Standard #: MA.912.AR.3.4

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.2.1
• MA.912.F.2.1

 

Terms from the K-12 Glossary

• Coordinate Plane  
• Domain 
• Function Notation 
• Quadratic Function 
• Range 
• $x$-intercept 
• $y$-intercept

 

Vertical Alignment

Previous Benchmarks

• MA.8.AR.3.3

Next Benchmarks

• MA.912.AR.3.9

 

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 $y$ = $a$$x$²  + $b$$x$ + $c$, where $a$, $b$ and $c$ are any rational number. This form can be useful when identifying the $y$-intercept. 
  • Factored Form can be described by the equation $y$ = $a$($x$ − $r$₁)($x$ − $r$₂), where  $r$₁ and  $r$₂ are real numbers and the roots, or $x$-intercepts. This form can be useful when identifying the $x$-intercepts, or roots. 
  • Vertex Form can be described by the equation $y$ = $a$($x$ − $h$)² + $k$, where the point ($h$, $k$) is the vertex. This form can be useful when identifying the vertex. 
• Instruction includes the use of $x$-$y$ 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 $a$ 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 $a$ = 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, $a$ = 1. 
  • • This process can be used for other values of $a$. When $a$ = 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 $a$ = $\frac{\text{1}}{\text{2}}$, points 1 unit to the left or right of the vertex are $\frac{\text{1}}{\text{2}}$(1²)  or $\frac{\text{1}}{\text{2}}$ a unit above or below the vertex. Points 2 units to the left or right of the vertex are $\frac{\text{1}}{\text{2}}$ (2)² or 2 units 2 above or below the vertex. 
  • Students can solve for a by substituting the values of $x$ and $y$ from a point on the parabola. 
    
    • Ask students to consider $y$ = $a$($x$ + 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 $x$ and for $y$, ask them if they know any combinations of $x$ and $y$ that are solutions. Students could use a table of values or a graph, if given, to determine values that could be used for $x$ and $y$. Have students pick one and substitute and solve for $a$. Ask for students who chose different points to share their value for a to help them see that all points, when substituted, produce $a$ = 1. 
• Instruction includes the use of graphing software or technology. 
  • For example, when determining the value of $a$, 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 $a$ 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 $h$. 
  • For example, students may see a vertex of (−1, −2) and an $a$ value of 3 and write the function as $y$ = 3($x$ − 1)²  − 2 instead of $y$ = 3($x$ + 1)² − 2. To address this, help students recognize that because $h$ is subtracted from $x$ 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, $y$ = $a$($x$ − $r$₁)($x$ − $r$₂). Instruction then includes reminding students of different methods that can be used to multiply binomials to convert the quadratic into standard form, $y$ = $a$$x$²  + $b$$x$ + $c$. 
• Instruction includes solving for a by substituting the values of $x$ and $y$ from a point on the parabola. Students may need to review the order of operations to ensure they correctly isolate $a$. Provide students with a review of the order of operations. 
  • Parenthesis
  • Exponents 
  • 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 $h$ is subtracted from $x$ 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 Tasks

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, $h$, at any point in time, $t$. 
  • Part B. How long will the tennis ball stay in the air? Round your answer to the nearest tenth second. 

Instructional Task 2 (MTR.4.1, MTR.5.1) 

• Part A. Create a quadratic function that contains the roots $\frac{\text{3}}{\text{5}}$ and 1.8.
• Part B. Compare your function with a partner. What do you notice?

 

Instructional Items

Instructional Item 1 

• Write the quadratic function that corresponds with the graph below.  

 

Instructional Item 2 

• 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.