M & M Ratio Activity/ Chapter 8/Flashlight Activity MA.912.G.2.2 MA.912.G.3.4
8.1 Ratio and Proportion MA.912.G.2.2 MA.912.G.3.4
8.1 Ratio and Proportion • Ratio of a to b – if a and b are two quantities that are measured in the same units then the ratio of a to b can be written as and as a:b. • Ratios are usually written in simplified form. The ratio 6:8 would be written 3:4.
8.1 Ratio and Proportion • The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB : BC is 3:2. Find the length and width of the rectangle.
8.1 Ratio and Proportion • The perimeter of a rectangle ABCD is 60 centimeters. The ratio of AB : BC is 3:2. Find the length and width of the rectangle. • Solution: Because the ratio of AB : BC is 3:2, you can represent the length AB as 3x and the width BC as 2x. • 2l + 2w = P (formula for perimeter) • 2(3x) + 2(2x) = 60 • 6x +4x = 60 • 10x = 60 • X = 6 • So, ABCD has a length of 18 centimeters and a width of 12 centimeters.
8.1 Ratio and Proportion • Proportion – an equation that has two ratios. IF the ratio of is equal to the ratio , then the following proportion can be written: • = • The numbers a and d are extremes of the proportion. The numbers b and c are the means of the proportion.
8.1 Ratio and Proportion • Properties of Proportions • 1. Cross Product Property – The product of the extremes equals the product of the means. IF =, then ad = bc • 2. Reciprocal Property – IF two ratios are equal, then their reciprocals are also equal. IF =, then =
8.1 Ratio and Proportion • = • =
8.1 Ratio and Proportion • Homework: Page 461 10-16, 26-28, 34-46 even
Steps to the activity • Open your package of M & Ms • Sort all your colored M & Ms • Count each color separately and record the amount on your piece of paper • Count the TOTAL number of M & Ms and record that amount
Let’s Begin • Write the answer to all the ratios as a fraction, using a colon, and the word to. • Example: What is the ratio of blue M & Ms to red M & Ms (In my bag: blue= 4 And red= 8) • 4/8 = ½ , 1:2 or 4:8, 1 to 2 or 4 to 8
Now answer the following: • What is the ratio of greenM & Ms to yellow M & Ms • What is the ratio of blue M & Ms to Red M & Ms • What is the ratio of brownM & Ms to the total number of M & Ms
Ready for some more? • What is the ratio of greenand redM & Ms to the total of M & Ms • Record the ratio of blueM & Ms to the total number of yellowand orangeM & Ms • Record your favorite color of M & Ms to the total number of M & Ms (You must write down your favorite color)
Need just a little more practice. • Record the ratio of orangeM & Ms to greenand yellowM & Ms • What is the ratio of your least favorite M & Ms to the total M & Ms • My favorite color is green. What is the ratio of my favorite color & your favorite color of M & Ms to the total number of M & Ms
Are you getting hungry? • Create a table showing your color of M & Ms and the amounts
You are almost finished! • Please put your name on your paper • Now you may eat your M & Ms!!!!! • I hope you had fun with ratios
8.3 Similar Polygons MA.912.G.3.4
8.3 Similar Polygons • Similar Polygons – the correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional. • Scale Factor – the ratio of the lengths of two corresponding sides of similar polygons.
8.3 Similar Polygons • Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. If KLMN ~ PQRS, then = = = =
8.3 Similar Polygons • Homework: Page 476 8-42 even
8.4 Similar Triangles MA.912.G.3.4
8.4 Similar Triangles • Postulate 22 Angle – Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. If JKL = XYZ and KJL = YXZ, then JKL ~ XYZ
8.4 Similar Triangles • Homework: Page 484 18-26, 34-46 even
8.5 Proving Triangles are Similar MA.912.G.3.4
8.5 Proving Triangles are Similar • Theorem 8.2 Side-Side-Side (SSS) Similarity Theorem If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. If = = Then ABC ~ PQR
8.5 Proving Triangles are Similar • Theorem 8.3 Side-Angle-Side (SAS) Similarity Theorem IF an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar If X = M and = Then XYZ ~ MNP
8.5 Proving Triangles are Similar • Homework: Page 492 6-26 even
8.6 Proportions and Similar Triangles MA.912.G.3.4
8.6 Proportions and Similar Triangles • Theorem 8.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If TU ǁ QS, then =
8.6 Proportions and Similar Triangles • Theorem 8.5 Converse of the Triangle Proportionality Theorem IF a line divides two sides of a triangle proportionally, then it is parallel to the third side. If = , then TU ǁ QS.
8.6 Proportions and Similar Triangles • Theorem 8.6 IF three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ǁ s and s ǁ t, and l and m Intersect r, s, and t, then = .
8.6 Proportions and Similar Triangles • Theorem 8.7 IF a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If CD bisects LABC, then =
8.6 Proportions and Similar Triangles • Homework: Page 502 12-30 even
End of Chapter Review • Homework: Page 519 1-18