Formative Instructional and Assessment Tasks

Transcription

1 Coin Collection 4.NBT.1 - Task 1 Domain Number and Operations in Base Ten Cluster Generalize place value understanding for multi-digit whole numbers. Standard(s) 4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Materials Paper and pencil Task Part 1: You have a collection of 826 coins. If the coins are pennies, what would be the value of your collection? If the coins are dimes, what would be the value of your collection? If they were dollars instead of coins, what would be the value of your collection? If they were ten dollar bills what would be the value of your collection? Look at the four values that you wrote for Part 1. What do you notice about the value of your collections? What pattern do you notice about the value of your collections? Explain your thinking in words, pictures, or numbers. Rubric Level I Level II Level III Not Yet Proficient Part 1: Student correctly identifies values for 3-4 coins. Explanation shows a developing understanding of place value patterns. Limited Performance Part 1: Student correctly identifies values for 0-2 coins. Explanation does not show consistent understanding of place value patterns. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning. Proficient in Performance Part 1: Student correctly identifies values for each coin. Value in pennies: $8.26 Value in dimes: $82.60 Value in dollars: $ Value in ten dollars: $8, Explanation demonstrates conceptual understanding of place value patterns.

2 Coin Collection Part 1: You have a collection of 826 coins. If the coins are pennies, what would be the value of your collection? If the coins are dimes, what would be the value of your collection? If they were dollars instead of coins, what would be the value of your collection? If they were ten dollar bills what would be the value of your collection? Look at the four values that you wrote for Part 1. What do you notice about the value of your collections? What pattern do you notice about the value of your collections? Explain your thinking in words, pictures, or numbers.

3 Domain Cluster Standard(s) Materials Task Adding Zeros 4.NBT.1 Task 2 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Paper and pencil Give students the following writing prompt: Gina said, Forty-six multiplied by 10 is 460 because when you multiply a number by ten, you add a zero on to the end of it. Do you agree or disagree with Gina? Explain your reasoning. Give at least one example to support your reason. Rubric Level I Level II Level III Not Yet Proficient The student may agree or disagree, but cannot clearly articulate that 46 x 10 = 460 because each digit is ten times as much and shifts place value one place to the left. The student cannot generate an example to support this idea. Limited Performance Students think that multiplying by ten is like adding a zero on to a number. They do not understand the relationship between multiplying by ten and the place value shift that digits make. Proficient in Performance The student may agree or disagree, and can clearly articulate that 46 x 10 = 460 because each digit is ten times as much and shifts place value one place to the left. The student is able to generate an example to support this idea. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning.

4 Adding Zeroes Gina said, Forty-six multiplied by 10 is 460 because when you multiply a number by ten, you add a zero on to the end of it. Do you agree or disagree with Gina? Explain your reasoning. Give at least one example to support your reason.

5 Domain Cluster Standard(s) Materials Task Packaging Soup Cans 4.NBT.1-Task 3 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Paper and pencil, activity sheet, base 10 blocks (optional), Virtual base ten blocks can be found here: Packaging Soup Cans There are 202 soup cans are in the factory. A crate will hold 200 cans. A case will hold 20 cans. The rest of the cans go into individual boxes. The factory wants to use as few packages as possible. 1. How many crates, cases, and individual boxes will you need to hold the 202 soup cans? 2. If you only had cases and individual boxes, how many of each would you need? 3. If you only had individual boxes, how many would you need? 4. What did you notice about the number of crates and cases in Part A compared to Part B? Explain your reasoning. Rubric Level I Level II Level III Not Yet Proficient Students provide correct answers on all but one of the parts above. Limited Performance Students provide correct answers on two or fewer of the parts above. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning. Proficient in Performance Students provide correct answers to all problems. Solutions: 1) 1 crate, 0 cases and 2 individual boxes, 2) 10 cases and 2 individual boxes, 3) 202 individual boxes,4) The explanation says something about the trading of 1 crate for 10 cases. In Part A we needed 1 crate and 0 cases. In Part B, there we needed 10 cases.

6 Packaging Soup Cans There are 202 soup cans are in the factory. A crate will hold 200 cans. A case will hold 20 cans. The rest of the cans go into individual boxes. The factory wants to use as few packages as possible. 1. How many crates, cases, and individual boxes will you need to hold the 202 soup cans? 2. If you only had cases and individual boxes, how many of each would you need? 3. If you only had individual boxes, how many would you need? 4. What did you notice about the number of cases in Part A compared to Part B? Explain your reasoning.

7 Value of the Bills 4.NBT.1-Task 4 Domain Numbers and Operations in Base Ten Cluster Generalize place value understanding for multi-digit whole numbers. Standard(s) 4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. Materials Paper and pencil, Activity sheet Task Part 1: Gina said, In my pocket I have 25 of the same amount of dollar bills. What is the value of Gina s money if she has: a) 25 one dollar bills b) 25 ten dollar bills c) 25 hundred dollar bills Gina reasoned, The value of the 2 when I have ten dollar bills is 200, but the value of the 2 when I have one dollar bills is only 20. Is Gina correct? Why or why not? Part 3: Consider Parts A, B, and C above if you had 260 of the same amount of dollar bills. What would the value of the bills be? Explain how you found your answer. Rubric Level I Level II Level III Not Yet Proficient The student is successful in 2 of the 3 parts of the task. Limited Performance Students get incorrect answers on all parts of the task. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning. Proficient in Performance Answers: Part 1: 25; 250, 2,500; Gina is correct. The explanation should talk about the idea that there are 25 groups of 1, 25 groups of 10, or 25 groups of 100. Part 3: 260; 2,600; 26,000. Explanation discusses the idea that the value of each digit is multiplied by 10 when the value of the dollar bills increases by 10.

8 Value of the Bills Part 1: Gina said, In my pocket I have 25 of the same amount of dollar bills. What is the value of Gina s money if she has: a) 25 one dollar bills b) 25 ten dollar bills c) 25 hundred dollar bills Gina reasoned, The value of the 2 when I have ten dollar bills is 200, but the value of the 2 when I have one dollar bills is only 20. Is Gina correct? Why or why not? Part 3: Consider Parts A, B, and C above if you had 260 of the same amount of dollar bills. What would the value of the bills be? Explain how you found your answer.

9 Formative Instructional and Assessment Tasks Arranging Students 4.NBT.2-Task 1 Domain Cluster Standard(s) Materials Task Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.2 Read and write multi-digit whole numbers using numerals, number names and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, = and < symbols to record the results of comparisons. Paper and pencil, Activity sheet, Base ten blocks, Virtual base ten blocks can be found here: Arranging Students For a field trip, 4th grade students will visit North Carolina State University s Millennial Campus. Students are in groups of 10. Each building can accommodate 10 groups at a time. Based on this information: 1) How many students would be in 4 buildings? 2) All of the students in 2 buildings and 13 other groups on the field trip visit the electron microscope before lunch. How many students saw the microscope? 3) There were 346 students from Hickory Elementary School. If they all were sent to the same buildings how many buildings were completely full of students from Hickory School? How many whole groups would go to another building? How many students from that school would be leftover without a group? 4) Students from New Hanover Elementary School take up 2 whole buildings, 14 whole groups in other buildings, and 9 students in different groups. Do they have more or less students than Hickory Elementary? How do you know? Level I Limited Performance Students do not provide correct answers to more than 2 parts Rubric Level II Not Yet Proficient Students do not provide correct answers to 1 or 2 parts. Level III Proficient in Performance Students provide correct answers for parts 1 through 4. Answers: 1) 400 students; 2) 330 students; 3) 3 whole buildings, 4 whole groups in another building, and 6 left over students; 4) New Hanover- 349 students. New Hanover has more students than Hickory. Standards for Mathematical Practice Makes sense and perseveres in solving problems. Reasons abstractly and quantitatively. Constructs viable arguments and critiques the reasoning of others. Models with mathematics. Uses appropriate tools strategically. Attends to precision. Looks for and makes use of structure. Looks for and expresses regularity in repeated reasoning.

10 Arranging Students For a field trip, 5 th grade students will visit North Carolina State University s Millennial Campus. Students are in groups of 10. Each building can accommodate 10 groups at a time. Based on this information: 1) How many students would be in 4 buildings? 2) All of the students in 2 buildings and 13 other groups on the field trip visit the electron microscope before lunch. How many students saw the microscope? 3) There were 346 students from Hickory Elementary School. If they all were sent to the same buildings how many buildings were completely full of students from Hickory School? How many whole groups would go to another building? How many students from that school would be leftover without a group? 4) Students from New Hanover Elementary School take up 2 whole buildings, 14 whole groups in other buildings, and 9 students in different groups. Do they have more or less students than Hickory Elementary? How do you know?

11 Domain Cluster Standard(s) Materials Task Juice Pouches 4.NBT.2-Task 2 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.2 Read and write multi-digit whole numbers using numerals, number names and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, = and < symbols to record the results of comparisons. Paper and pencil, Activity sheet Juice Pouches Juice pouches are packaged in different ways. A box holds 10 pouches. A case holds 10 boxes. A crate holds 10 cases. Some students bring in juice boxes for Field Day. The information is below. Miguel- 1 crates, 12 cases, 3 boxes and 6 pouches. Aaron- 1 crates, 13 cases, 17 boxes, and 2 pouches. Sarah- 1 crates, 12 cases, 2 boxes and 17 pouches. Vicky- 1 crates, 14 cases, 6 boxes, and 9 pouches. 1) If each person were going to reorganize their drink pouches to use as many of the larger containers as possible, and so that there are no more than 10 of each type of container, how many of each container would each of them need? 2) How many total drink pouches does each student have? Explain how you found your answer. 3) List in order from the student who had the most juice pouches to the student with the smallest number of juice pouches. Extension (4.NBT.6) If all of the boxes were going to be split evenly among the 6 grades at the school how many boxes would each grade get? Would there be any leftovers?

12 Rubric Level I Level II Level III Not Yet Proficient Students are able to create different representations of the numbers, but does not use place value as a strategy to compare them. Limited Performance Students are unable to create different representations of the numbers and does not use place value as a strategy to compare them. *Extension- 1,569 pouches per grade with no left overs. Proficient in Performance Students provide correct answers for parts 1, 2, and 3. Solutions: Part 1: Miguel- 2 crates, 2 cases, 3 boxes, and 6 pouches; Aaron- 2 crates, 4 cases, 7 boxes, and 2 pouches. Sarah- 2 crates, 2 cases, 3 boxes, and 7 pouches; Vicky- 2 crates, 4 cases, 6 boxes and 9 pouches; Part 2- Miguel- 2,236 pouches; Aaron- 2,472 pouches Sarah- 2,237 pouches; Vicky- 2,469.Explanation discusses adding up the number of pouches. Part 3- Aaron, Vicky, Sarah, Miguel. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning.

13 Juice Pouches Juice pouches are packaged in different ways. A box holds 10 pouches. A case holds 10 boxes. A crate holds 10 cases. Some students bring in juice boxes for Field Day. The information is below. Miguel- 1 crates, 12 cases, 3 boxes and 6 pouches. Aaron- 1 crates, 13 cases, 17 boxes, and 2 pouches. Sarah- 1 crates, 12 cases, 2 boxes and 17 pouches. Vicky- 1 crates, 14 cases, 6 boxes, and 9 pouches. 1) If each person were going to reorganize their drink pouches to use as many of the larger containers as possible, how many of each container would each of them need? 2) How many total drink pouches does each student have? 3) List in order from the student who had the most juice pouches to the student with the smallest number of juice pouches. Extension: If all of the boxes were going to be split evenly among the 6 grades at the school how many boxes would each grade get? Would there be any leftovers?

14 Domain Cluster Standard(s) Materials Task Open Number Lines 4.NBT.3-Task 1 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.3 Use place value understanding to round multi-digit numbers to any place. Paper and pencil Activity 1: Estimating sums and differences using an open number line. Specified as a tool for estimating by the CCSSM, an open number line is simply a blank number line. One way that it can be used as an estimation tool is by counting up from a given number to reach benchmark numbers, and then totaling the 'jumps.' For example, to find the difference between 46 and 100, you can jump 4 (from 46 to 50) and then 50 (from 50 to 100) to get a difference of 54. It is not necessary to draw ticks on the number line for each unit. Model using the open number line to find distances between numbers for each scenario. Molly needs to save $128 for a tablet. She received $47 for her birthday. About how much more does she need to save? I know that 47 is about 50. I'm trying to get to about 130. From 50 to 100 is 50. Then I need to go 30 more to 130. So 50 plus 30 is 80. She needs to save about 80 more dollars. Mr. Smart's class read 362 books in the Read A Thon. Mrs. Walter's class read 275 books. About how many more books did Mr. Smart's class read? I know that 362 is about 400 and 275 is about 300. That's a difference of about 100. Mrs. Collins' class read 446 books in the Read A Thon. That was about 100 more books than Mrs. White's class. How many books could Mrs. White's class have read? What are some exact numbers of books that would make sense? Since 446 is closer to 400 than 500, we can round 446 to 400 and Mrs. White's class could have read about 300 books. It would make sense to guess that Mrs. White's class could have read exactly 326 books since that rounds to 300. If you round 446 to 450, Mrs. White's class cold have read about 350 books. Activity 2 Give students the following problems to practice using open number lines. Ask them to use open number lines in at least two different ways for each problem. Find the difference between 429 and 216. Find the difference between 89 and 501. Find the difference between 350 and 1,050. Find the sum of 48 and 299. Find the sum of 12 and 372. After students have had time to think about their solutions, allow time for them to share their ideas, noticing similarities and differences in how they thought about the numbers and how they used the open number lines to find the sums or differences.

15 Rubric Level I Level II Level III Not Yet Proficient Students understand how to round a number to a given place value. They are able to estimate sums and differences using benchmark numbers and/or open number lines as a tool for computation, but may not be able to report more than one possible solution or way to find an answer. They are unable to explain why an estimate can include a range of exact numbers depending on the place value to which a number is rounded. Limited Performance Students do not understand how to round a number to a given place value. They are unable to estimate sums and differences using benchmark numbers and/or open number lines as a tool for computation. They are unable to make reasonable estimates of sums or differences and explain why an estimate can include a range of exact numbers depending on the place value to which a number is rounded. Proficient in Performance Students understand how to round a number to a given place value. They are able to estimate sums and differences using benchmark numbers and/or open number lines as a tool for computation, and can to report more than one possible solution for finding each sum or difference. They are able to explain why an estimate can include a range of exact numbers, and can justify their estimates using place value. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning.

16 Domain Cluster Standard(s) Materials Task Planning a Pizza Party 4.NBT.3-Task 2 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.3 Use place value understanding to round multi-digit numbers to any place. Pencil, paper, Activity sheet Planning a Pizza Party The following classes are having an end of the quarter pizza party for their good behavior. Teacher # of students participating Mr. Thomas 23 Mrs. Little 24 Mrs. Jones 17 Mrs. Gordon 24 Part 1: 1) About how many students are participating in the pizza party? 2) How close was your estimate in question 1 to the actual answer? 3) Explain why your estimate was different from your actual answer. 4) One pizza will feed 4 students. How many pizzas are needed for all of the students? 5) If each pizza costs $12.75 about how much money will be spent on pizza? 6) About $324 is spent on the cost of pizza and drinks. Based on your estimate in question 4, about how much money will be spent on drinks? Explain how you found your answer. Rubric Level I Level II Level III Not Yet Proficient Students cannot provide correct answers on one or two questions. Limited Performance Students cannot provide correct answers on more than two questions. Proficient in Performance Students provide correct answers on all questions. Answers: 1) = 80. 2) Actual: 88 students. 3) Possible answers could include: When we rounded to the tens place and added the rounded numbers we got 80 for the answer. 4) 88 divided by 4 is 22 pizzas. 5) We could round both numbers: 20x$13 = 260. We could round only the pizza 22x13 = 286. Either is acceptable. 6) 324 minus the answer to number 5. Answers could be 64 or 38.

17 Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning.

18 Planning a Pizza Party The following classes are having an end of the quarter pizza party for their good behavior: Teacher # of students participating Mr. Thomas 23 Mrs. Little 25 Mrs. Jones 16 Mrs. Gordon 24 Part 1: 1) About how many students are participating in the pizza party? 2) How close was your estimate in question 1 to the actual answer? 3) Explain why your estimate was different from your actual answer. 4) One pizza will feed 4 students. How many pizzas are needed for all of the students? 5) If each pizza costs $12.75 about how much money will be spent on pizza? 6) About $324 is spent on the cost of pizza and drinks. Based on your estimate in question 4, about how much money will be spent on drinks? Explain how you found your answer.

19 Domain Cluster Standard(s) Materials Task Filling the Auditorium 4.NBT.4 - Task 1 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. Activity sheet Filling the Auditorium On a field trip, three different schools send their fourth graders across town to the high school for a math competition. Each school sends between 120 and 170 students each. There are 417 students total. Part 1: How many students could have come from each school? Show your thinking. Find another possible solution to this task. Show your thinking. Part 3: If the number of students from each school was the same, how many students came from each school? Explain how you found your solution. Rubric Level I d) Level II Level III Not Yet Proficient The student has between two to four incorrect answers. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning Proficient in Performance The answers are correct. Part 1: All three numbers add up to 417. All three numbers add up to 417. Part 3: Each school had 139 fourth graders. The explanation is clear and accurate.

20 Filling the Auditorium On a field trip, three different schools send their fourth graders across town to the high school cafeteria. Each school sends between 120 and 170 students each. There are 417 students total. Part 1: How many students could have come from each school? Show your thinking. Find another possible solution to this task. Show your thinking. Part 3: If the number of students from each school was the same, how many students came from each school? Explain how you found your solution.

21 Domain Cluster Standard(s) Materials Task How Much Liquid? 4.NBT.4 - Task 2 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. Task handout How Much Liquid? The following amounts of juice were in separate containers after the school s parent breakfast. Container 1: 750 ml Container 2: 1,450 ml Container 3: 2 L Container 4: 299 ml Container 5: 476 ml Part 1: If all of the liquid was put into one large container how much liquid would be in the large container? The large container can hold 5 Liters. How much room is left in the container? Write an equation and an explanation about how you solved this problem. Rubric Level I Level II Level III Not Yet Proficient The student has between 1 and 2 errors. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning Proficient in Performance The answers are correct. Part 1: 4,975 ml. The equation is correct. 5,000 ml 4,975 ml = 25 ml. The explanation is clear and accurate.

22 How Much Liquid? The following amounts of juice were in separate containers after the school s parent breakfast. Container 1: 750 ml Container 2: 1,450 ml Container 3: 2 L Container 4: 299 ml Container 5: 476 ml Part 1: If all of the liquid was put into one large container how much liquid would be in the large container? The large container can hold 5 Liters. How much room is left in the container? Write an equation and an explanation about how you solved this problem.

23 Domain Cluster Standard(s) Materials Task Multiplication Strategies 4.NBT.5-Task 1 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.5 Multiply a whole number of up to 4 digits by a one digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Paper and pencil This standard calls for students to understand and use a variety of strategies for multiplying multidigit numbers. Strategies include the distributive property, doubling and halving, and drawing array area models. Part 1: Solve this word problem in at least three different ways. Show your thinking with pictures, numbers, and words. In the cafeteria, each row seats 22 students. There are 12 rows. How many students can be seated? As students work in pairs or groups, decide which strategies you would like them to share, and in which order. You may start with students who drew array models, and then move to those who used the distributive property in different ways. Possible strategies: Area array model: 20 x 10= x 2 = 40 2 x 10 = 20 2 x 2 = 4 Distributive Property: I broke 22 into 20 and 2. I multiplied 20 x 12 and that is or 240. Then I multiplied 2 x 12 to get 24. I added to get 264. I broke 22 into 11 and 11. I multiplied 11 x 12 to get 132, and then doubled it to get 264. I broke 12 into 10 and 2. I multiplied 10 x 22 to get 220 and 2 x 22 to get 44. I added to get 264. I broke 12 into 6 and 6. I multiplied 6 x 22 to get 132 and then doubled it to get 264. Connect the algorithm to student strategies Model using the standard algorithm for multiplication to solve 22 x 12. Ask students to explain how their strategies are the same as the algorithm. Look at your numbers and pictures and look at the way we solved this problem with the algorithm. What parts look the same? What parts look different? How are they related? Which strategy do you understand best, and why? What questions do you still have about any of these strategies?

24 Rubric Level I Level II Level III Not Yet Proficient Students can solve the multidigit multiplication problem accurately in one way, but do not provide a clear explanation of why it works or how it is related to the standard algorithm for multiplication. Limited Performance Students are unable to use a strategy to solve the multiplication problem. They may be able to use the algorithm to find an answer but cannot explain why it works. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning. Proficient in Performance Students can solve the multidigit multiplication problem accurately in at least two ways, and can provide a clear explanation of why each strategy works and how it is related to the standard algorithm for multiplication.

25 Multiplication Strategies Part 1: Solve this word problem in at least three different ways. Show your thinking with pictures, numbers, and words. In the cafeteria, each row seats 22 students. There are 12 rows. How many students can be seated? Look at your numbers and pictures and look at the way we solved this problem with the algorithm. What parts look the same? What parts look different? How are they related? Which strategy do you understand best, and why? What questions do you still have about any of these strategies?

26 Domain Cluster Standard(s) Materials Task Who Has a Bigger Garden? 4.NBT.5- Task 2 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Task handout Who Has a Bigger Garden? In eastern North Carolina, three farmers are having a discussion about who has the largest garden. Mr. Sanchez: My garden is 87 yards long and its width is 1/3 of its length. Mrs. Thompson: My garden s width is 18 yards less than the width of Mr. Sanchez garden. Its length is 8 yards longer than the length of Mr. Sanchez garden. Mr. Peterson: My garden is square and has a perimeter of 204 yards. Part 1: What are the dimensions of each garden? List the gardens in order from smallest to largest area. Rubric Level I e) Level II Level III Not Yet Proficient The student has between 1 and 2 errors. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Proficient in Performance The answers are correct. Part 1: Mr. Sanchez: 87x29, Mrs. Thompson: 95 x 11 Mr. Peterson: 51x51. Mrs. Thompson: 1,045 square yards; Mr. Sanchez: 2,523 square yards; Mr. Peterson: 2,601 square yards. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning

27 Who Has a Bigger Garden? In eastern North Carolina, three farmers are having a discussion about who has the largest garden. Mr. Sanchez: My garden is 87 yards long and its width is 1/3 of its length. Mrs. Thompson: My garden s width is 18 yards less than the width of Mr. Sanchez garden. Its length is 8 yards longer than the length of Mr. Sanchez garden. Mr. Peterson: My garden is square and has a perimeter of 204 yards. Part 1: What are the dimensions of each garden? List the gardens in order from smallest to largest area.

28 Domain Cluster Standard(s) Materials Task College Basketball Attendance 4.NBT.5- Task 3 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Activity sheet College Basketball Attendance Part 1: The college student sections at college basketball games vary UNC Chapel Hill has 1,197 seats reserve for their Carolina students. If students are only allowed to go to 1 game per season, how many different students can go to 6 games? Meanwhile, UNC Charlotte only has 874 seats reserved for college students. If students are only allowed to go to 1 game per season, how many different students can go to 6 games? Part 3: How many more students can go to Caroline games than UNC Charlotte games? Explain how you found your answer. Rubric Level I f) Level II Level III Not Yet Proficient The student has between 1 and 2 errors. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Proficient in Performance The answers are correct. Part 1:1,197 x 6 = 7, x 6= 5,244 Part 3: 7,182-5,244 = 1,938 Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning

29 College Basketball Attendance Part 1: The college student sections at college basketball games vary. UNC Chapel Hill has 1,197 seats reserved for their Carolina students. If students are only allowed to go to 1 game per season, how many different students can go to 6 games? Meanwhile, UNC Charlotte only has 874 seats reserved for college students. If students are only allowed to go to 1 game per season, how many different students can go to 6 games? Part 3: How many more students can go to Carolina games than UNC Charlotte games? Explain how you found your answer.

30 Domain Cluster Standard(s) Materials Task Dividing by Multiples of Ten 4.NBT.6 Task 1 Numbers and Operations in Base Ten Generalize place value understanding for multi-digit whole numbers. 4.NBT.6 Find whole number quotients and remainders with up to four digit dividends and one digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Paper and pencil One component of understanding the relationship between multiplication and division is understanding how multiples of 10, 100, or 1000 affect products and quotients. In these explorations, students work with multiples of 10 as divisors to understand how they relate to the quotient. Part 1: Use pictures and numbers to explain how 27 divided by 3 is related to 270 divided by 3 and 2,700 divided by 3. Prove that 23 divided by 4 is the same as 230 divided by 40. Rubric Level I Level II Level III Not Yet Proficient Students can divide multi-digit numbers using one or more strategies, but they are not consistently accurate. They are unable to explain how multiples of 10, 100, or 1000 affect products and quotients. Limited Performance Students are unable to divide multi-digit numbers using the standard algorithm or invented strategies. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning. Proficient in Performance Students use at least two different ways to divide multidigit numbers accurately. They are able to explain how multiples of 10, 100, or 1000 affect products and quotients.

31 Dividing by Multiples of Ten Part 1: Use pictures and numbers to explain how 27 divided by 3 is related to 270 divided by 3 and 2,700 divided by 3. Prove that 23 divided by 4 is the same as 230 divided by 40.

32 Domain Cluster Standard(s) Materials Task Packaging Cupcakes 4.NBT.6- Task 2 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Activity sheet Packaging Cupcakes The cupcake factory packages cupcakes into packages of 3, 6, and 9 cupcakes each. Part 1: They have 1,782 cupcakes to package. The company s leaders want to divide the cupcakes so that an equal number of cupcakes will be put into the 3 different types of packages. How many cupcakes will go into each type of package? How many packs of cupcakes will have 3 cupcakes in each pack? How many packs of cupcakes will have 6 cupcakes in each pack? How many packs of cupcakes will have 9 cupcakes in each pack? Part 3: Explain how you got your answer to Part 2 above. Rubric Level I g) Level II Level III Not Yet Proficient The student has between 1 and 2 errors. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning Proficient in Performance The answers are correct. Part 1:1,782 divided by 3 = 594 cupcakes per type of package. 3 packs: 594 divided by 3 = 198 packs; 6 packs: 594 divided by 6: 99 packs; 594 divided by 9: 66 packs. Part 3: The explanation is clear and accurate.

33 Packaging Cupcakes The cupcake factory packages cupcakes into packages of 3, 6, and 9 cupcakes each. Part 1: They have 1,782 cupcakes to package. The company s leaders want to divide the cupcakes so that an equal number of cupcakes will be put into the 3 different types of packages. How many cupcakes will go into each type of package? How many packs of cupcakes will have 3 cupcakes in each pack? How many packs of cupcakes will have 6 cupcakes in each pack? How many packs of cupcakes will have 9 cupcakes in each pack? Part 3: Explain how you got your answer to Part 2 above.

34 Domain Cluster Standard(s) Materials Task Dividing Resources 4.NBT.6- Task 3 Number and Operations-Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Task handout Dividing Resources The school s Parent Teacher Organization raised $7,326 through a fund raiser. Part 1: They divide the money evenly between the 6 grades at your school (Kindergarten through 5 th Grade). How much will each grade receive? With the amount of money that the fourth grade teachers received, they want to spend onethird of it on field trip fees. They will spend the rest on school supplies. How much money was spent on field trip fees? How much was spent on school supplies? Part 3: Explain how you found your answer to Part 2 above. Rubric Level I Level II Level III Not Yet Proficient The student has between 1 and 2 errors. Limited Performance The student is unable to use strategies to find correct answers to any aspect of the task. Proficient in Performance The answers are correct. Part 1:7326 divided by 6 = $1,221 One-third of $1,221 is $1,221 divided by 3 = $407. $407 was spent on field trip fees. The remaining $814 was spent on school supplies. Part 3: The explanation is clear and accurate. Standards for Mathematical Practice 1. Makes sense and perseveres in solving problems. 2. Reasons abstractly and quantitatively. 3. Constructs viable arguments and critiques the reasoning of others. 4. Models with mathematics. 5. Uses appropriate tools strategically. 6. Attends to precision. 7. Looks for and makes use of structure. 8. Looks for and expresses regularity in repeated reasoning

35 Dividing Resources The school s Parent Teacher Organization raised $7,326 through a fund raiser. Part 1: They divide the money evenly between the 6 grades at your school (Kindergarten through 5 th Grade). How much will each grade receive? With the amount of money that the fourth grade teachers received, they want to spend one-third of it on field trip fees. They will spend the rest on school supplies. How much money was spent on field trip fees? How much was spent on school supplies? Part 3: Explain how you found your answer to Part 2 above.