G8 U2 Constructed Response Rubric
Grade 8 Unit 2 Constructed Response Radicals and Integers Scoring Rubric Task
1. Combining Exponents
Common Core State Standard 8.EE.1 [m]: Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.2 [m]: Use square root and cube root symbols to represent solutions to equations of the form x2 =
2. Solving Square and Cubic Equations
p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.EE.3 [m]: 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.
3. Scientific Notation
Standards for Mathematical Practice SMP.1, SMP.3, SMP.4, SMP.6 SMP.1, SMP.2, SMP.4, SMP.6, SMP.8
SMP.1, SMP.4, SMP.6, SMP.8
8.EE.4 [m]: 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. Interpret scientific notation that has been generated by technology.
Note to Teacher: The following scoring rubric should be used as a guide to determine points given to students for each question answered. Students are required to show the process through which they arrived at their answers for every question involving problem solving. For questions involving a written answer, full points should be given to answers that are written in complete sentences, which address each component of the questions being asked. [m]: major work for grade level [a/s]: additional/supporting standard Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 1
Grade 8 Unit 2 Constructed Response Radicals and Integers Scoring Rubric Question 1. a. Student gives explanation: After analyzing her answer and her work, I was able to conclude that both were incorrect. The two errors I was able to identify were that she multiplied the bases together and the exponents. In this case she needed to either add the exponents together or write and count out the number of bases given in the problem. b. Student gives correct answer and shows work: 711 2. a. Student completes table and shows two methods for solving the problem: Answer: 67 . Order of methods may vary. Method 1: Add the exponents together. (Student should show understanding that 1/62 is equal to 6-2). Method 2: Write and count out the number of bases. b. Student gives correct answer and explanation: After simplifying the expression using the two methods I showed above I can conclude that the most efficient method was to add the exponents together. When you add the exponents together you save time because you do not have to write out the bases over and over again and then count them. If the bases are the same you simply add the exponents and get the same answer as in the other method. When I did this in the problem I had to add 65+4−2 which equaled 67 , on the other hand when I used the other method I had to write 6×6×6×6×6×6×6×6×6 out the problem which was and then I had to simplify it 6×6 7 to get 6 . t 3. a. Student gives correct answer and shows work: s3u2 b. Student gives explanation: If I were simplifying the reciprocal of the expression from above, the answer would also be the reciprocal, so instead t
s3 u2
of s3 u2 my answer would be t . The process for solving it would be the same but in this case the variables with negative exponents in the first expression would now be positive and the variables positive exponents would be negative. The resulting answer would be the reciprocal.
Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 2
Points 0.5
0.5 1.5
0.5
1.5 0.5
Total
Grade 8 Unit 2 Constructed Response Radicals and Integers 4. a. Student gives correct answer and shows work: √𝟐 b. Student gives explanation: After solving the equation I was able to conclude that my answer of √2 is an irrational number. I know that this is an irrational number because it is a non-terminating and non-repeating decimal that cannot be written in the form a/b. c. Student completes table: Mathematical process: Student should show √2 being plugged back into the original equation for x. Explanation of Steps: In order to prove that my answer is correct I need to plug in my answer of x =√2 back into the equation. In doing
0.25 0.25
0.5
2
so my new equation to solve is −14(√2) − 3 = −31. When I simplify this my answer is – 31 = −31, which means that both sides are balanced out and my answer of x = √2 is correct.
5. a. Student gives explanation: After reading the problem I was able to extract some important information that helped me develop a strategy for solving the problem. The first important pieces of information that I am given are that the living room is in the shape of a square and that each tile that they are placing on the floor measures 1 square foot. I am also told that there are 225 tiles covering the floor. We know that the area of that room is 225𝑓𝑡 2 since each tile measures 1 foot squared. Since the room is in the shape of a square, the length and the width are equal, so I must find the measurement. I am going to have to write and solve the equation 𝑥 2 = 225, where x is equal to the measurement I am trying to find. b. Student creates illustration and gives correct answer: 15ft; Sample illustration:
15ft
1
1
A= 225ft 2
15ft
6. a. Student gives explanation: After reading the problem I was able to extract some important information that helped me develop a strategy for solving the problem. The first important pieces of information that I am given are that the storage container is in the shape of a cube and that it has a volume of 64ft3. Since the container is in the shape of a cube, the length, the width, and the height are going to be equal, so to find the dimensions I Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 3
1
Grade 8 Unit 2 Constructed Response Radicals and Integers can write and solve the equation 𝑥 3 = 64, where x is equal to the measurement of one of the sides. Again since the container is in the shape of a cube, if I find the measurement of one side, I also have the measurements of the other sides. b. Student gives correct answer and shows work: length= 4ft; width= 4ft, height=4ft c. Student gives correct answer and shows work: length: 10ft; width= 10ft d. Student gives correct answer and gives explanation: After finding the dimensions of the room and the containers, I was able to determine that Stan can buy and place 4 containers in the room without needing to stack them. I know that the room has a length and width of 10 feet and each container has a length and width of 4 feet. So with this information I know that the two containers placed next to each other will take up 8 feet in length, which fits inside the room lengthwise. I can also place another two containers in a second row, which combined with the first row will take up 8 feet of the width of the room. So Stan will have about 2 feet extra on each side of the room. 7. a. Student gives correct answers: Examples given may vary but should make relative sense to the number. Decimal/Whole Number
Scientific Notation
Example
a. 40 m
4 × 101 𝑚
Height of a tree
b. 0.035
3.5 × 10−2 𝑚
Length of a cockroach
c. 60, 000,000 m
6 × 107 m
Distance between planets
d. 0.0009m
9 × 10−4m
Thickness of a coin
e. 3 m
3 × 100 m
Length of a truck
f. 9,000m
9 × 103 𝑚
Height of a mountain
8. a. Student gives correct answers: 6.53 × 104 − 7.89 × 103
4
b. Student gives correct answer and shows work: 5.741 × 10 c. Student gives explanation: In order to solve this problem I first had to figure out what the problem was asking me to find. I knew that the starting Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 4
1
1
0.25 0.75 1
Grade 8 Unit 2 Constructed Response Radicals and Integers population was 6.53 × 104 , and that it decreased by 7.89 × 103 , so that meant that to find the remaining population I would need to subtract these two values. However, since they were in scientific notation I had to make the power of ten in both numbers the same in order to subtract them. This was the most important step because the power of ten needs to be the same in both numbers in order to perform any operation with them. To do this I had to change 7.89 × 103 to . 789 × 104 . I could then subtract 0.789 from 6.53, which gave me 5.741. My answer for the population of the Honey Bees in the winter was then 5.741 × 104 . 9. a. Student gives correct answer and shows work: 3.626 × 100 b. Student completes table: Process: Student should multiply 2.03 × 107 by 3.626 × 100 Explanation of Steps: In order to prove that my answer is correct I have to multiply the size of the Southern Ocean, 2.03 × 107 , by my answer from part, 3.626 × 100 . When I do this the product is the size of the Indian Ocean that I was given in the table, which means that my answer for how many times larger this ocean is than the Southern Ocean is correct. c. Student gives correct answer and shows work: 2.52 × 108 d. Student gives explanation: The processes for finding my answers in Part A and C were different because of the operations that were necessary to solve for each. For example in Part A, I was trying to find how many times larger the Indian Ocean was than the Southern Ocean, so I had to divide the size of the Indian Ocean, by the size of the Southern Ocean. When it came to the exponents in the scientific notation, I had to subtract them to find the final power of ten in my answer. For part C I was told how many times larger the Pacific Ocean was than the Atlantic Ocean, and I was given the size of the Atlantic Ocean. Therefore, to find the size of the Pacific I had to multiply these two numbers. When I had to work with the exponents of the scientific notations, I had to add them together to find the final power of ten in my answer. The important operations necessary to solve for each part were the reciprocals of each other. Test Total
Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 5
1 0.25 0.25
15
1. Combining Exponents
Common Core State Standard 8.EE.1 [m]: Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.2 [m]: Use square root and cube root symbols to represent solutions to equations of the form x2 =
2. Solving Square and Cubic Equations
p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.EE.3 [m]: 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.
3. Scientific Notation
Standards for Mathematical Practice SMP.1, SMP.3, SMP.4, SMP.6 SMP.1, SMP.2, SMP.4, SMP.6, SMP.8
SMP.1, SMP.4, SMP.6, SMP.8
8.EE.4 [m]: 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. Interpret scientific notation that has been generated by technology.
Note to Teacher: The following scoring rubric should be used as a guide to determine points given to students for each question answered. Students are required to show the process through which they arrived at their answers for every question involving problem solving. For questions involving a written answer, full points should be given to answers that are written in complete sentences, which address each component of the questions being asked. [m]: major work for grade level [a/s]: additional/supporting standard Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 1
Grade 8 Unit 2 Constructed Response Radicals and Integers Scoring Rubric Question 1. a. Student gives explanation: After analyzing her answer and her work, I was able to conclude that both were incorrect. The two errors I was able to identify were that she multiplied the bases together and the exponents. In this case she needed to either add the exponents together or write and count out the number of bases given in the problem. b. Student gives correct answer and shows work: 711 2. a. Student completes table and shows two methods for solving the problem: Answer: 67 . Order of methods may vary. Method 1: Add the exponents together. (Student should show understanding that 1/62 is equal to 6-2). Method 2: Write and count out the number of bases. b. Student gives correct answer and explanation: After simplifying the expression using the two methods I showed above I can conclude that the most efficient method was to add the exponents together. When you add the exponents together you save time because you do not have to write out the bases over and over again and then count them. If the bases are the same you simply add the exponents and get the same answer as in the other method. When I did this in the problem I had to add 65+4−2 which equaled 67 , on the other hand when I used the other method I had to write 6×6×6×6×6×6×6×6×6 out the problem which was and then I had to simplify it 6×6 7 to get 6 . t 3. a. Student gives correct answer and shows work: s3u2 b. Student gives explanation: If I were simplifying the reciprocal of the expression from above, the answer would also be the reciprocal, so instead t
s3 u2
of s3 u2 my answer would be t . The process for solving it would be the same but in this case the variables with negative exponents in the first expression would now be positive and the variables positive exponents would be negative. The resulting answer would be the reciprocal.
Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 2
Points 0.5
0.5 1.5
0.5
1.5 0.5
Total
Grade 8 Unit 2 Constructed Response Radicals and Integers 4. a. Student gives correct answer and shows work: √𝟐 b. Student gives explanation: After solving the equation I was able to conclude that my answer of √2 is an irrational number. I know that this is an irrational number because it is a non-terminating and non-repeating decimal that cannot be written in the form a/b. c. Student completes table: Mathematical process: Student should show √2 being plugged back into the original equation for x. Explanation of Steps: In order to prove that my answer is correct I need to plug in my answer of x =√2 back into the equation. In doing
0.25 0.25
0.5
2
so my new equation to solve is −14(√2) − 3 = −31. When I simplify this my answer is – 31 = −31, which means that both sides are balanced out and my answer of x = √2 is correct.
5. a. Student gives explanation: After reading the problem I was able to extract some important information that helped me develop a strategy for solving the problem. The first important pieces of information that I am given are that the living room is in the shape of a square and that each tile that they are placing on the floor measures 1 square foot. I am also told that there are 225 tiles covering the floor. We know that the area of that room is 225𝑓𝑡 2 since each tile measures 1 foot squared. Since the room is in the shape of a square, the length and the width are equal, so I must find the measurement. I am going to have to write and solve the equation 𝑥 2 = 225, where x is equal to the measurement I am trying to find. b. Student creates illustration and gives correct answer: 15ft; Sample illustration:
15ft
1
1
A= 225ft 2
15ft
6. a. Student gives explanation: After reading the problem I was able to extract some important information that helped me develop a strategy for solving the problem. The first important pieces of information that I am given are that the storage container is in the shape of a cube and that it has a volume of 64ft3. Since the container is in the shape of a cube, the length, the width, and the height are going to be equal, so to find the dimensions I Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 3
1
Grade 8 Unit 2 Constructed Response Radicals and Integers can write and solve the equation 𝑥 3 = 64, where x is equal to the measurement of one of the sides. Again since the container is in the shape of a cube, if I find the measurement of one side, I also have the measurements of the other sides. b. Student gives correct answer and shows work: length= 4ft; width= 4ft, height=4ft c. Student gives correct answer and shows work: length: 10ft; width= 10ft d. Student gives correct answer and gives explanation: After finding the dimensions of the room and the containers, I was able to determine that Stan can buy and place 4 containers in the room without needing to stack them. I know that the room has a length and width of 10 feet and each container has a length and width of 4 feet. So with this information I know that the two containers placed next to each other will take up 8 feet in length, which fits inside the room lengthwise. I can also place another two containers in a second row, which combined with the first row will take up 8 feet of the width of the room. So Stan will have about 2 feet extra on each side of the room. 7. a. Student gives correct answers: Examples given may vary but should make relative sense to the number. Decimal/Whole Number
Scientific Notation
Example
a. 40 m
4 × 101 𝑚
Height of a tree
b. 0.035
3.5 × 10−2 𝑚
Length of a cockroach
c. 60, 000,000 m
6 × 107 m
Distance between planets
d. 0.0009m
9 × 10−4m
Thickness of a coin
e. 3 m
3 × 100 m
Length of a truck
f. 9,000m
9 × 103 𝑚
Height of a mountain
8. a. Student gives correct answers: 6.53 × 104 − 7.89 × 103
4
b. Student gives correct answer and shows work: 5.741 × 10 c. Student gives explanation: In order to solve this problem I first had to figure out what the problem was asking me to find. I knew that the starting Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 4
1
1
0.25 0.75 1
Grade 8 Unit 2 Constructed Response Radicals and Integers population was 6.53 × 104 , and that it decreased by 7.89 × 103 , so that meant that to find the remaining population I would need to subtract these two values. However, since they were in scientific notation I had to make the power of ten in both numbers the same in order to subtract them. This was the most important step because the power of ten needs to be the same in both numbers in order to perform any operation with them. To do this I had to change 7.89 × 103 to . 789 × 104 . I could then subtract 0.789 from 6.53, which gave me 5.741. My answer for the population of the Honey Bees in the winter was then 5.741 × 104 . 9. a. Student gives correct answer and shows work: 3.626 × 100 b. Student completes table: Process: Student should multiply 2.03 × 107 by 3.626 × 100 Explanation of Steps: In order to prove that my answer is correct I have to multiply the size of the Southern Ocean, 2.03 × 107 , by my answer from part, 3.626 × 100 . When I do this the product is the size of the Indian Ocean that I was given in the table, which means that my answer for how many times larger this ocean is than the Southern Ocean is correct. c. Student gives correct answer and shows work: 2.52 × 108 d. Student gives explanation: The processes for finding my answers in Part A and C were different because of the operations that were necessary to solve for each. For example in Part A, I was trying to find how many times larger the Indian Ocean was than the Southern Ocean, so I had to divide the size of the Indian Ocean, by the size of the Southern Ocean. When it came to the exponents in the scientific notation, I had to subtract them to find the final power of ten in my answer. For part C I was told how many times larger the Pacific Ocean was than the Atlantic Ocean, and I was given the size of the Atlantic Ocean. Therefore, to find the size of the Pacific I had to multiply these two numbers. When I had to work with the exponents of the scientific notations, I had to add them together to find the final power of ten in my answer. The important operations necessary to solve for each part were the reciprocals of each other. Test Total
Copyright © Swun Math Grade 8 Unit 2 Constructed Response Rubric, Page 5
1 0.25 0.25
15
