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Unit Title: Solving Systems of Equations/Inequalities
Grade Level/Subject Area(s): 8th Math
Stage 1: Desired Results Established Goals/Standards *9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Understandings • A system of equations consists of two or more
equations that have variables that represent the same items.
• Solving a system of equations can be useful in calculating the cost difference between various payment plans, or in figuring out when a business enterprise will break even.
• To solve a system means to find the x- and y-values for which both of the equations are true. Systems of linear equations can be solved using a variety of methods (graph, solve by substitution & elimination).
• A system of linear inequalities is an extension of a system of linear equations and consists of two (or more) linear inequalities that have the same variables.
• A system of inequalities can be useful in determining which combinations of products sold by a business will yield the maximum profit.
• To solve a system of linear inequalities, one must first determine the boundary lines by graphing each inequality as though it were an equation and then identifying the region where all of the inequalities would be shaded at the same time. The feasible region represents all the possible points that satisfy the system of inequalities.
Essential Questions • What is a system of equations/inequalities? • How is a system of equations/inequalities used in the real world?
• What methods can I use to solve a system of equations/inequalities?
• How are the methods to solve a system of equations/inequalities similar? Different?
Students will know… • Key vocabulary and terms involving systems of equations & inequalities (graph, represents, solution, system of equations/inequalities, ordered pair, intersect, parallel, feasible region, constraints)
• A system of equations/inequalities allows you to compare the relationship between various business plans in the real world
Students will be able to… • recognize situations that use a system of
equations/inequalities • be able to write a system of equations/inequalities
from a given set of information • solve a system of equations problem using various
methods • understand the meaning of, and solutions to,
systems of linear equations • solve a system of inequalities by graphing each of
the inequalities on the same grid and determining the region (if any) which satisfies all of the inequalities.
Stage 2: Assessment Evidence Performance Task Treasure Hunt (See Performance Task)
Other Evidence • White boards (daily quick check) • Weekly Pop Quizzes (teacher made) • Summative Assessment (teacher made) • Homework (assigned daily) • Response to the essential questions
Stage 3: Learning Plan
Monday Tuesday Wednesday Thursday Friday 1. Hook students with right hand/left hand activity 2. Go over what graphs of a system of equations can look like & what it means. 3. State objectives & essential questions. Go over packet for this unit H W O
4. What is an inequality? Go over real life situations that involve inequalities using sentences. 5. Write inequality statements numerically. 6. I do: example in notes (p.2) We do: Do an example together & with partner (p.2) You do: White boards & homework (p.3-‐6) Eq
7. Recap of yesterday w/ Recap WS (p.7) 8. I do: Define x & y-‐intercept. Review how to graph equations using x & y-‐intercepts. We do: Practice (p.8-‐9) You do: White boards & homework (p.10-‐11 Modify homework for lower level students (less problems & highlight key points) Eq T
9. Graphing linear inequalities using x & y-‐intercept I do: explain similarities & differences We do: Create double bubble map & do p.12-‐13 You do: White boards & p (14-‐15) 10.Pull a small group to work with for additional help Tailor HW for lower-‐ level students with less, but more focused problems Eq T
11. Pop Quiz (graphing equations & inequalities using x & y-‐intercept) Modify test for lower level students (fewer problems & allow use of notes for RSP) 12. Students write a reflection about what they learned this week, including their strengths & areas of growth. Ev T
13. I do: Quick review of how to graph using x & y-‐intercept. We do: White boards with table partners You do: White boards & homework (error analysis) 14. Students review the quiz they took on Friday & analyze their errors. Eq R Ev
15: I do: Ask students what other method they know for graphing equations? (Review slope intercept form p.16). We do: White boards practice with partners (p. 17) You do: White boards & homework (p.18) Eq
16. Quick write: How do you think you would graph inequalities using slope intercept form? I do: Go over how We do: p.19; double bubble map to compare You do: White boards & homework (p.20) Eq T
17. POP Quiz – Graphing equations & inequalities using slope intercept form 18. I do: Talk about systems & how they are just like prior lessons We do: Practice problems together You do: Study guide Tailor study guide for lower-‐ level students with less, but more focused problems Ev R
19. Continue practice on systems 20. Students write a reflection about what they learned this week, including their strengths & areas of growth. Ev T
21. Show students real life scenario with systems 22. I do: Talk about real life apps with systems We do: Practice together You do: White boards & homework Eq R Ev
23. I do: Show graph with 2 business We do: discuss what the graph means & groups create their own scenario for a system of equations You do: Create your own real life scenario Eq T
24. POP Quiz – grade & evaluate in class. Discuss with partner what you did wrong 25. Students present their scenarios from yesterday Eq
26. Spiral Review I do: Review for test, go over sample questions We do: Solve sample questions together You do: Study Guide Tailor study guide for lower-‐ level students with less, but more focused problem Eq T
27. Performance Task (Treasure Hunt) Modify PT for lower level students (notes, facts sheet) 28. Students write a reflection about what they learned this week, including their strengths & areas of growth. Ev R T
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An Inquiry Based Approach to Solving a System of Linear Inequalities
Lesson Summary This four day lesson will guide students through the necessary steps to successfully graph systems of linear inequalities.
Key words used Solving systems of inequalities. Background knowledge
Number sense and meanings of inequality symbols. OACS
Patterns, Functions and Algebra Standard - Benchmarks D, H – Grades 5-7 - Benchmarks F, H – Grades 8-10 - Indicators Grade 5 - #4, Grade 6 -#4, Grade 7 - #6, Grade 8 - #7, 9,10, Grade 9 - #6, 9, Grade 10 - #7, 11
Learning objectives
This lesson is designed to help the students discover and work with basic inequalities. The lesson is designed to guide the students through graphing inequalities on a number line and using X and Y-intercepts to graph linear equations on the coordinate grid. The lesson will then have the students graph linear inequalities and correctly shade the solution area on the coordinate grid. This will then lead into having the student use X and Y-intercepts to graph a system of linear inequalities on the coordinate grid and correctly shade the solution area. Finally, the lesson will conclude with the students having to locate a treasure chest by using a system of linear inequalities.
Materials needed Worksheets Colored pencils Procedures
Place students in groups, 2-4 per group. Have a class discussion after students have completed the activities.
Assessment Assessments will be included throughout and after each lesson. In addition, there will be a treasure hunt activity at the end of the lesson that will act as a final assessment of the lesson.
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Plotting Inequalities on a Number Line Name________________________________________ Goal: This lesson is designed to help you work with and understand basic inequalities.
1. Given the statement: The temperature will be higher than 70 degrees today. A) Give three possible temperatures that satisfy the statement. B) We could represent this by using an inequality. (x > 70) C) We could also represent this with a number line graph (Notice the open circle
above 70. This indicates that 70 is not included in the values represented on the graph since we only wanted temperatures larger than 70)
○ <-----|-----|-----|-----|-----|-----|-----|----->
40 50 60 70 80 90 100
2. Given the statement: To go on the Tea Cups ride at Disneland, you must be under 48 inches tall.
A) Give three possible heights that satisfy this statement B) Write an inequality to represent the statement. C) Notice that the number line graph is going the opposite direction since the
inequality is different. ○ <-----|-----|-----|-----|-----|-----|-----|----->
44 46 48 50 52 54 56
3. Given the statement: The temperature will be 70 degrees or warmer today. A) Give three possible temperatures that satisfy the statement. B) We could represent this by using an inequality. (x > 70) C) What is different between the graph of this statement and the statement in #1?(look closely at the circles) Why do you think this difference is necessary? ●
<-----|-----|-----|-----|-----|-----|-----|-----> 40 50 60 70 80 90 100
4. Given the statement: To go on the Tea Cups ride at Disneland, you must be 48
inches tall or shorter. A) Give three possible heights that satisfy this statement B) Write an inequality to represent the statement. C) If you were drawing the number line graph, would you use an open or closed circle? Why? D) Would your line go to the right or left? Why? E) Draw the number line graph (on the next page).
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<-----|-----|-----|-----|-----|-----|-----|----->
44 46 48 50 52 54 56 5. If x > 5, what does this mean? What x values make the statement true?
6. Why is 5 not included in your list?
7. If x < 2, what does this mean? What x values make the statement true?
8. Why is 2 included in this list?
9. Complete the following table
Symbol Meaning Open or closed Circle?
>
<
>
>
10. To graph x > 4 on a number line, we would need to show all the numbers that
are greater than but not equal to 4. Draw a number line graph for x > 4
<-----|-----|-----|-----|-----|-----|-----|-----> 0 1 2 3 4 5 6
11. Graph x < 2 on a number line.
<-----|-----|-----|-----|-----|-----|-----|-----> 0 1 2 3 4 5 6
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12. Plot the following ordered pairs on the same coordinate grid and then connect the points with a straight line. (2,6), (-3,6), (-7,6), (3,6)
What do you notice about your line? __________________________________ What do you notice about your ordered pairs? ___________________________ 13. Graph y = 6 on the coordinate grid.
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14. Compare your graphs from #5 and #6. What do you notice about them? _________________________________________________________
What type of line do you get when all your y values are the same? ____________ 15. Plot the following ordered pairs on the same coordinate grid and then connect
the points with a straight line. (3,7), (3,-4), (3, -5), (3,4)
What do you notice about your line? ___________________________________ What do you notice about your ordered pairs? ____________________________
16. Graph x = 3 on the coordinate grid.
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17. Compare your graphs from #7 and #8. What do you notice about them?
_________________________________________________________ What type of line do you get when all your x values are the same? ____________
Look at the following graphs and answer the questions. A) List 3 ordered pairs for each graph B) Write an equation of the line represented by the graph
18. 19.
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Day 1 Recap Worksheet
1. Write an inequality for the following statement. The temperature will be cooler than 50 degrees tonight.
2. Draw a number line graph to represent the statement in question 1.
<-----|-----|-----|-----|-----|-----|-----|----->
3. Draw a number line graph to represent the inequality x > -4
<-----|-----|-----|-----|-----|-----|-----|----->
4. On a coordinate grid, graph the line y = -2
5. Name three ordered pairs on your line. 6. The graph of x = -5 will be a horizontal/vertical (choose one) line.
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Graphing Linear Equations Using the X and Y-Intercepts. Name_______________________________________ Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph linear equalities on the coordinate grid. Intercepts are where one or more objects cross each other. When we talk about the X-intercept we are concerned with the point where the graph crosses the X-axis. Likewise, when we talk about the Y-intercepts of a graph, we are concerned with the point where the graph crosses the Y-axis.
The graph in the above picture crosses the X-axis at point (5,0) therefore we say the X-intercept is 5. It crosses the Y-axis at point (0,-4) and we say that the Y-intercept is -4. Using your knowledge of plotting points on the coordinate grid, plot (0,3) and (-2,0). Draw a line through these points.
What conclusions can you draw about (0,3) and (-2,0)? ___________________________
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Where is the graph crossing the X-axis?________ What is the X-intercept?_______ Where is the graph crossing the Y-axis?________ What is the Y-intercept?_______
Using the above picture, what do you notice about the coordinates where the graph crosses the X and Y-axis? You should have noticed that at the X-intercept the Y value is always zero and at the Y-intercept the X value is always zero. Now let's use this information to quickly construct graphs of linear equations that are in the written in the standard form. Take a look at 4x + 6y = 12. In order to find the X-intercept we know that the Y value must equal zero. Therefore, if we substitute zero in for y we now have: 4x + 6(0) = 12. This then gives us 4x = 12. Solving for x we get x = 3. That's our X-intercept! Let's find the Y-intercept. Remember that when we are looking for an intercept, one value must always equal zero. Since we are looking for the Y-intercept our X value must equal zero. Therefore, 4(0) + 6y = 12. This gives us 6y = 12, and y = 2. That's our Y-intercept. Now let's graph the line. Our X-intercept is 3 so we plot a point at 3 on the positive side of the X-axis and our Y-intercept is 2 so we plot a point at 2 on the positive side of the Y-axis. Now connect the dots (remember to extend your line through these points).
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Your Turn! Graph 10x + 5y = 20 What is the X-intercept? (hint: where does it cross the X-axis?) ______ What is the Y-intercept?______ Did you plot the points correctly and remember to extend your line through the points?
Graph 15x + 5y = 30 What is the X-intercept? (hint: where does it cross the X-axis?) ______ What is the Y-intercept?______ Did you plot the points correctly and remember to extend your line through the points?
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Graph the following equations using the X and Y-intercepts. 3x + 3y = 9 What is your X-intercept?_______ What is your Y-Intercept?_______
7x – 2y = 14 What is your X-intercept?_______ What is your Y-Intercept?_______
-4x – 8y = 24 What is your X-intercept?_______ What is your Y-Intercept?_______
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Graphing Linear Inequalities Using the X and Y-Intercepts. Name______________________________________ Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph linear inequalities on the coordinate grid and correctly shade the solution area. When we are graphing linear inequalities that are written in the standard form, we follow the same format of using the X and Y-intercepts. There are several very important differences with the actual graph though. Let's take a look! 3x + 2y > 6 On the right is the graph of the above linear inequality. What do you notice about the graph?
4x – 12y ≥ 24 On the right is the graph of the above linear inequality. What do you notice about the graph? What is different between this graph and the first graph? Just like when we graphed numbers on a number line and had closed and open circles depending on the inequality used, the same rule applies here. Instead of circles though, now we have lines. When we are using the < and > inequalities, our graph is a dotted
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line. When we are using ≤ and ≥ inequalities, our graph is a solid line. The solid line indicates that we include the points on the line. Now you try it! -2x + 4y < 12 What is your X-intercept?________ What is your Y-intercept?________ Did you remember to use the correct line style? Let's go take a look at the following inequality. 3x + 2y ≥ 6 Notice that graph cuts the coordinate plane into two pieces. One above the graph and one below the graph. Remember that when we are dealing with inequalities, only certain values actually "work" in the inequality. These values that work, make the inequality true and those values that don't work make the inequality false. In order to find out which work and which don't, we pick test points. Test points are any points that lie above or below the graph. Let's pick the point (0,0) and find out if it works or not in the inequality. If we substitute (0,0) into the inequality we get 3(0) + 2(0) ≥ 6. Solving we get 0 + 0 ≥ 6, which gives us 0 ≥ 6. Obviously this is false and therefore (0,0) does not work in the inequality. Notice that (0,0) is located below our graph. Now let's try a point above our graph. How about (4,4) (it doesn't matter what point just so long as it is above the graph). If we substitute (4,4) into our inequality we get 3(4) + 2(4) ≥ 6. Solving we get 12 + 8 ≥ 6 and this gives us 20 ≥ 6. Obviously true. What do you think will happen if you pick any point below the graph? What do you think will happen if you pick any point above the graph?
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What do you think will happen if you pick any point on the graph? The correct representation of the previous inequality is graphed below. Wow, now some of the coordinate plane is shaded. Why do you think just the area above the graph is shaded? If you said because that is where all the true value are, you're correct. When we are graphing inequalities, you must shade the area of the coordinate plane that contains the ordered pairs that work in the inequality. Let's try one more together. How about -2x + 6y ≤ 18. Based on the inequality, what type of line will we have (solid or dotted)?___________ Start with the X and Y-intercepts. X-intercept is _______________________ Y-intercept is _______________________ Now let's plot those points and connect the dots with a _____________ line (solid or dotted). Pick a test point. I always like to use (0,0) when it is available. If we substitute (0,0) into our inequality, do we end up with a true or false answer? What side of the coordinate plane should we shade, above or below? Did you shade it?
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Now it's your turn all by yourself. 6x – 3y > 24 4x + 2y ≥ -8
5x -15y < 30
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Review: Writing Equations in Slope Intercept Form & Graphing Goal: This lesson is designed to let you practice rewriting equations in slope-intercept form in order to graph. There are other methods of graphing besides finding the x & y-intercepts. Sometimes it’s easier to graph linear equations when they are in slope-intercept form. Slope-intercept form is y = mx + b, where m is the slope, and b is the y-intercept.
The graph in the above picture crosses the X-axis at point (5,0) therefore we say the X-intercept is 5. It crosses the Y-axis at point (0,-4) and we say that the Y-intercept is -4. Now to figure out what the slope is, we start at the y-intercept, and end at the x-intercept. From the y-intercept, we rise 4, and we run 5. Therefore, our slope is 4/5. If we were to write an equation for the graph above, it would be y = 4/5x – 4. Ex. Take a look at 4x + 6y = 12. Let’s rewrite it in slope-intercept form ( y = mx + b) 18x + 6y = 12 Problem -18x -18x Subtract 18x from each side 6y = -18x + 12 Rewrite the equation so that 18x on the other side 6 6 6 Divide everything by 6 to isolate y y = –3x + 2 Slope-intercept form & ready to graph!
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To graph y = –3x + 2, 1) Start with the y-intercept (the value for b, which is 2) 2) Then from the y-intercept, rise & run to plot the slope (the value of m, which is –3)
Your Turn! Rewrite 9x + 3y = 18 in slope intercept form, then graph. What is the equation in slope-intercept form? _______________________ What is the Y-intercept? (hint: the value of b in your equation) ______ What is the slope? (hint: count the times you would rise and run to get the next ordered pair on the graph) ______
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Rewrite the following linear equations in slope-intercept form, then graph. 1) 6x + 3y = 9 What is the equation in slope-intercept form? _______________________ What is the Y-intercept? (hint: the value of b in your equation) ______ What is the slope? (hint: count the times you would rise and run to get the next ordered pair on the graph) ______ 2) 8x – 2y = 14 What is the equation in slope-intercept form? _______________________ What is the Y-intercept? ______ What is the slope? ______
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Graphing and Solving a System of Linear Inequalities on the Coordinate Grid. Name__________________________________________ Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graph a system of linear inequalities on the coordinate grid and correctly shade the solution area. In the previous lesson you learned how to graph and shade a linear inequality. Now you will learn how to graph and shade a system of linear inequalities. A system of linear inequalities is more than one linear inequality that will have common solutions. Through graphing and shading you will be able to discover the area that represents the common solutions. Let's get started! As mentioned before, a system is more than one inequality. Let's start with two linear inequalities. 3x + 9y > 27 4x -2y ≤ 12 Approach these inequalities one at a time. Find the X and Y-intercepts of the first problem. X-intercept_______ Y-intercept_______ Plot those points and draw your graph. Did you remember to use the correct type of line? Now pick a test point and shade the correct portion of the coordinate plane with the blue colored pencil. Time to graph the next inequality. Start with the intercepts. X-intercept_______________ Y-intercept_______________ Plot the points, draw your graph with the correct type of line and pick a test point that is either above or below the linear inequality you just graphed, not the one you graphed earlier. Shade the correct side with the yellow colored pencil. Are there any areas that contain both blue and yellow?
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In order to solve a system, the solution must satisfy each equation in the system. In our case, the inequalities must both be true. Let's pick a test point (how about (0,0)) and substitute it into both inequalities. The first inequality will give us 0 > 27 and the second will give us 0 ≤ 12. The first is false and the second is true. What do you notice about the shading at (0,0)? Now let's try (1,8). The first inequality gives us 75 > 27 and the second inequality gives us -12 ≤ 12. Both of these are true! What do you notice about the shading at (1,8)? What conclusions can you draw about shading and solutions to the system? Do you think anything changes if there are three, four, five or more inequalities in our system? If you answered no, you're right. Besides a little more work, nothing changes. Let's try a system with 4 inequalities. 7x + 5y < 35 8x – 7y ≤ 56 2x + 3y > -12 5x – 3y ≥ -15
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An archeologist has discovered a series of coordinates that may represent the location of buried treasure. After deciphering the coded messages, it seems that the only true coordinates of the treasures are those coordinates (ordered pairs) that lie within the solution of the systems of inequalities. Potential Treasures may be located at the coordinates designated on the map below. Above each point, write the ordered pair.
Note: There are a total of 8 coordinates that you will need to label. Each one will be worth ½ a point. Get all of them correct, and you will earn 4 points! Score: ________ / 4
Name ___________________________ Date ___________ Block _______ Score ___________ Percentage _____________ Grade _____________ Parent Signature _______________________________________________
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Transfer your coordinates from the previous page onto the graph below. The solution to the following system of inequalities is the location of the treasure. You will need to graph the inequalities on the provided grid. The treasure will be located at the point that lies within the solution of the system of inequalities. 2x + 3y < 18 -4x – 4y < 8 -3x + 4y < 12 4x – 5y < 20
Note: Each inequality graphed correctly will be worth 1 point (4 inequalities = 4 points). Be sure to use the correct type of line & shade the correct area. Score: ________ / 4 Identify the location of the treasure (the ordered pair that lies within the solution of the inequalities) and earn 2 points! Score: ________ / 2
