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iCopyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

MathematicsNATIONALMATH + SCIENCEINITIATIVE

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LEVELGrade 8, Algebra 1, or Math 1 in a unit on solving systems of equations

MODULE/CONNECTION TO AP*Areas and Volumes*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

MODALITYNMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding.

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P – Physical V – VerbalA – AnalyticalN – NumericalG – Graphical

Solving Systems of Linear EquationsABOUT THIS LESSON

This lesson gives students practice in graphing linear equations; however, the primary purpose is to define regions in a coordinate

plane formed by the intersection of linear equations and to determine the area of a bounded region. Students will build on the work they did in middle grades plotting points to determine bounded regions as they graph linear equations on restricted domains and identify the intersections of systems of linear equations to determine areas.

This lesson focuses on solving systems of linear equations. At the same time the lesson reinforces the use of area formulas and enhances student understanding of this standard by developing coherence and connections among a variety of mathematical concepts, skills, and practices.

OBJECTIVESStudents will

● solve systems of linear equations graphically.● determine the area of a polygon formed by

the intersection of linear equations.

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Mathematics—Solving Systems of Linear Equations

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol (★) at the end of a specific standard indicates that the high school standard is connected to modeling.

Targeted Standards (if used in Grade 8)8.EE.8a-b: Analyze and solve pairs of simultaneous

linear equations. (a) Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (b) Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. See questions 1-9

Reinforced/Applied Standards (if used in Grade 8)6.G.1: Find area of right triangles, other

triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

See questions 1-8

6.G.3: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems scales. See question 9

Targeted Standards (if used in Algebra 1)A-REI.6: Solve systems of linear equations

exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. See questions 1-9

Reinforced/Applied Standards (if used in Algebra 1)G-GPE.7: Use coordinates to compute perimeters

of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★

See questions 1-8

A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ★

See question 9

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COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.1: Make sense of problems and persevere in solving them. Students graph equations, identify the planar region formed, use a variety of algebraic techniques to determine the dimensions necessary to calculate the area of the region, and determine the area.

MP.7: Look for and make use of structure. In question 5, students draw an auxiliary line to determine the height of the triangle.

FOUNDATIONAL SKILLSThe following skills lay the foundation for concepts included in this lesson:

● Graph linear equations● Calculate areas of rectangles, triangles,

and trapezoids

ASSESSMENTSThe following formative assessment is embedded in this lesson:

● Students engage in independent practice.

The following additional assessments are located on our website:

● Areas and Volumes – Algebra 1 Free Response Questions

● Areas and Volumes – Algebra 1 Multiple Choice Questions

MATERIALS AND RESOURCES● Student Activity pages● Graph paper● Straight edges● Applet to draw regions and calculate area

between curves:http://www.math.psu.edu/dlittle/java/calculus/areaBetweenCurves.html

http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

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TEACHING SUGGESTIONS

This lesson is richer if class time is allowed for a discussion of the various techniques that students use to determine the areas.

Emphasize to students that they cannot just guess the intercepts or intersection points. Questions 1, 3, and 7 require students to calculate the intercepts to obtain the dimensions of the triangles. For questions 2, 4, and 8, students may calculate the area of the rectangle in which each trapezoid is enclosed and then subtract the non-shaded triangle. Question 6 requires determining the intersection of two oblique lines. Question 5 has a variety of solutions. This is an excellent opportunity to let students investigate mathematics by asking them to find as many different methods to determine the area as possible. Some students “see” a large triangle and subtract out a trapezoid and a right triangle. Others subtract each of three non-shaded triangles from the 6 by 9 rectangle which encloses the shaded triangle, that is, 54 (27 9 6)− + + . Consider challenging students to calculate the shaded area using the smallest number of geometric figures. A solution using only two figures involves subtracting the area of the obtuse triangle with vertices (0, 0), (0, 6), and (2, 7) from the obtuse triangle with vertices (0, 0), (0, 6), and(6, 9) . Question 9 can be solved algebraically by writing the equations of the lines and finding their point of intersection or by using similar triangles to set up a proportion to solve for the height of the triangular region.

You may wish to support this activity with TI-Nspire™ technology. See Graphing a Function and Displaying a Table, Graphing Piecewise Functions, and Adjusting the Window in the NMSI TI-Nspire Skill Builders.

Suggested modifications for additional scaffolding include the following:2, 4, 8 Turn the paper a quarter turn to the right

so that the trapezoid is sitting on one of its bases.

6 Turn the paper a quarter turn to the left in order consider the base of the triangle to be on the y-axis.

Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

Solving Systems of Linear Equations

ANSWERS

1.

2.

3. Determine the x-intercept.

1 2 3 4 5 6 7 8 9 10

2

4

6

8

10

x

y

1 2 3 4 5 6 7 8 9 10

2

4

6

8

10

x

y

1 2 3

2

4

6

8

10

x

y

(9/5, 0)

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4.

5. Multiple solutions are possible, two are listed. Solution 1: Area of the bounded region, AB , is can be determined

from the difference of 2 obtuse triangles, A1 and A2

where the larger triangle, A1 , has (on the

y-axis) and h2 = 6units . The smaller triangle, A2 has

(same base) and

Solution 2: Right triangle (0, 0);(0, 9);(6,9) – Triangle (0, 6);(0, 9);(6,9) – Triangle (0, 0);(0, 6);(2,7)

6.

A= 12bh; base on the y-axis

A= 12

7units( ) 6units( )A= 21units2

1 2 3 4 5 6 7 8 9 10

4

8

12

16

20

x

y

10

8

6

4

2

1 2 3 4 5 6 7 8

2

4

6

8

10

x

y

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7.

2 0( )+4y= 25; y-intercept is 254

2x+4 0( )= 25;x-intercept is 252

A= 12bh

A= 12

254

units⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

252

units⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

A=62516

units2

8.

9. Algebraically:

1 2 3 4 5 6 7 8 9 10

2

4

6

8

10

x

y

(5, 17/3)

2 4 6 8 10 12 14

2

4

6

8

10

x

y

(0, 6.25)

(12.5, 0)

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Mathematics NATIONALMATH + SCIENCEINITIATIVE

Solving Systems of Linear Equations

For questions 1 – 8, graph and shade the region then calculate the area of the region. Show the work that leads to your answers in questions 1 – 9.

1. What is the area of the region in the first quadrant that is below the graph of ( ) 2( 3)f x x= − − ?

2. What is the area of the region enclosed by the graphs

of y = 0, x = 0, x = 6, and 1 ( 3) 43

y x= − + ?

3. What is the area of the region in the first quadrant that is below the graph of 5( 1) 4y x= − − + ?

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4. Let R be the region in the first quadrant under the graph of 2y x= for 4 9x≤ ≤ . What is the area of R?

5. What is the area of the region bounded by the graphs

of 1 62

y x= + , 72

y x= , and 32

y x= ?


of x = 0, 1 12

y x= + , and 2 ( 3) 63

y x= − − + ?

7. What is the area of the region in the first quadrant under the graph of 2x + 4y = 25?

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of y = 0, 2 93

y x= − + for 0 5x≤ ≤ ?

9. What is the area of the region R bounded by line m, line p, and the x-axis as shown in the graph?

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