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Formative Instructional and Assessment Tasks Equivalent Pizzas 4.NF.1 - Task 1 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper and pencil, Graph paper (optional) There is two-thirds of a pizza left. How many pieces of pizza are left if the original pizza had a total of 3 slices? 6 slices? 12 slices? Write a sentence to explain your thinking. Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: A 3 slice pizza explanation is unclear OR would have 2 slices left. A 6 work is logically shown but the slice pizza would have 4 slices student has made a calculation left. A 9 slice pizza would error. have 6 slices left. The sentence includes a clear explanation about finding equivalent fractions. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Equivalent Pizzas There is two-thirds of a pizza left. How many pieces of pizza are left if the original pizza had a total of 3 slices? How many pieces of pizza are left if the original pizza had a total of 6 slices? How many pieces of pizza are left if the original pizza had a total of 12 slices? Write a sentence to explain your thinking. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Comparing Ropes 4.NF.1- Task 2 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper and pencil, Graph paper (optional) Sally has a piece of rope that is 3/4 of a foot long. Tomas has a piece of rope that is 1/2 of a foot long. Mitch has a piece of a rope that is 1/3 of a foot long. How many inches is each piece of rope? Write a sentence explaining your thinking. Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: Sally- 9 inches. explanation is unclear OR Tomas- 6 inches. Mitch- 4 work is logically shown but the inches. student has made a calculation The sentence includes a clear error. explanation about finding equivalent fractions. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Comparing Ropes Sally has a piece of rope that is 3/4 of a foot long. Tomas has a piece of rope that is 1/2 of a foot long. Mitch has a piece of a rope that is 1/3 of a foot long. How many inches is each piece of rope? Write a sentence explaining your thinking. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Trading Blocks 4.NF.1 - Task 3 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Pattern blocks, Paper, Pencil, Activity sheet *Computer-based pattern blocks can be found herehttp://illuminations.nctm.org/ActivityDetail.aspx?ID=27 Task sheet (below). The 2 pattern blocks below have a value of 1 whole. Part One: If the 2 pattern blocks above have a value of 1 whole, then what is the fractional value of 3 trapezoids? What is the fractional value of 1 trapezoid? If you have 3 trapezoids, how many green triangles would it take to cover the same area? If the 2 hexagons have a value of 1 whole, what is the fractional value of all the green triangles? What is the fractional value of one triangle? Since the number of trapezoids and the number of green triangles covers the same space, they are equal. Write an equivalent fraction expressing the number of trapezoids and the number of green triangles. Part Two: If 2 hexagons have a value of 1 whole, what is the value of 4 blue rhombuses? What is the value of 1 blue rhombus? If you have 4 rhombuses how many green triangles would it take to cover the same area? If the 2 hexagons have a value of 1 whole, what is the fractional value of all the green triangles? What is the value of 1 green triangle? Write an equivalent fraction expressing the number of rhombuses and the number of green triangles. Conclusion: Write a sentence explaining how you found equivalent fractions. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Accurate solutions: Part Oneexplanation is unclear OR Trapezoids: 3/4, 1/4. Triangles work is logically shown but the 9/12, 1/12. Fraction: 3/4 = student has made a calculation 9/12. Part Two: Rhombuses: error. 2/3, 1/3. Triangles: 8/12, 1/12. Fraction: 2/3 = 8/12. AND Clearly and accurately explains their strategy. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Trading Blocks The 2 pattern blocks below have a value of 1 whole. Part One: If the 2 pattern blocks above has a value of 1 whole then what is the fractional value of 3 trapezoids? What is the fractional value of 1 trapezoid? If you have 3 trapezoids, how many green triangles would it take to cover the same area? If the 2 hexagons has a value of 1 whole, what is the fractional value of all the green triangles? What is the fractional value of one triangle? Since the number of trapezoids and the number of green triangles covers the same space, they are equal. Write an equivalent fraction expressing the number of trapezoids and the number of green triangles. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Part Two: If 2 hexagons have a value of 1 whole, what is the value of 4 blue rhombuses? What is the value of 1 blue rhombus? If you have 4 rhombuses how many green triangles would it take to cover the same area? If the 2 hexagons have a value of 1 whole, what is the fractional value of all the green triangles? What is the value of 1 green triangle? Write an equivalent fraction expressing the number of rhombuses and the number of green triangles. Conclusion: Write a sentence explaining how you found equivalent fractions. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Splitting to Make Equivalent Fractions 4.NF.1-Task 4 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/ (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper and pencil This standard addresses the idea that equivalent fractions can be made by multiplying the numerator and the denominator by the same number. It also introduces the idea that dividing or splitting the numerator and denominator by the same number results in an equivalent fraction. Students will make models to show that splitting the number in the whole also splits the number in a part of the whole. The resulting fraction is the same. Task 1: Jenna ate 1/3 of a cake and had 2/3 leftover for her friends. She split each of the remaining thirds into four pieces. How many pieces of cake did she have? What fraction of the whole was each piece? Each of her friends ate the same amount of cake as Jenna. How many pieces would each friend get to eat 1/3 of the whole cake? Write or draw this fraction in two different ways. Solution: There are 8 pieces of cake leftover, each piece is 1/12 of the whole, so 8/12 is leftover. Each friend will need to eat 4/12 to eat the same amount as Jenna (1/3). 1/3 = 4/12. Task 2: Ronoldo ate ¼ of a pizza for dinner and had ¾ of the pizza leftover. He cut the leftover pizza into 6 equal slices for his friends. What fraction of the whole pizza was each piece? Each of his friends ate the same amount of pizza as Ronoldo. How many pieces would each friend get in order to eat ¼ of the whole pizza? Represent (write or draw) the solution (fraction) in two different ways. Solution: There are 6 pieces leftover, each piece is 1/8 of the whole. If each friend ate ¼ of the whole, that would be 2 pieces that are eighths, or 2/8, so 2/8 = ¼. Task 3: 4/12 = 1/3 2/8 = ¼ Look at the equivalent fractions from the story problems. What relationships do you notice between the numerators and denominators in each equation? What is happening to the numbers? We see the numbers being split. How can we see this idea happening in the models that you drew? Can you think of additional examples that show the numerator and denominator of fractions being split by the same number? What is the result? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Students are unable to show with numbers or models that a/b = (𝑎 ÷ 𝑛)/(𝑏 ÷ 𝑛), or to explain why dividing the numerator and denominator of a fraction by the same number yields an equivalent fraction. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students can show with numbers or models that a/b = (𝑎 ÷ 𝑛)/(𝑏 ÷ 𝑛), but are unable to explain why dividing the numerator and denominator of a fraction by the same number yields an equivalent fraction. Level III Proficient in Performance Students can show with numbers or models that a/b = (𝑎 ÷ 𝑛)/(𝑏 ÷ 𝑛), and are able to explain why dividing the numerator and denominator of a fraction by the same number yields an equivalent fraction. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Splitting to Make Equivalent Fractions Task 1: Jenna ate 1/3 of a cake and had 2/3 leftover for her friends. She split each of the remaining thirds into four pieces. How many pieces of cake did she have? What fraction of the whole was each piece? Each of her friends ate the same amount of cake as Jenna. How many pieces would each friend get to eat 1/3 of the whole cake? Write or draw this fraction in two different ways. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Splitting to Make Equivalent Fractions Task 2: Ronoldo ate ¼ of a pizza for dinner and had ¾ of the pizza leftover. He cut the leftover pizza into 6 equal slices for his friends. What fraction of the whole pizza was each piece? Each of his friends ate the same amount of pizza as Ronoldo. How many pieces would each friend get in order to eat ¼ of the whole pizza? Represent (write or draw) the solution (fraction) in two different ways. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Splitting to Make Equivalent Fractions Task 3: 4/12 = 1/3 2/8 = 1/4 Look at the equivalent fractions from the story problems. What relationships do you notice between the numerators and denominators in each equation? What is happening to the numbers? We see the numbers being split. How can we see this idea happening in the models that you drew? Can you think of additional examples that show the numerator and denominator of fractions being split by the same number? What is the result? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Fraction Rectangles 4.NF.1-Task 5 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/ (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper and pencil, color tiles This standard addresses the idea that equivalent fractions can be made by multiplying the numerator and the denominator by the same number. Students will make models to show that doubling the number in the whole also doubles the number in a part of the whole. The resulting fraction is the same. Task 1: Use color tiles to make a rectangle or square that is one half red and one half blue. Students will make several different representations of one half. By looking at them in order, they can see that the denominator and numerator are being multiplied by the same number. Write the equations so that they can see the proper notation. Task 2: Use color tiles to make a rectangle or square that is one third red. Find at least three different ways to represent one third. Using pictures, numbers, and/or words, prove that the three models that you made are all equal to one third. Repeat the task with ¼ and 1/6 for additional practice. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Students are unable to show with numbers or models that a/b = (n x a)/(n x b), or to explain why multiplying the numerator and denominator of a fraction yields an equivalent fraction. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students can show with numbers or models that a/b = (n x a)/(n x b), but are unable to explain why multiplying the numerator and denominator of a fraction yields an equivalent fraction. Level III Proficient in Performance Students can show with numbers or models that a/b = (n x a)/(n x b), and can explain why multiplying the numerator and denominator of a fraction yields an equivalent fraction. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Fraction Rectangles Task 1: Use color tiles to make a rectangle or square that is one half red and one half blue. Task 2: Use color tiles to make a rectangle or square that is one third red. Find at least three different ways to represent one third. Using pictures, numbers, and/or words, prove that the three models that you made are all equal to one third. Repeat the task with ¼ and 1/6 for additional practice. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Tiling the Patio 4.NF.1 - Task 6 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper, Pencil Covering the Patio Part 1: Cover the patio below with the same kind of tile. Part 2: Now cover half of the patio. Complete the table below for how many tiles it would take to cover half of the tile. Part 3: For one of the fractions above explain how multiplication can help you find equivalent fractions. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Accurate solutions: Part 1: explanation is unclear OR Each region is accurately work is logically shown but the partitioned into equal sections. student has made a calculation Part 2: Tile A: 12, 6, 6/12. Tile error. B: 6, 3, 3/6. Tile C: 4, 2, 2/4. Tile D: 8, 4, 4/8. Part 3: The student writes something about, “Multiplying both the numerator and the denominator by the same number will result in an equivalent fraction.”. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Covering the Patio Cover the patio below with the same kind of tile. Part 1: Tile A: It takes 12 tiles Tile B: It takes 6 tiles Tile C: It takes 4 tiles Tile D: It takes 8 tiles Part 2: Cover half of the patio. Complete the table below for how many tiles it would take to cover half of the tile. Tile Tiles needed to cover the whole patio. Tile A Tile B Tiles needed to Fraction showing how much of the patio cover half the is covered. patio. 1 = 6 . 6 of the patio is covered by tiles. Tile C Tile D 2 12 12 1 2 1 2 1 2 Part 3: For one of the fractions above explain how multiplication can help you find equivalent fractions. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Weird Pieces of Cake 4.NF.1 - Task 7 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Paper, Pencil Weird Pieces of Cake Part 1: A baker makes square cakes and decides to cut the pieces different each day of the week. If she wants to make 8 dollars for the whole cake, how much money will each individual piece sell for? Part 2: While shopping on Wednesday, Martina says to the baker, “Buying 2 pieces of cake today will cost the same as one piece of cake on Monday. Is Martina correct? Explain why or why not. (Modified from the Unusual Baker, NCTM, 2012) Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Accurate solutions: Part 1: explanation is unclear OR Monday- $4 each. Tuesday- $4 work is logically shown but the for large piece. Small pieces student has made a calculation are $2 each. Wednesday- $2 error. each. Part 2: The explanation says something about, “Monday’s slices are ½ of the whole cake. Wednesday’s slices are 2/4 of the whole cake. ½ = 2/4.” NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Weird Pieces of Cake Part 1: A baker makes square cakes and decides to cut the pieces different each day of the week. If she wants to make 8 dollars for the whole cake, how much money will each individual piece sell for? Part 2: While shopping on Wednesday, Martina says to the baker, “Buying 2 pieces of cake today will cost the same as one piece of cake on Monday. Is Martina correct? Explain why or why not. (Modified from the Unusual Baker, NCTM, 2012) NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks The Whole Matters 4.NF.2-Task 1 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g. by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same size whole. Record the results of comparisons with symbols >, =, or <, and justify conclusions, e.g., by using a visual fraction model. Paper and pencil Task 1: Two friends each ate ½ of a pizza. Joselin says they must have eaten the same amount, but Donnie says they could have eaten different amounts. Who do you think is correct, and why? Explain your thinking in words, pictures, and numbers. Possible solution: The two friends could have each eaten half of two different size pizzas. Half of a large pizza is more than half of a medium pizza because the wholes are not the same size. Task 2: Mrs. Johnson and Mrs. Black each gave ½ of their students a pencil. Mrs. Johnson handed out 5 more pencils than Mrs. Black. What can we say about the number of students in each class? If Mrs. Johnson handed out 16 pencils and that was 5 more than Mrs. Black, how many students are in each class? Possible solution: Mrs. Black must have 22 students in her class. Mrs. Johnson must have 32 students in her class. Task 3: Jerry made one gallon of sweetened tea and one half gallon of lemonade for a picnic. If he drank ¼ of each container, how many cups of tea did he drink? How many cups of lemonade? *1 gallon = 16 cups If Jerry drank 2 cups of lemonade and 2 cups of tea, what fraction of the tea did he drink? What fraction of the lemonade did he drink? Possible solution: Question 1: Jerry drank 4 cups of tea and 2 cups of lemonade. Question 2: Jerry drank 1/8 of the gallon of tea and ¼ of the half gallon of lemonade. Connect the tasks by discussing with students how the size of the wholes matters in each context. Level I Limited Performance Students are unable to solve Task 1, 2, or 3. Rubric Level II Not Yet Proficient Students can solve 1 or 2 of the 3 tasks correctly with a complete explanation. NC DEPARTMENT OF PUBLIC INSTRUCTION Level III Proficient in Performance Students can solve and explain their answers to all three tasks. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks The Whole Matters Task 1: Two friends each ate ½ of a pizza. Joselin says they must have eaten the same amount, but Donnie says they could have eaten different amounts. Who do you think is correct, and why? Explain your thinking in words, pictures, and numbers. Task 2: Mrs. Johnson and Mrs. Black each gave ½ of their students a pencil. Mrs. Johnson handed out 5 more pencils than Mrs. Black. What can we say about the number of students in each class? If Mrs. Johnson handed out 16 pencils and that was 5 more than Mrs. Black, how many students are in each class? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks The Whole Matters Task 3: Jerry made one gallon of sweetened tea and one half gallon of lemonade for a picnic. If he drank ¼ of each container, how many cups of tea did he drink? How many cups of lemonade? *1 gallon = 16 cups If Jerry drank 2 cups of lemonade and 2 cups of tea, what fraction of the tea did he drink? What fraction of the lemonade did he drink? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Enough Soda 4.NF.2 - Task 2 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Paper and pencil, Graph paper (optional) You need 3/4 of a Liter of soda to make punch for a party. Which containers have enough soda in them to make punch? Write a sentence explaining your thinking. Container A- 2/4 of a Liter Container B- 2/3 of a Liter Container C- 5/6 of a Liter Container D- 11/12 of a Liter Container E- 7/12 of a Liter Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: Containers C and D. explanation is unclear OR The sentence shows a clear and work is logically shown but the logical explanation of their student has made a calculation strategy. error. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Enough Soda You need 3/4 of a Liter of soda to make punch for a party. Which containers have enough soda in them to make punch? Write a sentence explaining your thinking. Container A- 2/4 of a Liter Container B- 2/3 of a Liter Container C- 5/6 of a Liter Container D- 11/12 of a Liter Container E- 7/12 of a Liter NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Which is Bigger? 4.NF.2- Task 3 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Pattern blocks, Paper and pencil Optional: Graph paper *Computer-based pattern blocks can be found herehttp://illuminations.nctm.org/ActivityDetail.aspx?ID=27 Which is Bigger? Two joint hexagons have a value of 1 whole. Based on that, draw each fraction in terms of pattern blocks and determine which is bigger: 3/4 or 4/6 1/2 or 5/12 2/4 or 3/6 5/6 or 3/4 Write your own comparison question using fourths, sixths, or twelfths. Draw a picture to prove which is easier. Pick one of the questions above and write a sentence explaining how you know that you are correct. Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: 4/6, 1/2, they are explanation is unclear OR equal, 5/6. AND work is logically shown but the The sentence shows a clear and student has made a calculation logical explanation of their error. strategy. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Which is Bigger? If two hexagons have a value of 1 whole, think about the value of other pattern block. Draw each fraction below in terms of pattern blocks and determine which is bigger: 3/4 or 4/6 1/2 or 5/12 2/4 or 3/6 5/6 or 3/4 Write your own comparison question using fourths, sixths, or twelfths. Draw a picture to prove which is easier. Pick one of the questions above and write a sentence explaining how you know that you are correct. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pattern Blocks 4.NF.2-Task 4 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g. by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same size whole. Record the results of comparisons with symbols >, =, or <, and justify conclusions, e.g., by using a visual fraction model. Paper and pencil, pattern blocks, grid paper Task 1: Use pattern blocks. If a hexagon is one whole, which block represents ½? Which block represents 1/3? Which blocks represents 2/3? If a trapezoid if one whole, which block represents 1/3? Which block represents 2/3? If a blue rhombus is one whole, which block represents ½? If a blue rhombus is ½ of a whole, what would one whole look like? Find one half of a hexagon and one half of a blue rhombus. Why don't they make one whole altogether? If a green triangle is 1/3 of a whole, what would one whole look like? How many these wholes could you make with 3 hexagons? Task 2: Use grid paper. 1. Justin planted tomatoes in 1/3 of his 6' x 6' garden. Gina planted tomatoes in 1/3 of her 8' x 7' garden. How many square feet of the garden did each person use for tomatoes? If each person planted 1/3 of their garden with tomatoes, why did they use a different amount of square feet? 2. Deon used a 9 x 9 grid to represent 1 whole and Shawn used a 12 x 12 grid to represent 1. Each boy shaded in squares to show 1/3 of the whole. How many squares did Deon shade? How many squares did Shawn shade? Why did they shade different numbers of squares if they each shaded in 1/3? Level I Limited Performance Students are unable to solve Task 1 or 2. Rubric Level II Not Yet Proficient Students can solve 1 of the 2 tasks correctly with a complete explanation. NC DEPARTMENT OF PUBLIC INSTRUCTION Level III Proficient in Performance Students can solve and explain their answers to both tasks. Responses indicate that they understand that the size of the whole determines the amount in a fraction of that whole. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pattern Blocks Task 1: Use pattern blocks. If a hexagon is one whole, which block represents ½? Which block represents 1/3? Which blocks represents 2/3? If a trapezoid if one whole, which block represents 1/3? Which block represents 2/3? If a blue rhombus is one whole,which block represents ½? If a blue rhombus is ½ of a whole, what would one whole look like? Find one half of a hexagon and one half of a blue rhombus. Why don't they make one whole altogether? If a green triangle is 1/3 of a whole, what would one whole look like? How many these wholes could you make with 3 hexagons? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Task 2: Use grid paper. 1. Justin planted tomatoes in 1/3 of his 6' x 6' garden. Gina planted tomatoes in 1/3 of her 8' x 7' garden. How many square feet of the garden did each person use for tomatoes? If each person planted 1/3 of their garden with tomatoes, why did they use a different amount of square feet? 2. Deon used a 9 x 9 grid to represent 1 whole and Shawn used a 12 x 12 grid to represent 1. Each boy shaded in squares to show 1/3 of the whole. How many squares did Deon shade? How many squares did Shawn shade? Why did they shade different numbers of squares if they each shaded in 1/3? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who’s on the Bus? 4.NF.2 - Task 5 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Paper and pencil, Graph paper (optional) There are some children on the bus. 2/6 of the children are wearing tan pants. 6/10 of the children are wearing tennis shoes. 5/12 of the children are wearing a red shirt. 2/3 of the children are wearing a hat. For each item of clothing, are more than half or less than half of the children wearing that item? Write a sentence explaining how you know that you are correct. Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: Less than half- tan explanation is unclear OR pants, red shirt. More than work is logically shown but the half- tennis shoes, hat. student has made a calculation The sentence demonstrates a error. clear understanding of comparing fractions to the benchmark of 1/2. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who’s On The Bus? There are some children on the bus. 2/6 of the children are wearing tan pants. 6/10 of the children are wearing tennis shoes. 5/12 of the children are wearing a red shirt. 2/3 of the children are wearing a hat. For each item of clothing, are more than half or less than half of the children wearing that item? Write a sentence explaining how you know that you are correct. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who Has More Gum? 4.NF.2- Task 6 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Extend understanding of fraction equivalence and ordering. 4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Paper, pencil, sentence strips or paper to fold (optional) Who has more gum? A group of friends buys a big long strip of gum and tear it into pieces. Sally has 2/3 of a foot of gum. Josey has 3/4 of a foot of gum. Mitch has 4/6 of a foot of gum. Gary has 3/6 of a foot of gum. Part 1: Draw pictures and write an expression using the >, <, or = signs to show who has more gum between: Gary or Sally? Mitch or Sally? Josey or Mitch? Part 2: Taylor comes in and gets ½ of a foot of gum. Gary says, “We have the same amount.” Is Gary correct? Why or why not? Level I Limited Performance The student has not shown a clear understanding about how to find equivalent fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Level III Not Yet Proficient Proficient in Performance Answer is correct, but the Solutions: explanation is unclear OR Part 1: Sally. 3/6 < 2/3. The work is logically shown but the same. 4/6 = 2/3. Mitch. ¾ < student has made a calculation 4/6. error. Part 2: Gary is correct. 3/6 = ½. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who Has More Gum? A group of friends buys a big long strip of gum and tear it into pieces. Sally has 2/3 of a foot of gum. Josey has 3/4 of a foot of gum. Mitch has 4/6 of a foot of gum. Gary has 3/6 of a foot of gum. Part 1: Draw pictures and write an expression using the >, <, or = signs to show who has more gum between: Gary or Sally? Mitch or Sally? Josey or Mitch? Part 2: Taylor comes in and gets ½ of a foot of gum. Gary says, “We have the same amount.” Is Gary correct? Why or why not? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Sharing Cake 4.NF.3- Task 1 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Build fractions from unit fractions. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Other Standard: 4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Paper, pencil At a party you are giving out 8 pieces of cake. People will get different amounts of cake. Tom and Hal will both get 1 piece of cake. Mary will get 2 pieces of cake. Nancy and Bob share equally the remaining pieces of cake. What fraction of the cake will each person eat? Write an equation to match the situation. Write a sentence explaining the strategy used to solve the problem. Level I Limited Performance The student has not shown a clear understanding about how to represent the pieces of cake as fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Answer is correct, but the equation or explanation is incorrect OR work is logically shown but the student has made a calculation error. Level III Proficient in Performance Solutions: Tom and Hal: 1/8, Mary, Nancy and Bob: 2/8 Equation: 1/8 + 1/8 + 2/8 + 2/8 + 2/8 = 8/8 AND The sentence clearly describes an accurate strategy. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Sharing Cake At a party you are giving out 8 pieces of cake. People will get different amounts of cake. Tom and Hal will both get 1 piece of cake. Mary will get 2 pieces of cake. Nancy and Bob share equally the remaining pieces of cake. What fraction of the cake will each person eat? Write an equation to match the situation. Write a sentence explaining the strategy used to solve the problem. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Candy Bucket 4.NF.3 - Task 2 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Build fractions from unit fractions. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Other Standard: 4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Paper, pencil There are 12 pieces of candy in the bucket. Maria and Sam each get 2 pieces of candy. Tom gets 5 pieces of candy. Vinny gets the rest of the candy. What fraction does each student get? Write an equation to match this story. Write a sentence to explain the strategy used to solve the problem. Level I Limited Performance The student has not shown a clear understanding about how to represent the pieces of cake as fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Answer is correct, but the equation or explanation is incorrect OR work is logically shown but the student has made a calculation error. Level III Proficient in Performance Solutions: Maria and Sam: 2/12 or 1/6. Tom: 5/12. Vinny 3/12. Equation: 2/12 + 2/12 + 5/12 + 3/12 AND The sentence clearly describes an accurate strategy. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Candy Bucket There are 12 pieces of candy in the bucket. Maria and Sam each get 2 pieces of candy. Tom gets 5 pieces of candy. Vinny gets the rest of the candy. What fraction does each student get? Write an equation to match this story. Write a sentence to explain the strategy used to solve the problem. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Square Tiles 4.NF.3-Task 3 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Pencil, paper, square tiles Task 1: Use square tiles to build mixed numbers. Place 3 color tiles together as a whole on the table or draw on board or display however works best for your classroom. One way to understand mixed numbers is to look at them as groups of unit fractions. Model how to use square tiles to build the number 2 1/3. If this rectangle is a whole what does one tile represent? (thirds) How many square tiles does it take to make one whole? (three thirds) How many thirds will we need to build 2 whole? (six thirds) Build arrangement below, how many thirds in all did we use? (seven thirds or 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3) Continue to use the square tiles to connect unit fractions and mixed numbers. Now the square tiles represent ¼. If we wanted to build the number 3 ¼, how many tiles would we need? How do you know? Build 2 3/8. How many tiles do you need? How many unit fractions make one whole? If you have 17 square tiles and each one is an unit of a whole, how many fourths can you build? Students should make four groups of four fourths with one fourth left over. What mixed number does this make? 4 ¼ Task 2: Use mixed number fractions to compute. In Task 2, some students may be able to transition to drawing the mixed number models while others may continue to need the square tiles to count the unit fractions. Some may draw or build the models and cross out or combine pieces to find answers. One strategy of efficiently adding and subtracting mixed number is to convert them to improper fractions, and some students may be able to do this using models. Allow students to complete the problems and then share their strategies. Addition: Maria needs 6 1/3 feet of string for a solar system mobile. She has 2 2/3 feet of yellow string and 3 2/3 feet of green string. How much string does she have altogether? Will it be enough to complete the project? Explain why or why not. Subtraction: Leland has 5 1/8 pizzas left over from his birthday party. After giving some pizza to his friend, he has 3 3/8 pizzas left. How much pizza did Leland give away to his friend? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Students are unable to convert mixed numbers to improper fractions, use models to build mixed numbers, or add or subtract mixed numbers with like denominators. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students can build and draw mixed numbers, and can use models to show the equivalence of improper fractions and mixed numbers. Students are unable to solve mixed number word problems by breaking up mixed numbers to facilitate addition and subtraction. Level III Proficient in Performance Students can build and use models and equations with unit fractions to solve the problems. Students are able to use the idea of mixed numbers as groups of unit fractions, or as an improper fraction to solve addition and subtraction problems with like denominators. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Square Tiles Addition: Maria needs 6 1/3 feet of string for a solar system mobile. She has 2 2/3 feet of yellow string and 3 2/3 feet of green string. How much string does she have altogether? Will it be enough to complete the project? Explain why or why not. Subtraction: Leland has 5 1/8 pizzas left over from his birthday party. After giving some pizza to his friend, he has 3 3/8 pizzas left. How much pizza did Leland give away to his friend? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pattern Blocks & Unit Fractions 4.NF.3-Task 4 Domain Cluster Standard(s) Materials Task Number and Operation- Fractions Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3 Understand a fraction a/b with a> 1 as a sum of fractions 1/b. 4.NF.3a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Pattern blocks, Pencil, Paper Use unit fractions in the equations to show that a fraction can be thought of as the sum of several unit fractions. Task 1: Use pattern blocks. Establish equivalencies with students. If the yellow hexagon is one whole, what fraction of the whole is each block? If the green triangle is 1/6 of the whole, how many green triangles would we need to build 4/6? Use the pattern blocks to model and solve these problems. Write an equation to represent each situation. Use unit fraction to represent each pattern block in the equation. Scott and Zachary shared a sub. Scott ate 2/6 of the sub. Zachary ate 1/6 of the sub. How much of the sub did they eat together? 1/6 + 1/6 + 1/6 = 3/6 Three students were sharing 2 pies. Each student ate ½ of a pie. How much of the total amount of pie did they eat together? ½ + ½ + ½ = 3/2 or 1 ½ Trevor had 1 1/3 pizzas. His dad ate 2/3 of a pizza. How much pizza was left? 1 1/3 = 3/3 + 1/3 or 4/3. 4/3 = 1/3 + 1/3 + 1/3 + 1/3. If you subtract 2/3 from that there will be 2/3 leftover. Madeline had 7/6 yards of fabric. She cut off one yard to make curtains. How much fabric was left? 7/6 = 6/6 + 1/6 so she had 1/6 of one yard of fabric left. Lauren had 7/3 pans of brownies leftover after a party. Her brother ate 2/3 of a pan of brownies. What part of the total amount of brownies was left? 7/3 = 3/3 + 3/3 + 1/3 or 2 1/3. 2 1/3 = 1 + 1/3 + 1/3 + 1/3 +1/3 and if we subtract 2/3 there will be 1 2/3 pans of brownies leftover. Task 2: Use unit fractions to solve. Use what you know about unit fractions to solve the problems. Write an equation that includes unit fractions to show the answer for each problem. 2/3 + 2/3 = 1¼–¾= ½x5= 4/3 = 1/3 + x 2 1/5 – 3/5 = 1 3/5 + 4/5 = Rubric Level I Level II Limited Performance Not Yet Proficient Students are unable to use Students can use models to solve models or unit fractions to the problems in Task 1 but have solve the problems in Tasks 1 difficulty writing equations with and 2. unit fractions in Task 1 and/or 2. NC DEPARTMENT OF PUBLIC INSTRUCTION 5/7 + y = 12/7 8/9 – 3/9 = 2/3 x 4 = Level III Proficient in Performance Students can use models and equations with unit fractions to solve the problems in Tasks 1 & 2. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pattern Blocks & Unit Fractions Use the pattern blocks to model and solve these problems. Write an equation to represent each situation. Use unit fraction to represent each pattern block in the equation. Scott and Zachary shared a sub. Scott ate 2/6 of the sub. Zachary ate 1/6 of the sub. How much of the sub did they eat together? Three students were sharing 2 pies. Each student ate ½ of a pie. How much of the total amount of pie did they eat together? Trevor had 1 1/3 pizzas. His dad ate 2/3 of a pizza. How much pizza was left? Madeline had 7/6 yards of fabric. She cut off one yard to make curtains. How much fabric was left? Lauren had 7/3 pans of brownies leftover after a party. Her brother ate 2/3 of a pan of brownies. What part of the total amount of brownies was left? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Task 2: Use unit fractions to solve. Use what you know about unit fractions to solve the problems. Write an equation that includes unit fractions to show the answer for each problem. 2/3 + 2/3 = 4/3 = 1/3 + x 5/7 + y = 12/7 1¼–¾= 2 1/5 – 3/5 = 8/9 – 3/9 = ½x5= 1 3/5 + 4/5 = 2/3 x 4 = NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Dividing Up the Land 4.NF.3- Task 5 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Build fractions from unit fractions. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Other Standard: 4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Paper, pencil Pattern blocks Virtual pattern blocks can be found herehttp://www.mathplayground.com/patternblocks.html There is a plot of land shaped like the figure below. Each hexagon has a value of 1 whole unit. The plot of land, therefore, has a value of 3 whole units. Determine how to use pattern blocks to divide the shape up into the following ways. For each way, make a picture and write an equation. Part 1: The land owner will only sell the land in sections that are one-third of a unit. The following people buy land: Taylor: 2 sections Bill: 1 section Nick: 4 sections Use your pattern blocks to make a picture of how the land was divided up. Is there any land left? If so, how much? Write an equation to show how the land was split up by the land owner Part 2: The land owner will only sell the land in sections that are one-sixth of a unit. The following people buy land: Tom: 3 sections Susan: 2 sections Bob: 4 sections Mallory: 1 section Wes: 6 sections Use your pattern blocks to make a picture of how the land was divided up. Is there any land left? If so, how much? Write an equation to show how the land was split up by the land owner. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance The student has not shown a clear understanding about how to represent the pieces of cake as fractions. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Answer is correct, but the equation or explanation is incorrect OR work is logically shown but the student has made a calculation error. Level III Proficient in Performance Solutions: Part 1: There are 2 sections left or 2/3 of a unit left. 2/3 + 1/3 + 4/3 + 2/3 = 3. Part 2: There are 2 sections left or 2/6 of a unit. 3/6 + 2/6 + 4/6 + 1/6 + 6/6 + 2/6. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Dividing Up the Land There is a plot of land shaped like the figure below. Each hexagon has a value of 1 whole unit. The plot of land, therefore, has a value of 3 whole units. Determine how to use pattern blocks to divide the shape up into the following ways. For each way, make a picture and write an equation. Part 1 The land owner will only sell the land in sections that are one-third of a unit. The following people buy land: Taylor: 2 sections Bill: 1 section Nick: 4 sections Use your pattern blocks to make a picture of how the land was divided up. Is there any land left? If so, how much? Write an equation to show how the land was split up by the land owner. Include any unsold land. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Part 2 The land owner will only sell the land in sections that are one-sixth of a unit. The following people buy land: Tom: 3 sections Susan: 2 sections Bob: 4 sections Mallory: 1 section Wes: 6 sections Use your pattern blocks to make a picture of how the land was divided up. Is there any land left? If so, how much? Write an equation to show how the land was split up by the land owner. Include any unsold land. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks How Much Punch is Left? 4.NF.3- Task 6 Domain Cluster Standard(s) Materials Task Number & Operations- Fractions Build fractions from unit fractions. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Other Standard: 4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Paper, pencil Part 1 There are 2 gallons of punch left in the punch bowl. It gets divided between 8 students, with each getting a different amount. a) Micah takes 1/12 of a gallon of punch. b) Roberta takes three times as much Micah. c) Steve takes twice as much as Roberta. d) Yanni takes 2/12 of a gallon of punch less than Steve. e) Amy takes 1/12 of a gallon of punch less than Yanni. f) The remaining punch is divided between Tom, Jackie, and Henry. g) Tom and Jackie had the same amount of punch. h) Henry had less punch than both Tom and Jackie. How much punch did each person take? Draw a picture and write an equation to match this context. Part 2 At the next party, the amount of punch doubled to 4 gallons. Each person took the same fraction of the punch. How much would each person get? Level I Limited Performance The student has not shown a clear understanding about how to represent the pieces of cake as fractions. Rubric Level II Not Yet Proficient Answer is correct, but the equation or explanation is incorrect OR work is logically shown but the student has made a calculation error. NC DEPARTMENT OF PUBLIC INSTRUCTION Level III Proficient in Performance Solutions: Part 1: Micah: 1/12; Roberta, 3/12 or ¼; Steve: 6/12 or ½; Yanni: 4/12 or 1/3; Amy: 3/12 or ¼: Henry 1/12 while Tom and Jackie each get 3/12. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks How Much Punch is Left? Part 1 There are 2 gallons of punch left in the punch bowl. It gets divided between 8 students, with each getting a different amount. a) Micah takes 1/12 of a gallon of punch. b) Roberta takes three times as much Micah. c) Steve takes twice as much as Roberta. d) Yanni takes 2/12 of a gallon of punch less than Steve. e) Amy takes 1/12 of a gallon of punch less than Yanni. f) The remaining punch is divided between Tom, Jackie, and Henry. g) Tom and Jackie had the same amount of punch. h) Henry had less punch than both Tom and Jackie. How much punch did each person take? Draw a picture and write an equation to match this context. Part 2 At the next party, the amount of punch doubled to 4 gallons. Each person took the same fraction of the punch. How much would each person get? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Boxing Up Leftover Brownies 4.NF.3 Task 7 Domain Cluster Standard(s) Number and Operations - Fractions Build fractions from unit fractions. 4.NF3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 4.NF.3d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Materials Task Activity sheet Boxing Up Leftover Brownies Amaria has brownies at her birthday party. At the end of the party there are the following brownies left over: 5 brownies with cream cheese frosting 4 plain chocolate brownies 3 chocolate brownies with nuts 7 brownies with caramel frosting Part 1: After the party the brownies are put into boxes. A box can hold 8 brownies. If each type of brownie were packed into their own box, what fraction of a box does each type of brownie take up? Draw pictures below to show your work. Part 2: Amaria and her Mom want to use fewer boxes and put different types of brownies into the same box. How many whole boxes do they fill? Will there be a box partially filled? If so what fraction of the box is partially filled? Draw pictures to show your work. Part 3: Write an equation to match the picture that you drew in Part 2. Part 4: Is there space for any more brownies? If so how many more brownies do you have room for? Write an equation that shows your work. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Solutions include many errors and show limited understanding. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Solutions will include between 1 to 3 errors in various parts of the task. Level III Proficient in Performance Solutions include correct answers and show a deep understanding of concepts. Answers: Part 1: Pictures are correctly drawn and fractions are correctly labeled. Cream cheese: 5/8. Plain: 4/8. Nuts: 3/8. Caramel: 7/8. Part 2: Picture is correctly drawn. Answer is 2 and 3/8. Part 3: 5/8 + 4/8 + 3/8 + 7/8 = 2 and 3/8. Part 4: There is space for 5 more brownies or there is 5/8 of a box empty. Equation: 3 – 2 and 3/8 = 5/8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Boxing Up Leftover Brownies Amaria has brownies at her birthday party. At the end of the party there are the following brownies left over: 5 brownies with cream cheese frosting 4 plain chocolate brownies 3 chocolate brownies with nuts 7 brownies with caramel frosting Part 1: After the party the brownies are put into boxes. A box can hold 8 brownies. If each type of brownie were packed into their own box, what fraction of a box does each type of brownie take up? Draw pictures below to show your work. Part 2: Amaria and her Mom want to use fewer boxes and put different types of brownies into the same box. How many whole boxes do they fill? Will there be a box partially filled? If so what fraction of the box is partially filled? Draw pictures to show your work. Part 3: Write an equation to match the picture that you drew in Part 2. Part 4: Is there space for any more brownies? If so how many more brownies do you have room for? Write an equation that shows your work. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Going the Distance 4.NF.3 Task 8 Domain Cluster Standard(s) Number and Operations - Fractions Build fractions from unit fractions. 4.NF3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.3c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 4.NF.3d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Materials Task Activity sheet Going the Distance In order to train for the Girls on the Run 5K Race, the girls’ running team at Lincoln Elementary School runs the following distances: Week Week 1 Week 2 Week 3 Week 4 Distance 1 and 1/6 miles 1 and 3/6 miles 2 and 4/6 miles 2 and 5/6 miles Part 1: Draw a number line to show the distance that the girls ran each week. Part 2: How far did the girls run in all? Write an equation that matches the story. Part 3: The girls at Jefferson Elementary School ran 10 miles total during the same time. How much farther did they run than the girls at Lincoln Elementary School? Use a picture and an equation to find your answer. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Solutions include many errors and show limited understanding. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Solutions will include between 1 to 3 errors in various parts of the task. Level III Proficient in Performance Solutions include correct answers and show a deep understanding of concepts. Answers: Part 1: The number line matches the distance that the girls ran. Part 2: The girls ran 8 and 1/6 miles. Equation: 1 1/6 + 1 3/6 + 2 4/6 + 2 5/6 = 8 1/6. Part 3: Picture is correct. Equation: 10 – 8 1/6 = 1 5/6 miles. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Going the Distance In order to train for the Girls on the Run 5K Race, the girls’ running team at Lincoln Elementary School runs the following distances: Week Week 1 Week 2 Week 3 Week 4 Distance 1 and 1/6 miles 1 and 3/6 miles 2 and 4/6 miles 2 and 5/6 miles Part 1: Draw a number line to show the distance that the girls ran each week. Part 2: How far did the girls run in all? Write an equation that matches the story. Part 3: The girls at Jefferson Elementary School ran 10 miles total during the same time. How much farther did they run than the girls at Lincoln Elementary School? Use a picture and an equation to find your answer. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pasta Party 4.NF.4 - Task 1 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4 Understand a fraction a/b as a multiple of 1/b (i.e., 5/4 = 5 x ¼ = ¼ + ¼ + ¼ + ¼ + ¼ ); be able to express a multiple of a/b as 1/b and use this to multiply a fraction by a whole number (i.e., 3 x 2/5 = (3 x 2)/5 = 6 x 1/5), or generalize that n x a/b = (n x a)/b; Solve word problems involving multiplication of a fraction by a whole number. Paper and pencil Part 1: Katie makes 1/4 pound of pasta for each person at her dinner party. If seven people attend the party, how many pounds of pasta will be needed for her guests? Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Extension: Students can write their own word problem using ¼ x 7. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Part 1: Student is unable to write an addition equation or draw a model. Part 2: Student is unable to write a multiplication equation or draw a model. Rubric Level II Not Yet Proficient 1. 2. 3. 4. 5. 6. 7. 8. Part 1: Student writes a correct addition equation that totals 7/4 (i.e., ¼ + ¼ + ¼ + ¼ + ¼ + ¼ + ¼ = 7/4), but is unable to show the sum on a number line as seven ‘jumps’ of ¼, or as an area model, and does not clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation (¼ x 7= 7/4), but is unable to show the total 7/4 on a number line or area model, and does not clearly explain how the model matches their multiplication equation. Part 3: Students have some idea how they are alike and different. Level III Proficient in Performance Part 1: Student writes a correct addition equation that totals 7/4 (i.e., ¼+¼+¼+¼+¼+¼+¼= 7/4).They show the sum on a number line as seven ‘jumps’ of ¼, or as an area model, and clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation (¼ x 7= 7/4). They show the total 7/4 on a number line or area model, and clearly explain how the model matches their multiplication equation. Part 3: Students understand how they are alike and different and clearly states. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Pasta Party Part 1: Katie makes 1/4 pound of pasta for each person at her dinner party. If seven people attend the party, how many pounds of pasta will be needed for her guests? Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Extension: Write your own word problem using ¼ x 7. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Drawing a Model 4.NF.4 Task 2 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4 Understand a fraction a/b as a multiple of 1/b (i.e., 5/4 = 5 x ¼ = ¼ + ¼ + ¼ + ¼ + ¼ ); be able to express a multiple of a/b as 1/b and use this to multiply a fraction by a whole number (i.e., 3 x 2/5 = (3 x 2)/5 = 6 x 1/5), or generalize that n x a/b = (n x a)/b; Solve word problems involving multiplication of a fraction by a whole number. Paper and pencil Part 1: Kelly was making curtains for her living room. She bought four pieces of fabric that were each 2/3 yard long. How many yards of fabric did Kelly buy in all? Draw a picture and write an equation to show the total amount of fabric if each piece is 2/3 yard long. Part 2: With the fabric that she bought in part 1, Kelly cut each piece of fabric into a 1/3 yard long piece. Draw a picture and write an equation to show the total amount of fabric if each piece is 1/3 yard long. Part 3: Is the amount of fabric in Part 1 and Part 2 the same? Use pictures to prove it. Write a sentence to explain how you know that you are correct. Rubric Level I Level II Level III Limited Performance Part 1 Student is unable to create a correct equation and/or picture. Not Yet Proficient Part 1 addition equation 2/3 + 2/3 + 2/3 + 2/3 = 8/3 multiplication equation 4 x 2/3 = 8/3 Picture should show four groups of 2/3. Part 2 Addition equation 1/3 + 1/3 + 1/3 + 1/3 + 1/3 +1/3 + 1/3 + 1/3= 8/3 Multiplication equation 8 x 1/3 = 8/3 Picture should show eight groups of 1/3 Part 3 It is unclear from the student’s picture that they understand the relationship 8/3 = 4 x 2/3. Proficient in Performance Part 1 addition equation 2/3 + 2/3 + 2/3 + 2/3 = 8/3 multiplication equation 4 x 2/3 = 8/3 Picture should show four groups of 2/3. Part 2 Addition equation 1/3 + 1/3 + 1/3 + 1/3 + 1/3 +1/3 + 1/3 + 1/3= 8/3 Multiplication equation 8 x 1/3 = 8/3 Picture should show eight groups of 1/3 Part 3 Picture and explanation should show that eight individual copies of one third can be grouped as four groups of two-thirds. Equations: 8 x 1/3 = 4 x 2/3 Part 2 Student is unable to create a correct equation and/or picture. Part 3 The student cannot draw a model to show this relationship or explain it in words or numbers. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Drawing a Model Part 1: Kelly was making curtains for her living room. She bought four pieces of fabric that were each 2/3 yard long. How many yards of fabric did Kelly buy in all? Draw a picture and write an equation to show the total amount of fabric if each piece is 2/3 yard long. Part 2: With the fabric that she bought in part 1, Kelly cut each piece of fabric into a 1/3 yard long piece. Draw a picture and write an equation to show the total amount of fabric if each piece is 1/3 yard long. Part 3: Is the amount of fabric in Part 1 and Part 2 the same? Use pictures to prove it. Write a sentence to explain how you know that you are correct. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Chris’s Cookies 4.NF.4-Task 3 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4 Understand a fraction a/b as a multiple of 1/b (i.e., 5/4 = 5 x ¼ = ¼ + ¼ + ¼ + ¼ + ¼ ); be able to express a multiple of a/b as 1/b and use this to multiply a fraction by a whole number (i.e., 3 x 2/5 = (3 x 2)/5 = 6 x 1/5), or generalize that n x a/b = (n x a)/b; Solve word problems involving multiplication of a fraction by a whole number. Paper and pencil Students will work together or independently to show their solutions to the fraction word problems. Chris’s Cookies Chris is making cookies for his friend’s birthday party using the following recipe. Chocolate Chip Cookies: Makes 2 dozen cookies 2 cups flour 1/2 teaspoon baking soda 1 teaspoon salt 3/5 cups butter, softened 3/4 cups sugar 1/2 cup light brown sugar 1 egg 1 teaspoon vanilla extract 1 package (6 ounces) chocolate chips 1/2 cup chopped walnuts From http://www.mccormick.com/ 1. How much butter will he need to make 3 batches of cookies? Write an equation to show your answer. 2. How much butter will Chris need to make 6 batches of cookies? Write an equation to show your answer. 3. How much butter will Chris need to make 9 batches of cookies? Write an equation to show your answer. 4. What patterns do you notice in the amounts of butter needed for 3 batches, 6 batches, and 9 batches of cookies? 5. How can we use these patterns to predict the amount of butter needed for 18 batches? 36 batches? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Students may or may not be able to correctly compute answers to 1-3. They may still be relying on addition to compute. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient 1. 3 x 3/5 = 9/5 2. 6 x 3/5 = 18/5 3. 9 x 3/5 = 27/5 4. Students may notice patterns but they have difficulty using them to predict answers. Level III Proficient in Performance 1. 3 x 3/5 = 9/5 2. 6 x 3/5 = 18/5 3. 9 x 3/5 = 27/5 4. Students may notice patterns such as: Alternating odd and even totals (9/5, 18/5, 27/5) n x a/b = (n x a)/b Multiplying the numerator times the whole number always yields the answer. As the multiplier (3) grows by 3, the amount of butter increases by 9. Students may notice that this is because the numerator is 3. To predict the amount of butter for 18 batches, students may notice that they can double 3/5 x 9 since 9 doubled is 18 so that they have (3/5 x 9) x 2 = 27/5 x 2 = 54/2. To predict the amount of butter needed for 36 batches, students may double the amount for 18 batches, multiply the amount needed for 9 batches by 4, etc. (3/5 x 3) x 12 = 9/5 x 12 = 108/5 (3/5 x 6) x 6 = 18/5 x 6 = 108/5 (3/5 x 9) x 4 = 27/5 x 4 = 108/5 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Chris’s Cookies Chris is making cookies for his friend’s birthday party using the following recipe. Chocolate Chip Cookies: Makes 2 dozen cookies 2 cups flour 1/2 teaspoon baking soda 1 teaspoon salt 3/5 cups butter, softened 3/4 cups sugar 1/2 cup light brown sugar 1 egg 1 teaspoon vanilla extract 1 package (6 ounces) chocolate chips 1/2 cup chopped walnuts From http://www.mccormick.com/ 1. How much butter will he need to make 3 batches of cookies? Write an equation to show your answer. 2. How much butter will Chris need to make 6 batches of cookies? Write an equation to show your answer. 3. How much butter will Chris need to make 9 batches of cookies? Write an equation to show your answer. 4. What patterns do you notice in the amounts of butter needed for 3 batches, 6 batches, and 9 batches of cookies? 5. How can we use these patterns to predict the amount of butter needed for 18 batches? 36 batches? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Going the Distance 4.NF.4 - Task 4 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4 Understand a fraction a/b as a multiple of 1/b (i.e., 5/4 = 5 x ¼ = ¼ + ¼ + ¼ + ¼ + ¼ ); be able to express a multiple of a/b as 1/b and use this to multiply a fraction by a whole number (i.e., 3 x 2/5 = (3 x 2)/5 = 6 x 1/5), or generalize that n x a/b = (n x a)/b; Solve word problems involving multiplication of a fraction by a whole number. Paper and pencil During her first week of training for the Girls on the Run 5K, Molly runs ¾ of a mile each day for 6 days. Part 1: Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Part 4 Molly’s friend Tonya ran 1 and 2/4 miles each day for 3 days. Who ran more? Explain your reasoning. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Part 1: Student is unable to write an addition equation or draw a model. Part 2: Student is unable to write a multiplication equation or draw a model. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Part 1: Student writes a correct addition equation, but is unable to show the sum on a number line as seven ‘jumps’ of ¼, or as an area model, and does not clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation, but is unable to show the total on a number line or area model, and does not clearly explain how the model matches their multiplication equation. Part 3: Students communicate unclearly how they are alike and different. Part 4: Student does not clearly communicate that they need 4 containers that are 1 cup each. Level III Proficient in Performance Part 1: Student writes a correct addition equation that totals 18/4 (3/4+3/4+3/4+3/4+3/4+3/4= 18/4). They show the sum on a number line as five ‘jumps’ of 3/4 or as an area model, and clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation ¾ x 6 = 18/4). They show the total 18/4 on a number line or area model, and clearly explain how the model matches their multiplication equation. Part 3: Student understands how they are alike and different and clearly states. Part 4: Student communicates that Molly and Tanya have run the same amount. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Going the Distance During her first week of training for the Girls on the Run 5K, Molly runs ¾ of a mile each day for 6 days. Part 1: Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Part 4 Molly’s friend Tonya ran 1 and 2/4 miles each day for 3 days. Who ran more? Explain your reasoning. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Serving Ice Cream 4.NF.4 - Task 5 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.4 Understand a fraction a/b as a multiple of 1/b (i.e., 5/4 = 5 x ¼ = ¼ + ¼ + ¼ + ¼ + ¼ ); be able to express a multiple of a/b as 1/b and use this to multiply a fraction by a whole number (i.e., 3 x 2/5 = (3 x 2)/5 = 6 x 1/5), or generalize that n x a/b = (n x a)/b; Solve word problems involving multiplication of a fraction by a whole number. Paper and pencil Katie uses 2/3 of a cup of ice cream for each ice cream sundae that she makes. For a party she makes 5 sundaes. Part 1: Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Part 4 If ice cream were sold in 1 cup containers how many containers does she need to buy for her party? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Part 1: Student is unable to write an addition equation or draw a model. Part 2: Student is unable to write a multiplication equation or draw a model. Rubric Level II Not Yet Proficient 1. 2. 3. 4. 5. 6. 7. 8. Part 1: Student writes a correct addition equation, but is unable to show the sum on a number line as seven ‘jumps’ of ¼, or as an area model, and does not clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation, but is unable to show the total on a number line or area model, and does not clearly explain how the model matches their multiplication equation. Part 3: Students communicate unclearly how they are alike and different. Part 4: Student does not clearly communicate that they need 4 containers that are 1 cup each. Level III Proficient in Performance Part 1: Student writes a correct addition equation that totals 10/3 (2/3 + 2/3+2/3+2/3+2/3).They show the sum on a number line as five ‘jumps’ of 2/3 or as an area model, and clearly explain how the model matches their addition equation. Part 2: Student writes a correct multiplication equation 2/3 x 5 = 10/3). They show the total 10/3 on a number line or area model, and clearly explain how the model matches their multiplication equation. Part 3: Student understand how they are alike and different and clearly states. Part 4: Student communicates that in order to buy 10/3 cups of ice cream they must buy 4 containers that are 1 cup each. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Serving Ice Cream Katie uses 2/3 of a cup of ice cream for each ice cream sundae that she makes. For a party she makes 5 sundaes. Part 1: Write an addition equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows addition. Part 2: Write a multiplication equation to show this situation. Show your answer with a number line or an area model. Use numbers or words to explain how your model shows multiplication. Part 3 How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? Part 4 If ice cream were sold in 1 cup containers how many containers does she need to buy for her party? Write a sentence explaining your reasoning. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Karen’s Garden 4.NF.5 Task 1 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.5 Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100, and use this technique to add two fractions with respective denominators 10 and 100. Express 3/10 as 30/100 and add 3/10 + 4/100 =34/100. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil In this task, students will be using decimal grids (hundredths) to shade in tenths and hundredths as decimals and fractions, and find totals and differences. Introduce the manipulatives and task with the following practice problem. Karen is planting vegetables in her 10’ x 10’ garden. She wants 3/10 of the garden to be tomatoes. If Karen has already planted peas in 0.25 of the garden, how much space will she have left for other vegetables? Some students may be comfortable using hundredths grids while others may need the additional support of using base ten blocks (flat = 1 whole, ten stick = 1/10, unit cube = 1/100). Students may work on the Karen’s Garden sheet independently or in pairs if they need support. As they work, they should shade in hundredths grids or use base ten blocks to solve the addition or subtraction. Rubric Level I Limited Performance Solutions will include many errors in conversions and equivalencies, as well as addition and subtraction errors. Students may struggle to choose appropriate numbers from the problems for computation. Level II Not Yet Proficient Students should be able to convert decimals (tenths and hundredths) to fractions with 10 or 100 as a denominator. Their solutions should show that they can work flexibly with fractions and decimals. At this level, their solutions will include errors in several conversions and equivalencies. NC DEPARTMENT OF PUBLIC INSTRUCTION Level III Proficient in Performance At this level, their solutions will include few or no errors in conversions and equivalencies. Possible solutions: 1. 76/100 + 24/100 = 100/100 2. 0.6 + 0.23 = 0.83 6/10 + 23/100 = 83/100 60/100 + 23/100 = 83/100 3. 0.5 – 0.14 = 0.36 0.14 + 0.36 = 0.50 14/100 + 36/100 = 5/10 or 50/100 4. 2/10 + 0.2 + 20/100 = 0.6 or 6/10 or 60/100 5. Students should figure out that 0.6 + 0.36 = 0.96 of the original grid, so 0.04 would be leftover for radishes. If radishes are now taking up 22/100 of the garden, 0.22 – 0.04 = 0.18. The deer ate peas in 0.18 of the garden. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Karen’s Garden Use 10 x 10 grids to shade in your solutions to these problems. 1. Karen planted squash and watermelon in her 10’ x 10’ garden. She planted squash in 0.4 of her garden, but it grew so much that it took up 76/100 of the garden. How much space is left for watermelon? 2. Six tenths of the garden is planted with green beans. Twenty-three hundredths is planted with radishes. How much of the garden is planted? 3. Karen wants half of her garden to be planted with tomatoes. She has planted 0.14 of the garden with tomatoes so far. How much of the garden does she still need to plant with tomatoes? 4. Karen planted lettuce in 2/10 of her garden, peas in 0.2 of her garden, and peppers in 20/100 of the garden. How much of the garden is planted? 5. Six tenths of the garden was planted with peppers. Thirty six hundredths of the garden was planted with peas, but some of them were eaten by deer. Karen planted radishes in the leftover space and the empty space where the peas were eaten by deer. If radishes now take up 22/100 of the garden, how much of the garden did the deer eat? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Filling the Jar 4.NF.5 Task 2 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.5 Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100, and use this technique to add two fractions with respective denominators 10 and 100. Express 3/10 as 30/100 and add 3/10 + 4/100 =34/100. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil In this task, students will be using decimal grids (hundredths) to shade in tenths and hundredths as decimals and fractions, and find totals and differences. Part 1: A jar can hold 100 marbles. For each of the situations below, find the amount for the various colors of marbles. 3 2 19 Jar A: The jar contains blue, white, and red marbles. The rest are yellow. 10 10 100 What fraction represents the number of yellow marbles? _____ 7 1 4 green, purple, and brown. The rest are white. What fraction represents 10 10 100 the number of white marbles? ______ Jar B: 3 51 gray and black. The rest are pink or yellow. There are more pink than 10 100 yellow marbles. What fraction represents the number of pink and yellow marbles? ____ pink, ____ yellow. Jar C: 2 18 clear and orange. The rest are red and blue. There are 2 more red marbles 10 100 than blue marbles. Jar D: What fraction represents the number of red and blue marbles? ___ red, ____ blue. Part 2: Pick one of the tasks above. Explain how you worked with the different denominators to find your answer. Some students may be comfortable using hundredths grids while others may need the additional support of using base ten blocks (flat = 1 whole, ten stick = 1/10, unit cube = 1/100). NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Level I Limited Performance Solutions will include many errors in conversions and equivalencies, as well as addition and subtraction errors. Students may struggle to choose appropriate numbers from the problems for computation. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students should be able to convert decimals (tenths and hundredths) to fractions with 10 or 100 as a denominator. Their solutions should show that they can work flexibly with fractions and decimals. At this level, their solutions will include errors in several conversions and equivalencies. Level III Proficient in Performance Students correctly find the answer to each problem. Jar A: 31/100 yellow. Jar B: 16/100 white. Jar C: Multiple possible answers. Both fractions must add up to 19/100 and the fraction for pink must be greater than yellow. Jar D: Red: 32/100, Blue: 30/100. Part 2: The answer discusses that tenths can also be written as hundredths by multiplying the numerator and denominator both by 10. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Filling the Jar A jar can hold 100 marbles. For each of the situations below, find the amount for the various colors of marbles. Jar A: The jar contains 3 2 19 blue, white, and red marbles. The rest are yellow. 10 10 100 What fraction represents the number of yellow marbles? _____ Jar B: 7 1 4 green, purple, and brown. The rest are white. What fraction represents 10 10 100 the number of white marbles? ______ Jar C: 3 51 gray and black. The rest are pink or yellow. There are more pink than 10 100 yellow marbles. What fraction represents the number of pink and yellow marbles? ____ pink, ____ yellow. Jar D: 2 18 clear and orange. The rest are red and blue. There are 2 more red marbles 10 100 than blue marbles. What fraction represents the number of red and blue marbles? ___ red, ____ blue. Part 2: Pick one of the tasks above. Explain how you worked with the different denominators to find your answer. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Children’s Shirts 4.NF.5 Task 3 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.5 Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100, and use this technique to add two fractions with respective denominators 10 and 100. Express 3/10 as 30/100 and add 3/10 + 4/100 =34/100. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil Part 1: 2 A group of 100 fourth graders are on the playground. of the group are wearing black 10 11 3 shirts. of the group are wearing blue shirts. of the group are wearing white shirts. 100 10 The rest of the class is wearing green or yellow shirts. There are more children wearing yellow shirts than green shirts. What fraction of the group is wearing yellow shirts? Green shirts? Find at least three different correct solutions. Part 2: Describe your process for solving the task above. Rubric Level I Level II Limited Performance Not Yet Proficient Solutions will include Students should be able to many errors in convert decimals (tenths and conversions and hundredths) to fractions with equivalencies, as well 10 or 100 as a denominator. as addition and Their solutions should show subtraction errors. that they can work flexibly Students may struggle with fractions and decimals. to choose appropriate At this level, their solutions numbers from the will include errors in several problems for conversions and computation. equivalencies. 1. 2. 3. 4. 5. 6. 7. 8. Level III Proficient in Performance There are 39 students wearing a green or yellow shirt. Students should be able to find three correct solutions that show a that there are more yellow than green shirts and that add up to 39 of the 100 students. Students should be able to explain the process for finding solutions. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Children’s Shirts A group of 100 fourth graders are on the playground. wearing black shirts. 11 100 2 10 of the group are of the group are wearing blue shirts. 3 10 of the group are wearing white shirts. The rest of the class is wearing green or yellow shirts. There are more children wearing yellow shirts than green shirts. What fraction of the group is wearing yellow shirts? What fraction of the group is wearing green shirts? Find at least three different correct solutions. Part 2: Describe how you used equivalent fractions to solve the task above. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Where am I now? How much farther? 4.NF.6 Task 1 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil Where am I now? How much farther? Part 1: You have walked 0.32 of the way from your house to the school. A) What is that distance as a fraction? B) If you walk another 17/100 of the way to the school how far have you gone now in terms of both a fraction and a decimal? C) If you walk another 3/10 of the way to the school how far have you gone now in terms of both of a fraction and a decimal? D) Draw a number line from 0 to 1 and label each tenth on the number line. Plot the three distances above as both fractions and decimals. Part 2: Write an explanation explaining how you worked with equivalent fractions and decimals to solve this task. Level I Limited Performance Solutions will include many errors in conversions and equivalencies, as well as addition and subtraction errors. Students may struggle to choose appropriate numbers from the problems for computation. Rubric Level II Not Yet Proficient Students should be able to convert decimals (tenths and hundredths) to fractions with 10 or 100 as a denominator. Their solutions should show that they can work flexibly with fractions and decimals. At this level, their solutions will include errors in several conversions and equivalencies. NC DEPARTMENT OF PUBLIC INSTRUCTION Level III Proficient in Performance Students correctly find the answer to each problem. A) 32/100. B) 49/100, C) 79/100, D) all 3 fractions and decimals are correctly plotted on a number line. Part 2: The answer discusses that tenths can also be written as hundredths by multiplying the numerator and denominator both by 10, and that decimals are equivalent to fractions that have either a 10 or 100 in the denominator. FOURTH GRADE Formative Instructional and Assessment Tasks 1. 2. 3. 4. 5. 6. 7. 8. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Where Am I Now? How Much Farther? Part 1: You have walked 0.32 of the way from your house to the school. A) What is that distance as a fraction? B) If you walk another 17/100 of the way to the school how far have you gone now in terms of both a fraction and a decimal? C) If you walk another 3/10 of the way to the school how far have you gone now in terms of both of a fraction and a decimal? D) Draw a number line from 0 to 1 and label each tenth on the number line. Plot the three distances above as both fractions and decimals. Part 2: Write an explanation explaining how you worked with equivalent fractions and decimals to solve this task. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Is the Tire Full Yet? 4.NF.6 Task 2 Domain Cluster Standard(s) Materials Task Number and Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil Is the tire full? Part 1: You are about to go on vacation with your family. Your tire is 78/100 full of air. If you add air so that it 0.93 full, what fraction of the tire did you just fill with air? What fraction (and decimal) of the tire still needs to be filled with air? Part 2: Write an explanation explaining how you worked with equivalent fractions and decimals to solve this task. Level I Limited Performance Solutions will include many errors in conversions and equivalencies, as well as addition and subtraction errors. Students may struggle to choose appropriate numbers from the problems for computation. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students should be able to convert decimals (tenths and hundredths) to fractions with 10 or 100 as a denominator. Their solutions should show that they can work flexibly with fractions and decimals. At this level, their solutions will include errors in several conversions and equivalencies. Level III Proficient in Performance Students correctly find the answer to each problem. A) 15/100 or 0.15. B) 7/100 or 0.07. Part 2: The answer discusses that tenths can also be written as hundredths by multiplying the numerator and denominator both by 10, and that decimals are equivalent to fractions that have either a 10 or 100 in the denominator. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Is the Tire Full? Part 1: You are about to go on vacation with your family. Your tire is 78/100 full of air. If you add air so that it 0.93 full, what fraction of the tire did you just fill with air? What fraction (and decimal) of the tire still needs to be filled with air? Part 2: Write an explanation explaining how you worked with equivalent fractions and decimals to solve this task. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who Jumped Farther? 4.NF.7-Task 1 Domain Cluster Standard(s) Materials Number and Operations – Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil Who Jumped Farther? Part 1: Tom, Steve, and Peter each jump in a jumping contest. They measure their jumps and then discuss who jumped farther. Tom said, “I jumped 1 and 34 hundredths of a meter.” Steve said, “I jumped 1 and 43 hundredths of a meter.” Peter said, “I jumped 1 and 4 tenths of a meter.” For each ingredient, shade in the decimal grid and write a comparison statement using >, <, or =. Which boy jumped farther between Tom and Steve? How do you know? Which boy jumped farther between Tom and Peter? How do you know? Which boy jumped farther between Steve and Peter? How do you know? Part 2: Mitch jumped a distance between Steve and Peter. How far could he have jumped? Alex jumped a distance between Tom and Peter. How far could he have jumped? Level I Limited Performance Students are unable to accurately compare decimals. 1. 2. 3. 4. 5. 6. 7. 8. Rubric Level II Not Yet Proficient Students make 1 or 2 errors comparing decimals. OR students get correct answers but do not provide clear and accurate explanations. Level III Proficient in Performance The student provides correct answers. Part 1: 1.2 > 1.02, 1.2<1.23, 1.02<1.23. Part 2: You need either 1.21 or 1.22 Liters of fruit juice. All explanations are clear and accurate. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Who Jumped Farther? Part 1: Tom, Steve, and Peter each jump in a jumping contest. They measure their jumps and then discuss who jumped farther. Tom said, “I jumped 1 and 34 hundredths of a meter.” Steve said, “I jumped 1 and 43 hundredths of a meter.” Peter said, “I jumped 1 and 4 tenths of a meter.” Shade in the decimal grids for each of the boys’ distances. Which boy jumped farther between Tom and Steve? How do you know? Which boy jumped farther between Tom and Peter? How do you know? Which boy jumped farther between Steve and Peter? How do you know? Part 2: Mitch jumped a distance between Steve and Peter. How far could he have jumped? Alex jumped a distance between Tom and Peter. How far could he have jumped? NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Making Punch 4.NF.7-Task 2 Domain Cluster Standard(s) Materials Number and Operations – Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Decimal grids (hundredths grids), base ten blocks (optional), paper and pencil Making Punch Part 1: While making punch for a party the following ingredients are needed: 1.2 Liters of Ginger Ale 1.02 Liters of Sprite 1.23 Liters of Fruit Juice For each ingredient, shade in the decimal grid and write a comparison statement using >, <, or =. Do you need more Ginger Ale or Sprite? Explain your reasoning. Do you need more Ginger Ale or Fruit Juice? Explain your reasoning. Do you need more Sprite or Fruit Juice? Explain your reasoning. Part 2: For a different recipe, frozen yogurt can be added. The amount of frozen yogurt is between the amount of Ginger Ale and Fruit Juice. How much frozen yogurt is needed? Explain how you found your answer. Rubric Level I Limited Performance Students are unable to accurately compare decimals. 1. 2. 3. 4. 5. 6. 7. 8. Level II Level III Not Yet Proficient Proficient in Performance Students make 1 or 2 errors The student provides correct answers. comparing decimals. OR Part 1: 1.2 > 1.02, 1.2<1.23, 1.02<1.23. students get correct answers but Part 2: You need either 1.21 or 1.22 do not provide clear and accurate Liters of fruit juice. explanations. All explanations are clear and accurate. 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. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks Making Punch Part 1: While making punch for a party the following ingredients are needed: 1.2 Liters of Ginger Ale 1.02 Liters of Sprite 1.23 Liters of Fruit Juice For each ingredient, shade in the decimal grid. Do you need more Ginger Ale or Sprite? Explain your reasoning. Do you need more Ginger Ale or Fruit Juice? Explain your reasoning. Do you need more Sprite or Fruit Juice? Explain your reasoning. Part 2: For a different recipe, frozen yogurt can be added. The amount of frozen yogurt is between the amount of Ginger Ale and Fruit Juice. How much frozen yogurt is needed? Explain how you found your answer. NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE Formative Instructional and Assessment Tasks NC DEPARTMENT OF PUBLIC INSTRUCTION FOURTH GRADE