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S 101

by: Edwina Flatley
Edwina Flatley
GPA 3.8


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This 5 page Class Notes was uploaded by Edwina Flatley on Friday October 23, 2015. The Class Notes belongs to A at University of Kentucky taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/228263/a-university-of-kentucky in Studio Art at University of Kentucky.

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Date Created: 10/23/15
A amp S 101003 February 5 Class Notes Equivalence Relations Comment on Equivalence Equations from Dr Jones M During last class the discussion was made that algebraic equations are equivalent if their solution sets are identical Equations can also have equivalence by having the same domain Equations with the same domain still satisfy the three properties that equations with the same solution set do 1 Re exive property 2 Symmetric property 3 Transitive property Dr Seif said the he would make up a sheet with examples of when equivalence relations came up naturally Un nished business on equivalence relations If you have an equation and multiply it by a variable you usually get two different solution sets What equivalence relations do these two equations have For example Equation 1 xx l 5x If you multiply by lx then you get the second equation Equation 2 x l 5 The solution to this equation is x 4 but this answer does not give the only solution to equation Comments H The thing about this situation is when we multiply equation 1 by lx we are assuming x to be a variable but x in equation 1 is considered to be an unknown H In equation 1 0 is a possible solution but when we multiply equation 1 by lx we are multiplying the equation by lx where x is zero This cannot happen H When we multiply equation 1 by lx we are changing the limits of the equation as well H In this case we are making the solution set of equation 1 smaller by multiplying by lx H Sometimes when you multiply by an algebraic expression with a variable the solution set gets larger When teaching your class about solving equations you can show them examples of multiplying an equation by an algebraic expression with variables where the solution set of an equation is larger and smaller to help them understand about this topic Be sure you try to account for what you do to an equation This will help with mistakes that can be made What happens when you square both sides of an equation What happens when you take the square root of both sides of an equation What happens when you take the cube root of both sides of an equation ETC Quadratic Equations Take problem X2 2X 5 n Is isolating the variable helpful in this situation Can you solve it like you solve linear equations If you solved it like you do linear equations it looks like this X2 2X 5 Xl5 X This is a solution but it is not a number solution like when 2 you solve linear equations Another example X l2 4 n How would you go about solving this equation with or without using the quadratic equation x124 X124 2 2 7 X2Xl4 lX1 l4 x22x730 OR x 1x30 x12 x13 Xl3 n An algebra student with little experience might do this x 12 4 X lX l4 X l 4 X l 4 X 3 Equations should speak to you n How do you make equations interesting for students to do Just giving them a worksheet of lots of equations to solve is not interesting to students 1 Give students authentic eXamples of real world problems 2 Play jeopardy with problems Finding conteXt eXamples is not easy EXample Comparison between functions can be an eXample of an algebraic equation fX X2 Where do these graphs cross fX 2X 5 When you graph this there are two points of intersection so you need to make sure you have two solutions to your equation 2 7 X 7 2X 5 Homework Send an email to the group on any issues questions concerns or anything else you think of to ask or comment about minar 4 Equationsolving lV AS 101003 High school mathematics from a more advanced point of View 1 Welll proceed with quadratic equations today We7ll come back to Z12 stuff after quadraticsi 2 We need a volunteer to take notes and send them to the email list as an attachment 3 You received Amy Curlls wonderful notes on last weeks discussion If you need a hard copy ask Dr S Here are a couple of things Amy brought to light 4iAp o Multiplying both sides of an equation by an expression involving a variable can result in a change of the domain of the new equation Think of multiplying by i as in the example Brad put forth in Seminar 3 0 is lost to the domain of the resulting equation If 0 is a solution to the original equation we might a solution What to do in real class time about this Maybe put in parentheses next to the equation that the domain has been messed around with and when giving the solution set there will have to be accounting for that Students aren7t always clear about the meaning of solution to an equation Dr Jones7 example was interesting and students do this stuff all the time To solve 12 A 31 2 0 students isolate the variable a great tactic in linear equations and solve for z with z Is that wrong Hmm In the classroom when solving an equation it will help to provide students with a clear idea of your expectations roblem presented last time but slightly gussied up Here it is again Problem Let E be an equation Which of the following operations to E results in an equivalent equation If one uses the operation under consideration what notationscomments could be made next to the resulting equation A 93 V A n A 39D A l h V A m 27 V Adding the same constant to both sides of the equation V Adding the same algebraic expression to both sides assuming that involves only the variables in that algebraic expression are in V Adding the same algebraic expression to both sides of E assuming the alge braic expression contains variables not in E V Squaring both sides of El Cubing both sides of El Taking the square root of both sides of the equation V V Multiplying both sides of E by the same constant Multiplying both sides of E by the same algebraic expression assuming the algebraic expression involves only variables contained in i Quadratic equations in the real numbers The algorithm for solving a linear equation with one variable calls for isolating the variable Last time we brie y discussed the appropriateness of isolating variables in the quadratic equation case We saw that it might not lead to a numerical solution to a quadratic but that students could be expected to do stuff like 12 721 5 12 21 5 isolate the variable 11 2 5 factor out the variable I xi Voilal Of course there s not all that much wrong with the above procedure The only problem we expected to have a realnumber solutionsi Note We should expect to see such proceduressolutions from high school students As Dr J pointed out during the rst seminar we ask high school students to solve z 2y 7 for y in terms of I so a solution that involves z isnlt unheard of Of course here with the equation 12 721 5 we expect real number solutions it s important to clarify what is meant by solution to the equations i Okay so we assume that the student knows what we mean by solution set77 to the given quadratic equation Obviously isolating the variable is not the cleanest way to solve So how do we nd the numeric solutions to a quadratic So let s take up a simpler quadratic equation 12 4 This one talks to us It says to me a number squared is 4 Arithmetic facts lead me to the two solutions 74 4 No prob Should the square root word77 be invoked here Therels some confusion about the meaning of square root77nthese guys can be negative versus the square root functionna function with range the nonnegative realsi Letls leave that for another time A How about I 7 l2 4 Or 21 7 3 4 So how can be solve 12 721 5 Realworld problem You re designing a small rectangular city garden The length is one less than twice the width What dimensions should the garden be built so that its area is 6 square yards 1 leave you With the following questions Could you do a clean studentfriendly derivation of the quadratic formula In your most honest moment do you think presenting such a derivation or having students discover it With you might be of interest to you and to your students It might do some good to bring home to our students that ingenuity was required to solve quadratic equations Some humans somewhere quite cleverly nessed the isolating the variable doesn t workquot obstruction Do our students just want to see the formula and get it over with Might they be able to appreciate that formula if it was presented as a pretty decent accomplishment Or let them discover themselves how to do the nessing 9 Question Do you think it would be helpfuluseful to derive the quadratic formula perhaps jointly With your students Have you ever tried yourself to derive it andor thought about its derivation and its validity


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