Addition and Subtraction of Polynomials: Teacher Notes

Overview

In this activity students simplify algebraic expressions by adding and subtracting polynomials with and without algebra tiles. Students apply the same thinking they use when adding and subtracting integers.

Important Mathematical Ideas

  • The strategies used for the addition and subtraction of integers can be applied to the addition and subtraction of polynomials.
  • Only like terms can be combined through addition or subtraction.
  • The zero principle applies to algebra in the same way it applies to the addition and subtraction of integers.
  • Addition involves adding to a polynomial or combining polynomials and then combining zero pairs.
  • Subtraction involves taking away from a polynomial, or adding the opposite, and inserting zero pairs where appropriate.

Prior Knowledge

  • Familiarity with the vocabulary of algebra (e.g., variable, term, like terms, exponent, zero pairs, simplify, coefficients) as needed throughout the unit.
  • Representing phrases (words) as algebraic expressions.
  • Using the zero principle.
  • Adding, subtracting, multiplying and dividing integers.
  • Using algebra tiles (e.g., identification of tiles, representing the tiles algebraically, representing an algebraic expression concretely with algebra tiles).
  • Simplifying algebraic expressions using concrete materials as well as algebraically.

Common Misconceptions

  • 2x and x2 are the same (see Unit 7 Activity1).
  • When subtracting polynomials, not distributing subtraction to all terms inside the second bracket.

Information to Support/ Enhance/ Extend Learning

  • Students are asked to keep a journal for each unit in the course. It should contain notes of important mathematical ideas with examples and new vocabulary.
  • ePortfolio may be used for these journal entries.
  • Students can make individual choices whether this is a paper or digital personal resource.
  • Consider a variety of formats as alternatives to journal entries (e.g., student note, pair/share, group discussion, exit card, poster, electronic posting).
  • Develop a Word Wall and continue it throughout the unit as new vocabulary and terms arise that require clarification (e.g., polynomials, constant, variable, exponent, term, monomial, binomial, trinomial, degree of a polynomial, standard form, coefficient).
  • Additional resources
  • Paying Attention to Algebraic Reasoning
  • Paying Attention to Algebraic Reasoning Adobe Presenter

Materials

  • Algebra tiles
  • students can make their own algebra tiles
  • Algebra Tiles Template in Colour
  • Algebra Tiles Template in Black and White

Minds On

Task 1: Terminology

Students will:

  • read Polynomials
  • a summary of polynomial terminology
  • make notes using the Polynomial Terminology chart
  • posted with unit
  • students can use Mindomo to develop a Mind Map or a web for the terminology
  • Task can be organized as a Think/Pair/Share.

Journal Sample Response

Terminology / Meaning / Examples Given / Your Own
Examples
Monomial / polynomial with one term / 3xy2 / 12, -x, 3y÷5
Binomial / polynomial with two terms / 5x – 1 / 3x + 7y,
x - 5
Trinomial / polynomial with three terms / 3x + 5y2 - 3 / x2 – 3x + 6,
x – y +4
Polynomial / an algebraic expression with any number of terms / 5xy2 - 3x + 5y2 - 3 / x + y + xy + 2
2x - 1

Action

Task 2: Adding Polynomials - Example 1 Journal Prompt and Sample Response

Watch the Adding Polynomials Using Algebra Tiles video to simplify

(3x2 + 4 + 6x) + (5 - 4x2 + 3x)

  • Group like terms:

= 3x2 – 4x2 + 6x + 3x + 4 + 5

  • Simplify by looking for zero pairs.

= -x2 + 9x + 9

Task 3: Adding Polynomials - Example 2

Students will:

  • use algebra tiles to simplify
  • (3x2 – 2x +4) + (x2 + 3x – 2)
  • watch Polynomials Algebra Tiles Ex. 2 video to check their solution
  • the video describes how to add polynomials using algebra tiles
  • take screenshots of the solution for their journal

Journal Prompt and Sample Response

Watch the video again. This time, take notes or screenshots as you watch. Include these in your journal. Copy the algebraic solution given at the end of the video.

  • Represent the polynomials using algebra tiles

(3x2 – 2x +4) + (x2 + 3x – 2)

  • Simplify the expression by combining like terms
  • Simplify by looking for zero pairs
  • The solution to the problem

4x2+ x + 2

Task 4: Adding Polynomials - Example 3

Students will:

  • use their own algebra tiles or Virtual Algebra Tiles to simplify
  • (3x² + x – 1) + (–x² – 4x + 1).
  • watch either Add Polynomials Algebra Tiles Ex.3video to check their solution
  • a video describing how to add polynomials using virtual algebra tiles

Journal Prompt and Sample Response

Watch the video again. This time, take notes or screenshots as you watch. Include these in your journal. Copy the algebraic solution given at the end of the video.

  • Represent the polynomials using algebra tiles.

(3x² + x – 1) + (–x² – 4x + 1).

  • Group together the like terms.

3x2 – x2 + x – 4x –1 + 1

  • Look for zero pairs.
  • Simplify the representation.

= 2x2– 3x

  • An algebraic solution

(3x2 + x – 1) + (-x2 – 4x + 1)

= 3x2 – x2 + x – 4x - 1 + 1

= 2x2 – 3x

Task 5: Practice Adding Polynomials

Students will:

  • simplify given polynomial expressions using their own algebra tiles or Virtual Algebra Tiles
  • check solutions with feedback provided

Task 6: Subtracting Polynomials - Example 1Journal Prompt and Sample Response

Watch Subtracting Polynomials by Using a Take-Away Strategy video and take notes or screenshots to show how to simplify (5x2 – 4x + 3) – (2x2 – x +1)

  • On my workspace, I represent the first polynomial using algebra tiles.
  • From the workspace, I take-away algebra tiles that represent the terms in the second polynomial, 2x2, -x and +1.
  • What remains is the solution: 3x2 – 3x + 2.

Task 7: Subtracting Polynomials - Example 2

Students will:

  • watch Subtracting Polynomials Using Zero Pairs video to simplify (3x2 – 4x + 3) – (x2 + 2x – 4)

Journal Prompt and Sample Responses

In your journal, describe what is the same and what is different about the strategies that can be chosen to simplify (3x2 – 4x + 3) – (x2 + 2x – 4) from example 2 and (5x2 – 4x + 3) – (2x2 – x +1) from example 1.

In both examples I can take away algebra tiles from the first polynomial that represent the terms that are being subtracted. If there are not enough algebra tiles then I add enough zero pairs to take away the required tiles from the first polynomial.

Task 8: Subtracting Polynomials by Adding the Opposite

Students will:

  • watchSubtracting Polynomials by Adding the Opposite video

Journal Prompt and Sample Responses

Add notes to your journal about subtracting polynomials. Include the strategy of subtracting polynomials by adding the opposite.

1)Example: Subtract (3x2 – 4x + 3) – (x2 + 2x - 4) by adding the opposite using algebra tiles.

I model the first polynomial on the left side of the workspace.

To subtract the second polynomial, I can add it's opposite:

For x2 the opposite is –x2, for 2x, - 2x and for -4 the opposite is +4.


Next I combine like terms.

Simplify by looking for zero pairs.

The solution is 2x2 – 6x + 7.

2)An algebraic solution is

(3x2 – 4x + 3) – (x2 + 2x - 4)

= (3x2 – 4x + 3) + (-x2 - 2x + 4)

= 3x2 -x2 – 4x - 2x + 3 + 4 (add the opposite)

= 2x2 – 6x + 7

Task 9: Subtracting Polynomials - Example 3

Students will:

  • use their own algebra tiles to simplify (3x2 + 2x +1) – (–2x2 + 3x – 2) by inserting

enough zero pairs to make it possible to take-away

  • check their answer with Model Subtracting video
  • use their algebra tiles to simplify (3x2+2x+1) – (-2x2+3x-2) using the strategy of

adding the opposite

  • check their answer with the video solution
  • add notes to their journal about subtracting polynomials using zero pairs and adding the opposite

Task 10: Practise Subtracting Polynomials Questions and Sample Reponses

  • Simplify each of the following and explain the strategies you used.

1)(2x + 1) – (3x + 4)

I used the ‘add the opposite” strategy.

To subtract 3x + 4 I took the opposite of each term, which is -3x – 4. I added these terms to 2x + 1.My solution is - x – 3.

2)(–3x2 + x) – (2x2 – x)

I used the “take-away” and “zero pairs” strategy.

I added 2 zero pairs of x2tiles and 1 zero pair of x-tiles.

Next, I took away 2 x2 tiles and 1 negative x-tile.My solution is -5x2 + 2x

3)(2x2 – 3x – 1) – (3x2 + x – 2)

I added the opposite and simplified the expression algebraically.(2x2 – 3x – 1) – (3x2 + x – 2)

= (2x2 – 3x – 1) + (-3x2 – x + 2)

= 2x2 – 3x2 - 3x – x – 1 + 2

= - x2 – 4x + 1

4)(5x2+6x– 4) – (2x2 + 4x –1)

I used the “take-away” strategy.

From 5x2, I removed 2x2, from 6x, I removed 4x and from -4, I removed -1,

This leaves me with 3x2 + 2x – 3

  • Solutions should include:
  • a screenshot or drawing of algebra tiles, if used
  • a description of the strategy used to simplify the problem
  • a simplified answer
  • Common Errors:
  • modelling the polynomial incorrectly with algebra tiles
  • not adding enough zero pairs for the “take-away” strategy
  • not adding the opposite of all the terms in the polynomial being subtracted

Task 11: Assignment 1 Simplifying Polynomial Expressions Discussion Prompts

  • Share your answers to questions one to four.
  • Read through your classmates' responses and select two of them. Respond to each using the following three prompts:
  • Describe how your answers are similar and/or different.
  • Explain how or why you agree or disagree with their answers.
  • Describe any new learning or wonderings you have.

Consolidation

Task 12:Check Your Understanding Q & A: Multiple Choice

  • Students will:
  • simplify the given polynomials
  • select the correct answer from the choices given
  • check solution with feedback provided
  • Can be organized as a Think/Pair/Share.

Task 13:Assignment 2 Adding and Subtracting Polynomials

  • Assignment is posted with the unit.
  • Sample solution is posted in the Teacher Notes on the vLE.
  • Solutions can be shared through a Gallery Walk.

Task 14: Assignment 3: How to Add and Subtract Polynomials

  • Assignment is posted with the unit.
  • Sample solution is posted in the Teacher Notes on the vLE.

Task 15: Student Reflection

  • Students are asked to reflect on their understanding of this topic.
  • These reflections can be used as assessment for learning to help determine next steps for individual students.

Grade 9 Applied Blended Learning: Unit 7 Activity 3 Page 1 of 8