General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 3
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
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.
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.