Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Coordinate Plane
- Function Notation
- Quadratic Function
Purpose and Instructional Strategies
In middle grades, students wrote linear two-variable linear equations. In Algebra I, students write
quadratic functions from a graph, written description or table. In later courses, students will write
quadratic two-variable inequalities.
- Instruction includes making connections to various forms of quadratic equations to show
their equivalency. Students should understand when one form might be more useful than
other depending on the context.
- Standard Form
Can be described by the equation = 2 + + , where , and are any
rational number. This form can be useful when identifying the -intercept.
- Factored Form
Can be described by the equation = ( − 1)( − 2), where 1 and 2 are real
numbers and the roots, or -intercepts. This form can be useful when identifying
the -intercepts, or roots.
- Vertex Form
Can be described by the equation = ( − )2 + , where the point (, ) is
the vertex. This form can be useful when identifying the vertex.
- Instruction includes the use of - notation and function notation.
- Instruction includes the connection to completing the square and literal equations to
rewrite an equation from standard or factored form to vertex form.
- When determining the value of in a quadratic function, this can be done by two
methods described below.
- Students may notice a pattern from the points in the graph. When = 1, points 1
unit to the left or right of the vertex are 12 or 1 unit above or below the vertex.
Points 2 units to the left or right of the vertex are (2)2 or 4 units above or below the
vertex. Students may look at the table and notice this relationship exists,
therefore, = 1.
- This process can be used for other values of . When = 2, for example,
points 1 unit to the left or right of the vertex are 2(12) or 2 units above or
below the vertex. Points 2 units to the left or right of the vertex are 2(2)2 or 8 units above or below the vertex. Similarly, when = , points 1 unit to the left or right of the vertex are (12) or a unit above or below the vertex. Points 2 units to the left or right of the vertex are (2)2 or 2 units
above or below the vertex.
- Students can solve for a by substituting the values of and from a point on the
- Ask students to consider = ( + 4)2 − 2 and determine what
information they would need to be able to solve for a. As students express
the need to know values for and for , ask them if they know any
combinations of and that are solutions. Students could use a table of
values or a graph, if given, to determine values that could be used for and . Have students pick one and substitute and solve for . Ask for
students who chose different points to share their value for a to help them
see that all points, when substituted, produce = 1.
- Instruction includes the use of graphing software or technology.
- For example, when determining the value of , consider using graphing software
to allow students to use sliders to quickly observe this. This provides opportunity
for students to notice patterns regarding the value of and the concavity and
stretch of the parabola (MTR.5.1).
Common Misconceptions or Errors
- When writing functions in vertex form, students may confuse the sign of .
- For example, students may see a vertex of (−1, −2) and an value of 3 and write
the function as = 3( − 1)2 − 2 instead of = 3( + 1)2 − 2. To address this,
help students recognize that because is subtracted from in vertex form, it will
change the sign of that coordinate. Show students a graph of both functions to
confirm and make the connection to transformation of functions (MA.912.F.2.1).
Strategies to Support Tiered Instruction
- Teacher models substituting roots into the factored form of a quadratic, = ( − 1)( − 2). Instruction then includes reminding students of different methods that
can be used to multiply binomials to convert the quadratic into standard form, = 2 + + .
- Instruction includes solving for a by substituting the values of and from a point on
the parabola. Students may need to review the order of operations to ensure they correctly
isolate . Provide students with a review of the order of operations.
- Multiplication or Division (Whatever comes first left to right)
- Addition or Subtraction (Whatever comes first left to right)
- Instruction includes recall of knowledge demonstrating how when is subtracted from in vertex form, it will change the sign of that coordinate. A graph of both functions can
be used to confirm and make the connection to transformation of functions.
- Teacher provides a laminated cue card of the steps required to convert from factored
form to standard form.
Instructional Task 1 (MTR.7.1)
- Duane throws a tennis ball in the air. After 1 second, the height of the ball is 53 ft. After 2
seconds, the ball reaches a maximum height of 69 feet. After 3 seconds the height of the ball
is 53 ft.
- Part A. Write a quadratic function to represent the height of the ball, , at any point in
- Part B. How long will the tennis ball stay in the air? Round your answer to the nearest
Instructional Task 2 (MTR.4.1, MTR.5.1)
- Part A. Create a quadratic function that contains the roots and 1.8.
- Part B. Compare your function with a partner. What do you notice?
Instructional Item 1
Instructional Item 2
- Write the quadratic function that corresponds with the graph below.
- Given the table of values below from a quadratic function, write an equation of that function.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.