Warm Up Write down objective and homework in agenda Lay out homework (none) Homework (Distance & PT worksheet)

Unit 1 Common Core Standards 8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real- world and mathematical problems in two and three dimensions. 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Note: At this level, focus on linear and exponential functions. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R. Note: At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

Unit 1 Common Core Standards A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Note: At this level, focus on linear and exponential functions. A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. Note: At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions. G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. Note: Informal limit arguments are not the intent at this level. G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* Note: At this level, formulas for pyramids, cones and spheres will be given. G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Unit 1 Common Core Standards N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (5 1/3)3 must equal 5. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Note: At this level, focus on fractional exponents with a numerator of 1. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Warm Up

Vocabulary Distance Formula The distance d between any two points is given by the formula d = HypotenuseLongest side of a right triangle, opposite the right angle LegsThe two sides of a right triangle that make up the right angle Pythagorean Theorem The Pythagorean Theorem describes the relationship of the lengths of the sides of a right triangle where in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Pythagorean Triple Three positive integers that make up the lengths of the sides of a right triangle right angleAngle that measures 90 degrees

What is a right triangle? It is a triangle which has an angle that is 90 degrees. The two sides that make up the right angle are called legs. The side opposite the right angle is the hypotenuse. leg hypotenuse right angle

The Pythagorean Theorem In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a 2 + b 2 = c 2. Note: The hypotenuse, c, is always the longest side.

The Pythagorean Theorem “For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.” a 2 + b 2 = c 2 a 2 + b 2 = c 2

Proof

Find the length of the hypotenuse if 1. a = 12 and b = = c = c = c 2 Take the square root of both sides. 20 = c

= c = c 2 74 = c 2 Take the square root of both sides = c Find the length of the hypotenuse if 2. a = 5 and b = 7.

Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = b 2 = b 2 = 100 Solve for b b 2 = b 2 = 84 b = 9.17

Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7. a = 10 2 a = 100 Solve for a. a 2 = a 2 = 51 a = 7.14

Find the length of the hypotenuse given a = 6 and b =

Find the length of the missing side given a = 4 and c =

Pythagorean Triple A pythagorean triple is made up of three whole numbers that form the three sides of a right triangle No decimal answers!! Is 3, 4, and 5 a pythagorean triple? Is 5, 8, 12 a pythagorean triple? Is 2, 5, 9 a right triangle? Is 5, 12, 13 a right triangle?

Right Triangles Longest side is the hypotenuse, side c (opposite the 90 o angle) The other two sides are the legs, sides a and b Pythagoras developed a formula for finding the length of the sides of any right triangle

Applications The Pythagorean theorem has far-reaching ramifications in other fields (such as the arts), as well as practical applications. The theorem is invaluable when computing distances between two points, such as in navigation and land surveying. Another important application is in the design of ramps. Ramp designs for handicap-accessible sites and for skateboard parks are very much in demand.

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball Problem The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base?

Baseball Problem To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? a 2 + b 2 = c 2 Now use: a 2 + b 2 = c 2

Baseball Problem Solution The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution: x 2 = = 16,200 x = ft

Ladder Problem A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

Ladder Problem Solution First draw a diagram that shows the sides of the right triangle. Label the sides: – Ladder is 25 m – Distance from house is 7 m Use a 2 + b 2 = c 2 to solve for the missing side. Distance from house: 7 meters

Ladder Problem Solution b 2 = b 2 = 625 b 2 = 576

Word Problem Practice /AT1/PracPyth.htm

Extra Resources PT riangles/how-to-use-the-pythagorean- theorem.php riangles/how-to-use-the-pythagorean- theorem.php discovering-math-pythagorean-theorem- video.htm discovering-math-pythagorean-theorem- video.htm

Find the distance

Distance Formula Used to find the distance between two points

Example Find the distance between A(4,8) and B(1,12) A (4, 8)B (1, 12)

Find the distance between: – A. (2, 7) and (11, 9) – B. (-5, 8) and (2, - 4)

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Find the length of each side (d1, d2, etc) Find the area Find the perimeter

Find the length of each side (d1, d2, etc) Find the area Find the perimeter

Find the Distance between each 1) (7, 3), (−1, −4) 2) (3, −5), (−3, 0) 3) (6, −7), (3, −5) 4) (5, 1), (5, −6) 5) (5, −8), (−8, 6) 6) (4, 6), (−4, −3) 7) (−7, 0), (−2, −4) 8) (−4, −3), (1, 4) 9) (−2, 2), (−6, −8) 10) (6, 2), (0, −6)

Find the length of each side (d1, d2, etc) Find the area Find the perimeter

Find the length of each side (d1, d2, etc) Find the area Find the perimeter