Name: Karianne Randick
Topic: Section 3-1: “Solving Systems of Equations by Graphing”
Prerequisite Knowledge and Skills:
Students have learned how to graph linear equations.
Students know how to manipulate equations to be in slope-intercept form.
1) Objectives:
a) Students will use their previous knowledge of graphing linear equations to find the intersections of the two lines, if applicable.
b) Students will be able to solve systems of equations by graphing.
2) CCSS:
a) Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
b) Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
3) Academic Language Summary:
a) Intersection, slope-intercept form, systems of equations, solution, slope, y-intercept, independent, dependent, inconsistent, consistent, ordered pair
4) Required Materials:
a) Smart Board
b) Pencil and notebook
c) Video Clip from The Goonies
d) Graph Paper
5) Entry:
a) Slide One: Introduce the chapter: “Chapter 3: Systems of Equations and Inequalities”
i) Introduce the theme of the week: “Pirate Week”
b) Slide Two: Overview of systems of equations
i) Discuss what the definition of a system of equations is
ii) Reveal that there are three ways of solving systems of equations
(1) Stress that all three methods should reveal the same answer.
c) Slide Three: 3-1 Solving Systems of Equations by Graphing
i) Show the movie clip from The Goonies (15:26-17:17)
(1) This clip is of when the goonies find a treasure map in the attic
ii) Relate the movie clip to the lesson
(1) Ask the students: When looking at a treasure map, what usually represents the location of the treasure? – “X marks the spot”
(2) Relate this analogy to when looking at two graphed lines, their intersection point is the solution to the system of equations – where “X” is located
6) Procedure:
a) Slide 4: Example 1
i) Ask the students what form the equations would have to be in for them to be able to graph the lines– slope intercept form – Step One and Two
ii) Then after graphing the equations, ask the students where the intersection of the two graphs is: “X marks the spot” – Step Three
(1) stress that the students need to write the ordered pair to demonstrate that they understand where the solution is, rather than just having the graph shown – when finding the treasure on a map, it usually includes the latitude and longitude location which is depicted as an ordered pair so when finding the treasure it is located at a point!
iii) How can we check our answer to make sure it is correct? – Substitute the values of x and y into both original equations and determine whether the end result holds true – Step Four
b) Slide 5: Classifying Systems
i) Have the words consistent and inconsistent on the board. Reveal what the definition of consistent is and ask the students what they think the definition of inconsistent would be based off of the previously revealed definition.
ii) Have the words independent and dependent on the board. Reveal what the definition of independent is and ask the students what they think the definition of dependent would be based off of the previously revealed definition.
iii) Now look at the three graphs and ask the students what they think the classification of the first graph would be: Consistent and Independent
(1) Reveal the two pirates and use an analogy: Each pirate is on their own path to find the “treasure”, aka the intersection of the lines. Because they are on their own paths they are working independently of one another.
(2) Drag the pirates along the SMART Board to demonstrate how they can “reach the treasure where X marks the spot” independent of one another
iv) Ask the students what they think the classification of the second graph would be if there are two equations plotted and are the same graphed lines: Consistent and Dependent
(1) Reveal the pirates and use an analogy: The pirates now are on the same path to get to the “treasure. Moving the pirates, notice how they are working dependently on their search.
(2) Because the graphs are the same, that means that the intersection points are at every point on each graph thus there are infinitely many solutions
v) Ask the students what they think the classification of the third graph would be: Inconsistent
(1) Discuss how these are parallel lines, thus they will never intersect. The intersection is where the solution/ “treasure” is located, so therefore there are no solutions.
(2) Note that when a graph is inconsistent it is neither dependent nor independent because it has no solutions.
c) Slide Six: Example 2
i) Have the students recap the steps for solving systems of equations by graphing and walk through it with them
(1) Change the equations into slope-intercept form, graph the equations, find the intersections – solution, check their answers.
(2) The graph is the same line – What is the solution? What is the classification of this graph? How do we know this? – Infinite number of solutions, consistent and dependent
d) Slide Seven: Example 3
i) Have the students change the equations into slope-intercept form, graph the equations, find the intersections – solution, check their answers on their own.
ii) The graphs are parallel lines – What is the solution? What is the classification of this graph? How do we know this? – No solution, inconsistent
e) Slide Eight: Example 4
i) Have the students change the equations into slope-intercept form, graph the equations, find the intersections – solution, check their answers on their own.
ii) The graphs intersect at a point – What is the solution? What is the classification of this graph? How do we know this? – (-1,1) Consistent and independent
7) Reflection, Practice, and Feedback:
a) While going through the slides, students will be graphing the equations themselves. The lecture will be taught by myself but guided by the students so their participation and understanding is what moves the lesson along. Throughout the lesson probing questions will be asked and students may ask for understanding themselves. I will give the students some time before they answer questions and then refer to the class for agreement or disagreement.
8) Accommodations:
a) For those students who don’t register the notes and become lost, post the notes on CMS so they can reach the exact lecture notes from their home.
b) If necessary, provided guided notes in which there are helpful tips typed out (this shouldn’t be very necessary since the procedure is provided on the Smart Board).
9) Closure:
i) Slide Nine: Word Problem
(1) Relate solving systems of equations with a real world application
(a) Students will have to think back to their previous skills of creating equations from information provided in word problems. They then will practice by graphing and finding the intersection.
(b) This checks for their understanding and wraps everything up
10) Student Evaluation:
a) Informal Assessment: Examples
i) Students are doing the examples on their own and checking their progress with the answers on the board
b) Formal Assessment: HW Assignment
i) Book Work: pg. 120 (14-26 evens, 27-29 all, 34)