Unit 3

Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10

 

Grade 3

Math

Unit Length and Description:

 

25 days

Students learn the remaining multiplication and division facts as they continue to develop their understanding of multiplication and division strategies within 100 and use those strategies to solve one- and two-step word problems. The “2, 3, 4, 5 and 10 facts” unit (Unit 1) and the “0, 1, 6, 7, 8, 9 and multiples of 10 facts” unit (Unit 3) both provide important, sustained time for work in understanding the structure of rectangular arrays to prepare students for area in Unit 4.

 

Standards:

 

Major Cluster Standards

Represent and solve problems involving multiplication and division.

3.OA.2

Interpret wholenumber 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 = ?.

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

3.OA.5

Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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 unknownfactor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

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.

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

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.

Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of algorithms may be used.

3.NBT.3

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

Foundational 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.6 

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

Mathematical Practices

MP.1

Make sense of problems and persevere in solving them. Students engage in exploratory lessons to discover and interpret patterns, and apply their observations to solving multi-step word problems involving all four operations.

MP.3

 

Construct viable arguments and critique the reasoning of others.

As students compare solution strategies, they construct arguments and critique the reasoning of their peers. This practice is particularly exemplified in daily Application Problems and problem-solving specific lessons in which students share and explain their work with one another.

MP.4

Model with mathematics. Students use arrays, tape diagrams, and equations to represent word problem situations.

MP.5

Use appropriate tools strategically. Students analyze problems and select the appropriate tools and pathways to solutions. This is particularly evident as students select problem-solving strategies, and use arithmetic properties as simplifying strategies when appropriate.

MP.7

 

Look for and make use of structure. In this module, patterns emerge as tools for problem solving. Students make use of structure as they utilize the distributive property to establish the 9 = 10 – 1 pattern, for example, or when they check the solution to a fact using units of 9 by making sure the sum of the digits in the product adds up to 9. They make use of the relationship between multiplication and division as they determine unknown factors and interpret the meanings thereof.

Instructional Outcomes

 

3.OA.2: Interpret wholenumber 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.

  • I can find the quotient of whole numbers using equal groups.
  • I can tell what the numbers in a division problem mean.
  • I can explain what division means.
  • I can show division as equal sharing.

 

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.

  • I can multiply to solve word problems.
  • I can divide to solve word problems.
  • I can decide when to multiply or divide to solve word problems.

 

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 = ?.

  • I can find the missing number in a multiplication problem.
  • I can find the missing number in a division problem.

 

3.OA.5: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) 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.)

  • I can use the properties of multiplication and division to solve problems.
  • I can explain the commutative property of multiplication.
  • I can explain the associative property of multiplication.
  • I can explain the distributive property of multiplication.

 

3.OA.6: Understand division as an unknownfactor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8.

  • I can identify the multiplication problem related to the division problem.
  • I can use multiplication to solve division problems.
  • I can recognize and explain the relationship between multiplication and division.

 

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.

  • I can memorize all products within 100.
  • I can use strategies to solve a multiplication problem.
  • I can use strategies to solve a division problem.

 

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.

  • I can identify the order of operations of a problem.
  • I can identify different strategies for estimating.
  • I can construct and equation with a letter standing for the unknown quantity.
  • I can solve two-step word problems using the four operations.
  • I can justify my answer using estimation strategies and mental computation.

 

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.

  • I can identify patterns.
  • I can explain rules for a pattern using properties of operations.
  • I can explain relationships between the numbers in a pattern.

 

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

·        I can define “round or rounding” in relation to place value.

·        I can identify strategies to multiply one-digit numbers by multiples of 10.

·        I can round a whole number to the nearest 10.

·        I can round a whole number to the nearest 100.

 

Enduring Understandings:

 

·        Mental pictures help me remember facts and ideas.

·        Knowing and understanding multiplication helps me understand division.

·        I can solve problems by using multiplication and division.

·        Unfamiliar multiplication problems may be solved by using known multiplication facts and properties, e.g., 8 x 7 = (8 x 2) + (8 x 5)

 

 

Essential Questions:

 

·        Why is multiplication necessary?

·        Why do mental models help me remember?

·        How can multiplication and division help me solve problems?

·        How can working a problem help me better understand the answer?

·        What operation can we use to solve the problem and why?

·        How is multiplication related to division and other operations?