UNIT 6: CONJECTURE AND JUSTIFICATION WEEK 24: Student Packet


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1 Name Period Date UNIT 6: CONJECTURE AND JUSTIFICATION WEEK 24: Student Packet 24.1 The Pythagorean Theorem Explore the Pythagorean theorem numerically, algebraically, and geometrically. Understand a proof of the Pythagorean theorem. Use the Pythagorean theorem and its converse to solve problems Applications of the Pythagorean Theorem Use the Pythagorean theorem and its converse to solve problems. Find perimeters and areas of triangles and rectangles Statistics: Measures of Center Demonstrate knowledge of three commonly used measures of center. Review math concepts from prior lessons. Demonstrate competency in finding lengths and areas for polygons graphed in the coordinate plane (highlighted review) Week 24 SP
2 FOCUS ON VOCABULARY 24 Fill in the crossword puzzle using the clues below Across 6. The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. 7. This refers to the two sides of a right triangle that are adjacent to the right angle. 8. The middle number in the data set when the values are placed in order from least to greatest. 9. The value or values that occur most often. Down 1. A triangle that has a right angle 2. A data value that is unusually small or large compared to the overall pattern of values in the data set. 3. This refers to the side opposite the right angle in a right triangle. 4. The of a numerical data set is the difference between the greatest and least values in the data set. 5. The average of the value in the data set. Word Bank hypotenuse legs Pythagorean theorem mean median mode outlier range right triangle Week 24 SP0
3 24.1 The Pythagorean Theorem THE PYTHAGOREAN THEOREM Ready (Summary) We will explore the relationship between side lengths of right triangles and then look at a proof of the Pythagorean theorem. Then we will use this theorem to solve problems. Set (Goals) Explore the Pythagorean theorem numerically, algebraically, and geometrically. Understand a proof of the Pythagorean theorem. Use the Pythagorean theorem and its converse to solve problems. Find the area of each figure in square units. Go (Warmup) x y x y Simplify each expression. 5. a + a 6. ab + ab a+ a ab + ab 2 2 Week 24 SP1
4 24.1 The Pythagorean Theorem TWO RIGHT TRIANGLES = unit length = 1 square unit 1. Length of the shorter leg 2. Length of the longer leg 3. Area of the square on the shorter leg 4. Area of the square on the longer leg 5. Area of the square on the hypotenuse 6. Length of the hypotenuse Smaller triangle Larger triangle 7. Write a conjecture about the relationship between the area of square on the hypotenuse and the area of the squares of the legs. Week 24 SP2
5 24.1 The Pythagorean Theorem THE PYTHAGOREAN THEOREM: PART 1 Here is a right triangle with lengths a, b, and c: b c a a b a The two congruent squares on the right have been made using lengths a, b, and c. b 1. Label some right angles and some lengths. 2. Write the area of each polygonal piece inside of it. 3. Cut out both squares. Then cut them up into the polygons. Week 24 SP3
6 24.1 The Pythagorean Theorem This page is left intentionally blank. Week 24 SP4
7 24.1 The Pythagorean Theorem THE PYTHAGOREAN THEOREM: PART 2 a b a b b a = b a c c c c 1. Write the areas inside the polygonal pieces in the two square figures above. 2. Write an equation that equates the sum of the areas of the shaded polygons with the sum of the areas of the unshaded polygons. 3. Simplify your equation. 4. Use words to state the meaning of this equation as it refers to the legs and the hypotenuse of the original triangle. a c b 5. This relationship is called the Week 24 SP5
8 24.1 The Pythagorean Theorem PYTHAGOREAN THEOREM PRACTICE Pythagorean theorem: If a triangle is a right triangle, then the sum of the squares of the length of the two legs is equal to the square of the length of the hypotenuse. Converse of the Pythagorean theorem: If the sum of the squares of the lengths of the two shorter sides is equal to the square of the lengths of the hypotenuse, then the triangle is a right triangle. Use the Pythagorean theorem and its converse to answer these questions: 1. Draw the squares on the legs and the square on the hypotenuse of the right triangle below. Find the area of each square and the length of each side of the triangle. Using the correct numerical values, fill in the blanks to show the Pythagorean relationship. Area equation: ( ) + ( ) = ( ) Side length equation: ( ) 2 + ( ) 2 = ( ) 2 2. A classmate suggests that a triangle with side lengths of 4, 5, and 9 units is a right triangle. Use the Pythagorean theorem to show that this must be incorrect. Week 24 SP6
9 24.1 The Pythagorean Theorem PYTHAGOREAN THEOREM PRACTICE (continued) 3. A right triangle has legs of lengths 5 and 12 units. What is the length of its hypotenuse? 4. Notice that the answer to problem #3 is a whole number. When all three sides of a right triangle have whole number side lengths, the three numbers are called a Pythagorean triple. Are the side lengths of the triangle in problem #1 a Pythagorean triple? 5. Find another Pythagorean triple in this lesson and write the three side lengths of the corresponding right triangle. 6. One triangle has sides of length 4, 6 and 8 centimeters. Another triangle has sides of length 6, 8 and 10 centimeters. Is either of these triangles a right triangle? Why? 7. Latonya said, The third side of triangle T is = 61 inches long. Is Latonya correct? Explain. 6 in. T 5 in. Week 24 SP7
10 24.2 Applications of the Pythagorean Theorem APPLICATIONS OF THE PYTHAGOREAN THEOREM Ready (Summary) We will use the Pythagorean theorem to solve problems. Set (Goals) Use the Pythagorean theorem and its converse to solve problems. Find perimeters and areas of triangles and rectangles. Go (Warmup) 1. A triangle has side lengths of 3, 5, and 6 units. Use the Pythagorean theorem to show why this cannot be a right triangle. 2. Find the length of the hypotenuse in the right triangle. 8 in x 6 in Week 24 SP8
11 24.2 Applications of the Pythagorean Theorem FIND THE MISSING PART Find the missing length in each right triangle. If needed, use fractions to write square root approximations. Example: x + 6 = 10 2 x + 36 = x = 64 x = x cm 13 ft v 5 cm w 5 ft Solve. 3. To get from home to work every day, Samos drives 7 miles east on Avenue A, and then drives north on Avenue B. He knows that the straightline distance from his home to his place of work is about 25 miles. How many miles is his drive north on Avenue B? Picture: Symbols/Numbers: Words: Week 24 SP9
12 24.2 Applications of the Pythagorean Theorem MORE GEOMETRIC FIGURES For each problem, use fractions to write square root approximations. 1. Find the diagonal of a rectangle whose sides are 15 mm and 20 mm long. Picture: Symbols/Numbers: Words: 2. Find the diagonal of a square whose side is 10 cm long. Picture: Symbols/Numbers: Words: Week 24 SP10
13 24.2 Applications of the Pythagorean Theorem MORE GEOMETRIC FIGURES (continued) For each problem, use fractions to write square root approximations. 3. Find the height of an isosceles triangle with two congruent sides of 12 inches each and a base that is 18 inches long. Picture: Symbols/Numbers: Words: 4. Find the height of an equilateral triangle whose sides are 4 feet in length. Picture: Symbols/Numbers: Words: Week 24 SP11
14 24.3 Statistics: Measures of Center SKILL BUILDER 1 Complete each sentence with the correct word. mean median mode 1. The is/are the number(s) that appear(s) most often in a group of numbers. 2. The is the sum of a group of numbers divided by the number of addends. 3. The is the middle number in a group of numbers arranged in numerical order. 4. Ken rolled five number cubes labeled 16 and found that the sum was 15. What was the mean (average) of these rolls? 5. Suppose none of Ken s rolls were sixes. What could the five number cubes rolls be?,,,, 6. What is the mode(s) of your numbers from problem #5? 7. Six students recorded the number of hours they watched TV over the weekend. The minimum number of hours watched was 4 and maximum was 9. Their mean time was 7 hours. How much TV might each student have watched?,,,,, 8. Jackie played in 5 basketball games. She always scored more than 5 points. She scored in doubledigits once. Her median score was 8. What might her 5 scores be so that her mean is also 8?,,,, 9. What is the range for Jackie s scores from your answer to problem #8? Week 24 SP12
15 24.3 Statistics: Measures of Center SKILL BUILDER 2A The test scores of five students in three different math classes are shown below. Find the mean, median and mode for each class. 1. Test Scores from Mrs. Adewole s class Mean: Median: Mode: 2. Test Scores from Mr. Lopez s class Mean: Median: Mode: 3. Test Scores from Ms. Tran s class Mean: Median: Mode: 4. Which class had the highest mean score? 5. Which class had the highest median score? 6. Which class had the highest mode score? Week 24 SP13
16 24.3 Statistics: Measures of Center SKILL BUILDER 2B Find the perimeter and area for each figure yd 28 yd Perimeter: Area: 8. C 16 ft A This rectangle has a vertical height of 5 ft. and a horizontal length of 16 ft. Name its coordinates. 5 ft C (, ) A (, ) T (, ) Find the perimeter and area. S (3, 2) T Perimeter: Area: m 18.3 m Perimeter: Area: Week 24 SP14
17 24.3 Statistics: Measures of Center SKILL BUILDER 2C Find percents for each amount. Amount Find 5% Find 10% Find 15% Find 20% 10. $ $ $ $ Find the missing length of the right triangle using the Pythagorean theorem. 4 ft x Pythagorean theorem: 3 ft A bag contains 1 green, 1 purple and 1 orange colored marble. Without looking in the bag, you choose a marble, replace it, and then choose another marble. 15. Make an outcome grid to show all the possible outcomes. 16. What is the probability of drawing at least one green marble? 17. What is the probability of drawing an orange marble twice? Week 24 SP15
18 24.3 Statistics: Measures of Center SKILL BUILDER 3A 1. Find the mean, median and mode for the heights below. Students heights (in inches) Mean: Median: Mode: 2. What is the range of the students heights? Simplify Verbal Expression Algebraic Expression less than x 7. The product of n and times x divided by more than 2 times x Week 24 SP16
19 24.3 Statistics: Measures of Center SKILL BUILDER 3B Find the perimeter and area for each figure m 6 m Perimeter: Area: cm 15 cm 16 cm Perimeter: Area: 12. A rectangle is formed by connecting the coordinates (0, 1), (0, 4.5), (9.5, 4.5), and (9.5, 1). Perimeter: Area: Week 24 SP17
20 24.3 Statistics: Measures of Center TEST PREPARATION 24 Show your work on a separate sheet of paper and choose the best answer. 1. What is the length of the missing side of the right triangle? 15 cm 17 cm A. 8 cm B. 9 cm C. 10 cm D. 11 cm x 2. What is the diagonal of a rectangle whose sides are 6 m and 8 m? E. 8 m F. 9 m G. 10 m H. 11 m 3. Jessica had the following scores for five rounds of a board game: 78, 80, 69, 75, 73. What is the mean of the scores? A. 50 B. 71 C. 74 D What is the area of triangle XYZ? W Y 3 m X Z 4 m E. 5 m F. 6 m G. 7 m H. 8 m 5. Simplify A. 1 B. 1 C D A right triangle has legs of 9 cm and 12 cm. Find the length of the hypotenuse. E. 13 cm F. 14 cm G. 15 cm H. 16 cm Week 24 SP18
21 24.3 Statistics: Measures of Center This page is left intentionally blank for notes. Week 24 SP19
22 24.3 Statistics: Measures of Center This page is left intentionally blank for notes. Week 24 SP20
23 24.3 Statistics: Measures of Center KNOWLEDGE CHECK 24 Show your work on a separate sheet of paper and write your answers on this page The Pythagorean Theorem 1. Determine whether a triangle with side lengths 4 m, 6 m, and 7 m is a right triangle. 2. A right triangle has legs of 5 mm and 12 mm. Find the length of its hypotenuse Applications of the Pythagorean Theorem 3. A square has a perimeter of 20 cm. Find the length of its diagonal rounded to the nearest tenth. 4. Find the height of an equilateral triangle whose side is 8 ft. Round your answer to the nearest tenth Statistics: Measures of Center 5. Compute the mean, median, mode, and range of the following set of numbers: 5, 5, 5, 5, 5, 5, 5, 64, 70, 55, 60, 60, 80, 45, 5 6. There are seven whole numbers in a group. The minimum number is 5 and the maximum number is 15. The mean and the median are 11 and the mode is 15. What might be the possible numbers in the group? Highlighted Review: Finding Lengths and Areas 7. The points A (3, 3), B (3, 4), C (1, 4), and D (1, 3) form rectangle ABCD. Find the lengths of side AB and side BC. 8. If the area of the triangle formed by points A, B, and C is 14 square units, then what is the area of the rectangle ABCD? Week 24 SP21
24 24.3 Statistics: Measures of Center HomeSchool Connection 24 Here are some questions from this week s lessons to review with your young mathematician. 1. Determine whether a triangle with side lengths 3 m, 4 m, and 5 m is a right triangle. 2. Find the height of an isosceles triangle with two congruent sides of 5 m and a third side that is 6 m (use the long side as the base). 3. There are five whole numbers in a group. The minimum number is 7 and the maximum number is 14. The mode is 9 and the median is 9. The mean is 10. What might be the possible numbers in the group? Parent (or Guardian) signature Selected California Mathematics Content Standards AF Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = 1/2bh, C = pd  the formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). MG Use formulas routinely for finding the perimeter and area of basic twodimensional figures and the surface area and volume of basic threedimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. MG Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. SDP Compute the range, mean, median, and mode of data sets. MR Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed. Week 24 SP22
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