Archdiocese of New York Practice Items Mathematics Grade 6 Teacher Sample Packet Unit 1 NY MATH_TE_G6_U1.indd 1
NY MATH_TE_G6_U1.indd 2
1. For every 3 cats Alex had, his sister Alexis had 6 fish. What ratio describes this relationship? A 1:2 B 1:3 C 2:1 D 2:3 Key A Measured CCLS: 6.RP.1 Standard Description: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak. For every vote candidate A received, candidate C received nearly three votes. Commentary: The item measures Standard 6.PR.1 because it requires the student to correctly find the ratio between the 2 objects mentioned. Extended Rationale A This answer is the correct one as it was stated that For every 3 cats Alex had, his sister Alexis had 6 fish. The original ratio would be 3:6 which when divided by 3 results in the final answer of 1:2. B This incorrect answer is a result of reducing the ratio incorrectly and thinking that while 3 divided by 3 is 1, 6 divided by 3 is 3 (instead of 2). The student probably correctly divided the first number but not the second. C This incorrect answer is a result of the student not recognizing the importance of order within ratios and switching around the numbers. The student probably correctly divided the second number but not the first. D This incorrect answer is a result of inaccurately dividing 3 by 2 and getting 2 (instead of dividing by 3 and getting 1) and then dividing 6 by 2 and getting 3. Mathematics Assessment, Unit 1 1 NY MATH_TE_G6_U1.indd 1
2. There are 12 ounces in one can of soda pop. If one can of soda pop costs 84 cents, how much is one ounce of soda pop? A B C D 6 cents per ounce 7 cents per ounce 10 cents per ounce 14 cents per ounce Key B Measured CCLS: 6.RP.2 Standard Description: Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. For example, This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar. We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. Commentary: The item measures Standard 6.PR.2 because it requires the student to correctly find the per ounce cost of a can of soda pop. Per ounce is a unit rate. Extended Rationale A This answer is incorrect and is probably a result of the student performing the right operation (division) on the numbers but most likely miscounting wrong when dividing. It is possible there is a weakness in knowing the multiplication tables. B This correct answer is a result of dividing 84 (the cost of the can of soda pop) by 12 (the number of ounces in the can), resulting in the per ounce cost (unit rate) of the soda pop. C This incorrect answer is a result of the student not understanding unit rates or the process in finding them. D This incorrect answer is a double the correct answer. The student may have divided incorrectly or somehow miscalculated the numbers involved. 2 Grade 6 NY MATH_TE_G6_U1.indd 2
3. Bart was excited to take out his new car. He decided to visit his grandmother who lived 120 miles away. If the trip to her house took 3 hours, how far could he travel in 12 hours? What was his average rate of speed on the trip to his grandmother s house? A B C D He could cover 480 miles and his rate of speed was 25 miles per hour. He could cover 400 miles and his rate of speed was 25 miles per hour. He could cover 400 miles and his rate of speed was 40 miles per hour. He could cover 480 miles and his rate of speed was 40 miles per hour. Key D Measured CCLS: 6.RP.3 (a or b) Standard Description: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? Commentary: The item measures Standard 6.PR.2 because it requires the student to correctly find the per ounce cost of a can of soda pop. Per ounce is a unit rate. Extended Rationale A This answer is incorrect. While the student got the distance that could be traveled in 12 hours correct, the rate of speed was incorrect and was probably a result of dividing 3 into 12. B This answer is incorrect. The student probably got the first part incorrect because he did not divide properly, and the rate of speed was incorrect and was probably a result of dividing 3 into 12. C This answer is incorrect. While the student got the rate of speed correct, she did not get the distance that could be traveled in 12 hours correct, probably due to a result of inaccurate division (or multiplication). D This is the correct answer. Once it is determined that the rate of speed is 40 miles per house (120 miles divided by 3 hours), then that is multiplied by 12 to find the number of miles that can be covered in 12 hours. Mathematics Assessment, Unit 1 3 NY MATH_TE_G6_U1.indd 3
4. Below is a table that indicates how much gas was purchased by 5 people. Use the table to find: (1) How much gas did Bonnie purchase? (2) What was the cost of Elvin s gas? GAS PURCHASED Customer Alan Bonnie Carlos Denise Elvin Amount Purchased 7 gallons 12 gallons 11 gallons 8 gallons? Amount Paid $28.00? $44.00 $32.00 $56.00 A B C D Bonnie s gas cost $56.00 and Elvin bought 14 gallons. Bonnie s gas cost $48.00 and Elvin bought 14 gallons. Bonnie s gas cost $48.00 and Elvin bought 15 gallons. Bonnie s gas cost $52.00 and Elvin bought 15 gallons. Key B Measured CCLS: 6.RP.3d Standard Description: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Commentary: The item measures Standard 6.PR.3d because it requires the student to correctly find a missing amount and a missing cost based upon rates. Extended Rationale A This answer is incorrect and the student probably knew the unit rate because he determined that $56.00 bought 14 gallons but miscalculated when determining that 12 gallons would cost $48.00. Could also be a result of not being competent in multiplying by 12. B This answer is correct because the unit ratio is $4/gallon so 12 gallons would cost $48.00 and $56.00 could buy 14 gallons. C This incorrect answer is a result of the student not finding the correct unit rate to divide into $56.00. He did however, correctly determine that 12 gallons of gas would cost $48.00. D This incorrect answer is a result of not being able to determine the unit rate. Without this information the student was unable to determine how many gallons of gas could be purchased with $56.00 nor could she determine how much 12 gallons of gas would cost. 4 Grade 6 NY MATH_TE_G6_U1.indd 4
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. 10 5 x 10 5 5 10 5 10 y What is the anticipated change in temperature 4 days from now? Answer Measured CCLS: 6.NS.8 Standard Description: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Mathematics Assessment, Unit 1 5 NY MATH_TE_G6_U1.indd 5
Commentary: The item measures Standard 6.NS.8 because it requires the student to graph and find distances. It also requires the finding of the distance between 2 points. Extended Rationale: The item measures Standard 6.NS.8 because the first question asks the student to graph 4 points on the Cartesian coordinate system. The second question asks the student to find distances between 2 points using absolute value. The appropriate answer will have all 4 points plotted correctly on the Cartesian coordinate system so that the first coordinate in the ordered pair represents the x-value and the second coordinate in the ordered pair is the y-value. The change in temperature will need to be determined from the points (4, 6) and (4, 1). The other 2 points ( 3, 8) and ( 3, 5) do have a change in temperature but are for a time in the past, not the expected change in temperature in 4 days as asked by the question. To find the change in temperature a variation of the Distance Formula is used. Beginning with either one of the y-coordinates is acceptable. Beginning with the (4, 6) point would look like this: 6 ( 1) = 7 = 7 degrees F. Starting with the point (4, 1) would result in the same answer but the work would look like this: 1 6 = 7 = 7 degrees F. 6 Grade 6 NY MATH_TE_G6_U1.indd 6
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 2 (out of 2 points) This response is correct and indicates a thorough understanding of the Cartesian coordinate system including, but not limited to, the x- and y-coordinates, the x- and y-axes on the coordinate system as all 4 points were correctly graphed in the correct quadrant. Furthermore, this response is consistent with one who knows how to correctly find the distance between 2 points on the coordinate system. This student chose to begin with the point (4, 6) to find the difference between it and (4, 1). The student also had an awareness to recognize that without using the coordinates of the 2 points, one could also find the distance merely by counting the spaces between the 2 points concerned. Mathematics Assessment, Unit 1 7 NY MATH_TE_G6_U1.indd 7
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 2 (out of 2 points) This response is correct and indicates a thorough understanding of the Cartesian coordinate system including, but not limited to, x- and y-coordinates, the x- and y-axes on the coordinate system as all 4 points were correctly graphed in the correct quadrant. Furthermore, this response is consistent with one who knows how to correctly find the distance between 2 points on the coordinate system. This student chose to begin with the point (4, 6) to find the difference between it and (4, 1). 8 Grade 6 NY MATH_TE_G6_U1.indd 8
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 2 (out of 2 points) This response is correct and indicates a thorough understanding of the Cartesian coordinate system including, but not limited to, the x- and y-coordinates, the x- and y-axes on the coordinate system as all 4 points were correctly graphed in the correct quadrant. Furthermore, this response is consistent with one who knows how to correctly find the distance between 2 points on the coordinate system. This student chose to begin with the point (4, 1) to find the difference between it and (4, 6). Mathematics Assessment, Unit 1 9 NY MATH_TE_G6_U1.indd 9
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 1 (out of 2 points) This response is partially correct. The answer indicates a complete lack of understanding of the Cartesian coordinate system including, but not limited to, the x- and y-coordinates, the x- and y-axes on the coordinate system as all 4 points were incorrectly graphed. However, this student was able to overcome the graphing deficiency and still manage to find the distance between the 2 points relevant to finding the change in temperature 4 days from now. 10 Grade 6 NY MATH_TE_G6_U1.indd 10 10/8/13 1:21 PM
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 1 (out of 2 points) This response is partially correct. While the graphing showed a complete understanding of the Cartesian coordinate system including the x- and y-coordinates, the x- and y-axes on the coordinate system as all 4 points were correctly graphed in the correct quadrant. However, while this student knew how to find the distance between 2 points, they did not understand that 4 days in the future would be a positive number (on a number line) and therefore, this response does not contain the correct change in temperature. Mathematics Assessment, Unit 1 11 NY MATH_TE_G6_U1.indd 11 10/8/13 1:21 PM
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 0 (out of 2 points) This response is incorrect and indicates a complete lack of understanding of the Cartesian coordinate system as ad have the x- and y-coordinates confused and/or the x- and y-axes reversed. Furthermore, this response indicates an inadequate understanding of finding the distance between 2 points. While the work shows 2 incorrect points were used, it also shows a deficiency in subtracting signed numbers. 12 Grade 6 NY MATH_TE_G6_U1.indd 12 10/8/13 1:21 PM
5. Today will be the starting point (origin on the coordinate system). The x-axis represents days. Days before today will lie to the left of the origin. Days in the future will lie to the right. The y-axis represents temperature. All temperature is given in degrees Fahrenheit. Three days ago the low temperature was ( 5) degrees and the high temperature was 8 degrees. In four days it is expected that the low temperature will be ( 1) degrees and the high will be 6 degrees. Graph these 4 points on the coordinate system below. What is the anticipated change in temperature 4 days from now? Answer Score Point 0 (out of 2 points) This response is incorrect and indicates a complete lack of understanding of the Cartesian coordinate system as ad have the x- and y-coordinates confused and/or the x- and y-axes reversed. While the student knew how to successfully use the distance formula for this problem, the wrong points were used and therefore an incorrect change in temperature was the result. Mathematics Assessment, Unit 1 13 NY MATH_TE_G6_U1.indd 13 10/8/13 1:21 PM
Teaching Tips for Item 5 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. Study: Clue Words º Points the location based upon an ordered pair º Ordered pair 2 numbers that relate to one another and are written as (x, y) º Graph find the location on a coordinate system and place a dot at that location º Coordinate system use of an x- and y-axis º Change in temperature the distance between 2 ordered pairs y values Facts º Temperatures (low and high) for 2 days are being used º The lows were 5 and 1 º The highs were 6 and 8 º Only the anticipated temperatures for a future date (4 days) are to be used in finding the answers º The temperature statistics can be represented by points on a coordinate system º The difference in temperatures on a given day can be found by utilizing the relevant ordered pairs Discovery º What day s temperatures are being measured? º What 2 ordered pairs are relevant? º What is the difference between the y values of those 2 pairs? Ask yourself: Have I ever seen a problem similar to this one? º If so, how is it similar? º What did I need to do? Strategize: Game Plan º Have students graph the given points. º Ask the students which 2 ordered pairs are relevant. º Find the change between the y values of the 2 relevant pairs. Strategies º Change in y values, distance formula, counting along a y-axis Solve: Use your strategies to solve the problem. 1 6 = 7 = 7 Ruminate: Look over your solution Does it seem probable? Did you answer the question? Are you certain? Did you answer using the language in the question? Is the answer in the same units? 14 Grade 6 NY MATH_TE_G6_U1.indd 14 10/8/13 1:21 PM