Archdiocese of New York Practice Items

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1 Archdiocese of New York Practice Items Mathematics Grade 8 Teacher Sample Packet Unit 1 NY MATH_TE_G8_U1.indd 1

2 NY MATH_TE_G8_U1.indd 2

3 1. Which choice is equivalent to ? A B 25 1 C 2 1 D 25 Key D Measured CCLS: 8.EE.1 Standard Description: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1 = Commentary: The item measures Standard 8.EE.1 because it involves knowing and applying the properties of integer exponents to generate equivalent numerical expressions. Extended Rationale A This response is an expression that is not equivalent to the given expression. The student likely did not understand that it is possible to simplify only the numerator. B This response is an expression that is not equivalent to the given expression. The student likely did not understand that with negative exponents they can be written as a positive by putting them in the denominator. C This response is an expression that is not equivalent to the given expression. The student likely did not understand that the fraction could not be reduced by canceling out the fives and just simplifying the exponents. D The student correctly identified the expression that is equivalent to the given expression. The student who selects this response determined that 52 is equivalent to = = 1 25 Mathematics Assessment, Unit 1 1 NY MATH_TE_G8_U1.indd 1

4 2. The volume of a cube-shaped building can be found by cubing the length of a side. The volume of this building can be found with the expression If there are 2 buildings the same size, what would the total volume be? A 2,744 B 5,488 C 14 5 D 28 3 Key B Measured CCLS: 8.EE.1 Standard Description: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1 = Commentary: The item measures Standard 8.EE.1 because it involves knowing and applying the properties of integer exponents to generate equivalent numerical expressions. Extended Rationale A This response is an expression that is not equivalent to the given expression. The student likely did not understand that after solving the expression 14 3, the product then needed to be multiplied by 2. B The student identified the expression that is equivalent to the given expression. The student that selects this response determined that the expression to solve was ( ) 2 2, ,488 C This response is an expression that is not equivalent to the given expression. The student likely did not understand that the exponent 3 would not have the number 2 added to it to find the total volume of both buildings. D This response is an expression that is not equivalent to the given expression. The student likely did not realize that the total volume could not be found by first multiplying 14 by 2 and then cubing this total. 2 Grade 8 NY MATH_TE_G8_U1.indd 2

5 3. A state has two main political parties. In preliminary elections, Party A has voters turn out. Party B has voters present. How do the number of voters present for each party compare in size? A Party A has 1 2 as many voters show up as Party B. B Party A has ten times as many voters show up as Party B. C Party A has five times as many voters show up as Party B. D Party A has 100 more voters show up than Party B. Key C Measured CCLS: 8.EE.3 Standard Description: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10 8 and the population of the world as 7 times 10 9, and determine that the world population is more than 20 times larger. Commentary: The item measures Standard 8.EE.3 because it involves using single digit numbers multiplied by an integer power of ten to estimate very large quantities. To make this multiplicative comparison, students need to divide the two expressions, including the single digits and powers of ten. Extended Rationale A This response reflects a comparison of the single digits only. The student may have divided 4 by 8, but did not take the powers of ten into account. A student who selects this response may not understand how the single digit relates to the power of ten in each expression. B This response reflects a comparison of the powers of ten only. The student may have looked at the powers of ten and recognized that Party A was multiplied by one power larger than Party B and mistaken this as ten times the total number of people. C The student correctly determined multiplicative comparison of Party A with respect to Party B. The student who selects this response likely divided the given expressions using one of these methods = 1 10 = = = = 5 D This response reflects students trying to subtract the expressions from one another and not applying the order of operations first. A student who selects this response may not understand how to divide expressions in the form of a single digit times a power of ten. Mathematics Assessment, Unit 1 3 NY MATH_TE_G8_U1.indd 3

6 4. A business owner has two large fishing boats. Over a week, Boat A catches pounds of fish. Boat B catches pounds of fish. How does the amount of fish caught by each boat compare in size? A Boat A catches about 1 3 as many pounds of fish as Boat B. B Boat A catches three times the amount of fish as Boat B. C Boat A catches 2 3 fewer pounds of fish than Boat B. D Boat A catches 1,400 pounds less than Boat B. Key A Measured CCLS: 8.EE.3 Standard Description: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10 8 and the population of the world as 7 times 10 9, and determine that the world population is more than 20 times larger. Commentary: The item measures Standard 8.EE.3 because it involves using single digit numbers multiplied by an integer power of ten to estimate very large quantities. To make this multiplicative comparison, students need to divide the two expressions, including the single digits and powers of ten. Extended Rationale A The student correctly determined multiplicative comparison of Boat A with respect to Boat B. The student who selects this response likely divided the given expressions using one of these methods = = 3 10 = = = 0.3 B This response reflects a comparison of the single digits only. The student may have divided 6 by 2, but did not take the powers of ten into account. A student who selects this response may not understand how the single digit relates to the power of ten in each expression. C This response reflects a comparison of the powers of ten only. The student may have looked at the powers of ten and recognized that Boat A was multiplied by one power less than Boat B and mistaken this as 2 3. D This response reflects students multiplying the single digits by the power of ten and then subtracting the expressions from one another. A student who selects this response may not understand how to divide expressions in the form of a single digit times a power of ten. 4 Grade 8 NY MATH_TE_G8_U1.indd 4

7 Measured CCLS: 8.EE.4 Standard Description: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology Commentary: The item measures Standard 8.EE.4 because it involves performing operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. This item specifically assesses multiplication of expressions represented in scientific notation. The word coefficient may be used to name the number being multiplied by the power of ten (e.g., 2.4 is the coefficient in ). Extended Rationale: This questions asks students to solve operations with numbers expressed in scientific notation. Multiplication of expressions represented in scientific notations are used to solve this problem. Mathematics Assessment, Unit 1 5 NY MATH_TE_G8_U1.indd 5

8 Part A. 240,000,000 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 240,000,000 Part B. The statement on the billboard is not correct. Brand X sells about 240,000,000 cups of coffee a day. Other coffee shops sell about 900,000,000 cups of coffee a day. 9 x 10 8 = 9 x = 900,000, ,000,000 IV 240,000,000 Score Point 2 (out of 2 points) This response answers the question correctly and demonstrates a thorough understanding of the mathematical concepts. The equation correctly solved scientific notation using mathematically sound procedures. 6 Grade 8 NY MATH_TE_G8_U1.indd 6

9 Part A. 24 x 10 7 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 240,000,000 = 24 x 10 7 Part B. The statement on the billboard is incorrect. Brand X sells about 240,000,000 cups of coffee a day. Other coffee shops sell about 900,000,000 cups of coffee a day. 9 x 10 8 = 9 x = 900,000, ,000,000 IV 240,000,000 9 x 10 8 IV 24 x 10 7 Score Point 2 (out of 2 points) This response answers the question correctly and demonstrates a thorough understanding of the mathematical concepts. The equation correctly solved scientific notation using mathematically sound procedures. Mathematics Assessment, Unit 1 7 NY MATH_TE_G8_U1.indd 7

10 Part A. 24 x 10 7 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 240,000,000 = 24 x 10 7 Part B. The statement on the billboard is incorrect. Brand X sells about 24 x 10 7 cups of coffee a day. Other coffee shops sell about 9 x 10 8 cups of coffee a day. The power of ten of Brand X is smaller, 9 x 10 8 = 9 x = 900,000, ,000,000 IV 240,000,000 9 x 10 8 IV 24 x 10 7 Score Point 2 (out of 2 points) This response answers the question correctly and demonstrates a thorough understanding of the mathematical concepts. The equation correctly solved scientific notation using mathematically sound procedures. 8 Grade 8 NY MATH_TE_G8_U1.indd 8

11 Part A. 24 x 10 7 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 240,000,000 = 24 x 10 7 Part B. The statement on the billboard is correct. Brand X sells about 24 x 10 7 cups of coffee a day. Other coffee shops sell about 9 x 10 8 cups of coffee a day. The power of ten of Brand X is larger. 9 x 10 8 = 9 x = 900,000, ,000,000 IV 240,000,000 9 x 10 8 IV 24 x 10 7 Score Point 1 (out of 2 points) This response is only partially correct. The equation solved scientific notation using mathematically sound procedures, but compared the numbers incorrectly. Mathematics Assessment, Unit 1 9 NY MATH_TE_G8_U1.indd 9

12 Part A. 24 x 10 7 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 240,000,000 = 24 x 10 7 Part B. The statement on the billboard is correct. Brand X sells about 24 x 10 7 cups of coffee a day. Other coffee shops sell about 9 x 10 8 cups of coffee a day. The power of ten of Brand X is larger. 9 x 10 8 = 9 x = 90,000,000 90,000,000 IV 240,000,000 9 x 10 8 IV 24 x 10 7 Score Point 1 (out of 2 points) This response is only partially correct. The equation solved scientific notation using mathematically sound procedures, but compared the numbers incorrectly. 10 Grade 8 NY MATH_TE_G8_U1.indd 10

13 Part A. 24 x 10 6 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40,000 x 6,000 = 24,000,000 = 24 x 10 6 Part B. The statement on the billboard is correct. Brand X sells about 24 x 10 6 cups of coffee a day. Other coffee shops sell about 9 x 10 8 cups of coffee a day. Brand X sells a larger amount than other stores. 9 x 10 8 = 9 x = 90,000,000 90,000,000 IV 24,000,000 9 x 10 8 IV 24 x 10 6 Score Point 0 (out of 2 points) This response is incorrect. Computational and procedural errors initially show the incorrect number of coffee cups sold; it applies the scientific notation incorrectly and the comparisons are incorrect. Mathematics Assessment, Unit 1 11 NY MATH_TE_G8_U1.indd 11

14 Part A. 24 x 10 6 cups of coffee sold in a day by Brand X. 4 x 10 4 = 4 x = 40,000 6 x 10 3 = 6 x 1000 = 6,000 40, ,000 = 46,000 Part B. The statement on the billboard is correct. Brand X sells about 24 x 10 6 cups of coffee a day. Other coffee shops sell about 9 x 10 8 cups of coffee a day. Brand X sells a larger amount than other stores. 9 x 10 8 = 9 x 8 x 10 = IV 46,000 Score Point 0 (out of 2 points) This response is incorrect. Computational and procedural errors initially show the incorrect number of coffee cups sold; it applies the scientific notation incorrectly and the comparisons are incorrect. 12 Grade 8 NY MATH_TE_G8_U1.indd 12

15 Teaching Tips for Item 5 8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology Study: Clue Words º Powers of ten: the number of times ten is multiplied by itself º Scientific notation: a way to write large or small numbers multiplied by a power of ten Facts Discovery How many total cups of coffee are sold by Brand X coffee shops in one day? Using the information from the stem and Part A, is the statement on the billboard 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 Find the solutions to each of the given facts. º Multiply the number of coffee shops by the number of cups of coffee sold daily. º Compare the total number of cups to the number of cups sold by other shops. Strategies º Formulas º Guess & check Solve: Use your strategies to solve the problem. Writer: Take one of the strategies you listed above, and show the work below 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? Mathematics Assessment, Unit 1 13 NY MATH_TE_G8_U1.indd 13

Fractions with Frosting

Fractions with Frosting Activity- Fractions with Frosting Sources: http://www.mybakingaddiction.com/red- velvet- cupcakes- 2/ http://allrecipes.com/recipe/easy- chocolate- cupcakes/detail.aspx http://worksheetplace.com/mf/fraction-