# MAFS.912.A-APR.2.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

General Information
Subject Area: Mathematics
Domain-Subdomain: Algebra: Arithmetic with Polynomials & Rational Expressions
Cluster: Level 1: Recall
Cluster: Understand the relationship between zeros and factors of polynomials. (Algebra 1 - Supporting Cluster) (Algebra 2 - Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications
Assessed with:
MAFS.912.F-IF.3.8

## Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1207310: Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1201300: Mathematical Analysis Honors (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200335: Algebra 2 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2019 (course terminated))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 and beyond (current))
1200387: Mathematics for Data and Financial Literacy (Specifically in versions: 2016 - 2022 (current), 2022 and beyond)

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.A-APR.2.AP.3a: Find the zeros of a polynomial when the polynomial is factored (e.g., If given the polynomial equation y = x2 + 5x + 6, factor the polynomial as y = (x + 3)(x + 2). Then find the zeros of y by setting each factor equal to zero and solving. x = -2 and x = -3 are the two zeroes of y.).
MAFS.912.A-APR.2.AP.3b: Use the zeros of a function to sketch a graph of the function.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Assessments

Sample 4 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 3 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 4 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 1 State Interim Assessment:

This is the State Interim Assessment for high school.

Type: Assessment

## Formative Assessments

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.

Type: Formative Assessment

Zeros of a Cubic:

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

Type: Formative Assessment

Use Zeros to Graph:

Students are given the factored form of a cubic polynomial and are asked to use the zeros to sketch the graph between two given points on the coordinate plane without the use of technology.

Type: Formative Assessment

## Lesson Plan

Representing Polynomials:

This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials as well as recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x).

Type: Lesson Plan

## Perspectives Video: Expert

Type: Perspectives Video: Expert

## Tutorials

Graphs of second degree polynomials:

In this tutorial, students will look at input and output values of quadratic functions to help them understand why the graph of a second degree polynomial curves.

Type: Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

## Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

### Using this CMAP

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

Type: Unit/Lesson Sequence

## Virtual Manipulatives

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

## MFAS Formative Assessments

Use Zeros to Graph:

Students are given the factored form of a cubic polynomial and are asked to use the zeros to sketch the graph between two given points on the coordinate plane without the use of technology.

Zeros of a Cubic:

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

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.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Perspectives Video: Expert

Type: Perspectives Video: Expert

## Tutorials

Graphs of second degree polynomials:

In this tutorial, students will look at input and output values of quadratic functions to help them understand why the graph of a second degree polynomial curves.

Type: Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

## Virtual Manipulatives

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial