UNIT FOUR COORDINATE GEOMETRY MATH 421A 23 HOURS 71
UNIT 4: Coordinate Geometry Previous Knowledge With the implementation of APEF Mathematics at the Intermediate level, students should be able to: - Grade 7 - analyze graphs to determine a rate of change (slope) and base amount (y intercept) - Grade 8 and 9 - construct and analyze tables and graphs to describe how change in one quantity affects a related quantity - Grade 8 - link visual characteristics of slope with its numerical value - Grade 9 - represent patterns using a table of values, a graph and writing an equation describing the relationship - Grade 9 - given slope and y intercept, determine the equation of a line - Grade 9 - determine the equation of a line by obtaining its slope and y intercept from a graph - Grade 9 - sketch graphs using slopes and y intercepts Overview: - simplify radicals - operations with radicals - lengths of line segments - midpoints of line segments - slope - point-slope form and standard form of a linear equation - slope-y intercept form of a linear equation - parallel and perpendicular lines - graph linear equations using - any 2 points - the intercepts - the slope and y intercept 72
SCO: By the end of grade 10 students will: A7 demonstrate and apply an understanding of discrete and continuous number systems Elaboration - Instructional Strategies/Suggestions Real Number System Review number systems which students are familiar with (natural, whole, integer, rational, irrational and real). Design activities that have students discover properties of rational and irrational numbers. Explain how these sets are nested within each other. D7 apply the Pythagorean Theorem Pythagorean Theorem (p.2) Investigate the Pythagorean Theorem as a means of demonstrating the need for irrational numbers. Students can actually see lengths representing irrational numbers. 73
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Pythagorean Theorem (p.2) Activity/Manipulatives Mathpower has a good activity which can include the use of geoboards to demonstrate the need for irrational numbers. See Math 10 for problems relating to the Pythagorean Theorem. Find the length of a roof where the ratio of height to length is 4:12. (4 in 12 pitch) Pythagorean Theorem Mathpower 10 p.2 # 1, 2 Math 10 p.96-100 See an interactive applet at: http://www.ies.co.jp/math/java/ samples/pytha2.html Pencil/Paper Draw a line that is cm in length using centimetre grid paper. Once it is drawn as accurately as you can, measure it with a ruler. Then use a calculator to find the length to two places of decimals. Geoboard Assuming the distances between the pins on a geoboard is 1 unit, use an elastic to represent a length of units. 74
SCO: By the end of grade 10 students will Elaboration - Instructional Strategies/Suggestions Estimation of Square Roots (1.3) Estimation skills with radicals is a good indicator as to a student s understanding of square root values on a real number line. B2 develop algorithms and perform operations on irrational numbers Simplifying Square Roots (1.4) a) Entire to Mixed Radicals For square roots only convert radicals which are not perfect squares into simplest radical form. Determine the highest perfect square factor of the radicand. Note: Students should completely simplify answer C12 solve linear, simple radical, exponential, or absolute value equations or linear inequalities Simplifying square roots can also be looked at pictorially. Examining the diagrams below, one notices that each small square has an area of 2. The lengths of the sides of the three diagrams are: Look at this formula and the below diagrams, each small square has an area of 2: 75
Worthwhile Tasks for Instruction and/or Assessment Journal Ask students to explain, using Pythagorean Theorem, how they could draw a line segment which has the exact value of the square root of 13. Research Have students prepare a brief presentation, paper or poster about Pythagoras Radicals Pencil/Paper/Technology Evaluate with and without using a calculator. Suggested Resources http:/aleph0.clarku.edu/~djoyce /mathhist/chronology.html Estimation of Square Roots (1.3) Pencil/Paper Estimate the following: a) b) Estimation of Square Roots Mathpower 10 p.14 # 2-13 Simplifying Square Roots (1.4) Pencil/Paper Write each of the following in simplest form: Simplifying Square Roots Mathpower 10 p. 19#1-15 76
SCO: By the end of grade 10 students will be expected to: E20 develop and apply formulas for distance and midpoints Elaborations - Instructional Strategies/Suggestions Distance between two points (6.1) Allow time for student groups to explore methods of finding the length of a horizontal, vertical or diagonal line segment on a Cartesian Coordinate Plane. Enrichment - collinearity Invite students to construct a triangle using a compass and straight-edge with sides of 7cm, 9cm and 12cm 5cm, 8cm and 12cm 5cm, 7cm and 12cm Challenge students to see the trend here. The 3 points become collinear when AB + BC = AC. The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle is greater than the third side. The limiting case of this theorem is when the points become collinear and no longer form a triangle. Midpoint of a line segment (6.2) Invite students to work on problems on p.258 where midpoints of horizontal and vertical line segments are to be calculated. By studying p.259, students can understand the midpoint formula. Challenge students to attempt mathpower 10 p.261 #29. Students must research the term geometric median discuss it in their groups and apply that understanding to the problem. 77
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Distance between two points (6.1) Pencil/Paper/Activity Find the distance from one corner of your classroom to the corner diagonally opposite. Pencil/Paper How long must a cable be to go from the top of a 15m power pole to the ground at a point 5m from the base of the pole? (Assume the pole and ground meet at right angles). Distance between two points Mathpower 10 p.252 # 1-3 p.256 #1,3,5,11,13,14 16,20,24,25 Pencil/Paper/Presentation Find the distance from one corner of your classroom to the diagonally opposite corner of the room at the ceiling. Journal Write a few sentences explaining how three points can be shown to be collinear or not. Midpoint of a line segment (6.2) Midpoint of a line segment Mathpower 10 p.261#1,5,6,11,13,14 21, 27,29 a) Find the coordinates of the midpoints of AB and AC above. Label these points as D and E respectively. b) Find the lengths of DE and BC. How do they compare? Note to teachers: DE is known as a mid-segment. You may want to research this term and find out two properties of a mid-segment. 78
SCO: By the end of grade 10 students will be expected to: B24 analyze rates of change from graphs and calculate slope at various points B25 develop and apply a procedure to calculate the rate of change Elaborations - Instructional Strategies/Suggestions Slope (6.3) Student groups can brainstorm their ideas on the concept of slope. The main ideas to be brought out are: - the number represents steepness - the sign represents direction Any diagonal motion is broken down into 2 standard components( just as with vectors) - vertical or rise - horizontal or run the slope formula is: B27 interpret positive, negative and zero slopes other ways of expressing slope are: C58 demonstrate an understanding of the concept rate of change in a variety of situations Slope as a rate of change (6.4) In the real world the graphical axes are not x and y but represent real life quantities such as velocity, time, flow of electricity, productivity of farmlands, efficiencies of assembly lines, EEG s and many other quantities from numerous fields. Most fields of study do graphical analysis of data. Even your car is computerized and mechanics must be able to hook your engine up to a computer and analyze the graphs generated by the engine. One of the most important ways to analyze a graph is to investigate its slope or rate of change. 79
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Slope (6.3) Pencil/Paper Determine the slope of each side of the triangle below. Slope Mathpower 10 p.267 #1-4,9,11,13,18, 24,30,31,41,42 Slope as rate of change Slope as rate of change (6.4) Communication/Presentation Give three examples of rates of change that you have experienced in your everyday life. Mathpower 10 p.272 #5,6,8,9 Research Project/Pencil/Paper Use your research skills to determine the number of family farms on PEI in 1981, 1986, 1991, and 1996. Plot the data and draw a line of best fit. What is the rate of change(number of farms lost per year) If this rate of change continues, how many family farms will there be in 20 years time? Do you have any evidence in your community or family to support this data? Note to teachers: As a wrap-up this data could be graphed on the TI-83, then determine a regression line and get the slope using the TI-83. A discussion could ensue on the agreement between the students results and those on the TI-83. 80
SCO: By the end of grade 10 students will be expected to: C17 determine if a graph is linear by plotting in a given situation F9 determine and apply a line of best fit using linear regression with technology Elaborations - Instructional Strategies/Suggestions Point - Slope Linear Equation (6.5) Allow student groups time to read the section and discuss the point - slope formula and its applications. Hopefully, through collaboration, students will see that the objective is to get the equation of a straight line using the point - slope formula y! y 1 = m(x! x 1 ) when given the slope m and the coordinates of any point on the line (x 1,y 1 ) or given two ordered pairs on the line. The two most common ways of writing the answer are: 1) slope - y-intercept formula y = mx + b 2) Standard form Ax + By + C = 0 Where A, B, C, I A > 0 A and B are not both zero 81
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Point - Slope Linear Equation (6.5) Pencil/Paper/Technology Write an equation of the line that passes through the points (1,1) and (3,!3). Express the answer in standard form. Point - Slope Linear Equation Mathpower 10 p.28#1,4,5,10,11,18, 21,25,40,41,47, 49 Pencil/Paper/Technology Surface temperatures have warmed over the past century. For the following graph draw the line of best fit, determine two ordered pairs on the line, calculate the slope and get the equation of the line. Use that equation to predict what the surface air temperature will be in the year 2050. Transparency at end of unit A web-site for stock market graphs is below and a sample graph is at the end of the unit. http://www.imoney.com/bm_apps/ graphs/cgi-bin/stocks.egi amilies of lines F Families of lines Mathpower 10 p.285 #3,4 Journal Explain the two characteristics that determine families of lines. Group Activity Use the following graphs to get the equation for each of the lines shown. 82
SCO: By the end of grade 10 students will be expected to: C14 determine the slope and y-intercept of a line from a table of values or a graph Elaborations - Instructional Strategies/Suggestions Slope - y intercept Linear Equation (6.6) Challenge student groups to develop definitions for x and y intercepts. If the general coordinates of the y - intercept are represented as (0,b), invite student groups to modify the point - slope formula to obtain the slope - y intercept linear equation. y! y 1 = m(x! x 1 ) y! b = m(x! 0) y! b = mx C15 determine the equation of a line using the slope and y-intercept C54 sketch graphs from words and tables, and from real data collected in experiments Allow students to investigate the problems in Math Power 10 p.288 to discover the applications of the formula. They are: -given an equation - re-arrange in y = mx + b form to determine m and b. Then use these to draw the graph of the equation. - given the m and b values - write the equation of the line and re-arrange in standard form - draw the graph 83
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Slope - y intercept Linear Equation (6.6) Pencil/Paper/Technology If you climb a mountain the atmosphere gets thinner and the air temperature drops. Assuming the air temperature at sea level is 20 0 C, re-arrange the formula below in slope - y intercept form, determine the m and b values and use these values to draw the graph. Using the graph complete the following table. a + 150t = 3000 a = altitude in metres a is the independent variable t = temperature in 0 C t is the dependent variable Slope - y intercept Linear Equation Mathpower 10 p.288 #9-14,19,21,23, 24,28,30,31,35 40,42-44 Written assignment/communication Jane accepts a position where she gets a base salary of $300 and commission at the rate of 5% on all sales. Write an equation describing her pay, p, in relation to sales, s. Sketch the graph and describe what the $300 and the 5% actually represent on the graph. Communicate this to the rest of the members in your group or to the class as a whole. Group activity Write an equation for the graph below and describe a realworld scenario it could be picturing. Possible scenario could be: The x axis represents the price of a suit and the y axis the number of suits sold in a month at a certain store. 84
SCO: By the end of grade 10 students will be expected to: C14 determine the slope and y-intercept of a line from a table of values or a graph Elaborations - Instructional Strategies/Suggestions Parallel and Perpendicular lines (6.7) The activity in Math Power 10 p.291 is a simple way to draw parallel or perpendicular lines. A second method could be as follows: For the given equations, draw the graph, which lines appear to be parallel or perpendicular. Calculate the slope of each line and make a conjecture about the slopes of parallel and perpendicular lines. 1) 5x! 2y + 4 = 0 2) 2x! y + 1 = 0 3) x + 2y! 4 = 0 4) y = 2x + 3 5) 2x + 5y! 5 = 0 6) x + 2y! 2 = 0 85
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Parallel and Perpendicular lines (6.7) The activity in Math Power 10 p.291 is a simple way to draw parallel or perpendicular lines. In Junior High students would have learned about slides or translations. These are a type of transformation or mapping. Perhaps students could graph a line then slide it up or down by changing the y coordinates of the ordered pairs. Example: (1,1) and (4,3) yield a line with slope m = 2/3. If we map (x,y) ± (x,y!2) we get the ordered pairs (1,!1) and (4,1) yielding again a slope of m = 2/3. Parallel and Perpendicular lines Mathpower 10 p.294 #1,3,5,7,9,17,19 21, 27(a),(f),41 43-48,60,63 Note to teachers: An alternate activity could be to use the Corel Presentations Cartesian Coordinate template to duplicate ( click the line to select it, then Ctrl C and Ctrl V to copy it. Then click and drag the line, where ever you choose to place it, the line will be parallel to the first line drawn. A transformation called rotation rotates a figure. Using the Corel Presentations Cartesian Coordinate template students could draw a line segment. Use Ctrl C and Crtl V to copy the line and then select the second line segment and use edit rotate; to rotate the line 90 0 right click one of the corner handles, enter 90 0 and OK. Students can then get the slopes of the two lines and see the negative reciprocal relationship. 86
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Parallel and Perpendicular lines (cont d) (6.7) Pencil/Paper Find the slope of line segments AB and BC. How do they compare? Note to teachers: An angle inscribed in a semicircle is a right angle(ie. Sides are perpendicular) Parallel and Perpendicular lines Problem Solving Strategies Math Power 10 p.303 #5 Pencil/Paper Find the slope of line segments AD and BC; how do they compare? Then find the slopes of line segments AB and CD; compare them. Can you guess the name of this quadrilateral? Note: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. 87
SCO: By the end of grade 10 students will be expected to: C15 determine the equation of a line using the slope and y-intercept C16 graph by constructing a table of values, by using graphing technology and, when appropriate, by the slope y-intercept method C36 describe real world relationships depicted by graphs, tables of values and written descriptions Elaborations - Instructional Strategies/Suggestions Graphing linear equations (6.8) This section is a recap of work done in this and the previous unit. Various methods are reviewed here: Given the formula for a linear relationship ( equation of a line), graph; < using the x and y intercepts < using any two points < using the slope and y intercept Given linear data or a linear graph; determine the linear formula. Perhaps the problems could be real world examples of linear situations and not just problems with x and y. An experiment is conducted to examine rates of water loss in plant leaves. A student applied petroleum jelly to the underside of a plant and used a second plant as a control. Use the graph below to answer these questions: < which plant may have petroleum jelly on its underside < determine the equation of that line < what would the water loss be after 6 hours C54 sketch graphs from words and tables, and from real data collected in experiments Source: Biology Living Systems p.447 88
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Graphing linear equations (6.8) Pencil/Paper/Technology Graph the data below, get the line of best fit, determine the equation of the line and extrapolate to get the height of a seedling after 30 days. Graphing linear equations Mathpower 10 p. #35,36,40,41 Pencil/Paper/Technology Below is data on the effects of nicotine. The effects seem to indicate a linear relationship. Determine a formula for this relationship. Interpolate to find the heart rate for 1.5 drops and extrapolate for the heart rate for 6 drops of nicotine. Source: Biology Living Systems p.727 Internet Research/Journal Try to find data on the amount of nicotine in various products such as a pack of cigarettes. 89
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Art with a Graphing Calculator The design below consists of seven lines. Determine the equations of these lines using the screens below. Then check your work by graphing the lines on a TI-83 using the same window dimensions. 91