**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Standard:**Write, solve and graph quadratic equations, functions and inequalities in one and two variables.

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Quadratic Function
- $x$-intercept
- $y$-intercept

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students multiplied two linear expressions to obtain a quadratic expression. In Algebra I, students transform a quadratic function to highlight and interpret its vertex or its zeroes. In later courses, students will determine key features of higher degree polynomials.- Instruction includes making connections to various forms of quadratic equations to show
their equivalency. Students should understand and interpret 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.

- Standard Form
- Instruction includes the use of $x$-$y$ notation and function notation.
- Most contexts for this benchmark will present functions in standard form. Depending on
their perspectives, students might take several approaches to determine vertices and
zeros. Have students discuss strategies they might use. Let students know they should use
the approach that is most efficient for them
*(MTR.3.1).*- To determine zeros, students could use the quadratic formula or convert the function into factored form, or complete the square, or use Loh’s method.

- To determine the vertex, students could convert the function into vertex form or determine the axis of symmetry ($x$ = $\frac{\text{\u2212b}}{\text{2a}}$). Calculate this value from one of the previous functions discussed and guide students to see that the vertex of each parabola falls on the line of symmetry. Considering this, they can substitute that $y$-value into the function to determine the corresponding $y$-value of the vertex.

### Common Misconceptions or Errors

- Some students may have difficulty interpreting the meaning of the zeros and vertex.

### Strategies to Support Tiered Instruction

- Teacher provides a graphic organizer for key terms (zeros and vertex) that can be created using information provided in a given problem.

- Teacher co-creates a graphic organizer to compare real-world and mathematical contexts
related to quadratics.
- For example, the chart below can be used to compare mathematical terms with
real-world context.

- For example, the chart below can be used to compare mathematical terms with
real-world context.

- Teacher provides a visual aid, including using graphing software, to interpret the zeros
and vertex of a quadratic function. Students can compare definitions and examples to
determine the zeros and vertex of the parabola.
- For example, the roots of the graph shown are (2, 0) and (−4, 0). The locations on the graph that intersect the $x$-axis are the roots. The roots are also the solutions of the quadratic. The vertex of the graph shown is (−1, −9). The vertex is the minimum or maximum point (sometimes called the extrema) of the parabola. The vertex is also the point where the quadratic changes from decreasing to increasing or from increasing to decreasing, and is halfway between the roots.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.6.1, MTR.7.1)*

- A diver jumps off a cliff 5 meters high into a lake. The diver’s position can be represented by
the function $h$($t$) = −4.9$t$
^{2}+ 1.5$t$ + 5, where $h$ represents the diver’s height relative to the lake’s surface and $t$ represents the time in seconds.- Part A. Determine the roots of the function described.
- Part B. Interpret each with respect to the situation.

Instructional Task 2 (

Instructional Task 2 (

*MTR.3.1*,*MTR.7.1*)- A local campground charges $23.50 per night per campsite. They average about 32 campsites
rented each night. A recent survey indicated that for every $0.50 decrease, the number of
campsites rented increases by five.
- Part A. Write a quadratic equation that describes this situation.
- Part B. Determine the vertex and zeroes of the function described.
- Part C. Interpret each with respect to the situation.
- Part D. What price will maximize revenue?

### Instructional Items

*Instructional Item 1*

- Marcus just purchased a Super Bouncy Ball from a local toy store. Once outside, he throws it
down to see how high the ball will bounce. The function $h$($x$) = − 2$x$
^{2}+ 24$x$ − 31.5 represents the height, $h$ in inches, above the top of his head of the ball as it relates to its horizontal distance from Marcus $x$, in inches. Find the zeros and vertex of this function and interpret the meaning of each.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Original Student Tutorials

## Perspectives Video: Expert

## Tutorials

## MFAS Formative Assessments

Students are asked to rewrite a quadratic function in an equivalent form by completing the square and to use this form to identify the vertex of the graph and explain its meaning in context.

Students are asked to compare two quadratic functions, one given by a table and the other by a function.

Students are asked to factor and find the zeros of a polynomial function given in context.

Students are asked to identify the zeros of polynomials, without the use of technology, and then describe what the zeros of a polynomial indicate about its graph.

## Original Student Tutorials Mathematics - Grades 9-12

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.

Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 1 of a 2 part series. Click **HERE **to open Part 2.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click **HERE** to open part 1.

## Student Resources

## Original Student Tutorials

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click **HERE** to open part 1.

Type: Original Student Tutorial

This is part 1 of a 2 part series. Click **HERE **to open Part 2.

Type: Original Student Tutorial

Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.

Type: Original Student Tutorial

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Type: Original Student Tutorial

Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!

Type: Perspectives Video: Expert

## Tutorials

This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.

Type: Tutorial

This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.

Type: Tutorial