Multiple Representations for Algebraic Expressions and

Equations

This is a five day unit with the goal to teach students different ways to solve and create algebraic expressions and equations using inquiry-based learning. This lesson is appropriate for grades 8 and 9. Throughout the unit students use a variety of tools to accomplish this goal. Students will use a graphing calculator in several lesson, the computer (internet with virtual manipulatives), algebra tiles and toothpicks.

Created by Andrea Koralewski

Overall objectives for the unit with NCTM and New York State Standards are addressed in the unit. Objectives addressed throughout the unit: By the end of the unit, students will be able to: 1. By the end of the class, students will be able to: 2. differential between algebraic expressions and equations 3. evaluate algebraic expression with more than one variable 4. evaluate algebraic expressions with two different values for the same variable in a

graphing calculator 5. balance an equation scale 6. solve algebraic equations using addition, subtraction, multiplication and division 7. create and solve algebraic equations 8. work cooperatively in pairs 9. balance/solve algebraic equations using a balance 10. solve algebraic equations using arrow drawings 11. create an algebraic expression or equation based on a steady increasing model 12. analyze if an algebraic expression is correct based on the situation 13. apply algebraic equations to real life situations 14. create algebraic equations based on pictorial patterns 15. analyze algebraic equations to see if they are correct 16. enter algebraic equations into a graphing calculator to find the x and y values NCTM Standards addressed throughout the unit:

Understand patterns, relations, and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules

• relate and compare different forms of representation for a relationship

Represent and analyze mathematical situations and structures using algebraic symbols • use symbolic algebra to represent situations and to solve problems, especially

those that involve linear relationships

• recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

Use mathematical models to represent and understand quantitative relationships

• model and solve contextualized problems using various representations, such as graphs, tables, and equations.

New York State Standards addressed throughout the unit:

Standard 1: Analysis, Inquiry, and Design. Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions. Standard 3: Mathematics Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, and data analysis.

UNIT OVERVIEW

DAY OVERVIEW

1-5 The students will learn about the difference of algebraic expressions and equations. They will then learn how to solve them using a different methods along the standard way of teaching mathematics (give examples on the board and students do some on their own.) Students will also learn how to create algebraic expressions and equations using inquire-based learning and talk about how this is useful in everyday life. This unit would be best for students who have already touched upon learning algebra and solving equations so it would be more of a review or teaching for students of different learning abilities. This lesson can also be used for higher up algebra with modifications (extensions). Students will be using the graphing calculator quite a bit and other technology and tools to foster their learning. All students learn a different way, and this unit will help students of most learning styles.

1 Students will start to look at just algebraic expressions and how to solve them with paper and pencil as a refresher. They will then learn how to solve algebraic expressions with one variable that has two different values on a graphing calculator. Then, they will learn how to solve an algebraic expression with different variables with different values using a graphing calculator.

2 Students will learn how to solve algebraic equations using algebra tiles and an equation scale. The day will mostly concentrate on solving positive equations (addition) and then move on to negative (subtraction) numbers.

3 Students will go on the internet and use the Virtual Manipulative Library and use the equation scale on their web-site to practice solving algebraic equations with tiles and a balance/equation scale. After that, students will learn how to solve algebraic equations using arrow drawings.

4 Students will play “Guess my rule” using a graphing calculator to open up the class period. This will get students thinking more in-depth about algebraic equations and look at them from a different view point. They will then start to create their own algebraic expressions and equations in real-life situations. They will start by developing an equation for the height of styrofoam cups. This will also have students thinking at a higher level and implement inquiry-based learning even more.

5 Students will continue to look at life situations that need algebra to be solved. Students will participate in inquire-based learning and then look more in-depth into the process of creating algebraic equations and how to analyze other people’s algebraic equations that they created and how they came about their answer.

Day 1: Objectives: • By the end of the class, students will be able to: • differential between algebraic expressions and equations • evaluate algebraic expression with more than one variable • evaluate algebraic expressions with two different values for the same variable in a

graphing calculator Materials: Math book Graphing calculator Math notebooks Opening activity: Explain to students the difference between and algebraic expression and an algebraic equation. Review algebraic expressions and explain to students that they are going to learn how to solve algebraic expressions by substituting in a value for their variable using a graphing calculator. Procedure:

1. Review the order of operations. 2. Evaluate (9+6)/5*4+(2³-3) to review order of operations. 3. Solve on board while students write in their notebooks, 3x+5*2, substituting x

for 7. Express how it is important to show step by step work. 4. Evaluate a-b+7, when a=15 and b=9 as a whole class. 5. Evaluate 3a+4b if a=5 and b=17 as a whole class. 6. Have students evaluate the following problems on their own in their notebooks:

m³ 2n if m=6 and n=9

6(ab) ³ if a=3 and b=2

7. Have two students come to the board to solve the problems. 8. Write: 3(x-6)/2+(x²-15) for x=8 and x=12 9. Pass out graphing calculators. 10. Show students how to store numbers for the x value. 11. Walk students through on how to evaluate the problem. 11. Evaluate problem on board for x=8. 12. Have students change the stored value for 8 to 12. Explain how to do this and that this will now evaluate our problems for x=12 instead of x=8.

13. Put up 6(ab)3 if a=3 and b=2 on the board again (from procedure 6). Show students how to solve this problem with the graphing calculator by storing the values for a and b. 14. Keeping the values for a and b the same, show students how to change the equation so it will solve for the same values for a and b (3,2).

Closure: Explain to students that expressions can contain more than 1 or 2 variables and they would evaluate them the same way. Assign page 14, #24-31 odd for homework. Explain homework and how credit will only be given if all work is shown. Go over first problem with students. (3x+4y-2w if w=4, x=7, and z=3). (If there is extra time, students will be allowed to start the homework).

Answer key for problems done in class and homework: (procedure 2) (9+6) )5*4+(2³-3) 15)5*4+(8-3)

15)5*4+5 3*4+5 12+5

17 (procedure 3) 3x+5*2 when x=7 =3(7)+5*2 =14+10 =24 (procedure 4) a-b+7, when a=15 and b=9 =15-9+7 =6+7 =13 (procedure 5) 3a+4b if a=5 and b=17 =3(5)+4(17) =15+68 =83 (procedure 6) m³

2n if m=6 and n=9 =3³ =216 =216)18 =12 2(9) 18 2nd problem in procedure 6 (ab) ³ if a=3 and b=2 =6(3*2) ³ =6(6) ³ =6(216) =1296 (procedure 8-12) 3(x-6)/2+(x²-15) for x=8 and x=12 when x=8, the value of the expression is 52. when x=12, the value of the expression is 138.

Homework answer key: #23 = 37 #25 = 48 #27 = 96 #29 = 13.5 #21= 9

Day 2: Objectives: By the end of the class, students will be able to: • balance an equation scale • solve algebraic equations using addition, subtraction, multiplication and division • create and solve algebraic equations • work cooperatively in pairs Materials: Algebra tiles (overhead set) Equation scale-transparency Class set of laminated algebra tiles Class set of laminated equation scale Dry erase markers Math Book Math notebooks Opening activity: Students will be told that they are going to play a game with tiles that ties into algebra. Teacher will discuss that they are now going to work with algebra equations that include an equals sign instead of just an algebraic expression. Procedure: 1. Teacher will set up an equation scale with algebra tiles on the overhead. 2. Teacher will give out the goal to the game to the students and post them on the board.

The goal is to get a single x (green rectangle tile) on one side of the scale with how many ever yellow squared tiles on the other side to balance it. Students will be told that what they do to one side they MUST do the same thing to both sides of the scale.

3. Teacher will set up two x tiles and two yellow tiles on the left side of the scale and 10 yellow tiles on the other side of the scale. Students will be told that they have to find what is in common on both sides of the scale.

4. Teacher will remove or take away two yellow tiles from both sides. 5. Teacher will then show how there are only x tiles on the left side of the scale and 8

yellow tiles on the right side of the scale. The students are reminded that they want to get only one x tile on one side. Ask students for suggestions on what to do.

6. Students will be told that they can then arrange the tiles into two groups (an x tile on one side with 4 yellow tiles and the other x tile with 4 yellow tiles). This would be dividing both sides by two, or in half. So, have of 2 x tiles is one and half of 8 yellow tiles is 4. Since they want to leave an x tile and have to do the same to both sides, they can remove one group and be left with an x tile on one side of the scale and 4 yellow tiles on the other side of the scale. Students will be showed how they now have one x tile on one side and 4 yellow tiles on the other. The game is then complete.

7. Teacher will then demonstrate another problem. 8. Teacher will place 3 x tiles on the left with three yellow tiles and place 2 x tiles on the

right with four yellow tiles. 9. Teacher will ask students what is in common on both sides. 10. Depending on student responses, teacher will ultimately remove 3 yellow tiles from

both sides then 2 x tiles from both sides. 11. The equation scale will leave one x tile on one side and 1 yellow tile on the other

side. 12. Teacher will do this again with 3 x tiles and 4 yellow tiles on one side and 10 yellow

tiles on the other side. The scale will be left with one x tile and two yellow tiles on the other side.

13. Teacher will do the same set up in step 12 but this time add in numbers and x values for what is being done.

14. The teacher will first take away or subtract 3 (yellow tiles) from both sides of the equation (3x+4=10), then divide both sides by 3, leaving x=2.

15. Teacher will write 2x+4=x+7 on the board. Students will be instructed to set up their scales so it looks like this problem. Teacher will demonstrate on the overhead (2 x tiles and 4 yellow tiles on one side with one x tile and 7 yellow tiles on the other side.)

16. Student will be asked to take away what is in common on both sides. (4 yellow tiles). Teacher will write steps on board with students are too.

17. They will be asked to do this until they can’t anymore (next take away an x tile from both sides).

18. Students will be left with an x tile on one side and 3 yellow tiles on the other side. 19. Students will be given 4x+3=x+6 on the board and asked to solve this with their tiles

while showing the steps for what they are doing. 20. Choose a student to come to the overhead to explain what they did when done. 21. Teacher will place 6x+3=15 on the board and asks students to solve it without the

tiles. 22. Call on a student to come to the board and solve the problem. The students

should’ve removed 3 from both sides then divided by 6 and got x=2. 23. Teacher will then describe what to do if they have a negative and divided numbers in

the equation. Students will be asked to notice how they did the inverse operation for addition and multiplication to solve the previous problems, so they would do the inverse of subtraction or division to solve problems with these operations.

24. As a class, teacher will solve 4c-3=25 on the board. The answer will be c=7

25. Teacher will put 8c-3=13 and x/8-6=2 on the board. Students will be asked to do them in their math notes then call on two students to come to the board to solve. The answers will be c=2 and x=64.

26. Students will be assigned page 13-29 odd on page 35 for homework. They will have to show their work and are allowed to use the tiles if they can or want.

Closure:

The class will solve a few more problems on the board using numbers with decimals and other variables. Students will then be given the remainder of the period time to create their own equation with a partner and have them solve it using the algebra tiles.

Answers for homework page 35 #13-29 odd: #13=3 #15=4 #17=152 #18=192 #19=192 #21=0.5 #23=0.3 #25=0.25 #27=12 #29=18.9

Day 3: Objectives: By the end of the class, students will be able to: 27. balance/solve algebraic equations using a balance 28. solve algebraic equations using arrow drawings Materials: Computers and internet for each student math notebooks previous days homework Opening activity: Students will be asked if they had any questions on the homework and these questions will be gone over. Students will then be told that they are going to learn how to do the subtraction problems with algebra tiles and play on-line a little more with this approach to mastering algebraic equations. Procedure: 1. Students will be instructed to the National Library of Virtual Manipulative

(http://nlvm.usu.edu/en/nav/topic_t_2.html) 2. Students will be asked to scroll to Algebra (Grades 9-12) and click on Algebra Balance

Scales-Negatives. 3. Students will be given an introduction on how to use this. 4. Students will be shown how to get to Algebra balance scale for plosives also and are given the

first half of the class to work through the equations given in the two programs. 5. For the second half of class students will be taught how to solve algebraic equations using

arrow drawings. 6. Students will be given 2x+13=57 and a line. On one side of the line will be x and the other

will be the 57 because we want to solve for x. We want to know what we did to x to get to 57, so if we do the opposite, or inverse of that to 57 we will get the value for x.

7. Students will be asked what is the first thing done to x to equal 57 (multiplied by 2). *2 will then be drawn on the top of the line on the left (1st) half.

8. Students will be asked what was done next to 2x to equal 57 (added 13). +13 will be drawn on the top of the line on the right (2nd) half.

9. Students will be shown how we go from x and end up with 57 using the arrow line drawing. (x*2=2x, then +13 and we have 2x+13=57)

10. Students will draw arrows on the line from x to 57. Next to the line they will write 2x+13=57. 11. Students will be told how they are going to now work backwards from 57 by finding the

inverse of when they went from x to 57. 12. Students will be asked what the inverse of addition is (subtraction). So, the inverse of +13 is -

13, this will be written on the bottom of the line, under +13.

13. Students will be asked what the inverse of multiplication is (division). So, the inverse of *2 is )2, this will be written on the bottom of the line, under *2.

14. Since they are now going from the 57 to the x, they will draw dotted lines to represent this and arrows going the opposite way.

15. Students will now solve for x working from 57. 57-13=44. 44/2=22. X=22. 16. Teacher will demonstrate this again, but write down what was done under the written equation.

2x+13=57 -13 -13 2x = 44 2 2

x=22 17. Students will then be asked to draw a arrow drawing for 5x-7=28 and solve for x. Teacher will choose a

student to come to the board to solve. (x=7) 18. Teacher will write ½ x-27=42 on the board and have students solve it. Students will be reminded to think

of the inverse, or opposite of ½ along with finding the inverse of the operation. Closure: The teacher will have the students try to solve bx+5=24 using arrow drawing. A student will be chosen to come to the board to show their work. Students will be instructed to copy it in their notebooks. Students will then be assigned to do last nights homework again, but this time use arrow drawings.

Answer key to in class activities:

Day 4: Objectives: By the end of the class, students will be able to: • create an algebraic expression or equation based on a steady increasing model • analyze if an algebraic expression is correct based on the situation • apply algebraic equations to real life situations • work cooperatively in groups Materials: Graphing calculator and overhead hook-ups overhead projector 5 styrofoam cups for each pair of students 1 ruler for each pair of students styrofoam cup activity worksheet for each student Opening activity: Teacher will introduce “guess my rule” using a graphing calculator. The students then have to give different values of x to see what the algebraic expression equals for each value. Then, the students have to try to figure out what the expression is. For example: when x=1 y(the equation)=3

when x=5 y=15 when x=-2 y=-6 The expression is 3x.

Procedure: 1. After the teacher puts in one or two different equations, the teacher will call on a few students to put in an

equation of their own and try to get the class to guess it. 2. The class will play this for most of the 1st half of class. Then, they will be told that they are going to create

algebraic expressions for given situations. 3. Students will be grouped into pairs. Each pair will be given 5 styrofoam cups, a ruler and a styrofoam cup

worksheet. 4. Teacher will explain how to fill in the worksheet and let the students work in their groups to come up with

an expression for the total height (to the nearest cm). 5. If students need help: the total height = initial height + (# of cups*increase per cup). 6. Have students give their expressions (or equations for those who gave the total height a variable) and write

them on the board (if more than one exact equation/expression was given only write the first one given). 7. Have various groups discuss how they created their expression and the relationships they used to achieve

this answer. 8. Discuss if their expressions are correct and why (or why not).

Closure: Discuss how coming up with expressions or equations will be helpful in real life. For instance, if you needed to fill a cup holder and wanted to know how many cups can fit in it, you can find an equation that will tell you how tall the nth cup is (like we just did) and see if this would be enough cups or if you have to add or take away some cups.

Name__________________

Styrofoam cups activity

Height of first cup, not including the top lip = _______cm.

Number of styrofoam cups Height (cm.) to top of cup

Create an algebraic (linear) expression to determine the height of x amount of cups: ______________________________ What would the height be for the 10th cup using your expression? _____________________________ What would the height be for the 22nd cup using your expression? _____________________________

Day 5: Objectives: By the end of the class, students will be able to: • create algebraic equations based on pictorial patterns • analyze algebraic equations to see if they are correct • enter algebraic equations into a graphing calculator to find the x and y values • work cooperatively in groups. Materials: Britannica student pages 10-13 30 toothpicks per pair of students Graphing calculators (for each pair). Opening activity: Students will continue to discuss patterns and create equations/expressions. Students will receive student page 10 and given a few minutes to think about the questions. The class will then have a class discussion about #18 and #19/ Procedure: 1. Students will be paired up. 2. Every student will receive student pages 11-13 and 30 toothpicks for each pair. 3. Students will do #20-27 in their group first then there will be a class discussion after each question is done. 4. After students complete 22c and it is discussed as a class, they will be instructed how to enter this into y= in

their graphing calculator. 5. Students will then go to the table and see if their formula works. 6. Students will check their answers in the table on the calculator to the table created in 20. 7. After students complete #23, the suggestions of each person will be put into the calculators to check and see

if they are correct, this is to be done as a class. Closure: Students will work on number 28 and have a few minutes to solve it. It will then be discussed as a whole class. The student hand-outs and answer key can be found at http://www.mmmproject.org/vp/tp11.htm

Resources: Mathematics book: McGraw-Hill, Mathematics, Applications and Connections, Collins, William, 2001. Chapter , pages 11-25. Internet Sources: http://www.mmmproject.org/, Modeling Middle School Mathematics, Bolster Education. http://nlvm.usu.edu/, National Library of Virtual Manipulatives. Utah State University.