Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Purpose and Instructional Strategies
In grade 8, students rewrote binomial algebraic expressions as a common factor times a
binomial. In Algebra I, students rewrite polynomials, up to 4 terms, as a product of polynomials
over the real numbers. In later grades, students will rewrite polynomials as a product of
polynomials over the real and complex number systems.
- Instruction includes special cases such as difference of squares and perfect square
- Instruction builds upon student prior knowledge of factors, including greatest common
- Instruction includes the student understanding that factoring is the inverse of multiplying
- Instruction includes the use of models, manipulatives and recognizing patterns when
- The expression 2 + 7 + 10 can be written as ( + 5)( + 2) since 5 + 2 = 7
and 5(2) = 10.
- Factor by Grouping
- The expression 3+ 72+ 2 + 14 can be grouped into two binomials and
rewritten as (3+ 72) + (2 + 14). Each binomials can be factored and
rewritten as 2 ( + 7) + 2( + 7) resulting in the same factor and the factored
form as (2 + 2)( + 7)
- A-C Method
- When factoring trinomials 2 + + , multiply and , then determine factor
pairs of the product. Using the factor pair that add to and multiply to , rewrite
the middle term and then factor by grouping.
- For example, given 22 + −
6 and that = −12, one can determine
that two numbers that add to 1 and multiple to -12 are 4 and -3. This
information can be used to rewrite the given quadratic as 22 + 4 − 3 − 6 . Then, using factor by grouping the expression is equivalent to (22 + 4) - (3 + 6) which is equivalent to 2( + 2) − 3( + 2)
which is equivalent to the factored form (2 − 3)( + 2).
- Box Method
- To factor 2 + + the general box method is shown below.
- For example, to factor 22 - 9 - 5 the box method is shown below.
- Area Model (Algebra tiles)
- The factorization of 22 - 9 - 5 using algebra tiles is shown below.
Common Misconceptions or Errors
- Students may not identify the greatest common factor or factor completely.
Strategies to Support Tiered Instruction
- Instruction includes providing a flow chart to reference while completing examples.
- Instruction includes providing definition of greatest common factor and strategies for
identifying the greatest common factor of numerical or algebraic terms.
- For example, the expression 83 - 42 has common factors of 2 and , but these
are not greatest common factors. The greatest common factor of the coefficients is
4 and the greatest common factor of the variable terms is 2. So, the greatest
common factor of the two terms is 42. The expression 83 - 42 can be
rewritten as 42 (2 -1).
Instructional Task 1 (MTR.3.1, MTR.4.1, MTR.5.1)
- Part A. Given the polynomial 4 – 164 8, rewrite it as a product of polynomials.
- Part B. Discuss with your partner the strategy used. How do your polynomial factors
compare to one another?
Instructional Task 2 (MTR.3.1, MTR.5.1)
- Part A. What are the factors of the quadratic 162 - 48 + 36?
- Part B. Determine the roots of the quadratic function () = 162 - 48 + 36.
- Part C. What do you notice about your answers from Part A and Part B?
- Part D. Graph the function () = 162 - 48 + 36.
Instructional Item 1
Instructional Item 2
- Given the polynomial 4 – 164 8, rewrite it as a product of polynomials.
Instructional Item 3
- Given the polynomial 2 - 10 + 24, rewrite it as a product of polynomials.
Instructional Item 4
- Given the polynomial 3 - 32 - 9 + 27 rewrite it as a product of polynomials.
- What is one of the factors of the polynomial 213 2 - 14 2?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.