Multiple Representations for Algebraic Expressions and Equations

Multiple Representations forAlgebraic Expressions andEquationsThis is a five day unit with the goal to teach students different ways to solve andcreate algebraic expressions and equations using inquiry-based learning. This lesson isappropriate for grades 8 and 9. Throughout the unit students use a variety of tools toaccomplish this goal. Students will use a graphing calculator in several lesson, thecomputer (internet with virtual manipulatives), algebra tiles and toothpicks.

Created by Andrea Koralewski

Overall objectives for the unit with NCTM and New York StateStandards 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 equations3. evaluate algebraic expression with more than one variable4. evaluate algebraic expressions with two different values for the same variable in agraphing calculator5. balance an equation scale6. solve algebraic equations using addition, subtraction, multiplication and division7. create and solve algebraic equations8. work cooperatively in pairs9. balance/solve algebraic equations using a balance10. solve algebraic equations using arrow drawings11. create an algebraic expression or equation based on a steady increasing model12. analyze if an algebraic expression is correct based on the situation13. apply algebraic equations to real life situations14. create algebraic equations based on pictorial patterns15. analyze algebraic equations to see if they are correct16. enter algebraic equations into a graphing calculator to find the x and y valuesNCTM 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 relationshipRepresent and analyze mathematical situations and structures using algebraic symbols• use symbolic algebra to represent situations and to solve problems, especiallythose that involve linear relationships• recognize and generate equivalent forms for simple algebraic expressions andsolve linear equationsUse mathematical models to represent and understand quantitative relationships

• model and solve contextualized problems using various representations, such asgraphs, 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 engineeringdesign, as appropriate, to pose questions, seek answers, and develop solutions.Standard 3: MathematicsStudents will understand mathematics and become mathematically confident bycommunicating and reasoning mathematically, by applying mathematics in real-worldsettings, and by solving problems through the integrated study of number systems,geometry, algebra, and data analysis.

DAYUNIT OVERVIEWOVERVIEW1-5 The students will learn about the difference of algebraic expressions andequations. They will then learn how to solve them using a different methodsalong the standard way of teaching mathematics (give examples on the boardand students do some on their own.) Students will also learn how to createalgebraic expressions and equations using inquire-based learning and talkabout how this is useful in everyday life. This unit would be best for studentswho have already touched upon learning algebra and solving equations so itwould be more of a review or teaching for students of different learningabilities. This lesson can also be used for higher up algebra withmodifications (extensions). Students will be using the graphing calculatorquite a bit and other technology and tools to foster their learning. All studentslearn 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 themwith paper and pencil as a refresher. They will then learn how to solvealgebraic expressions with one variable that has two different values on agraphing calculator. Then, they will learn how to solve an algebraicexpression with different variables with different values using a graphingcalculator.2 Students will learn how to solve algebraic equations using algebra tiles and anequation 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 anduse the equation scale on their web-site to practice solving algebraic equationswith tiles and a balance/equation scale. After that, students will learn how tosolve algebraic equations using arrow drawings.4 Students will play “Guess my rule” using a graphing calculator to open up theclass period. This will get students thinking more in-depth about algebraicequations and look at them from a different view point. They will then start tocreate 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 inquirybasedlearning 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-depthinto the process of creating algebraic equations and how to analyze otherpeople’s algebraic equations that they created and how they came about theiranswer.

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 agraphing calculatorMaterials:Math bookGraphing calculatorMath notebooksOpening activity:Explain to students the difference between and algebraic expression and analgebraic equation. Review algebraic expressions and explain to students that they aregoing to learn how to solve algebraic expressions by substituting in a value for theirvariable 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 xfor 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:m³2n if m=6 and n=96(ab) ³ if a=3 and b=27. Have two students come to the board to solve the problems.8. Write: 3(x-6)/2+(x²-15) for x=8 and x=129. 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 andthat 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). Showstudents how to solve this problem with the graphing calculator by storing the valuesfor a and b.14. Keeping the values for a and b the same, show students how to change theequation 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 andthey 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 firstproblem with students. (3x+4y-2w if w=4, x=7, and z=3). (If there is extra time, studentswill 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+53*4+512+517(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) m³2n if m=6 and n=9=3³ =216 =216)18 =122(9) 182nd 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=12when 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 pairsMaterials:Algebra tiles (overhead set)Equation scale-transparencyClass set of laminated algebra tilesClass set of laminated equation scaleDry erase markersMath BookMath notebooksOpening activity: Students will be told that they are going to play a game with tiles thatties into algebra. Teacher will discuss that they are now going to work with algebraequations 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 howmany ever yellow squared tiles on the other side to balance it. Students will be toldthat 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 10yellow tiles on the other side of the scale. Students will be told that they have to findwhat 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 8yellow tiles on the right side of the scale. The students are reminded that they want toget 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 onone side with 4 yellow tiles and the other x tile with 4 yellow tiles). This would bedividing both sides by two, or in half. So, have of 2 x tiles is one and half of 8 yellowtiles is 4. Since they want to leave an x tile and have to do the same to both sides, theycan remove one group and be left with an x tile on one side of the scale and 4 yellowtiles on the other side of the scale. Students will be showed how they now have one xtile 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 theright 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 fromboth 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 otherside.12. Teacher will do this again with 3 x tiles and 4 yellow tiles on one side and 10 yellowtiles on the other side. The scale will be left with one x tile and two yellow tiles onthe other side.13. Teacher will do the same set up in step 12 but this time add in numbers and x valuesfor what is being done.14. The teacher will first take away or subtract 3 (yellow tiles) from both sides of theequation (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 theirscales so it looks like this problem. Teacher will demonstrate on the overhead (2 xtiles and 4 yellow tiles on one side with one x tile and 7 yellow tiles on the otherside.)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 fromboth 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 tileswhile 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 thetiles.22. Call on a student to come to the board and solve the problem. The studentsshould’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 inthe equation. Students will be asked to notice how they did the inverse operation foraddition and multiplication to solve the previous problems, so they would do theinverse 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 dothem in their math notes then call on two students to come to the board to solve. Theanswers will be c=2 and x=64.26. Students will be assigned page 13-29 odd on page 35 for homework. They will haveto 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 decimalsand other variables. Students will then be given the remainder of the period time tocreate 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 balance28. solve algebraic equations using arrow drawingsMaterials:Computers and internet for each studentmath notebooksprevious days homeworkOpening activity:Students will be asked if they had any questions on the homework and these questions willbe gone over. Students will then be told that they are going to learn how to do the subtractionproblems with algebra tiles and play on-line a little more with this approach to masteringalgebraic 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 BalanceScales-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 thefirst 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 usingarrow drawings.6. Students will be given 2x+13=57 and a line. On one side of the line will be x and the otherwill 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 willthen be drawn on the top of the line on the left (1 st ) half.8. Students will be asked what was done next to 2x to equal 57 (added 13). +13 will be drawn onthe top of the line on the right (2 nd ) 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 theinverse 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, thiswill 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 arrowsgoing 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 -132x = 442 2x=2217. Students will then be asked to draw a arrow drawing for 5x-7=28 and solve for x. Teacher will choose astudent 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 thinkof 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 bechosen 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 groupsMaterials:Graphing calculator and overhead hook-upsoverhead projector5 styrofoam cups for each pair of students1 ruler for each pair of studentsstyrofoam cup activity worksheet for each studentOpening activity:Teacher will introduce “guess my rule” using a graphing calculator. The students then have to givedifferent values of x to see what the algebraic expression equals for each value. Then, the students have totry to figure out what the expression is.For example: when x=1 y(the equation)=3when x=5 y=15when x=-2 y=-6The 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 anequation of their own and try to get the class to guess it.2. The class will play this for most of the 1 st half of class. Then, they will be told that they are going to createalgebraic expressions for given situations.3. Students will be grouped into pairs. Each pair will be given 5 styrofoam cups, a ruler and a styrofoam cupworksheet.4. Teacher will explain how to fill in the worksheet and let the students work in their groups to come up withan 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 writethem 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 achievethis 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 youneeded to fill a cup holder and wanted to know how many cups can fit in it, you can find an equation that willtell you how tall the n th cup is (like we just did) and see if this would be enough cups or if you have to add ortake away some cups.

Name__________________Styrofoam cups activityHeight of first cup, not including the top lip = _______cm.Number of styrofoam cupsHeight (cm.) to top of cupCreate an algebraic (linear) expression to determine the height of x amount of cups:______________________________What would the height be for the 10 th cup using your expression?_____________________________What would the height be for the 22 nd 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 10-1330 toothpicks per pair of studentsGraphing calculators (for each pair).Opening activity:Students will continue to discuss patterns and create equations/expressions. Students will receive studentpage 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= intheir 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 seeif 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 wholeclass.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.