Standard #: MA.3.AR.2.1


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Restate a division problem as a missing factor problem using the relationship between multiplication and division.


Examples


The equation 56÷7=? can be restated as 7×?=56 to determine the quotient is 8.

Clarifications


Clarification 1: Multiplication is limited to factors within 12 and related division facts.

Clarification 2: Within this benchmark, the symbolic representation of the missing factor uses any symbol or a letter.



General Information

Subject Area: Mathematics (B.E.S.T.)
Grade: 3
Strand: Algebraic Reasoning
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Equation 
  • Factor

 

Vertical Alignment

Previous Benchmarks

 

Next Benchmarks

 

Purpose and Instructional Strategies

The purpose of this benchmark is to build students’ fluency with division facts by relating them to known multiplication facts. Division is often more challenging for students than multiplication, so relating division to multiplication helps to determine quotients. Students learned a similar strategy when relating subtraction to addition in Grade 1 (MA.1.AR.2.1). 
  • Instruction should have students build and use fact families to relate division and multiplication equations. It is important for students to understand that multiplication and division are inverse operations. During instruction, students should have practice with solving and explaining division problems that can also be represented as an unknown factor in multiplication problems (MTR.3.1, MTR.5.1). 
  • To help students understand the relationships between division problems and unknown factor problems conceptually (and to build understanding about fact families), teachers should utilize arrays that show 4 related multiplication and division facts. In addition to arrays, instruction of this standard pairs well with MA.3.AR.1.2 while students solve one- and two-step real-world problems. When students translate problem contexts to division equations, this benchmark helps students find solutions (MTR.3.1, MTR.5.1).

 

Common Misconceptions or Errors

  • Students may have difficulty understanding that the quotient of a division equation will become a factor in a multiplication equation. Allowing students to use an array model and/or reinforcing fact families may help to clarify the relationship.

 

Strategies to Support Tiered Instruction

  • Instruction includes opportunities to use array models to practice relating multiplication and division as inverse operations. The teacher shows an array model and guides students to identify the factors and the product, having them assist in writing the corresponding equation. The teacher guides students to complete the fact family using prompts as needed, reminding them that multiplication and division are inverse operations. After practicing with several examples, students practice completing fact families without arrays, solving for an unknown factor. 
    • For example, students draw an array model to show 3 × 7. 

3 × 7 = 21 
7 × 3 = 21 
21 ÷ 7 = 3 
21 ÷ 3 = 7
    • For example, the students write the fact family and solve for 42 ÷ 6. 
42 ÷ 6 = __  42 ÷ 6 = 7
42 ÷ __ = 6  42 ÷ 7 = 6
6 × __ = 42  6 × 7 = 42
__ × 6 = 42  7 × 6 = 42 
  • Teacher provides opportunities to use manipulatives to practice relating multiplication and division as inverse operations. The teacher guides students to develop a model using manipulatives (e.g., counters or base-ten blocks) and uses explicit instruction and questioning to help students identify the related equation. Additionally, the teacher guides students to complete the fact family using explicit instruction, verbal prompts, and nonverbal cues while reminding students that multiplication and division are inverse operations. After practicing with several examples, students practice completing fact families without arrays, solving for an unknown factor with the support of number cards. 
    • For example, the teacher uses counters to show an array to represent 4 × 8 and asks guiding questions to help students build the array. With prompting, the teacher guides students to identify the product and write the complete fact family. 

counters

4 × 8 = 32 
8 × 4 = 32 
32 ÷ 8 = 4 
32 ÷ 4 = 8
    • For example, students use number cards to rearrange equations to create all four parts to the fact family and solve for a missing factor. Students may also write on notecards for this activity. One card should have the multiplication symbol on one side and the division symbol on the other. The teacher uses a blank card for the missing factor until the students solve it. Students move each card to a different location to build the entire fact family and record each equation on a sheet of paper or mini whiteboard as they manipulate the cards.

 

Instructional Tasks

Instructional Task 1 

  • Part A. Write a multiplication equation that can be used to find the quotient 48 ÷ 12. Use n to represent the unknown factor. 
  • Part B. What is the quotient?

 

Instructional Items

Instructional Item 1 

Which of the following equations can be used to find the quotient 72 ÷ 8 ? 
  • a. 8 × ? = 72 
  • b. 72 × 8 = ? 
  • c. 72 − 8 = ? 
  • d. ? + 8 = 72 

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.




Related Courses

Course Number1111 Course Title222
5012050: Grade Three Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712040: Access Mathematics Grade 3 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012055: Grade 3 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))


Related Access Points

Access Point Number Access Point Title
MA.3.AR.2.AP.1 Explore division as multiplication with a missing factor using the relationship between multiplication and division.


Related Resources

Formative Assessments

Name Description
Using Multiplication to Solve Division Problems

Students are asked to solve a division problem using a calculator but without using the division key.

Changing Division Equations into Multiplication Equations

Students consider a division fact that they are likely to know and are asked to turn it into a multiplication fact. If successful, they are asked to rewrite a basic division fact that they are not likely to know and which has a symbol for the unknown number.

Multiplication as the Inverse of Division

Students are given a word problem and asked to write an equation for the problem. Then the students are to select a multiplication equation that can also be used to solve the problem.

Alien Math

Students are told of a visiting alien from a planet where division is not taught, and asked to rewrite four division problems as multiplication problems so the alien can do them. The students are also asked to explain why it might be easier to do the multiplication problems than the given division problem.

Lesson Plans

Name Description
Spin Beyblades

Students will get to evaluate data and create a process for which bey blade would be the "best" for Mr. Brown's toy store. Data will include customer feedback, price, style and revolutions per minute. Students will apply understanding of division in problem-solving. They will write a letter explaining their procedure using grade-appropriate language conventions.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Three is NOT a Crowd!

This lesson will provide students with practical application activities to help them understand how division is simply solving a problem with an unknown factor. This activity includes opportunities for students to use fact families when identifying unknown factors and a tic-tac-toe game to provide whole group practice as well as to be used at a center, or for independent reinforcement of the skills.

Original Student Tutorial

Name Description
Fly Me to the Moon: Multiplication & Division

Explore how multiplication can help you solve division problems during this moon-themed, interactive tutorial.

Student Resources

Original Student Tutorial

Name Description
Fly Me to the Moon: Multiplication & Division:

Explore how multiplication can help you solve division problems during this moon-themed, interactive tutorial.



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