulty adding and subtracting within 1,000 or one such adding or subtracting with one step of multiplication or division can help to maintain fluency with addition and subtraction while giving the needed time to the major Grade 3 multiplication and division standards. (See p. 28 in the OA Progressions.)


The unit fraction is ¼. There are 4 parts with equal

areas. 2/4 means that there are 2 one-

fourths. 4 equal parts make 1 whole.

Topic 12: Understand Fractions as Numbers Pacing: Jan. 27- Feb. 25


Understand a fraction 1𝑏𝑏 as the quantity formed by 1 part when a whole is partitioned into b equal parts;

understand a fraction 𝑎𝑎𝑏𝑏 as the quantity formed by a parts of size 1

𝑏𝑏.

MAFS.3.NF.1.1 area data denominator distance endpoint equal length equal parts equivalent fractions fraction greater than 1 interval line plot number line numerator partition unit fraction whole

NOTE: For all lessons in this topic, the emphasis should be that fractions are numbers.

Students will (for fractions less than one):

• determine the number of equal parts that make a whole from a given model. • identify the area of one of the equal parts of a partitioned shape as a unit fraction represented as 1

𝑏𝑏.

• understand a fraction 𝑎𝑎𝑏𝑏 as the quantity formed by a groups of 1

𝑏𝑏 ( 48 is 4 groups of 1

8 and parts this size can be counted as 1

8, 28, 38,

48

).

E.g.,

NOTE: Set models (e.g., There are two boys and three girls. 2

5 of the group are boys.) are not explored in Grade 3.

Students will (for fractions greater than one):

• determine the number of equal parts that make one whole from a given model showing two or more same sized wholes. E.g.,

This picture shows 2 identical wholes. 4 equal parts are needed to compose each of the wholes.


• identify the area of one of the equal parts of a partitioned shape as a unit fraction represented as 1𝑏𝑏.

E.g.,

• understand a fraction 𝑎𝑎𝑏𝑏 as the quantity formed by a groups of 1

𝑏𝑏

E.g.,

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Items are limited to combining or putting together unit fractions rather than formal addition or subtraction of fractions. Maintain concept of a whole as one entity that can be equally partitioned in various ways when working with unit fractions. Fractions a/b can be fractions greater than 1. Items may not use the term “simplify” or “lowest terms” in directives. Items may not use number lines. Shapes may include: quadrilateral, equilateral triangle, isosceles triangle, regular hexagon, regular octagon, and circle.

Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and

partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

MAFS.3.NF.1.2

Students will: • partition the interval from 0 to 1 on a number line (i.e., linear model) into 2, 3, 4, 6, or 8 equal-length segments.

The area of each of the 4 equal parts needed to compose a whole is represented by the unit fraction 1

4.

There are six 14 size parts shaded.

These parts can be counted as 14, 24, 34, 44

, 54

, 64

. 6 groups of 1

4 is 6

4.


E.g.,

• identify the length of each of the equal-length segments as a unit fraction represented as 1𝑏𝑏.

E.g.,

• locate the number 1

𝑏𝑏 on a number line by labeling the endpoint of the length from 0 as 1

𝑏𝑏.

• understand that on a number line, each number’s location is based on its distance from 0.

E.g.,

• represent a number 𝑎𝑎

𝑏𝑏 (including fractions greater than 1) on a number line by making 𝑎𝑎 iterations of the length 1

𝑏𝑏 from 0 and

labeling the endpoint of the accumulated length as 𝑎𝑎𝑏𝑏.

E.g.,


Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Number lines in MAFS.3.NF.1.2b items may extend beyond 1. Only whole number marks may be labeled on number lines.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

MAFS.3.NF.1.3

Students will:

• use models to show and explain whole numbers as fractions, and fractions as whole numbers.

E.g., locate 44 and 1 at the same point of a number line

• express whole numbers as fractions

E.g., 7 = 7

1 3 = 12

4 1 = 6

6

• recognize fractions that are equivalent to whole numbers

E.g., 5

1 = 5 8

2 = 4

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Fractions must reference the same whole entity that can be equally partitioned. Items may not use the term “simplify” or “lowest terms” in directives. Visual models may include number lines and area models. Only whole number marks may be labeled on number lines.


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 number, halves, or quarters.

MAFS.3.MD.2.4

Students will: • use a ruler marked in halves and fourths of an inch to measure lengths of objects in whole, half, and quarter inches. • make a line plot with the horizontal scale marked off in whole number, half, or quarter units to display the data that is collected or

given. E.g.,

NOTE: Since students in Grade 3 are also working with categorical data and bar graphs, a student might find it natural to summarize a measurement data set by viewing it in terms of categories—the categories in question being the nine distinct length values which appear on the number line above. For example, the student might want to say that there are two observations in the “category” of 32

4

inches. However, it is important to recognize that 324 inches is not a category like “blue, yellow or red” Unlike these colors, 32

4 inches is a

numerical value with a measurement unit. That difference is why the data in this table are called measurement data and presented on a line plot rather than a bar graph. A display of measurement data must present the measured values with their appropriate magnitudes and spacing on the number line of the line plot. NOTE: On a number line, each tick mark should be labeled with the value that represents that tick mark’s distance from 0. Therefore, when labeling number lines that include units greater than 1, tick marks should be labeled with the mixed number that represents each point’s location (e.g., rather than label the tick marks after 3 on the number line above as 1

4, 24, 34, they are labeled precisely as 31

4, 32

4,

3 34).

Assessment Limits Standard rulers may not be used; only special rulers that are marked off in halves or quarters are allowed. Measurements are limited to inches.


Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1

4 of the area of the

shape. MAFS.3.G.1.2

Students will:

• partition shapes into 2, 3, 4, 6, or 8 parts with equal-sized areas.

NOTE: Shapes may include: quadrilateral, equilateral triangle, isosceles triangle, hexagon, octagon, and circle. (Students do not need to identify triangles as equilateral or isosceles).

• describe the area of each part as a unit fraction of the whole.

E.g., This figure was partitioned/divided into four equal parts. Each part is 14 of the total area of the figure. 1

4 is the unit fraction of

the whole (e.g., because four 14 sizes parts, or 4

4, are required to form one whole).

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Maintain concept of a whole as one entity that can be equally partitioned in various ways when working with unit fractions. Shapes may include: quadrilateral, equilateral triangle, isosceles triangle, regular hexagon, regular octagon, and circle.



Applicable information from the progression document: Grade 3 students start with unit fractions (fractions with numerator 1), which are formed by partitioning a whole into equal parts and taking one part, e.g., if a whole is partitioned into 4 equal parts then each part is 1

4 of the whole, and 4 copies of that part make the whole. Next, students build fractions from unit fractions, seeing the

numerator 3 of 34 as saying that 3

4 is the quantity you get by putting 3 of the 1

4 's together. They read any fraction this way, and in particular there is no need to introduce

"proper fractions" and "improper fractions" initially; 53 is the quantity you get by combining 5 parts together when the whole is divided into 3 equal parts.

As students experiment on number line diagrams they discover that many fractions label the same point on the number line, and are therefore equal; that is, they are equivalent fractions. For example, the fraction 1

2 is equal to 2

4 and also to 3

6. Students can also use fractions strips to see fraction equivalence.


Previously, in Grade 2, students compared lengths using a standard measurement unit. In Grade 3, they build on this idea to compare fractions with the same denominator. They see that for fractions that have the same denominator, the underlying unit fractions are the same size, so the fraction with the greater numerator is greater because it is made of more unit fractions. For example, [a] segment from 0 to 3

4 is shorter than the segment from 0 to 5

4 because it measures 3 units of 1

4 as

opposed to 5 units of 14. Therefore 3

4 < 5

4.

To construct a unit fraction on a number line diagram, e.g., 1

3, students partition the unit interval into 3 intervals of equal length and recognize that each has length 1

3.

They locate the number 13 on the number line by marking off this length from 0, and locate other fractions with denominator 3 by marking off the number of lengths

indicated by the numerator.

The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5

3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1

3.

Students also develop competence in the composition and decomposition of rectangular regions, that is spatially structuring rectangular arrays. They learn to partition a rectangle into identical squares by anticipating the final structure and thus forming the array by drawing rows and columns. In Grade 3, students are beginning to learn fraction concepts (3.NF). They understand fraction equivalence in simple cases, and they use visual fraction models to


represent and order fractions. Grade 3 students also measure lengths using rulers marked with halves and fourths of an inch. They use their developing knowledge of fractions and number lines to extend their work from the previous grade by working with measurement data involving fractional measurement values. For example, every student in the class might measure the height of a bamboo shoot growing in the classroom, leading to the data set shown in the table. (Again, this illustration shows a larger data set than students would normally work with in elementary grades.) To make a line plot from the data in the table, the student can ascertain the greatest and least values in the data: 131

2 inches and 143

4 inches. The student can draw a segment of a number line diagram that includes these

extremes, with tick marks indicating specific values on the measurement scale. This is just like part of the scale on a ruler. Having drawn the number line diagram, the student can proceed through the data set recording each observation by drawing a symbol, such as a dot, above the proper tick mark. As with Grade 2 line plots, if a particular data value appears many times in the data set, dots will “pile up” above that value. There is no need to sort the observations, or to do any counting of them, before producing the line plot. Students can pose questions about data presented in line plots, such as how many students obtained measurements larger than 141

4

inches.

(See p. 3-4, in the NF Progressions. See p. 13 in the Geometry Progressions. See p. 10-11 in the MD Progressions.)


Unit 4 PACING: Feb. 26 – May 29

Topic 13: Fraction Equivalence and Comparison Pacing: Feb 26- Mar 31


Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols

MAFS.3.NF.1.3 denominator equal parts equivalent fractions numerator partition unit fraction whole < (less than) > (greater than) = (equal to)

NOTE: For all lessons in this topic, the emphasis should be that fractions are numbers. Students will:

• identify and represent equivalent fractions using linear models and area models. E.g.,

• recognize and generate simple equivalent fractions. • explain fraction equivalence (i.e., same area of the whole or same point on a number line).

NOTE: Grade 3 students only explore equivalent fractions using continuous models, rather than using: -algorithms: 1

2 × 3

3 = 3

6 (multiplying by a fraction equivalent to 1)

-procedures: such as cross multiplying (the butterfly method)

• use models to show and explain whole numbers as fractions, and fractions as whole numbers. • express whole numbers as fractions, and fractions as whole numbers.

E.g., 51 = 5 7 = 7

1 82 = 4 3 = 12

4 1 = 6

6

• recognize that fraction comparisons are valid only when the two fractions refer to the same size whole.

E.g., When comparing the pieces of two cakes, the cakes need to be the same size and shape.


• compare two fractions with the same denominator with and without visual models.

E.g., 38 of a pizza is less than 7

8 of the same size pizza. The pieces are the same size, so 3 of the 1

8 size pieces is less than 7 of the

18 size pieces.

• compare two fractions with the same numerator with and without visual models.

E.g., 2

6 of a candy bar is more than 2

8 of the same size candy bar. Cutting the candy bar into 6 pieces results in bigger pieces than

cutting it into 8 pieces. Therefore 2 of the 16 sized pieces is more candy than 2 of the 1

8 sized pieces.

• record the results of comparisons with the symbols >, =, or <, and justify the conclusion.

Assessment Limits Denominators are limited to 2, 3, 4, 6, and 8. Fractions must reference the same whole entity that can be equally partitioned, unless item is assessing MAFS.3.NF.1.3d. Items may not use the term “simplify” or “lowest terms” in directives. Visual models may include number lines and area models. Only whole number marks may be labeled on number lines.



Applicable information from the progression document: As students experiment on number line diagrams they discover that many fractions label the same point on the number line, and are therefore equal; that is, they are equivalent fractions. For example, the fraction 1

2 is equal to 2

4 and also to 3

6. Students can also use fractions strips to see fraction equivalence.

Previously, in Grade 2, students compared lengths using a standard measurement unit. In Grade 3, they build on this idea to compare fractions with the same denominator. They see that for fractions that have the same denominator, the underlying unit fractions are the same size, so the fraction with the greater numerator is greater because it is made of more unit fractions. For example, [a] segment from 0 to 3

4 is shorter than the segment from 0 to 5

4 because it measures 3 units of 1

4 as

opposed to 5 units of 14. Therefore 3

4 < 5

4.

(See p. 4 in the NF Progressions.)


Topic 14: Solve Time, Capacity, and Mass Problems Pacing: Apr. 1-14


Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

MAFS.3.MD.1.1 a.m. analog clock attributes estimate increments liquid volume mass measure p.m. time interval

Students will: • tell and write time to the nearest minute. • measure intervals of time in minutes. • solve word problems involving addition and subtraction of time intervals measured in minutes.

E.g., Tonya wakes up at 6:45 a.m. It takes her 5 minutes to shower, 15 minutes to get dressed, and 15 minutes to eat breakfast. What time will she be ready for school? NOTE: See Common Addition and Subtractions Table on page 48.

Assessment Limits Clocks may be analog or digital. Digital clocks may not be used for items that require telling or writing time in isolation.

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

MAFS.3.MD.1.2

Students will: • measure masses of solid objects (grams and kilograms). • estimate masses of solid objects (grams and kilograms). • measure volumes of liquids (liters). • estimate volumes of liquids (liters). • solve one-step word problems involving masses or volumes using addition, subtraction, multiplication, or division.



Assessment Limits Items may not contain compound units such as cubic centimeters (cm3) or finding the geometric volume of a container. Items may not require multiplicative comparison (e.g., “times as much/many”). Unit conversions are not allowed. Units are not limited to grams, kilograms, and liters.



Applicable information from the progression document: Students in Grade 3 learn to solve a variety of problems involving measurement and such attributes as length and area, liquid volume, mass, and time.3.MD.1, 3.MD.2 Many such problems support the Grade 3 emphasis on multiplication (see Table 1) and the mathematical practices of making sense of problems (MP1) and representing them with equations, drawings, or diagrams (MP4). Such work will involve units of mass such as the kilogram. A few words on volume are relevant. Compared to the work in area, volume introduces more complexity, not only in adding a third dimension and thus presenting a significant challenge to students’ spatial structuring, but also in the materials whose volumes are measured. These materials may be solid or fluid, so their volumes are generally measured with one of two methods, e.g., “packing” a right rectangular prism with cubic units or “filling” a shape such as a right circular cylinder. Liquid measurement, for many third graders, may be limited to a one-dimensional unit structure (i.e., simple iterative counting of height that is not processed as three-dimensional). Thus, third graders can learn to measure with liquid volume and to solve problems requiring the use of the four arithmetic operations, when liquid volumes are given in the same units throughout each problem. Because liquid measurement can be represented with one-dimensional scales, problems may be presented with drawings or diagrams, such as measurements on a beaker with a measurement scale in milliliters. (See pp. 18-19 in the MD Progressions.)


Topic 15: Attributes of Two-Dimensional Shapes Pacing: Apr. 15-21


Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

MAFS.3.G.1.1 angle attribute category polygon side subcategory vertex/vertices

Students will: • analyze, compare, and classify quadrilaterals by their attributes. • understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category

(e.g., all closed plane figures with four straight sides belong to the category of quadrilaterals). • recognize rhombuses, rectangles, and squares as examples of quadrilaterals. • draw examples of quadrilaterals that do not belong to any of these subcategories; rhombus, rectangle, or square.

Assessment Limits Shapes may include two-dimensional shapes and the following quadrilaterals: rhombus, rectangle, square, parallelogram, and trapezoid. Items may reference and/or rely on the following attributes: number of sides, number of angles, whether the shape has a right angle, whether the sides are the same length, and whether the sides are straight lines. Items may not use the terms “parallel” or “perpendicular.” Items that include trapezoids must consider both the inclusive and exclusive definitions. Items may not use the term "kite" but may include the figure.


• Conceptual understanding: The Standards call for conceptual understanding of key concepts. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures.

Applicable information from the progression document: Students analyze, compare, and classify two-dimensional shapes by their properties (3.G.1). They explicitly relate and combine these classifications. Because they have built a firm foundation of several shape categories, these categories can be the raw material for thinking about the relationships between classes. For example, students can form larger, superordinate, categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as squares, rectangles, rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of those subcategories…The Standards do not require that such representations be constructed by Grade 3 students, but they should be able to draw examples of quadrilaterals that are not in the subcategories.


Similarly, students learn to draw shapes with prespecified attributes, without making a priori assumptions regarding their classification.MP1 For example, they could solve the problem of making a shape with two long sides of the same length and two short sides of the same length that is not a rectangle. Students investigate, describe, and reason about decomposing and composing polygons to make other polygons. Problems such as finding all the possible different compositions of a set of shapes involve geometric problem solving and notions of congruence and symmetry (MP7). They also involve the practices of making and testing conjectures (MP1), and convincing others that conjectures are correct (or not) (MP3). Such problems can be posed even for sets of simple shapes such as tetrominoes, four squares arranged to form a shape so that every square shares at least one side and sides coincide or share only a vertex. More advanced paper-folding (origami) tasks afford the same mathematical practices of seeing and using structure, conjecturing, and justifying conjectures. Paper folding can also illustrate many geometric concepts. For example, folding a piece of paper creates a line segment. Folding a square of paper twice, horizontal edge to horizontal edge, then vertical edge to vertical edge, creates a right angle, which can be unfolded to show four right angles. Students can be challenged to find ways to fold paper into rectangles or squares and to explain why the shapes belong in those categories. Students also develop more competence in the composition and decomposition of rectangular regions, that is, spatially structuring rectangular arrays. They learn to partition a rectangle into identical squares by


anticipating the final structure and thus forming the array by drawing rows and columns (see the bottom right example on p. 11; some students may still need work building or drawing squares inside the rectangle first). They count by the number of columns or rows, or use multiplication to determine the number of squares in the array. They also learn to rotate these arrays physically and mentally to view them as composed of smaller arrays, allowing illustrations of properties of multiplication (e.g., the commutative property and the distributive property).

Students need to learn to conceptualize area as the amount of two-dimensional space in a bounded region and to measure it by choosing a unit of area, often a square. A two-dimensional geometric figure that is covered by a certain number of squares without gaps or overlaps can be said to have an area of that number of square units. (See pp. 13-14 in the Geometry Progressions. See p. 16 in the MD Progressions.)


Topic 16: Solving problems involving shapes Pacing: Apr. 22- May 1


Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

MAFS.3.MD.4.8 angle attribute distance length polygon side vertex/vertices width

Students will: • explore the concept of perimeter as a measure of distance, using a variety of tools (e.g., geoboards, paper clips, graph paper,

string, etc.). • find the perimeter of a polygon (regular and irregular) that is located on a grid by counting the units. • find the perimeter of a polygon when given the lengths of all sides. • identify and use properties of polygons to find the unknown side length(s) of a polygon when given the perimeter. • solve real world and mathematical problems involving perimeter. • demonstrate how rectangles with the same perimeter can have different areas. • demonstrate how rectangles with the same area can have different perimeters. • distinguish between linear (perimeter) and area measures and when each measure is appropriate to use.

NOTE: Students will not use the formula p= 2l + 2w until Grade 4.

Assessment Limits For items involving area, only polygons that can be tiled with square units are allowable. Dimensions of figures are limited to whole numbers. All values in items may not exceed whole number multiplication facts of 10 x 10. Items are not required to have a graphic, but sufficient dimension information must be given.



Applicable information from the progression document: Third graders focus on solving real-world and mathematical problems involving perimeters of polygons (3.MD.8). A perimeter is the boundary of a two-dimensional shape. For a polygon, the length of the perimeter is the sum of the lengths of the sides. Initially, it is useful to have sides marked with unit length marks, allowing students to count the unit lengths. Later, the lengths of the sides can be labeled with numerals. As with all length tasks, students need to count the length-units and not the end-points. Next, students learn to mark off unit lengths with a ruler and label the length of each side of the polygon. For rectangles, parallelograms, and regular polygons, students can discuss and justify faster ways to find the perimeter length than just adding all of the lengths (MP3). Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent sides and add those numbers or add lengths of two adjacent sides and double that number. A regular polygon has all sides of equal length, so its perimeter length is the product of one side length and the number of sides. Perimeter problems for rectangles and parallelograms often give only the lengths of two adjacent sides or only show numbers for these sides in a drawing of the shape. The common error is to add just those two numbers. Having students first label the lengths of the other two sides as a reminder is helpful. Students then find unknown side lengths in more difficult “missing measurements” problems and other types of perimeter problems.


Students can be taught to multiply length measurements to find the area of a rectangular region. But, in order that they make sense of these quantities (MP2), they first learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the number of rows.

They also learn to understand and explain that the area of a rectangular region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 x 5 or by adding two products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property. (See pp. 16-18 in the MD Progressions.)


Topic 17: Demonstrating Computational Fluency in Problem Solving Pacing: May 4-29



MAFS.3.OA.3.7 difference dividend divisor equal groups/shares equation estimation expression factor groups of multiply product quotient reasonableness sum unknown quantity

Students will:





NOTE: By the end of Grade 3, students are to know “from memory” all products of two one-digit numbers. Accurate, efficient,

and flexible strategies come from memory, not from mere memorization. Memorization via rote drills and repetitive timed tests is void of strategy while learning facts “from memory” relies on strategy. Automaticity comes through learning, repetition, and practicing strategies throughout the course of the year, until it becomes a natural response.

Students are NOT expected to identify the properties of operations by name.


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.

MAFS.3.OA.4.8

Students will: • use the four operations to solve two-step word problems representing the problems using equations with a letter standing for

the unknown quantity. • use mental computation and estimation strategies, including rounding (prior to solving), to determine the reasonableness of

answers to two-step word problems.



NOTE: Present word problems to students and encourage them to use the models and strategies that are most helpful to them. Avoid recurring a specific model, strategy, or “first step” for solving. Record student work with equations using variables and ask students to do the same. Prompt students to assess the reasonableness of their answers and their classmates’ answers.

Assessment Limits Adding and subtracting is limited to whole numbers within 1,000. All values in multiplication or division situations may not exceed whole number multiplication facts of 10 x 10 or the related division facts. Students may not be required to perform rounding in isolation. Equations may be provided in items.

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. MAFS.3.NBT.1.2

Students will: • add and subtract within 1000, using strategies based on place value, properties of operations, and/or the relationship between

addition and subtraction.

• fluently add (up to 3 addends) and subtract within 1000, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Strategy Clarification Example

Place Value Strategy

• student breaks each addend into its place value (expanded form) and like place value amounts are combined

Subtract by Adding Up

• student adds up from the number being subtracted (subtrahend) to the whole (minuend)

• the larger the chunks added, the more efficient the strategy

380 – 241 241 + 9 = 250 250 + 50 = 300 300 + 80 = 380 80 + 50 + 9 = 139 therefore 380 – 241 = 139

NOTE: The Standards distinguish strategies from algorithms. From the Standards glossary:


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. 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. Work with computation begins with use of strategies and “efficient, accurate, and generalizable methods.” For addition and subtraction, the culmination of this work is signaled in the Grade 4 Standards by use of the term “standard algorithm.”

Assessment Limits Addends and sums are less than or equal to 1,000. Minuends, subtrahends, and differences are less than or equal to 1,000. Items may not require students to name specific properties.






Critical Areas for Mathematics in Grade 3

In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

(1) Students develop 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. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to and less than. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

(4) Students describe, analyze, and compare properties of two-dimensional shapes (i.e., quadrilaterals). They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.


Grade 3 Major, Supporting, and Additional Work

Topic Title Major Work Supporting Work Additional Work

1 Understanding Multiplication and Division of Whole Numbers 3.OA.1.1

3.OA.1.2 3.OA.1.3

2

Multiplication Facts: Use Patterns 3.OA.3.7 3.OA.4.9

3 Apply Properties: Multiplication Facts for 3, 4, 6, 7, 8 3.OA.2.5 3.OA.3.7

4 Use Multiplication to Divide: Division Facts 3.OA.1.4 3.OA.2.6 3.OA.3.7

5 Fluently Multiply and Divide within 100 3.OA.3.7 3.OA.4.9

6 Connect Area to Multiplication and Addition 3.MD.3.5 3.MD.3.6 3.MD.3.7

7 Represent and Interpret Data 3.MD.2.3

8/9 Use Strategies and Properties to Add and Subtract and Fluently Add and Subtract within 1,000

3.OA.4.9

3.NBT.1.1 3.NBT.1.2

10 Multiply by Multiples of 10 3.NBT.1.3

11 Use Operations with Whole Numbers to Solve Problems 3.OA.4.8

12 Understand Fractions as Numbers 3.NF.1.1 3.NF.1.2 3.NF.1.3.c

3.MD.2.4 3.G.1.2

13 Fraction Equivalence and Comparison 3.NF.1.3

14 Solve Time, Capacity, and Mass Problems 3.MD.1.1 3.MD.1.2

15 Attributes of Two-Dimensional Shapes 3.G.1.1

16 Solve Perimeter Problems 3.MD.4.8


Common Addition and Subtraction Situations Table

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?

2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?

2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?

? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now?

5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?

5 - ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?

? – 2 = 3

Total Unknown Both Addends Unknown1 Addend Unknown2

Put Together/

Take Apart

Three red apples and two green apples are on the table. How many apples are on the table?

3 + 2 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?

5 = ? + ? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 + 4 + 1 5 = 2 + 3, 5 = 3 + 2

Five apples are on the table. Three are red and the rest are green. How many apples are green?

3 + ? = 5 5 – 3 = ?

Difference Unknown Bigger Unknown Smaller Unknown

Compare

“How many more?” version:

Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

“How many fewer?” version:

Lucy has two apples. Julie has five apples. How may fewer apples does Lucy have than Julie?

2 + ? = 5 5 – 2 = ?

“More” version suggests operation:

Julie has 3 more apples than Lucy. Lucy has two apples. How many apples does Julie have?

“Fewer” version suggests operation:

Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

“Fewer” version suggests wrong operation:

Lucy has three fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

2 + 3 = ? 3 + 2 = ?

“More” version suggests wrong operation: Lucy has three fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

5 – 3 = ? ? + 3 = 5

Darker shading indicates the four Kindergarten problem subtypes. Grade 1 and 2 students work with all subtypes and variants. Unshaded (white) problems are the four difficult subtypes or variants that students should work with in Grade 1 but need not master until Grade 2. Adapted from CCSS, p. 88, which is based on Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, National Research Council, 2009, pp. 32–33.

1 This can be used to show all decompositions of a given number, especially important for numbers within 10. Equations with totals on the left help children understand that = does not always mean “makes” or “results in” but always means “is the same number as.” Such problems are not a problem subtype with one unknown, as is the Addend Unknown subtype to the right. These problems are a productive variation with two unknowns that give experience with finding all the decompositions of a number and reflecting on the patterns involved.

2 Either addend can be unknown; both variations should be included.


Common Multiplication and Division Situations Table1

Unknown Product Group Size Unknown (“How many in each group?” Division)

Number of Groups Unknown (“How many groups?” Division)

3 × 6 = ? 3 × ? = 18 and 18 ÷ 3 = ? ? × 6 = 18 and 18 ÷ 6 = ?

Equal Groups

There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

Arrays2, Area3

Unknown Product Unknown Factor Unknown Factor There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example. What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

Compare4

Larger Unknown Smaller Unknown Multiplier Unknown A blue hat costs $6. A red hat cost 3 times as much as the blue hat. How much does the red hat cost? Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does the blue hat cost? Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

Smaller Unknown 13 x 18 = ?

Larger Unknown 13 x ? = 6

Multiplier Unknown ? x 18 = 6

A red hat costs $18. A blue hat costs 13 as

much as the red hat. How much does the blue hat cost?

A blue hat costs $6 and that is 13 of the cost of a

red hat. How much does a red hat cost? A red hat costs $18 and a blue hat costs $6. What fraction of the cost of the red hat is the cost of the blue hat?

General a × b = ? a × ? = p and p ÷ a = ? ? × b = p and p ÷ b = ?

1 The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. 2 The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. 3 Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. 4 Multiplicative Compare problems appear first in Grade 4, with whole-number values for A, B, and C, and with the “times as much” language in the table. In Grade 5, unit fractions language such as “one third as much” may be used. Multiplying and unit fraction language change the subject of the comparing sentence, e.g., “A red hat costs A times as much as the blue hat” results in the same comparison as “A blue hat costs 1

𝑎𝑎 times as much as the red hat,” but has a different subject.

Documents

Third Grade MATHEMATICS - Volusia · 2019-10-10 · Third Grade MATHEMATICS Curriculum Map 2019 – 2020 Volusia County Schools Mathematics Florida Standards . Grade 3 Math Instructional