Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Area Model
- Composite Number
- Greatest Common Factor (GCF)
- Prime Number
- Rectangular Array
Purpose and Instructional Strategies
In grade 4, students determined factor pairs for whole numbers from 0 to 144 and determined if numbers are prime, composite, or neither. In grade 5, students multiplied and divided using products and divisors greater than 144. In future grades, students will use their understanding of factors, factorization, and the distributive property to generate equivalent expressions and solve equations.
- This benchmark builds upon student understanding of the distributive property and supports the decomposition of numbers in earlier grades and extends to future learning in algebraic reasoning in future grade levels.
- Students should not be using the multiplication symbol “×” when rewriting composite numbers as a common factor multiplied by the sum of two whole numbers. In grade 3, students have seen the multiplication symbol “×” when using the distributive property. Students should move away from this practice in grade 6 to writing the common factor directly outside the parentheses with an understanding that a number directly next to parentheses means you multiply (MTR.3.1).
- For example, 24+12 = 6(4+2).
- Instruction includes the use of manipulatives, models, drawings and equations to rewrite the sum.
- If two numbers have multiple common factors, there will be multiple equivalent common factors multiplied by the sum of two whole number possibilities (MTR.2.1, MTR.5.1).
- For example the equations below are equivalent:
42+96=2(21+48) 42+96=3(14+32) 42+96=6(7+16)
- Provide opportunities for students to identify the common factor and the factors to sum in the parenthesis as well as given the common factor multiplied by the sum of two whole numbers and determine the two composite whole numbers to be summed.
- For this benchmark, all terms are numerical, not algebraic.
- Students should use their basic multiplication facts to do this benchmark and think through number relationships and patterns (MTR.5.1).
Common Misconceptions or Errors
- When working with the distributive property, some students may incorrectly multiply only the common factor by only one of the terms inside the parenthesis. Students need to make sure to multiply the common factor by both terms in the parentheses.
- Students may incorrectly think that they always must factor out the greatest common factor from the addends, but is not necessarily the case. Students must read the task or question carefully to determine if any common factor is allowable or if it must be the greatest common factor.
Strategies to Support Tiered Instruction
- Instruction includes the use of area models to visually represent an application of the distributive property.
- Teacher provides review of prior knowledge of the distributive property using an area model. For example, the teacher can begin with the expression 5(2 + 3) and the area model below.
Then the teacher can co-construct the area model to demonstrate the distributive property.
- Teacher models using an area model to determine a common factor. For example, the teacher can begin with the expression 9 + 21 and the area model below.
- Then the teacher can co-construct the area model to determine a common factor.
- Instruction includes providing students with incomplete area models they must complete and use to write equivalent expressions.
Instructional Task 1 (MTR.3.1, MTR.4.1)
Rewrite 121 + 66 in the form
is the greatest common factor of 121 and 66 and
are whole numbers. Does your expression have the same value as 121 + 66? Justify your thinking.
Instructional Item 1
Rewrite the following numerical expression in an equivalent form using the distributive property: 24 + 36.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.