This lesson gives students practice in graphing

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1 NATIONAL MATH + SCIENCE INITIATIVE Mathematics 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 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 b the darkened points of the star to introduce, eplore, and reinforce mathematical concepts and to enhance conceptual understanding. Solving Sstems of Linear Equations ABOUT THIS LESSON This lesson gives students practice in graphing linear equations; however, the primar purpose is to define regions in a coordinate plane formed b the intersection of linear equations and to determine the area of a bounded region. Students will build on the work the did in middle grades plotting points to determine bounded regions as the graph linear equations on restricted domains and identif the intersections of sstems of linear equations to determine areas. This lesson focuses on solving sstems of linear equations. At the same time the lesson reinforces the use of area formulas and enhances student understanding of this standard b developing coherence and connections among a variet of mathematical concepts, skills, and practices. OBJECTIVES Students will solve sstems of linear equations graphicall. determine the area of a polgon formed b the intersection of linear equations. G N P A V P Phsical V Verbal A Analtical N Numerical G Graphical Copright 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 appl each of these standards rather than providing the initial introduction to the specific skill. The star smbol ( ) 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: Analze and solve pairs of simultaneous linear equations. (a) Understand that solutions to a sstem of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisf both equations simultaneousl. (b) Solve sstems of two linear equations in two variables algebraicall, and estimate solutions b graphing the equations. Solve simple cases b inspection. For eample, + = 5 and + = have no solution because + cannot simultaneousl 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 polgons b composing into rectangles or decomposing into triangles and other shapes; appl these techniques in the contet of solving real-world and mathematical problems. See questions 1-.G.: Draw polgons 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. Appl these techniques in the contet of solving real-world and mathematical problems scales. See question 9 Targeted Standards (if used in Algebra 1) A-REL.: Solve sstems of linear equations eactl and approimatel (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 polgons 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 Copright 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 variet 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, identif the planar region formed, use a variet of algebraic techniques to determine the dimensions necessar 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 auiliar line to determine the height of the triangle. Foundational Skills The following skills la 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 Activit pages Graph paper Straight edges Applet to draw regions and calculate area between curves: areabetweencurves.html Copright 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 the cannot just guess the intercepts or intersection points. Questions 1,, and 7 require students to calculate the intercepts to obtain the dimensions of the triangles. For questions,, and, students ma 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 variet of solutions. This is an ecellent opportunit to let students investigate mathematics b asking them to find as man 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 b 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 onl 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 algebraicall b writing the equations of the lines and finding their point of intersection or b using similar triangles to set up a proportion to solve for the height of the triangular region. You ma wish to support this activit with TI- Nspire technolog. See Graphing a Function and Displaing 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 -ais. iv Copright 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 sith 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. th Grade Skills/Objectives Given 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. Given or coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or clinder formed b revolving the bounded region about either of the lines. 7th Grade Skills/Objectives Given 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. Given or coordinate points that form a triangle or a rectangle with one side on a horizontal or vertical line, calculate the surface area and/or volume of the cone or clinder formed b revolving the bounded region about either of the lines. Algebra 1 Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, or composite of these three figures formed b linear equations and/ or determine the equations of the lines that bound the figure. Given the equations of lines (at least one of which is horizontal or vertical. that bound a triangular, rectangular, or trapezoidal region, calculate the surface area and/or volume of the solid formed b revolving the region about the line that is horizontal or vertical. Geometr Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed b linear equations or equations of circles and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed b revolving the region about a horizontal or vertical line. Algebra Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed b linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed b revolving the region about a horizontal or vertical line. Pre-Calculus Skills/Objectives Calculate the area of a triangle, rectangle, trapezoid, circle, or composite of these figures formed b linear equations, linear inequalities, or conic equations and/or determine the equations of the lines and circles that bound the figure. Given the equations of lines or circles or a set of inequalities that bound a triangular, rectangular, trapezoidal, or circular region, calculate the volume and/or surface area of the solid formed b revolving the region about a horizontal or vertical line. Copright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at v

NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Sstems of Linear Equations ANSWERS 1. 9 units. units..1 units 1 5 7 9 1 5 7 9 (9/5, 0) 1. 5 units 5. 1 units. 1 units 0 7. 1 1 1 5 7 9 9.

7 NATIONAL MATH + SCIENCE INITIATIVE Mathematics Solving Sstems of Linear Equations ANSWERS 1. 9 units. units..1 units (9/5, 0) 1. 5 units 5. 1 units. 1 units units. units (0,.5) (5, 17/) 9. (1.5, 0) = ( ) intersects = at the point, The area is square units. 1 Alternativel, use similar triangles to set up the proportion: = h h ; h = 0 1 The area = 1 0 ( 1 )= 1 1 units. Copright 01 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at vii

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

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

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

This lesson gives students practice in graphing

This lesson gives students practice in graphing NATIONAL MATH + SCIENCE INITIATIVE 9 Mathematics Solving Systems of Linear Equations 7 5 3 1 1 3 5 7 LEVEL Grade, Algebra 1, or Math 1 in a unit on solving systems of equations MODULE/CONNECTION TO AP*