Expressions Content Module
Plot the Course
http://www.worthwhilesmile.com/air-balloons-kaleidoscope/
The rationale
The act of simplifying complex tasks into smaller, simpler steps is a skill that has implications both inside and outside the classroom. Whether you're trying to determine if you have enough money to see a movie and buy snacks or you need to buy materials to build a flowerbed, these complex tasks often require the combining or simplification of similar terms.
Module Goal
The goal of this module is to provide detailed instruction on the more difficult concepts of simplifying expressions to teachers of students with disabilities at the middle and high school level. This module promotes a mathematical understanding of these concepts so that a teacher can begin to plan how to teach the concepts to students. Additionally, this module will provide instructors with potential adaptations and modifications to consider when designing materials and instruction for students with severe disabilities.
Module Objectives
After viewing the content module, teachers will:
- Identify similar terms within an expression
- Complete all the steps to simplify an expression in its simplest form using the commutative, associative, and distributive properties
- Translate word problems into an expression
Time for Take Off
Understanding the vocabulary used within simplifying expressions is important for both teachers and students in planning and implementing math lessons. As a teacher, knowing and using the mathematical terms not only ensures your instruction stays true to the math content, but also will help with collaborating with other math teachers or content experts. When choosing which vocabulary to teach, it is most important that the teacher selects the most salient, important, or most frequently used vocabulary for each lesson. Below you will find a list of vocabulary included within this module. It may or may not be necessary to provide instruction for all terms as students may have learned them previously. Expressions are mostly covered in middle school so vocabulary for this content module has been combined. If you are a high school teacher and are not confident your students know some of these vocabulary terms, you may want to review and teach some unknown terms in the focus and review part of your lesson plan. While providing vocabulary instruction, you may consider including pictures or objects to make the instruction more concrete for students with disabilities (See ideas to support vocabulary learning below).
Vocabulary
- Terms- expressions that are separated by a plus or minus sign (e.g., 2t-3b=)
- Like terms- terms that have the same variable (e.g., 2t and 3t are like terms)
- Coefficients- a number multiplied by a variable (e.g., In the term 6y, 6 is the coefficient).
Floating on Air
Before you can begin teaching simplifying expressions, you need a deep understanding of these mathematical concepts. Some of these concepts may be familiar to you. Below is a list of skills that should be covered at each grade level. For concepts that you need more information about, please view the accompanying PowerPoint presentation that will walk you through an example as well as make some suggestions for instruction.
Middle and High School
In middle and high school skills include:
- 6.PRF.2a2 Use variable to represent numbers and write expressions when solving real world problems
- 6.NO.1i2 Solve numerical expressions involving whole number exponents
- 6.SE.1a3 Write expressions for real-world problems involving one unknown number
Insert simplifying expressions PowerPoint presentation here
- H.PRF.2a1 Translate an algebraic expression into a word problem
- H.NO.2c1 Simplify expressions that include exponents
- H.NO.2c2 Rewrite expressions that include rational exponents
Great! Now that you have viewed the PowerPoint presentation most useful to you, the next section will provide some ideas to consider when planning for Universal Design for Learning.
Simplifying Expressions PowerPoint
Sharing the Sky
UNIVERSAL DESIGN FOR LEARNING
Some examples of options for teaching expressions to students who may present instructional challenges due to: | ||||
Visual Impairment or Deaf/Blind | Physical Impairment:
Little/No Hand Use |
Lacks Basic Numeracy Concepts | Motivational/Attention Issues | |
Representation | Add corresponding textures (e.g., Velcro) to manipulatives representing each term in the expression. | Student scans an array of possible options and uses a switch to select the appropriate terms, coefficients, or exponents. | Use objects to represent numbers in the expression; color code similar terms within the expression. | Create personally-relevant word problems or stories to pair with expressions. |
Expression | Student states answer or scans raised numbers to select correct answer; use voice output devices for student to select the correct answer. | Uses a switch to indicate correct answers; uses an eye gaze board to select answer; "yes/no" response, these can easily be answered using an eye gaze, head turn, two switches, etc. | Student selects numbers or terms versus writing them; selection of correct answer is done after a model. | Student simplifies expressions using computer software or other technology. |
Engagement | Add corresponding textures (e.g., Velcro) to manipulatives representing each term in the expression. | Use a computer with AT where the student can click to answer; use manipulatives that are large and easily manipulated; pair student with another student without a physical impairment and have them work together to simplify expressions. | Use objects to represent numbers in the expression; color code similar terms within the expression. | Include personally-relevant contexts for simplifying the expressions. |
Prepare for Landing
Below you will find ideas for linking simplifying expressions to real-world applications, the college and career readiness skills addressed by teaching these concepts, module assessments for teachers, sample general education lesson plans incorporating Universal Design for Learning framework, blog for teachers to share their ideas, and a place to upload and share lesson plans from teachers who completed this module. One way to help assist in a special educator's development within this curricular area is through collaboration with other teachers in your building. Some activities with real world connection include:
- Estimating costs at the grocery store by combining same priced objects; and
- When counting money (25 quarters, 10 dimes, 15 nickels 25 pennies) and then when you come across 15 more nickels, you would put the nickels together when counting the coins.
25 Q + 10 D + 15 N + 25 P + 15 N... combine N so now you have 30 N In addition to the real-world applications of these concepts, skills taught within this content module also promote the following college and career readiness skills.
Communicative competence Students will increase their vocabulary to include concepts related to "simplification" and "expressions". In addition, they will be learning concepts such as: "terms", "coefficient", and "exponent".
Fluency in reading, writing, and math Students will have an opportunity to increase their numeracy and sight word fluency while participating in problem solving related to "expressions" such as number recognition, counting, and grouping similar things.
Age appropriate social skills Students will engage in peer groups to solve problems related to "simplifying expressions" that will provide practice on increasing reciprocal communication and age appropriate social interactions.
Independent work behaviors By working with real life problems related to "simplifying expressions" students will improve work behaviors that could lead to employment such as marketing or any job that has to analyze sales rates, stock clerks, order fillers, and other construction based professions. When providing opportunities for real life problems leave some materials out and prompt/teach the students to determine who they should ask and what they should ask for to be able to solve the problem. Skills in accessing support systems At times, students will need to ask for assistance to complete activities related to "simplifying expressions" which will give them practice in accessing supports. Students will gain practice asking for tools such as graphing calculators, or other manipulatives. They can ask a peer to complete the physical movements of the tasks they are not about to do themselves. Be sure to teach students to ask versus having items or supports automatically given to them.
In addition to collaborating with other educational professionals in your building, the following list of resources may also help provide special educators with ideas for activities or support a more thorough understanding of the mathematical concepts presented in this content module.
Additional Resources
- http://www.algebrahelp.com/calculators/expression/oops/ - Allows students to enter an expression and the website will simplify automatically. Great place for students to check their answers.
- http://www.ncpublicschools.org/acre/standards/common-core-tools/ - This website provides an "unpacking document" for the Mathematics Common Core Standards that helps teachers identify what is most important and the essential skills for each standard.
- http://mathforum.org/ - Website specifically for teachers which provides a variety of ideas and activities to use in your classroom.
- www.teachertube.com - Youtube for teachers! Simply search for your content area and this websites provides a variety of videos including videos of math experts working through math problems step by step (free registration required).
- http://www.purplemath.com/modules/simpexpo.htm - This website provides examples and extra resources simplifying expressions with exponents.
- http://www.google.com/url?q=http://www.ksde.org/LinkClick.aspx%3Ffileticket%3DVq9AjrFFWzE%253D%26tabid%3D3763%26mid%3D11170&sa=U&ei=8lB3Try4CJOltwfmq5DfDA&ved=0CBIQFjAA&usg=AFQjCNE_DzuxI_rhYkU0H1qpjuqmM9sjng - This website provides a webinar about how to adapt materials for students who have visual impairments.
Module Assessments
Insert assessment here
Sample General Education lesson plans
Insert developed lesson plans here
Have an idea: Upload the lesson plans you've created here
Insert link for teachers to upload lesson plans
Teacher's Corner: Blog with other teachers
Insert forum or blog for teachers to share ideas
Adapt the following general education lesson plan; adapt, and upload. These lesson plans may be shared with higher education professionals developing strategies to provide meaningful academic instruction in mathematics to students with severe disabilities. Insert blank lesson plan form with UDL chart here Insert link for teachers to upload lesson plans
Simplifying Expressions Assessment
- Combine the like terms for the expression: 14p-5p
- 19p
- 9p
- -9p
- 11p
- Combine the like terms for the expression: 3(4y+5) +8
- 17y +23
- 7y+13
- 127+40
- 12y + 23
- Combine the like terms for the expression: x^3+y+ 〖3x〗^3+ 7y
〖4x〗^3+8y 〖3x〗^3+7y 〖4x〗^3+7y 〖3x〗^3+8y
- When the expression 3(2x+7) +10x is simplified, what is the coefficient for x?
- 6
- 16
- 21
- 12
- How many terms are in the following expression: 6y+4r-3x+y
- 4
- 3
- 7
- 6
- How many like terms are in the following expression: 6y+4r-3x+y-3y-2h+10
- 8
- 5
- 7
- 3
- Write an expression for the following word problem: Bethany has quarters (q) and pennies (p). Her sister has 3 times as many quarters as Bethany and 5 times as many pennies.
q-p-3q-5p q+p+5q+3p q-p+5q+3p q+p+3q+5p
3^5 is the equivalent of which expression? 3×3×3×3×3 3×3×3 5×5×5 15
- In the expression 𝑥−6′′, the coefficient for x is?
- 0
- There is no coefficient
- 1
- -6
- How many factors does the expression 2(x+4) have?
3 2 1 5
Simplifying Expressions Assessment: Answer Key
- Combine the like terms for the expression: 14p-5p
- 19p
- 9p
- -9p
- 11p
Correct feedback: Yes, the answer is 9p. Incorrect feedback: Sorry, the answer is 9p. Please review the simplifying expressions PowerPoint.
- Combine the like terms for the expression: 3(4y+5) +8
- 17y +23
- 7y+13
- 12y+40
- 12y + 23
Correct feedback: Yes, the answer is 12y + 23. Incorrect feedback: Sorry, the answer is 12y + 23. Please review the simplifying expressions PowerPoint.
- Combine the like terms for the expression: x^3+y+ 〖3x〗^3+ 7y
〖4x〗^3+8y 〖3x〗^3+7y 〖4x〗^3+7y 〖3x〗^3+8y
Correct feedback: Yes, the answer is 〖4x〗^3+8y. Incorrect feedback: Sorry, the answer is 〖4x〗^3+8y. Please review the simplifying expressions PowerPoint.
- When the expression 3(2x+7) +10x is simplified, what is the coefficient for x?
- 6
- 16
- 21
- 12
Correct feedback: Yes, the answer is 16. Incorrect feedback: Sorry, the answer is 16. Please review the simplifying expressions PowerPoint.
- How many terms are in the following expression: 6y+4r-3x+y
- 4
- 3
- 7
- 6
Correct feedback: Yes, the answer is 4. Incorrect feedback: Sorry, the answer 4. Please review the simplifying expressions PowerPoint.
- How many like terms are in the following expression: 6y+4r-3x+y-3y-2h+10
- 8
- 5
- 7
- 3
Correct feedback: That's right, the answer is 5. Incorrect feedback: Sorry, the answer is 5. Please review the simplifying expressions PowerPoint.
- Write an expression for the following word problem: Bethany has quarters (q) and pennies (p). Her sister has 3 times as many quarters as Bethany and 5 times as many pennies.
q-p-3q-5p q+p+5q+3p q-p+5q+3p q+p+3q+5p
Correct feedback: The answer is 𝑞+𝑝+3𝑞+5𝑝′′ Incorrect feedback. The answer is q+p+3q+5p. Please review the simplifying expressions PowerPoint. 3^5 is the equivalent of which expression? 3×3×3×3×3 3×3×3 5×5×5 15
Correct feedback, Great! The answer is 3×3×3×3×3′′. Incorrect feedback: Sorry, the answer is 3×3×3×3×3. Please review the simplifying expressions PowerPoint.
- In the expression x-6, the coefficient for x is?
- 0
- There is no coefficient
- 1
- -6
Correct feedback: That's correct, the answer is 1. Incorrect feedback. Sorry, the answer is 1. Please review the simplifying expressions PowerPoint.
- How many factors does the expression 2(x+4) have?
3 2 1 5
Correct feedback: That's right, the answer is 2. Incorrect feedback: Sorry, the answer is 2 ′′. Please review the simplifying expressions PowerPoint.
General Education Math Lesson Plan – Simplifying Expressions
Source: Doing What Works: http://dww.ed.gov Assignment: Simplifying Expressions—Twin Groves Middle School, Illinois Lesson Plan: Simplifying Expressions Objective: Students will simplify expressions by combining like terms. Procedure – Opener: Review distributive property 3( x + 2) –2(x – 4) –(5x + 3) –(4 – 2x) 3x (x + 5) Vocabulary: Discuss what an expression is and when we write them. Discuss parts of expression – coefficient, terms, like terms, constant.
Prior Knowledge: Display both a simplified expression and one that is not simplified to see if students can see a difference. Discuss why you want to simplify.
Sample Problems: Go through sample problems, showing on board and then give students one to try on their white board, send students to board in front of room. Problems should build up in difficulty. Discuss proper answer form.
- one like variable
- 8a + 2a 9p – 5p
- two different variables
- 4x + 8y – 6x + 2y
- different variables and constants (be sure to include terms with coefficient of 1 and -1 in both the problem and in answers)
- 8c – 7a + 3 – 2c + 6a
- distribute and simplify
- 5(2x + 6) – 8x
- multiple distributive and simplify
- 6(2x – 3) – 5(3x + 6) + 7x
- same base variable, different exponents
- 8 + 4x + 3x2 – x + 5x3 – x2 + 2x3 – 1
- if time allows application with perimeter
- Find the perimeter of a rectangle with the length of 3x – 5 and the width of x + 3.
\*\*Helpful hints for struggling students\*\*
- use different colors to highlight like terms
- circle or box off like terms
Generate set of rules with student: - scan problem, distribute/combine like terms - order answer descending powers alphabetical constant term last Pass out worksheet to do in pairs. Discuss pair answers.
Pass out practice worksheet.
Simplifying Expressions – Problem Worksheet
1. 4x + 7x
2. 9y – 12y
3. 3x + 7y – x – 4y
4. 8r – 3q + q – 7r + 5q
5. 5f – 4 - 3d + 2 – 8f – 6 + d
6. –7k + 8m - 2 + 6k + 2 – 6m
7. 4 (3n + 5) – 10n
8. k + 2 – 3(6k – 5)
9. 8(3x + 4) + 5(7x – 8) + 2x
10. –2(5x – 4) – 3(2x + 8) – 7x
11. 5x + 6x2 + 9x + 7x3 – 3x2
12. 4x(x + 6) + 7x – 3(x – 5) + 6x2
\*\*13. –x + 4x(7 – 3x) + 3x(2x + 4) – (4 – 2x) + 5
\*\*14. Find the perimeter of a rectangle with the length of 2y – 4 and the width of x + 3.
\*\* Challenge problems
Simplifying Expressions – Homework 1. 11x + 7x 2. 15y – 20 y 3. 7x + 5y – x – 2y 4. 9x – 4m + m – 6x + 5m 5. 8e – 6 -2c + 4 – 10e – 5 + c 6. –8x + 7m – 5 + 7x + 3 – 3m 7. 5(2x + 7) – 8x 8. n + 4 – 5(3n – 6) 9. 3(7x + 6) + 4( 9x – 3) + 4x 10. –3 (7x – 6) – 5(4x + 7) – 9x 11. 4x + 7x2 + 3x + 9x3 – 4x2 12. 5x (x + 7) + 8x – 4 (x – 3) + 5x2 13. Write an expression that needs to be simplified and then simplify it.
Activity: Create a universally designed version of the above lesson
UDL Planning | My Ideas |
Representation - adaptations in materials (e.g., adapt for sensory impairments) | |
Expression - how will student show learning (e.g., use of assistive technology; alternative project) | |
Engagement - how will student participate in the activity |
General Education Math Lesson Plan – Simplifying Expressions
Source: Bennett, J.M., Burger, E. B., Chard, D. J., Hall, E., Kennedy, P. A…Waits, B. W. (2011). Mathematics. Austin, TX: Holt McDougal. Standard: H.NO.2c1 Simplify expressions that include exponents. H.NO.2c2 Rewrite expressions include rational exponents. Activities:
- Focus and Review: Talk about the phrase "compare apples to oranges" and what that means. Explain how similar terms in expressions are combined.
- Lecture: Teacher works through a variety of problems simplifying expressions with and without exponents. During this lecture, the teacher highlights common mistakes made. For example, in the expression 4x +3 +x, commonly students will not combine all three terms correctly because they forget that if no coefficient is given the coefficient is 1. Therefore x is 1x.
- Guided Practice: Students simplify a variety of expressions from their textbook.
- Independent Practice: Students complete activity sheet
Activity: Create a universally designed version of the above lesson
UDL Planning | 'My Ideas' |
Representation - adaptations in materials (e.g., adapt for sensory impairments) | Include manipulatives with each expressions representing terms (squares for x terms, circles for y terms, etc.) to help students pick terms that should be simplified. |
Expression - how will student show learning (e.g., use of assistive technology; alternative project) | Provide student a term and an expressions and ask them to identify the terms that could be combined. |
Engagement - how will student participate in the activity | Student can work in a pair during independent practice; include personally relevant word problems or stories to put the expression in context. |