Mathematics Investigating Area Under a Curve About this Lesson This lesson is an introduction to areas bounded by functions and the x-axis on a given interval. Since the functions in the beginning of the lesson are linear, or piecewise linear, the enclosed regions form rectangles, triangles, or trapezoids. Within the lesson, the concept of accumulation is extended to approximating areas enclosed by non-linear functions and the x-axis on a given interval using rectangles and trapezoids. Prior to the lesson, students should be familiar with evaluating linear equations, as well as computing areas of plane figures. This lesson is included in Module Accumulation. Objectives Students will determine the area of a bounded region formed by linear functions. use rectangles to approximate the area of a bounded region formed by non-linear functions. Level Algebra Common Core State Standards for Mathematical Content This lesson addresses the following Common Core State 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. Explicitly addressed in this lesson Code Standard Level of Thinking G-GPE.7 Use coordinates to compute perimeters of Analyze polygons and areas of triangles and rectangles, e.g., using the distance formula. F-IF.7a Graph functions expressed symbolically and Analyze show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph linear and quadratic functions and show intercepts, maxima, and minima. Depth of Knowledge III III Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. i
Teacher Overview Investigating Area Under a Curve Code Standard Level of Thinking F-IF. Relate the domain of a function to its graph Apply and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Depth of Knowledge II 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. LTF 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. Implicitly addressed in this lesson Code Standard Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. ii
Teacher Overview Investigating Area Under a Curve LTF Content Progression Chart In the spirit of LTF s goal to connect mathematics across grade levels, the Content Progression Chart demonstrates how specific skills build and develop from sixth grade through pre-calculus. Each column, under a grade level or course heading, lists the concepts and skills that students in that grade or course should master. Each row illustrates how a specific skill is developed as students advance through their mathematics courses. th Grade Skills/Objectives Calculate and/or approximate area by counting squares or by using left-hand rectangles, righthand rectangles, or trapezoids in graphical contexts. (00_0.AC_E.0) Compare approximation methods and/or determine and justify whether an approximation is an overestimate or an underestimate. (00_0.AC_E.0) 7th Grade Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area by counting squares or by using left-hand rectangles, right-hand rectangles, or trapezoids in graphical contexts. (00_07.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_07.AC_E.0) Algebra Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using left-hand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_A.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_A.AC_E.0) Geometry Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using left-hand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_GE.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an underestimate, and/or interpret the meaning of accumulated area. (00_GE.AC_E.0) Algebra Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using left-hand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_A.AC_E.0) Compare approximation methods, determine and justify whether an approximation is Pre-Calculus Skills/Objectives Calculate total area by summing areas of connected geometric figures and/or approximate area using left-hand rectangles, righthand rectangles, midpoint rectangles, or trapezoids in verbal, graphical, numerical, and analytical contexts including rate graphs. (00_PC.AC_E.0) Compare approximation methods, determine and justify whether an approximation is an overestimate or an an overestimate or an underestimate, underestimate, and/or interpret the and/or interpret the meaning of meaning of accumulated area. accumulated area. (00_A.AC_E.0) (00_PC.AC_E.0) Connection to AP* AP Calculus Topic: Accumulation *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. Materials and Resources Student Activity pages Graphing calculators Assessments The following types of formative assessments are embedded in this lesson: Students engage in independent practice. Students apply knowledge to a new situation. The following additional assessments are located on the LTF website: Accumulation Algebra Free Response Questions Accumulation Algebra Multiple Choice Questions Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. iii
Teacher Overview Investigating Area Under a Curve Teaching Suggestions In question, all of the functions are linear, so the students should calculate the exact area of the bounded region. Demonstrate how to drop the vertical boundaries from the function to the x-axis at the endpoints of the interval, and then discuss using only vertical lines to divide the figure into triangles, rectangles, and/or trapezoids. Calculate the areas of the geometric figures then sum (accumulate) the values to determine the total area of the region. In questions and, students are asked to approximate the area of the region bounded by a curve and the x-axis on a given interval. Since the functions are curved, the exact area cannot be determined using only algebraic techniques. Explain that the vertical lines drawn from known points of the function to the x-axis will be used to create the sides of rectangles, These vertical line segments are not the same length; therefore, the students must determine where to locate the tops of the rectangles. One of two rules is generally applied in this lesson: () draw the top of the rectangle from the top of the vertical segment on the left (left-hand rectangles), or () draw the top of the rectangle from the top of the vertical segment on the right side of the rectangle. The directions in the questions tell the students to use left-hand rectangles or right-hand rectangles. Work question a with the students to demonstrate this concept. Students who are successful with questions and will also enjoy the following lessons: Applying Accumulation (Geometry) Adaptation of AP Calculus 00 AB (Geometry) Accumulation with a Quadratic Function (Algebra ) Accumulation (Pre-Calculus) Modality LTF 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 these representations to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding. G N P A V P Physical V Verbal A Analytical N Numerical G Graphical Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. iv
Answers. a. trapezoid: b b h A ()( ) square units b. trapezoid: b b h Teacher Overview Investigating Area Under a Curve A ()( ) 8 square units 9 c. triangle height = () units 9 7 A () square units 8 d. trapezoid: b ( ) b () h A () 9 square units e. rectangle: b h trapezoid: b b h A ()() ()( ) 7 square units f. rectangle: b h trapezoid: b b h A ()() ()( ) square units g. trapezoid(): b 0 b h rectangle: b h trapezoid(): b b 8 h A ()( ) ()() ()( ) 0 square units h. trapezoid: b 0 b h rectangle: b h triangle: b h A ()( ) ()() ()() square units Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. v
Teacher Overview Investigating Area Under a Curve. a. i. Left-hand rectangles (0)(0) (0)(0) (0)(0) (0)() (0)() 700 square units ii. Right-hand rectangles (0)(0) (0)(0) (0)() (0)() (0)(70) 0 square units iii. Average 700 0 square units The average is a better approximation because the average gives the sum of areas of trapezoids whose upper edges are closer to the graph of f than the upper edges of the rectangles. b. i. Left-hand rectangles (0)(0) (0)(00) (0)(80) (0)(0) 8800 square units ii. Right-hand rectangles (0)(00) (0)(80) (0)(0) (0)(0) 00 square units iii. Average 8800 00 70 square units The average is a better approximation because the average gives the sum of areas of trapezoids whose upper edges are closer to the graph of f than the upper edges of the rectangles. c. i. Left-hand rectangles ()(0) (0)() (0)() (0)(0) (0)() ()() 7 square units ii. Right-hand rectangles ()() (0)() (0)(0) (0)() (0)() ()(0) 7 square units iii. Average 7 7 7 square units Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. vi
Teacher Overview Investigating Area Under a Curve. a. 0 0 0-0 - b. x f(x) 0 9 0 c. ()() ()(9) ()() 9 square units d. ()(9) ()() ()(0) 90 square units e. 9 90 9. square units. Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org. vii
Mathematics Investigating Area Under a Curve. For the given interval, determine the area between the graph of the given function and the x-axis: a. Interval: x b. Interval: x Function: y x Function: y x - - 0 - - - 0 - - - c. Interval: x d. Interval: x Function: y x Function: y x - - 0 - - - 0 - - - Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org.
Student Activity Investigating Area Under a Curve e. Interval: 0 x f. Interval: x Function:, x f( x) x, x Function:, x f( x) x, x - - 0 - - - 0 - - - g. Interval: 0 x h. Interval: 0 x 8 Function: x, x f( x), x 8 x, x x, x Function: f( x), x 8 x, x - - - 0 7 8 9 - - - - - 0 7 8 9 - - Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org.
Student Activity Investigating Area Under a Curve. Approximate the area between the function and the x-axis for the interval shown on the graph and in the table: i. by calculating the sum of left-hand rectangles with widths determined by the data in the table. ii. by calculating the sum of right-hand rectangles with widths determined by the data in the table. iii. by averaging the sums of the left-hand and right-hand rectangles and explaining why this may be a better approximation for the actual area under the curve. a. 80 70 0 0 0 0 0 0-0 0 0 0 0 0 0 0 70 80 90-0 x f(x) 0 0 0 0 0 0 0 70 90 70 b. 0 0 00 80 0 0 0-0 0 0 0 0 0 0 0 70 80 90 00 x f(x) 0 0 0 00 0 80 80 0 00 0 Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org.
Student Activity Investigating Area Under a Curve c. 0 0 0 0 0 0-0 - 0 0 0 0 0 0 0 70-0 x f(x) 0 0 0 70 0. a. Use a graphing calculator to sketch the graph of the function 0 x. y x on the interval 0 0 0-0 - b. Complete the table of values for the function. x f(x) 0 Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org.
Student Activity Investigating Area Under a Curve c. Use the data in the table to approximate the area under the graph of the function by calculating the sum of left-hand rectangles with widths indicated by the data table. d. Use the data in the table to approximate the area under the graph of the function by calculating the sum of the right-hand rectangles with widths indicated by the data table. e. Calculate the average of your answers to parts (c) and (d). Copyright 0 Laying the Foundation, Inc., Dallas, T. All rights reserved. Visit us online at www.ltftraining.org.