This lesson gives students practice in graphing

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1 NATIONAL MATH + SCIENCE INITIATIVE 9 Mathematics Solving Systems of Linear Equations LEVEL Grade, 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 Eamination Board. The College Board was not involved in the production of this product. MODALITY NMSI 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, eplore, and reinforce mathematical concepts and to enhance conceptual understanding. ABOUT 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. OBJECTIVES Students will solve systems of linear equations graphically. determine the area of a polygon formed by the intersection of linear equations. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at i

2 COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This 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 ).EE.a-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 eample, 3 + y = 5 and 3 + y = have no solution because 3 + y cannot simultaneously be 5 and. See questions 1-9 Reinforced/Applied Standards (if used in Grade ).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 contet of solving real-world and mathematical problems. See questions 1-.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 contet of solving real-world and mathematical problems scales. See question 9 Targeted Standards (if used in Algebra 1) A-REI.: Solve systems of linear equations eactly and approimately (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- A-CED.: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. See question 9 ii Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at

3 COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These 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 auiliary line to determine the height of the triangle. FOUNDATIONAL SKILLS The following skills lay the foundation for concepts included in this lesson: Graph linear equations Calculate areas of rectangles, triangles, and trapezoids ASSESSMENTS The 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: areabetweencurves.html Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at iii

4 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,, and, students may calculate the area of the rectangle in which each trapezoid is enclosed and then subtract the non-shaded triangle. Question requires determining the intersection of two oblique lines. Question 5 has a variety of solutions. This is an ecellent 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 by 9 rectangle which encloses the shaded triangle, that is, 5 ( ). 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,),and(,7) from the obtuse triangle with vertices (0,0),(0,),and (, 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:,, Turn the paper a quarter turn to the right so that the trapezoid is sitting on one of its bases. Turn the paper a quarter turn to the left in order consider the base of the triangle to be on the y-ais. iv Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at

5 NMSI CONTENT PROGRESSION CHART In the spirit of NMSI s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades, 7,, and Algebra 1 are compacted into three courses. Grade includes all of the Grade content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade, and Algebra 1 includes the remainder of the content from Grade and all of the Algebra 1 content. The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence. 3rd Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with whole number side lengths by counting unit squares or multiplying side lengths. th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures where one side is a fraction and one side is a whole number or a fraction. (Computations involving addition and subtraction of fractions are limited to like denominators.) 5th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles and rectilinear figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles on the coordinate plane.) th Grade Skills/ In a problemsolving situation with a real-world application, determine the area of rectangles, triangles, and composite figures with fractional and decimal side lengths. (These situations can include calculating area of rectangles and triangles on the coordinate plane.) 7th Grade Skills/ Given 3 or coordinate points that form a triangle or a rectangle with one side on a horizontal line and one side on a vertical line, calculate the area of the figure. Algebra 1 Skills/ Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed by linear equations and/ or determine the equations of the lines that bound the figure. Geometry Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Algebra Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Pre-Calculus Skills/ Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed by linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. AP Calculus Skills/ Calculate the area between curves. Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at v

7 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Systems of Linear Equations ANSWERS 1. y y 3. Determine the -intercept. y (9/5, 0) 1 3 Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at vii

. y 0 1 1 1 3 5 7 9 5. Multiple solutions are possible, two are listed.

8 . y Multiple solutions are possible, two are listed. Solution 1: Area of the bounded region, A B, is can be determined from the difference of obtuse triangles, A 1 and A where the larger triangle, A 1, has (on the y-ais) and h = units. The smaller triangle, A has (same base) and Solution : Right triangle (0, 0);(0, 9);(,9) Triangle (0, );(0, 9);(,9) Triangle (0, 0);(0, );(,7). A = 1 bh; base on the y-ais A = 1 ( 7units )( units) A = 1units y viii Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at

9 7. ( 0)+ y = 5; y-intercept is 5 + ( 0)= 5;-intercept is 5 A = 1 bh A = 1 5 units 5 units A = 5 1 units y (0,.5) (1.5, 0) 1 1. y (5, 17/3) 9. Algebraically: Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at i

11 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Systems of Linear Equations For questions 1, graph and shade the region then calculate the area of the region. Show the work that leads to your answers in questions What is the area of the region in the first quadrant that is below the graph of f( ) = ( 3)?. What is the area of the region enclosed by the graphs of y = 0, = 0, =, and 1 y = ( 3) +? 3 3. What is the area of the region in the first quadrant that is below the graph of y = 5( 1) +? Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 1

12 . Let R be the region in the first quadrant under the graph of y = for 9. What is the area of R? 5. What is the area of the region bounded by the graphs of y = +, y =, and y =?. What is the area of the region enclosed by the graphs 1 of = 0, y = + 1, and y = ( 3) +? 3 7. What is the area of the region in the first quadrant under the graph of + y = 5? Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at

13 . What is the area of the region enclosed by the graphs of y = 0, y = + 9 for 0 5? 3 9. What is the area of the region R bounded by line m, line p, and the -ais as shown in the graph? Copyright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 3

This lesson gives students practice in graphing

This lesson gives students practice in graphing NATIONAL MATH + SCIENCE INITIATIVE Mathematics 9 7 5 1 1 5 7 LEVEL Grade, Algebra 1, or Math 1 in a unit on solving sstems of equations MODULE/CONNECTION TO AP* Areas and Volumes *Advanced Placement and