Metro RESA...Building leaders of teaching and learning Digging Deeper Into Mathematics Clayton County Summer Math Academy Algebra I – Solving Equations & Systems Sarah Ledford June 9,

Metro RESA...Building leaders of teaching and learning Solve it! 2 Solve the following equation for t: 105 = 100(1 +.05t) Explain how you solved it & how you know your solution is correct.

Metro RESA...Building leaders of teaching and learning Who is Correct? Students were asked to solve the future simple interest equation (A = P(1 + rt)) for t. Sam and Ella got two different answers. Who is correct? Sam’s solution: t = (A – P)/(Pr) Ella’s solution: t = (A/P – 1) ÷ r 3

Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 4

Metro RESA...Building leaders of teaching and learning Algebra I GSE A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. 5

Metro RESA...Building leaders of teaching and learning 6-8 GSE 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.3 Apply the properties of operations to generate equivalent expressions. 6.EE.4 Identify when two expressions are equivalent. 6.EE.5 Understand solving an equation… 6

Metro RESA...Building leaders of teaching and learning 6-8 GSE 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.3 Solve multistep … mathematical problems posed with positive and negative rational numbers in any form by applying properties of operations… 7

Metro RESA...Building leaders of teaching and learning 6-8 GSE 7.EE.4 Use variables to represent quantities in a real- world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 8.EE.7 Solve linear equations in one variable. 8

Metro RESA...Building leaders of teaching and learning Equivalent? These two equations are equivalent: 2x + 5y = 400 & x = -5/2 y Explain how you know that they are equivalent. 9

Metro RESA...Building leaders of teaching and learning Same two equations… 2x + 5y = 400 & x = -5/2 y Liam says that both equations model this situation: I have some $5 bills and two identical piles of $1 bills. Altogether, I have $400. Which equation do you think is a better description? Why? From More Great Questions – Differentiating pg 39 10

Metro RESA...Building leaders of teaching and learning Same Solutions? Which of the following equations have the same solution? Give reasons for your answer that do not depend on solving the equations. A.x + 3 = 5x − 4B. x − 3 = 5x + 4 C.2x + 8 = 5x − 3D. 10x + 6 = 2x − 8 E. 10x − 8 = 2x + 6F x = 0.5x − 0.4 From Illustrative Mathematics 11 REFERENCE THE source (GaDOE

Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 12

Metro RESA...Building leaders of teaching and learning Solving Equations Let’s solve this very basic equation: 3x + 4 = 7 Let’s turn this multi-step equation into a system of equations. Seems strange, I know… Go to “Pan Balance” applet Type 3x + 4 & 7 into the balance pans Play with the app & try to figure out what it does! 13

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Metro RESA...Building leaders of teaching and learning TI-84 The app is fun to play with, but we can accomplish the same thing with our TI-84s. To graph our equation in 2 parts: “y = ” Type in “y 1 = ” 3x + 4 Type in “y 2 = ” 7 “graph” 15

Metro RESA...Building leaders of teaching and learning TI-84 If you can’t see what you want to see, you may need to change the window. “zoom” Scroll down to “Zstandard” which will show a window from -10 to 10 on each axis OR “window” Input whatever window you want to view your graphs on 16

Metro RESA...Building leaders of teaching and learning TI-84 “Trace” allows you to move your keys in the top right to move a point along your graph. This will give you ordered pairs at the bottom of your screen that satisfy the equation. Note in the top left of your screen which equation is shown – that tells you which graph you are tracing points on. Hitting the up OR down key changes the equation/graph. 17

Metro RESA...Building leaders of teaching and learning TI-84 “Trace” allows you to get an estimate for the point of intersection of the two lines, which is what we want to find. “Calc”  “2 nd ” “trace” “intersect” This will calculate the exact (if possible) intersection. Other options: value, zero, minimum, maximum, … 18

Metro RESA...Building leaders of teaching and learning TI-84 We have solved the equation, turned it into a system of equations, and found the point of intersection of the system. We can also look at the table of values. “Table”  “2 nd ” “Graph” What do you see? 19

Metro RESA...Building leaders of teaching and learning Graphing Solution We can use this graphing method to solve any equation: 1.Turn the equation into a system 2.Graph each equation in the system 3.Find the point of intersection 4.Write the solution to the original equation 20

Metro RESA...Building leaders of teaching and learning Try to solve these |2x + 3|= 5 x 2 – 4x + 3 = 3 x 2 – 4x + 3 = x – 1 (3x + 1) ½ = 2 21

Metro RESA...Building leaders of teaching and learning System of Linear Equations Now, let’s look at a system of equations. y = 2x – 3 y = 5 – 2x Find the solution & explain what it means. 22

Metro RESA...Building leaders of teaching and learning Substitution Both equations are written as “y = ” so we can substitute one equation into the other for y. 2x – 3 = 5 – 2x Now, simply solve for x. Once you have x, you can solve for y. Make sure to write your solution in terms of (x, y) because our original equations had both variables! 23

Metro RESA...Building leaders of teaching and learning Elimination y = 2x – 3 y = 5 – 2x I am going to rewrite these equations into the format that you are used to for the elimination method. I do NOT have to do this!! 24

Metro RESA...Building leaders of teaching and learning Elimination – Line it up!! 2x – y = 3 -2x – y = -5 We can add the two equations together to eliminate the x-values. WHY can we do this? We can multiply either of the original equations by -1 in order to eliminate the y-values. WHY can we do this? 25

Metro RESA...Building leaders of teaching and learning Linear & Quadratic System Solve: y = x 2 – 4x + 3 y = 3 We have already solved this using the graphing method on the previous slide. Using the substitution method, this becomes a simple quadratic to factor and solve! 26

Metro RESA...Building leaders of teaching and learning Linear & Quadratic System Solve: y = x 2 – 4x + 3 y = x - 1 We already did this one, too! It looks hard but it’s not!! 27

Metro RESA...Building leaders of teaching and learning Another System Solve: (x – 3) 2 + (y – 3) 2 = 25 y = x + 5 This one REALLY looks hard! You have a circle and a line. All you have to do is… 28

Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 29

Metro RESA...Building leaders of teaching and learning Alg I GSE A.REI.5 Show and explain why the elimination method works to solve a system of two-variable equations. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A.REI.11 Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same. 30

Metro RESA...Building leaders of teaching and learning 6-8 GSE All of the standards about solving equations in one variable and writing equivalent expressions. 8.EE.7 Solve linear equations in one variable. 8.EE.8 Analyze and solve pairs of simultaneous linear equations (systems of linear equations). 31

Metro RESA...Building leaders of teaching and learning Babysitting Sabina earns money by baby-sitting. She offers her clients two different payment options and presents them in the graph on the next slide. What do you think each payment option is? How did you figure it out? Explain which option you would choose and why. From Good Questions Grades 5-8 (pg 110) 32

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Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 34

Metro RESA...Building leaders of teaching and learning Elimination Method Proof Handout Teacher key at SarahLikesMath.com 35