Standardized Tasks. Eighth Grade

Transcription

1 Standardized Tasks Eighth Grade Problem 1 (from AIMS: The Pythagorean Relationship) Objective 3.02 Apply geometric properties and relationships, including the Pythagorean theorem to solve problems. Objective 5.04 Solve equations using the inverse relationships of squares and square roots. Quilting Question: A quilting design calls for 12-inch blocks made with a square in the center as shown in drawing. The quilter needs to figure out how big to cut the center square. If the outside square block is 12-inches, what is the length of the side of the center square? (Use what you have learned about right triangles!) Solution: One solution would be to find the length of the diagonal between the midpoints of the adjacent sides (the length of the side of the blue square). The leg of the yellow triangle is 6 since the side of the square block is 12. a 2 + b 2 = = inches. Another approach: The blue area is ½ of the square block. The square block is 12 by 12 with an area of 144 sq inches. ½ of 144 is 72. If 72 sq inches is the area of the blue square, then the length of the side of the blue square is the square root of 72 or approximately 8.49 inches.

2 Problem 2 (from NCTM: Mathematics Assessment Sampler) Recognize the base and height of a triangle regardless of its orientation. Use those measurements to find the general values of area and perimeter. Objective 3.01 Represent problem situations with geometric models. Four identical triangles are arranged inside a rectangle as shown. The figure is not drawn to scale. Let b represent the length of the base of a triangle and h represent the height of a triangle. 1. Find the area of the rectangle in terms of b and h. Show all of your work. 2. Explain how you would find the perimeter of the rectangle in terms of b and h. Solution: One length is (h + b + h) = 2h + b One width is 2h, therefore A = (2h + b) (2h) or 4 h 2 +2bh P = 2(2h + b) + 2(2h) or 8h + 2b

3 Problem 3 (from NCTM: Mathematics Assessment Sampler) Imagine the effect that multiplication will have on a graphical representation (Dilation). Objective 3.03 Identify, predict, and describe dilations (note: This problem is on a number line, not in the coordinate plane. But it is a good problem to check transfer of knowledge.) If each data point in the data set is multiplied by the same number, which of the following statements will be true? a. The range is unaffected. b. The median is doubled regardless of the number used. c. The mean is equal to the constant times the original mean. d. The interquartile range remains the same. Solution: The answer is c. All data points will increase by being multiplied by the constant, affecting all values but not their differences. The range will be equal in length to the original multiplied by the constant. The median will be doubled only if the constant is 2 or -2. The interquartile range will be equal to the constant times the original IQR. Problem 4 (from NCTM: Mathematics Assessment Sampler) Compound discounts with % Objective 1.02 Develop flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil. Walmart is having a half-price sale on every item in the store. Clearance items are marked down an additional 25%. Susan plans to use a store coupon worth 25% off any item including clearance items. Susan says, This means I don t have to pay anything to buy a clearance item! Do you agree with Susan? Explain your thinking. Correct response: Student should disagree with Susan. Student realizes that each successive discount is taken off the new discounted price. Sample explanations: The clearance item Susan purchases originally costs $20. Half-off makes the price $10. Since it s a clearance item take 25% off the $10: $10-$2.50=$7.50. Using her coupon she saves 25% of $7.50 or an addition $1.88. Final sale price is $7.50-$1.88 = $5.62. Or Whenever one finds a fractional part of a whole number the answer will never come out to be zero. ½ * 10= 5; ½*5=2.5; ½*2.5= 1.25; etc.

4 Problem 5 (from NCTM: Mathematics Assessment Sampler) Linear relationship/equation, negative slope, table and graph Objective 4.01 Collect, organize, and analyze and display data to solve problems. Objective 5.01 Develop an understanding of a function. Objective 5.03 Solve problems using linear equations; justify symbolically and graphically. (This problem does not contain inequality.) Ruth has 2 candles one tall and thin, the other short and thick. The tall, thin candle is 40 cm tall and loses 3 cm in height for each hour it burns. The short, thick candle is 15 cm tall and loses a cm in height for each hour it burns. Create a double line graph and a table to show the relationship between number of hours the candles burn and the height of the candles. Let x represent the number of hours that the candle burns, and h represent the height of the candle. Write an equation or formula to compute the height of each candle after it burns for a given number of hours. Based on your equation, chart and graph, which of the 2 candles will last longer? Will the candles ever be the same height? If your answer is yes, tell when the candles will be the same height. If your answer is no, explain why the candles will never be the same height. Explain your thinking. Solutions: After 4 hours the tall, thin candle will be 28 cm tall. The short, thick candle lasts longer than the tall, thin candle (approximately one minute longer). Based on the graph, the candles will be the same height at the point where the lines cross (after 12.5 hours, at a height of 2.5 cm.) Tall, Thin Candle h=40-3x Short, Thick Candle h=15-x See graph on the following page. Hours Tall, Thin Short, Thick h=height in cm x=time in hours h=40-3x h=15-x

5 Hours Tall, Thin Short, Thick Height in centimeters Candle Height Time in hours Tall, Thin Short, Thick

6 Problem 6 (from NCTM: Teaching with the Curriculum Focal Points) Objective 4.01 Collect, organize, analyze and display data (including scatter plots) to solve problems. (Also 5.01 and 5.03). Technology 8.TT.1.2 Use appropriate technology tools and other resources to organize information (e.g. graphic organizers, databases, spreadsheets, and desktop publishing). The Video Arts store has made some modifications in its rental plans: Plan A has a $20 annual membership fee, and all videos rent for $2 per day. Plan B has no member ship fee, and videos rent for $2.50 per day. For what number of video rentals will these two plans cost the same? Include a table, graph, and equation to explain your thinking. The table and graph might be created on either a graphing calculator or spreadsheet. Solution: $ $ $ $80.00 $60.00 $40.00 Plan A Plan B $20.00 $ # of Videos Plan A Plan B 0 $20.00 $ $30.00 $ $40.00 $ $50.00 $ $60.00 $ $70.00 $ $80.00 $ $90.00 $ $ $ $ $ $ $ X = number of videos rented. Cost for plan A is 2x + 20 Cost for plan B is 2.5x 2.5x = 2x x 2x = 2x 2x x = x 0.5 = X = 40