Standardized Tasks. Seventh Grade. Four identical triangles are arranged inside a rectangle as shown. The figure is not drawn to scale.

Transcription

1 Standardized Tasks Seventh Grade Problem 1 (from NCTM: Mathematics Assessment Sampler) Objective 5.04 Develop fluency in the use of formulas to solve problems Four identical triangles are arranged inside a rectangle as shown. The figure is not drawn to scale. The length of the rectangle is 36 inches and the width is 27 inches. What is the area of one of the triangles? Solution: 2h = 27 inches, h = 13.5 inches h+ b + h = 36, 2h + b = 36, 27 + b = 36, b = 9 inches A = ½ bh, A = ½ (9 * 13.5) = square inches

2 Problem 2 (from NCTM: Mathematics Assessment Sampler) Objective 2.02 Solve problems involving volume of cylinders A farm has a vertical cylindrical oil tank that has an inside diameter of 2.5 feet. The depth of the oil in the tank is 2 feet. If 1 cubic foot of space holds 7.48 gallons, about how many gallons of oil are in the tank? Show all of your work in solving this problem. Solution: Calculate volume of the cylinder: 3.14 * * 2 = or approximately 10 cubic feet. 10 * 7.48 = 75. There are about 75 gallons of oil in the tank. Problem 3 (from NCTM: Mathematics Assessment Sampler) Objective 4.02 Calculate, use, and interpret the mean and median for a set of data Objective 4.04 Identify outliers and determine their effect on the mean and median of a set of data. The following table lists the selling prices of 28 homes in Milwaukee, Wisconsin. You work for a realtor, and are asked to report the average: selling price of a home for a newspaper article. Would you use the mean or the median? Explain your reasoning. $229,900 $980,000 $975,000 $219,900 $149,900 $1,095,900 $91,900 $579,000 $84,900 $219,900 $353,900 $299,500 $264,500 $112,900 $389,450 $264,900 $749,000 $99,000 $189,900 $ $218,000 $892,500 $242,000 $544,900 $624,000 $538,440 $449,900 $220,000 Solution: The mean is $415,142.50, and the median is $282,200. The median price is more realistic because it gives the middle price, indicating that 50 percent of the houses cost more than $282,200 and 50 percent cost less. Students choosing the mean will not have accounted for the outlier in the data set.

3 Problem 4 (from NCTM: Mathematics Assessment Sampler) Objective 4.02 Calculate, use, and interpret the mean, median, range and interquartile range for a set of data If the same number is added to each data point in a data set, which of the following statements will be true? a. The range is increased by double the number added. b. The median is unchanged. c. The interquartile range is unchanged. d. The mean is unchanged. Solution: The answer is c. The values of the mean and median will increase by the value of the constant that is added to each point, but the range and interquartile range will remain unchanged. Problem 5 (from NCTM: Mathematics Assessment Sampler) Comparing costs including % Objective: 1.03 Develop Flexibility in solving problems by selecting strategies and using mental computation, estimation, calculators or computers, and paper and pencil. Tim needs to buy 4 new tires for his car. Using the information from advertisements in the local newspaper contained in the chart below, determine where Tim would get the best deal. Show all of your work and explain how you decided which store would give him the best deal. Meyer s Tire Tire City Phillip s Tire Each tire: $63.55 Each tire: $46.25 Each tire: $56.75 Buy 3 and get one free 20% off the total purchase Solution: Meyer s Tire: 3 * $63.55 = $ Tire City: $4 * $46.25 = $ Phillip s Tire: 0.8 ($56.75* 4) = $ Tim should go to Phillip s Tire.

4 Problem 6 (from NCTM: Mathematics Assessment Sampler) Linear relationship, negative slope, table and graph Objective 4.01 Collect, organize, analyze and display data to solve problems. Objective 5.01 Identify, analyze and create linear relations, sequences and functions using symbols, graphs, tables, diagrams and written descriptions. Objective 5.03 Use and evaluate algebraic expressions and linear equations to solve problems (this problem does not use 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. What is the height of the tall, thin candle after it has burned for 4 hours? Based on your chart and graph, which of the 2 candles will last longer? Will the candles ever 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 (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.) Hours Tall, Thin Short, Thick

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

6 Problem 7 (from NCTM: Teaching with the Curriculum Focal Points/Principles and Standards for School Mathematics 2000, pg 275) Objective 1.01 Develop and use ratios, proportions and percents to solve problems. Southwest Middle School Band is hosting a concert. The 7th grade class is in charge of refreshments. One of the items to be served is punch. The school cook has given the students four different recipes calling for sparkling water and cranberry juice. Recipe A: Recipe B: 2 cups of cranberry juice 4 cups of cranberry juice 3 cups sparkling water 8 cups sparkling water Recipe C: Recipe D: 3 cups of cranberry juice 1 cups of cranberry juice 5 cups sparkling water 4 cups sparkling water 1. Which recipe will make punch that has the strongest cranberry flavor? Explain your answer. 2. Which recipe will make punch that has the weakest cranberry flavor? Explain your answer. 3. The band director says that 120 cups of punch are needed. For each recipe, how many cups of cranberry juice and how many cups of sparkling water are needed? Explain your answer. Solution: A is 2 parts out of 5 or.4 cranberry (40% cranberry) B is 4 out of 12 or.33 C is 3 out of 8 or.375 D is 1 out of 5 or.2 A is the strongest, and D is the weakest in cranberry taste. A = * 2 = 48 cups of cranberry juice, 24 * 3 = 72 cups of sparkling water B 40c cranberry and 80c s. water; C 45c cranberry and 75c s. water; D 24c cranberry and 96c s. water

7 Problem 8 (from NCTM: Teaching with the Curriculum Focal Points) Objective 5.01 Identify, analyze and create linear relations, and sequences and functions using symbols, graphs, tables, diagrams and written descriptions. Technology 7.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