Last updated: 8/15/2021 ## Grade 3 - Numbers and Operations of Algebra

 Subject: Mathematics (NYS P-12 Common Core)
 Unit Title: Grade 3 - Numbers and Operations of Algebra
 Approx. Number of Weeks: 11-20 weeks

#### Unit Summary:

Develop understandings of and strategies for multiplication and division w/in 100.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (Add and subtract within 1,000)

Represent and solve problems involving multiplication and division.

Understand properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100. - Math facts speed drill, 30 problems in 2 minutes

Solve problems involving the four operations, and identify and explain patterns in arithmetic. Collaboration & Communication Creativity & Innovation Critical Thinking & Problem Solving Research & Information Fluency Social & Emotional Intelligence

### 1. What will students know and be able to do?

#### Standards:

 3.OA.1 - Interpret products of whole numbers, e.g., interpret 5 * 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 * 7. 3.OA.2 - Interpret whole-number quotients of whole numbers, e.g., interpret 56 / 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, 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. 3.OA.3 - Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.4 - Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 * ? = 48, 5 = 􀃍 / 3, 6 * 6 = ?. 3.OA.5 - Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 * 4 = 24 is known, then 4 * 6 = 24 is also known. (Commutative property of multiplication.) 3 * 5 * 2 can be found by 3 * 5 = 15, then 15 * 2 = 30, or by 5 * 2 = 10, then 3 * 10 = 30. (Associative property of multiplication.) Knowing that 8 * 5 = 40 and 8 * 2 = 16, one can find 8 * 7 as 8 * (5 + 2) = (8 * 5) + (8 * 2) = 40 + 16 = 56. (Distributive property.) 3.OA.6 - Understand division as an unknown-factor problem. For example, find 32 / 8 by finding the number that makes 32 when multiplied by 8. 3.OA.7 - Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 * 5 = 40, one knows 40 / 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. 3.OA.8 - Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 3.OA.9 - Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. 3.NBT.1 - Use place value understanding to round whole numbers to the nearest 10 or 100 3.NBT.2 - Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 3.NBT.3 - Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.

#### Essential Understandings:

Number relationships can be used to represent and solve problems.

#### Essential Questions:

What kind of problem is this?

What relationships do I see that can help me solve this problem?

#### Students will know:

Division is repeated subtraction.

Addition and multiplication have properties (rules to solve problems).

Multiplication and division are related operations.

#### Students will be able to:

Number and Operations in Base Ten

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

FLUENCY  STANDARD !

Fluently add and subtract within 1000 using strategies (number lines, 100 charts, number decomposition) and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (3.NBT.2)

Use place value understanding to round whole numbers to the nearest 10 or 100. (3.NBT.1)

Operations and Algebraic Thinking

Represent and solve problems involving multiplication and division.

Interpret products of whole numbers e.g., interpret 3 x 4 as the total number of objects in 3 groups of 4 objects each. (3.OA.1)

For example, describe a context in which a total number of objects can be expressed as 3x4 using a variety of strategies.

3x4 array

****

****

****

Skip counting by a factor 4,8,12

Draw a picture; 3 groups of 4

Interpret whole-number quotients of whole numbers, e.g., interpret 12 ÷ 4 as 12 objects to be partitioned into 4 equal shares. (3.OA.2)

For example, describe a context in which a number of shares or a number of groups can be expressed as 12 ÷ 4 using a variety of strategies.

Repeated subtraction

12-4-4-4=0

Draw a picture of equal groups or an array.

Skip count to figure out how many groups   4,8,12

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. (3.OA.3)

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. (3.OA.4)

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

8 x n = 48

5 = n x 3

6 x 6 = n

Understand properties of multiplication and the relationship between multiplication and division.

Apply properties of operations as strategies to multiply and divide. (3.OA.5)

Commutative property of multiplication.

If 6 x 4 = 24 is known, then  4 x 6 = 24 is also known.

6 x 4= 􀀀 x 6

Associative property of multiplication.

3 x 5 x 2 can be found by grouping factors in any order

(3x5) x 2= 30

3 x 5 = 15, then 15 x 2 = 30,

or by (5 x 2) x 3=30

5 x 2 = 10, then 3 x 10 = 30.

Distributive property.

8 x 7 can be solved by knowing

8 x 5 = 40 and 8 x 2 = 16,

8 x (5 + 2) = (8 x 5) + (8 x 2)

(8 x 5) + (8 x 2 )=  40 + 16

Understand division as an unknown-factor problem. (3.OA.6)

For example, solve 32 ÷ 8 by finding the missing factor 8 x _ = 32.

FLUENCY  STANDARD !

Multiply and divide within 100. (3.OA.7)

Fluently multiply and divide within 100, using the relationship between multiplication and division and properties of operations. (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8)

*** By the end of Grade 3, know products of basic facts within 100.

(30  in 3 min.)

Solve problems involving the four operations and identify and explain patterns in arithmetic.

Solve two-step word problems using the four operations. (3.OA.8)

Represent these problems using equations with a symbol standing for the unknown quantity.

Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Identify arithmetic patterns (including patterns on the 100 chart and multiplication table), and explain them using properties of operations. (3.OA.9)

For example,

Multiples of a number;

Multiples of 2: are always even

Multiples of 5:  end in 5 or 0

Multiples of 4 are doubles of multiples of 2

Also, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.  (4x3) = (2x3) +(2x3)

Number and Operations in Base Ten

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.

Color Code Key: Gaps, Major Clusters, Supporting Clusters, Additional Clusters

### 3. What learning activities will students participate in?

#### Discipline Specific Considerations:

Vocabulary – Specialized and High Frequency

Algorithm, repeated addition, repeated subtraction, multiply, divide, addend, digit, compare, tens, ones, hundreds, thousand, decompose, doubling, halving, factor, product, multiples, strategies, relationship, models, symbol, unknown, equation, skip count, operation, addition, subtraction, <, >, =, value, sum, difference, equal, commutative property, associative property, distributive property, pattern, round, estimate, array, row, column, quotient, partitioning, equal groups, reasonableness of answers

An unknown is any missing value in an equation.  It has a constant value and is not variable. (true)

A multiple of ten does not mean a zero added to a single digit number. (5x50 is NOT 5x5 with a zero added at the end. A factor of ten is being applied. 5x50 is NOT 25 + 0  It is 25 x10) (true)

That 700 represents 7 hundreds, 0 tens, 0 ones, but also represents: 70 tens or 700 ones. (true)

Although 4x5 and 5x4 have the same product their representations are different and have different meanings in word problems. (5 cars with 4 wheels each looks very different from 4 cars with 5 wheels each!) (true)

Common Core State Standards:  Standards for Mathematical Practice

Standard 1:    Make sense of problems and persevere in solving them

Standard 2:    Reason abstractly and quantitatively

Standard 3:    Construct viable arguments and critique the reasoning of others

Standard 4:    Model with Mathematics

Standard 5:    Use appropriate tools strategically

Standard 6:   Attend to precision

Standard 7:  Look for and make use of structure

Standard 8:  Look for and express regularity in repeated reasoning 