Proportional shapes

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  • This PowerPoint was made to teach primarily 8 th grade students proportions. This was in response to a DLC request (No. 228).
  • Due to the math it does not make a difference whether the smaller side is the numerator or denominator. The only thing which matters is that it is consistent on both sides of the equation.
  • Knowing the two figures are similar the proportion between the two stick figures is 8 feet:12 feet. Once written as a fraction 8/12 reduces to 2/3. So the proportion between the two stick figures is 2:3 . If the proportion is 2:3 then the student should set up this equation and solve for x: 2 / 3 = 4 / x 2 * x = 3 * 4 x = 12 / 2 x = 6 feet
  • The right angles are equal, and the angles the shadow makes with the ground can assumed to be equal. They can be assumed to be equal because for objects close in distance the sun is the same angle from the ground. Thus the shadows have similar angles, so the triangles are similar. Also the hypotenuses do not matter in these triangles. You could solve for them using Pathagorean’s Theorem, but it isn’t required to solve the problem so we will leave them alone.
  • Proportional shapes

    1. 1. Proportions
    2. 2. ProportionsWhat are proportions? - If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures. 1 4 = 1:3 = 3:9 2 8What do we mean by similar? - Similar describes things which have the same shape but are not the same size.
    3. 3. ExamplesThese two stick figures aresimilar. As you can see bothare the same shape. However,the bigger stick figure’sdimensions are exactly twicethe smaller. 8 feetSo the ratio of the smaller 4 feetfigure to the larger figure is 1:2(said “one to two”). This canalso be written as a fraction of½. 2 feetA proportion can be made 4 feetrelating the height and the 4 ft 8 ftwidth of the smaller figure to = 2 ft 4 ftthe larger figure:
    4. 4. Solving Proportional ProblemsSo how do we useproportions and similar 8 feetfigures? 4 feetUsing the previousexample we can showhow to solve for an 2 feetunknown dimension. ? feet
    5. 5. Solving Proportion ProblemsFirst, designate the unknown sideas x. Then, set up an equationusing proportions. What does thenumerator represent? What doesthe denominator represent? 8 feet 4 ft 8 ft = 4 feet 2 ft x ftThen solve for x by crossmultiplying: 2 feet 4x = 16 ? feet X=4
    6. 6. Try One Yourself Knowing these two stick figures are similar to each other, what is the8 feet 12 feet ratio between the smaller figure to the larger figure? 4 feet x feet Set up a proportion. What is the width of the larger stick figure?
    7. 7. Similar ShapesIn geometry similar shapes are very important.This is because if we know the dimensions of oneshape and one of the dimensions of another shapesimilar to it, we can figure out the unknowndimensions.
    8. 8. Triangle and Angle ReviewToday we will be working withright triangles. Recall that one ofthe angles in a right triangleequals 90o. This angle isrepresented by a square in thecorner. 90o angleTo designate equal angles wewill use the same symbol for bothangles. equal angles
    9. 9. Proportions and Triangles What are the unknown values on these triangles? First, write proportions relating the two triangles. 20 m 4m 3m 4m ym = =16 m 16 m xm 16 m 20 m Solve for the unknown by cross multiplying. xm 4x = 48 16y = 80 ym x = 12 y=5 4m 3m
    10. 10. Triangles in the Real WorldDo you know how tall your school building is?There is an easy way to find out using right triangles.To do this create two similar trianglesusing the building, its shadow, asmaller object with a known height(like a yardstick), and its shadow.The two shadows can be measured,and you know the height of the yardstick. So you can set up similartriangles and solve for the height ofthe building.
    11. 11. Solving for the Building’s HeightHere is a sample calculation for buildingthe height of a building: x ft 48 ft x feet = 3 ft 4 ft 48 feet 4x = 144 yardstick x = 36 3 feetThe height of the building is 36 4 feet feet.
    12. 12. Accuracy and ErrorDo you think using proportions to calculate theheight of the building is better or worse thanactually measuring the height of the building?Determine your height by the same techniqueused to determine the height of the building. Nowmeasure your actual height and compare youranswers.Were they the same? Why would there be adifference?
    13. 13. Cool Proportions• Measure wrist to fingertip. Measure top of shoulder to wrist. Write the ratio of your hand length to your arm length. What whole number ratio is it close to?• Measure fingertip to heart. Double this length. Is this equal to your height?• Measure your foot. Measure forearm (wrist to elbow). What is the ratio of your foot to your arm?
    14. 14. Similar Figure Activity• On loose-leaf, record your proportions to solve for x. You don’t need to draw the figures over (just make sure to write the number of the problem)• How many can you finish in 15 min?
    15. 15. Similar Figure Word Problems• On loose-leaf, write down key information (not entire problem). Show proportion and solve.• Green paper- solve to nearest whole number• Pink paper- solve to nearest tenth

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