Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Purpose and Instructional Strategies
In grade 5, students translated written real-world and mathematical descriptions into numerical expressions, evaluated multi-step numerical expressions used order of operations involving combinations of the four arithmetic operations and parentheses with whole numbers, decimals and fractions. They also determined and explained whether an equation involving any of the four operations is true or false. In grade 6, students will use substitution to evaluate algebraic expressions, including exponents and integer coefficients. The values being substituted will also be integers. This benchmark extends to grade 7 where students will evaluate more complex numerical expressions with rational coefficients, apply laws of exponents and generate equivalent linear expressions.
- Substitution is the process in which a symbol or variable is replaced by a given value. In prior grades students have found missing terms in equations and then substituted the value back in to check it they are correct.
- For example, students may have seen something such as 2+? =10 and decided that 8 is the value that should replace the ?.
- This prior experience can be used to connect prior student understanding with new learning. In grade 6, instead of seeing a symbol like a question mark or a box for a missing value, we use letters called variables.
- An algebraic expression is built from integer constants, variables and operations whereas an equation is the statement of two equivalent expressions. This benchmark is specifically addressing substitution.
- Depending on the given expression, students may see opportunities to start generating equivalent expressions before substituting the value of the variable(s). This is a way students can demonstrate flexible thinking and the understanding of patterns and structure in mathematical concepts (MTR.5.1).
Common Misconceptions or Errors
- Students may incorrectly generate equivalent expressions using the order of operations.
- If more than one variable is present in a given expression, students may incorrectly substitute one value in for all given variables or apply the wrong value to each of the variables. To address this misconception, students can use colors (pens, pencils, markers) to keep track of which variable and location correspond to each given value.
- If students try to generate an equivalent expression before substituting integer values, they may try to combine unlike terms (constants with variables or unlike variable terms).
Strategies to Support Tiered Instruction
- Instruction includes building a foundation for the meaning of substitution by introducing algebraic expressions in a single variable with an exponent of 1 and representing the expression with algebra tiles. Then the variable tile can be replaced with the appropriate number of unit tiles to represent the provided value and the expression can be evaluated. The algebraic expression should be represented simultaneously to draw connections between the concrete and abstract representations. Use this foundation to build towards using algebraic strategies to evaluate expressions that cannot be represented with algebra tiles.
- For example, when evaluating 3 + 5 when = −4, students can represent this as shown below.
- Teacher provides opportunities for students to use different colored pencils to represent different variables and use the coordinating color to replace the variable with its assigned value before utilizing the order of operations to evaluate.
- For example, if evaluating the expression −52 + , where = −3 and =−12, the teacher can color coordinate as shown below.
−52 + ; = −3 and = −12
−5(−3)2 + (−12)
−5(9) + (−12)
−45 + (−12)
- Teacher models how students can use colors (pens, pencils, markers) to keep track of which variable and location correspond to each given value.
Instructional Task 1 (MTR.5.1)
To compute the perimeter of a rectangle you add the length, l
, and width, w
, and double this sum.
- Part A. Write an expression for the perimeter of a rectangle.
- Part B. Use the expression to find the perimeter of a rectangle with length of 15 feet and width of 8 feet.
Instructional Task 2 (MTR.6.1)
To determine the distance traveled by a car you multiply the speed the car traveled by the amount of time the car was traveling at that speed. The scenario can be represented as st
, where s
is speed, and t
is time. What is the distance traveled by a car that travels at an average speed of 75 miles per hour for 20 minutes?
Instructional Item 1
= 3, find 3x
Instructional Item 2
= 8, find 1 p
Instructional Item 3
Evaluate the expression −5 a2
, where a
=−3 and c
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.