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The product 4×72 can be found by rewriting the expression as 4×(70+2) and then using the distributive property to obtain (4×70)+(4×2) which is equivalent to 288.Clarifications
Clarification 1: Within this benchmark, the expectation is to apply the associative and commutative properties of multiplication, the distributive property and name the properties. Refer to K-12 Glossary (Appendix C).Clarification 2: Within the benchmark, the expectation is to utilize parentheses.
Clarification 3: Multiplication for products of three or more numbers is limited to factors within 12. Refer to Properties of Operations, Equality and Inequality (Appendix D).
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Expression
- Equation
- Distributive property
- Factors
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
- Students are introduced to the distributive property of multiplication over addition as a strategy for using products that they know in order to solve products that they do not know. For example, if students are asked to find the product of 6 x 9, they might decompose 6 into 4 and 2 and then multiply 4 x 9 and 2 x 9 to arrive at 36 + 18, which equals 54. Because of the distributive property, students use parentheses to show how to decompose two-digit numbers by the value of their tens and ones, and then multiply the one-digit number by both the values of the two-digit number’s tens and ones values and find the sum of those products. The application of the commutative and associative properties of multiplication allows for two-digit numbers to be decomposed and multiplication expressions reorganized so that the distributive property can work (MTR.2.1).
- During instruction, teachers should model where the properties are applied while multiplying and expect students to explain how they work during explanations of their strategies and solutions. Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty (MTR.2.1, MTR.4.1).
- Building understanding of the distributive property in Grade 3 will help students decompose larger numbers as they continue to multiply multi-digit numbers with procedural reliability and procedural fluency in Grade 4. Splitting arrays can help students understand the distributive property. They can use a known fact to learn other facts that may cause difficulty.
Common Misconceptions or Errors
- Students can be confused about how to write expressions using the distributive property. One common mistake that students make is writing an expression 4 × 72 as (4 × 70) × (4 × 2) instead of (4 × 70) + (4 × 2). Instruction should show concrete models (e.g., base ten drawings) along with equations so students can understand the relationship between multiplication and addition while applying the property and writing expressions.
Strategies to Support Tiered Instruction
- Instruction includes opportunities to use concrete models and drawings along with equations to increase understanding of the relationship between multiplication and addition when applying the distributive property and writing equations. The teacher begins by modeling a one-digit number multiplied by a one-digit number, guiding students to decompose one of the factors, and using models or drawings to demonstrate the reorganization of the multiplication expression using parentheses. Next, the teacher models the multiplication of a one-digit number by a two-digit number, guiding students to decompose the two-digit number into the value of the tens and the ones using models or drawings. The teacher clarifies that the decomposed factor can be represented in expanded form by adding the tens and the ones, repeating with additional one-digit by two-digit multiplication equations.
- For example, the teacher uses a model or drawing to use the distributive property to solve 3 x 24.
- 3 × (20 + 4)
- (3 × 20) + (3 × 4)
- (3 × 20) + (3 × 4) = 60 + 12
- (3 × 20) + (3 × 4) = 72
so 3 × 24 = 72
- Teacher provides opportunities to apply the distributive property to solve one-digit by two-digit multiplication equations using base-ten blocks or place value disks. The teacher provides the equation and guides students to decompose the two-digit number into the value of the tens and the ones using manipulatives. If needed, the teacher prompts students to count by 10s and 1s using the base-ten blocks or place value disks.
- For example, the teacher uses base-ten blocks to solve 3 × 24 while asking guiding questions such as “How many tens are in 24?” “How many ones are in 24?” “How would we write 24 in expanded form?”
Instructional Tasks
Instructional Task 1
- Part A. 4 × 52 = (4 × 50) + (4 × )
- Part B. × 3 = (20 × 3) + (9 × 3)
- Part C. 8 × 36 = ( × 30) + ( × 6)
- Part D. 48 × 6 =
Instructional Task 2
- Tory tried to use the associative and commutative properties to create the following equations. Using pictures and/or words, explain why Tory is incorrect.
- 4 x (11 + 6) = (4 x 11) + 6
- 4 x (11 + 6) = 11 x (4 + 6)
Instructional Items
Instructional Item 1
- a. (8 × 30) × (8 × 9) = 24 + 72 = 96
- b. (8 × 30) + (8 × 9) = 240 + 56 = 296
- c. (8 × 30) + (8 × 9) = 38 + 17 = 45
- d. (8 × 30) + (8 × 9) = 240 + 72 = 312
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Formative Assessments
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Perspectives Video: Teaching Ideas
STEM Lessons - Model Eliciting Activity
In this Model Eliciting Activity, MEA, students will help the Sweet Donut Shop determine what the newest donut will be. Students are given the cost to make each batch along with the selling price and are asked to determine the profit for each batch. Students create a procedure for ranking the donuts and write a letter explaining the procedure and the ranking. In the “twist” students are provided the starting and finishing times for each batch. They must determine the total amount of time, decide if their procedure should change based on the new information, and write a letter explaining whether the procedure changed and the new ranking of the donuts.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
MFAS Formative Assessments
Students are given two arrays, one representing the equation 7 x 9 = 63 and the other representing the equation (5 + 2) x 9 = 63, to see if they recognize a relationship between the two.
Students are given two problems to solve, one represented by the equation 4 x 6 = 24 and the other by the equation 6 x 4 = 24, to see if they recognize the answer to the second problem based on the Commutative Property.
Students are asked to find the product of three numbers and are observed to see if they use the Associative Property to find the product more easily.
Original Student Tutorials Mathematics - Grades K-5
Learn strategies, like the commutative property, to help you become better at multiplying in this interactive tutorial.
Student Resources
Original Student Tutorial
Learn strategies, like the commutative property, to help you become better at multiplying in this interactive tutorial.
Type: Original Student Tutorial
