Math 7 Notes - Unit 8B (Chapter B) Proportions in Geometr Sllabus Objective: (6.23) The student will use the coordinate plane to represent slope, midpoint and distance. Nevada State Standards (NSS) limits finding distance with horizontal and vertical lines onl at this grade level. Since students have NOT been taught the Pthagorean Theorem at this point, the will onl be epected to find the distance of horizontal and vertical segments. This standard also states that the distance will be shown on a coordinate plane. So teach students to correctl count the boes or horizontal or vertical grid lines from one point to another. This skill will make finding midpoints easier if students can do this mentall. Remind students that the distance will be a positive value. Eample: Find the distance of the given segment below. Students should be taught to count the boes/grid lines from one point to the other. It does not change the answer if ou count left to right or right to left. Answer: 7 units - - Eample: Find the distance of the given segment below. Students should be taught to count the boes/grid lines from one point to the other. The answer will be the same if ou count from bottom to top or top to bottom. Answer: 7 units - - Math 7, Unit 8B NOTES Holt: Chapter B Page 1 of 18 Revised 213-CCSS
Midpoint Students ma be given a graph with a segment or two ordered pairs and asked to find the midpoint. The segment ma be a horizontal, vertical or an oblique segment. Be sure students understand that the midpoint is the average of the s for the -coordinate, and the average of the s for the -coordinate. 1 2 1 2 The formula is +, + 2 2 Eample: Find the midpoint of a horizontal segment. Using mental math Students could find the length of this segment b counting the boes (the distance). 8 Then the could divide that number b 2. 8 2 = 4 Add that to the smaller coordinate -3 + 4 =1 The coordinate is not changing so = 3 Answer (1, 3) OR - - Using the formula, 3+ 3+ 3 2 6, =, = 1, 3 2 2 2 2 ( ) Eample: Find the midpoint of a vertical segment. Using mental math Students could find the length of this segment b counting the boes (the distance). Then the could divide that number b 2. 2 = 2. Add that to the smaller coordinate -1 + 2. =1. The coordinate is not changing so = 2 Answer (-2, 1.) - - OR Using the formula, 2+ 2 1+ 4 4 3, =, = 2, 1. 2 2 2 2 ( ) Math 7, Unit 8B NOTES Holt: Chapter B Page 2 of 18 Revised 213-CCSS
Eample: Find the midpoint of a segment that is not vertical and not horizontal. Using mental math Students could find the length of this segment b counting the boes but this time students must count both the rise and the run. Up 2, Right 6 Then the could divide b 2. Up 1, Right 3 Starting with the far left point and moving up 1 and right 3, Answer (1, 2) OR - - Using the formula, 2+ 4 1+ 3 2 4, =, = 1, 2 2 2 2 2 ( ) Slope The idea of slope is used quite often in our lives. However outside of school, it goes b different names. People involved in home construction might talk about the pitch of a roof. Sometimes on mountain roads and highwas we see signs warning truck drivers about steep hills and long inclines/declines. Man times the signs use a percent to show how steep (% would be flat and 1% would be a cliff) and an illustration to whether it s steep going up or down. If ou were riding in our car, ou might have seen a sign on the road indicating a grade of 6% up or down a hill. Both of those cases refer to what we call slope in mathematics. Kids use slope on a regular basis without realizing it. We covered these ideas earlier in the school ear in Unit but this is a good reminder and great linkage. A student bus a cold drink for $.. If two cold drinks were purchased, the student would have to pa $1.. I could describe that mathematicall b using ordered pairs: (1, $.), (2, $1.), (3, $1.), and so on. The first element in the ordered pair represents the number of cold drinks; the second number represents the cost of those drinks. Eas enough, don t ou think? Now if I asked the student how much more was charged for each additional cold drink, hopefull the student would answer $.. So the difference in cost from one cold drink to adding another is $.. The cost would change b $. for each additional cold drink. The change in price for each additional cold drink is $.. Another wa to sa that is the rate of change is $.. We call the rate of change slope. In math, the rate of change is called the slope and is often described b the ratio rise run. Math 7, Unit 8B NOTES Holt: Chapter B Page 3 of 18 Revised 213-CCSS
The rise represents the change (difference) in the vertical values (the s), the run represents the change in the horizontal values, (the s). Mathematicall, we write m = 2 1 2 1 Let s look at an two of those ordered pairs from buing cold drinks: (1, $.) and (3, $1.). Find the slope. Substituting in the formula, we have: 1.. m = 3 1 1. = 2 =. We find the slope is $.. The rate of change per drink is $.. At this level students should begin to identif if the slope is positive (line on the graph goes uphill from left to right), negative (line on the graph goes downhill from right to left), zero or undefined and state the slope of a line on a graph. Eamples: Determine if the slope is positive, negative, zero or undefined. 1 - An line that rises or goes uphill from left to right has positive slope -1-1 - 1 Math 7, Unit 8B NOTES Holt: Chapter B Page 4 of 18 Revised 213-CCSS
1 - An line that falls or goes downhill from left to right has negative slope -1-1 - 1 1 A horizontal line has zero slope. - -1-1 - 1 1 - A vertical line has undefined slope sometimes called no slope. -1-1 - 1 Math 7, Unit 8B NOTES Holt: Chapter B Page of 18 Revised 213-CCSS
Knowing that slope is rate of change has some implications for the graphs. We will see that the slope of a line must be constant rate of change so as we look at the graph it must be a line. If we have a graph of a variable rate of change, the graph is not a line. Eamples: Tell whether each graph shows a constant or variable rate of change. - - - - - - variable constant variable Eample: Find the slope of the line that connects the points on the graph. In this case, I will start at the point furthest to the left in Quadrant II, ( 4, ). To get from this point to the point on the right in Quadrant IV (3, 2), I need to go down 7, ( 7) and go to the right 7, (+7). - 7 So the slope is = 1. 7 - Note that we could have started with the point furthest to the right and counted up 7 units (+7) and then to the left 7 units ( 7) to arrive at the same value for the slope: 7 7 = 1. Math 7, Unit 8B NOTES Holt: Chapter B Page 6 of 18 Revised 213-CCSS
Eample: Find the slope of the line that connects the points in the graph. 8 4 up 4 over 4-4 -8-8 -4 4 8 The first point is (2, 1), the second point is (6, ). How man units did I go up? Count them, up 4. Then we went over 4 to the right. So the slope is 4/4 = 1. Eample: Find the slope of the line that connects the points ( 2, 1) and (4, 2). - - Slope = up 3 + 3 1, which is =. right 6 + 6 2 Math 7, Unit 8B NOTES Holt: Chapter B Page 7 of 18 Revised 213-CCSS
Eamples: Find the slope of the line that connects the points on the graph. 1 In this case, I will start at the point furthest to the left in Quadrant II, ( 3, ). To get to the point on the right on the ais (, 6). I need to rise 1, and - go right 3, so the slope is 1/3 = 1 3-1 -1-1 8 4-4 In this case, I will start at the point furthest to the left in Quadrant II, ( 4, ). To get to the point on the right in Quadrant IV (3, 2), I need to go down 7, ( 7), and go over 7, so the 7 slope is = 1. 7-8 -8-4 4 8 Math 7, Unit 8B NOTES Holt: Chapter B Page 8 of 18 Revised 213-CCSS
Eample: Find the slope of the line. 1 - -1-1 - 1 In this case no points are identified. Students must find two points that intersect the horizontal and vertical grid lines. Then count the rise and the run. Using the points (-,) and (,3) I count a rise of 3 and a run of. So the slope is 3. 1 - -1-1 - 1 Again no points are identified. Students must find an two points that intersect the horizontal and vertical grid lines. Then count the rise and the run. 2,, 4, I Using the points ( ) and ( ) count a fall of 4, ( 4) and a run of 2. 4 So the slope is = 2. 2 Eample: Use the given slope and point to graph each line. 2; (3, 4) 1/4; (3, 1) Begin at the point (3, 4). Begin at the point (3,1). 2 Since slope is 2 =, 1 1 Since slope is, go up 1 unit 4 go down 2 units and right 1. and right 4 units. Repeat for several Repeat for several points. points. Also show down 1 unit and left 4 units because 1 1 =. 4 4 Math 7, Unit 8B NOTES Holt: Chapter B Page 9 of 18 Revised 213-CCSS
1 1 - - -1-1 - 1-1 -1-1 Sllabus Objective: (6.17) The student will solve problems involving congruence and similarit using simple ratios and proportions. Applications Proportions Congruent and Similar Polgons Congruent Polgons have the eact same shape and size. Here are several eamples: Triangles ABC and XYZ shown below are congruent. B Y Z A C X We could write that as ABC XYZ. The smbol is read is congruent to. If we could lift ABC and place it on top of XYZ, A would fall on X, B on Y, and C on Z. Math 7, Unit 8B NOTES Holt: Chapter B Page 1 of 18 Revised 213-CCSS
These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles, and the sides joining corresponding vertices are corresponding sides. Polgons are congruent if these two statements are true: 1. Corresponding angles of congruent figures are congruent AND 2. Corresponding sides of congruent figures are congruent. When we name two congruent figures we list corresponding vertices in the same order. Thus when we see ABC XYZ or CAB ZXY, we know that: A X, B Y, C Z AB XY, BC YZ, CA ZX Knowing corresponding parts can help us to determine missing measurements in congruent figures. Eample: Find the values of and, given: ABCD FGHJ A D B cm 11 C G H 7 2 cm J F Because the figures are congruent, the corresponding angles are congruent and the corresponding sides are congruent. So, B G, so m B = m G = 7 ; = 7 BC GH, so BC = GH = cm; = cm Similar Polgons have the same shape but not necessaril the same size. is similar to is similar to Math 7, Unit 8B NOTES Holt: Chapter B Page 11 of 18 Revised 213-CCSS
With the figures below, we could write ABCDEFG is similar to RSTUVWX as ABCDEFG RSTUVWX. The notation means is read is similar to. R A B C F G D E X W T S V U Corresponding parts of polgons are in the same relative position. For instance, in the similar figures below, ABCD EFGH. A B G H D C F E AB corresponds to EF BC corresponds to FG CD corresponds to GH AD corresponds to EH Two polgons are similar if: A corresponds to E B corresponds to F C corresponds to G D corresponds to H a) the measure of their corresponding angles are equal, AND b) the ratio of the lengths of their corresponding sides are proportional The notation means is similar to. For instance, in the above eample we can write ABCD EFGH which we would read, rectangle ABCD is similar to rectangle EFGH. Remember, the ratios of the lengths of corresponding sides of similar figures are equal. This propert can be used to help us determine missing lengths in similar figures. Eample: Find the value of given these two rectangles are similar. 4 in Math 7, Unit 8B NOTES Holt: Chapter B Page 12 of 18 1 in Revised 213-CCSS in
Since the two rectangles are similar, their sides must be in proportion. That is, the left side is to the bottom of the first rectangle, as the left side is to the bottom of the second rectangle. Another wa of saing that is the width is to the length as the width is to the length. 2 4 = or 1 4 = 1 = 4 1 1 = 2 2 = 2 = 2 Sometimes two polgons are contained in one figure. When that occurs, it might be helpful to redraw the two polgons so the don t overlap so ou can see the corresponding parts more readil. Eample: Given ABC ~ DEC, find DE. A D 2 B 12 E 36 C Taking DEC out of the ABC, we have the similar triangles shown without an overlapping. Writing the proportion, we have the left side is to the bottom as the left side is to the bottom. A D 2 B 48 C E 36 C AB DE = BC EC 2 = 48 36 Math 7, Unit 8B NOTES Holt: Chapter B Page 13 of 18 Revised 213-CCSS
2 = OR = 48 36 12 36 2 36 = 48 36 = 12 72 = 48 18 = 12 1= 1= We can use similar figures to find lengths that are difficult to measure. We call this indirect measurement. We will use indirect measurement to determine the height of the tree in the following problem. Eample: A student whose height is 6 feet is standing near a tree. The length of the student s shadow is 2 feet. If the tree casts a shadow of feet, how tall is the tree? Since the student and the tree are perpendicular to the ground, the sun s ras strike the student and the tree at the same angle, creating two similar figures. A sketch will help us to see the similar triangles. h 6 ft So, ft 2ft height of the tree length of the tree's shadow = height of the student h length of the student's shadow = 6 2 2h = 6 The height of the tree is 1 feet. 2h = 3 h = 1 Scale Drawings 7.G.1-1: Solve problems involving scale drawings of geometric figures b including computing actual lengths and areas from a scale drawing. Math 7, Unit 8B NOTES Holt: Chapter B Page 14 of 18 Revised 213-CCSS
7.G.1-2: Solve problems involving scale drawings of geometric figures b reproducing a scale drawing at a different scale. Another application of proportions is in the use of scale drawings. A scale drawing is a twodimensional drawing that is similar to the object it represents. A scale model is a threedimensional model that is similar to the object it represents. The scale of a scale drawing or scale model gives the relationship between the drawing or model s dimensions and the actual dimensions. For eample, if a map shows a scale of 1 cm : m, it means that 1 centimeter on the scale drawing represents an actual distance of meters. The scale of a scale drawing or scale model can be written without units if the measurements have the same unit. To write the scale from our eample without units, write meters as centimeters. 1 cm 1 cm 1 cm : m 1: m cm So, we can write the scale without units as 1:. Eample: On a map, the distance from our house to school is centimeters. The scale is 1 cm : m. What is the actual distance from our house to school? map distance 1 cm cm = actual distance m d m 1d = d = 2 The distance from our house to school is 2, meters. Eample: You have a scale model of an airplane, scale of 1:9. The length of the model airplane from nose to tail is 1.8 feet. Determine the length (from nose to tail) of the actual airplane. model length 1 1. 8 = airplane length 9 The actual airplane is 162 feet. 1 = 162 Eample: The figure at the right is a scale drawing of a large rectangular room. What is the area of the actual room? 1 cm 7 cm 2 cm:6 m Math 7, Unit 8B NOTES Holt: Chapter B Page 1 of 18 Revised 213-CCSS
Solution: length = width = scale ( cm) 2 1 = = = 3 3m actual m 6 ( ) scale ( cm) 2 7 = = = 21 21m actual m 6 ( ) Area = length width = 3 m 21 m = 63 sq. m 63 square meters Eample: Julie showed ou the scale drawing of her room. If each 2 cm on the scale drawing equals ft, what are the actual dimensions of Julie s room? Reproduce the drawing at 3 times its current size..6 cm 1.4 cm 4 cm 1.4 cm 2.6 cm 4.2 cm This eample is given for ou to see the scope of the CCSS, the solution is left to ou. Sllabus Objective: (6.16) The student will create scale drawings using ratio and proportion. Here is a perfect opportunit for students to be given a project to (create a scale drawing) appl all the skills and concepts taught in this unit. Math 7, Unit 8B NOTES Holt: Chapter B Page 16 of 18 Revised 213-CCSS
SBAC eample: Standard 7.G.1, 7.RP.2 Difficult: Medium Item Tpe :SR (selected response) A compan designed two rectangular maps of the same region. These maps are described below. Map 1: The dimensions are 8 inches b 1 inches. The scale is 3 mile to 1 inch. 4 Map 2: The dimensions are 4 inches b inches. Which ratio represents the scale on Map 2? A. 1 2 mile to 3 4 inch B. 3 4 mile to 1 2 inch C. 1 mile to 1 inch 4 D. 3 mile to 1 inch 8 Ke and Distractor Analsis: A. Found correct relationship but reversed order B. Correct C. Subtracted the first term of ratio b scale factor D. Multiplied the first term of ratio b scale factor SBAC eample: Standard 7.RP.2 Difficult: Medium Item Tpe :SR (selected response) Helen made a graph that represents the amount of mone she earns,, for the numbers of hours she works,. The graph is a straight line that passes through the origin and the point (1, 12.). Which statement must be true? A. The slope of the graph is 1. B. Helen earns $12. per hour. C. Helen works 12. hours per da. D. The -intercept of the graph is 12.. Math 7, Unit 8B NOTES Holt: Chapter B Page 17 of 18 Revised 213-CCSS
Ke and Distractor Analsis: A. Reverses the meaning of the coordinates. B. Correct C. Focuses on the vertical ais. D. Thinks 12. is the initial value. SBAC eample: Standard 7.RP.2 Difficult: Medium Item Tpe : TE (This will be a technolog question.) The value of is proportional to the value of. The constant of proportionalit for this relationship is 2. On the grid below, graph this proportional relationship. - - [Create two points b clicking on the intersections of the gridlines. When ou create the second point, a line will automaticall be drawn through the two points. If ou make a mistake, use the Clear button to begin again.] Ke and Distractor Analsis: Student must select two of these points: (-4, -8), (-3, -6), (-2, -4), (-1, -2), (, ), (1, 2), (2, 4), (3, 6), (4, 8). Math 7, Unit 8B NOTES Holt: Chapter B Page 18 of 18 Revised 213-CCSS