Mathematics Task Arcs


 Gillian McCormick
 2 years ago
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1 Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number of standards within a domain of the Common Core State Standards for Mathematics. In some cases, a small number of related standards from more than one domain may be addressed. A unique aspect of the task arc is the identification of essential understandings of mathematics. An essential understanding is the underlying mathematical truth in the lesson. The essential understandings are critical later in the lesson guides, because of the solution paths and the discussion questions outlined in the share, discuss, and analyze phase of the lesson are driven by the essential understandings. The Lesson Progression Chart found in each task arc outlines the growing focus of content to be studied and the strategies and representations students may use. The lessons are sequenced in deliberate and intentional ways and are designed to be implemented in their entirety. It is possible for students to develop a deep understanding of concepts because a small number of standards are targeted. Lesson concepts remain the same as the lessons progress; however the context or representations change. Bias and sensitivity: Social, ethnic, racial, religious, and gender bias is best determined at the local level where educators have indepth knowledge of the culture and values of the community in which students live. The TDOE asks local districts to review these curricular units for social, ethnic, racial, religious, and gender bias before use in local schools. Copyright: These task arcs have been purchased and licensed indefinitely for the exclusive use of Tennessee educators.
2 mathematics Grade 2 Put Together & Compare Situational Tasks: Missing Addend Addition as Subtraction A SET OF RELATED S UNIVERSITY OF PITTSBURGH
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4 Table of Contents 3 Table of Contents Introduction Overview... 7 Identified CCSSM and Essential Understandings... 8 Tasks CCSSM Alignment... 9 Lesson Progression Chart Tasks and Lesson Guides TASK 1: Picking Flowers Lesson Guide TASK 2: Fruit Pies Lesson Guide TASK 3: What s the Unknown? Lesson Guide TASK 4: Addition and Subtraction Lesson Guide TASK 5: Pencils Lesson Guide TASK 6: Basket of Fruit Lesson Guide TASK 7: Beads Lesson Guide TASK 8: Comparisons Lesson Guide... 54
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6 mathematics Grade 2 Introduction Put Together & Compare Situational Tasks: Missing Addend Addition as Subtraction A SET OF RELATED S
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8 Introduction 7 Overview This set of related lessons provides a study of story situations that explore the use of subtraction to solve missing addend addition problems. The related tasks are aligned to 2.OA.A.1 and 2.NBT.B.7 of the Content Standards of the CCSSM. Task 1 begins with a put together situation with an unknown sum to connect to students prior knowledge and then builds upon that knowledge with another put together situation, but with an unknown addend in the second position instead of an unknown sum. Task 2 continues the exploration of put together situations with unknown addends in the second position. Task 3 further develops an understanding of put together situations with increased cognitive demand because the situations now have the unknown addend in the first and second position. Subtraction is introduced as a means for solving missing addend problems in Task 1, reinforced in Tasks 2 and 3, and solidified in Task 4. Task 5 introduces compare situations and solving for an unknown difference as how many more. Task 6 continues with compare situations and solving for an unknown difference, but increases in cognitive demand by introducing the phrase how many fewer in addition to the phrase how many more. Task 7 further increases in cognitive demand by asking students to make claims of comparison. Task 8 solidifies understanding of compare situations with unknown differences. Students are challenged to set up and use both a missing addend addition equation and a subtraction equation when solving compare problems. Students will discuss the reason why either operation can be used to solve compare problems. The prerequisite knowledge necessary to enter these lessons is an understanding of sets, onetoone correspondence, and counting. Students will also benefit from being familiar with add to situational problems, as well as the format of addition equations and subtraction equations. Through engaging in the lessons in this set of related tasks, students will: solve put together situational tasks using a variety of strategies and partpartwhole maps, diagrams, and/or physical models; determine if the unknown in put together situational tasks represents a part in the situation or the whole amount in the situation; solve compare situational tasks (unknown difference only) using a variety of strategies and partpartwhole maps, and comparison models; determine the unknown difference as being how many more or how many fewer ; and utilize subtraction to solve missingaddend problems. By the end of these lessons, students will be able to answer the following overarching questions: What strategies can be used to solve story problems with an unknown addend? What strategies can be used to solve compare story problems? The questions provided in the guide will make it possible for students to work in ways consistent with the Standards for Mathematical Practice. It is not the Institute for Learning s expectation that students will name the Standards for Mathematical Practice. Instead, the teacher can mark agreement and disagreement of mathematical reasoning or identify characteristics of a good explanation (MP3). The teacher can note and mark times when students independently provide an equation and then recontextualize the equation in the context of the situational problem (MP2). The teacher might also ask students to reflect on the benefit of using repeated reasoning, as this may help them understand the value of this mathematical practice in helping them see patterns and relationships (MP8). In study groups, topics such as these should be discussed regularly because the lesson guides have been designed with these ideas in mind. You and your colleagues may consider labeling the questions in the guide with the Standards for Mathematical Practice.
9 8 Introduction Identified CCSSM and Essential Understandings CCSS for Mathematical Content Operations and Algebraic Thinking Essential Understandings Represent and solve problems involving addition and subtraction. 2.OA.A.1 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. When sets are compared there is a onetoone correspondence between items within the sets, and the unaligned items indicate the amount that the sets are different (amount less or amount more). 2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting threedigit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations; = 45, = 28) Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative Property; = ) The CCSS for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. *Common Core State Standards, 2010, NGA Center/CCSSO
10 Introduction 9 Tasks CCSSM Alignment Task 2.OA.A.1 2.NBT.B.7 MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8 Task 1 Picking Flowers Developing Understanding Task 2 Fruit Pies Developing Understanding Task 3 What s the Unknown? Developing Understanding Task 4 Addition and Subtraction Solidifying Understanding Task 5 Pencils Developing Understanding Task 6 Basket of Fruit Developing Understanding Task 7 Beads Developing Understanding Task 8 Comparisons Solidifying Understanding
11 10 Introduction Lesson Progression Chart Overarching Questions What strategies can be used to solve story problems with an unknown addend? What strategies can be used to solve compare story problems? Task 1 Picking Flowers Developing Understanding Task 2 Fruit Pies Developing Understanding Task 3 What s the Unknown? Developing Understanding Task 4 Addition and Subtraction Solidifying Understanding Content Solves put together situational problems, total unknown, addend unknown (first position). Explores using subtraction to solve an unknown addend addition problem. Solves put together situational problems with 2 and 3 addends, addend unknown (first position). Uses subtraction to solve an unknown addend addition problem. Solves put together situational problems with two and three addends, addend unknown (first and second position). Uses subtraction to solve an unknown addend addition problem. Solidifies understanding of subtraction as an unknown addend addition problem (unknown addend in first and second position).. Strategy Counting: on back Partpartwhole map diagram Counting: on back Partpartwhole map diagram. Counting: on back Partpartwhole map diagram Counting: on back Partpartwhole map diagram Representations Starts with context and asks students to construct a model or use the partpartwhole map to show the partpartwhole relationship and write addition and subtraction equations. Starts with context and asks students to construct a model or use the partpartwhole map to show the partpartwhole relationship and write addition and subtraction equations. Starts with context and asks students to construct a model or use the partpartwhole map to show the partpartwhole relationship and write addition and subtraction equations. Starts with a story problem and two diagrams and asks students to notice similarities and differences, asks students to use the strategic representations of others to solve situational problems, and then asks students to solve missing addend addition equations with subtraction.
12 Introduction 11 Task 5 Pencils Developing Understanding Task 6 Basket of Fruit Developing Understanding Task 7 Beads Developing Understanding Task 8 Comparisons Solidifying Understanding Content Solves compare situational problems, difference unknown (how many more). Uses subtraction to solve an unknown addend addition problem. Solves compare situational problems, difference unknown (how many more, how many fewer). Uses subtraction to solve an unknown addend addition problem. Solves compare situational problems, difference unknown (how many more, how many fewer). Uses subtraction to solve an unknown addend addition problem. Solidifies understanding of compare situational problems, difference unknown (how many more, how many fewer). Strategy Counting: on back Partpartwhole map diagram Comparison model Counting: on back Partpartwhole map diagram Comparison model Counting: on back Partpartwhole map diagram Comparison model Counting: on back Partpartwhole map diagram Comparison model Representations Starts with context and asks students to construct a model, make a diagram, or use the partpartwhole map to show relationships between quantities, and/ or write addition and subtraction equations. Starts with context and asks students to construct a model, make a diagram, or use the partpartwhole map to show relationships between quantities, and/ or write addition and subtraction equations. Starts with context and asks students to make claims, then construct a model, make a diagram, or use the partpartwhole map to show relationships between quantities to support the claim, and write addition and/ or subtraction equations. Starts with a story problem and two diagrams and equations and asks students to select equations that describe the diagram, then asks students to solve a series of situational problems and make noticings about the structure of the problems.
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14 mathematics Grade 2 Tasks and Lesson Guides Put Together & Compare Situational Tasks: Missing Addend Addition as Subtraction A SET OF RELATED S
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16 Tasks and Lesson Guides 15 Name Picking Flowers Read about the flowers that Evie and Maria pick. Evie picks 8 red flowers and 9 yellow flowers. Maria picks 15 flowers. She picks 9 red flowers and the rest are yellow. TASK 1 1. Who picks more flowers, Evie or Maria? Make a diagram and write an equation to explain how you know who has more flowers. 2. Who picks more yellow flowers, Evie or Maria? Make a diagram and write an equation to explain how you know she has more flowers. 3. Look at your diagrams and equations for Evie s flowers and Maria s flowers. What do you notice about these two situations? How are they alike? How are they different?
17 16 Tasks and Lesson Guides 1 Picking Flowers Rationale for Lesson: Students use prior knowledge to solve the put together situation with an unknown whole in part 1, and then they explore how to solve put together situations with an unknown addend in part 2. Students develop an understanding of the relationship between addition with a missing addend and subtraction. After solving part 1 and part 2, students determine how the two situations are similar and different. Task 1: Picking Flowers Read about the flowers that Evie and Maria pick. Evie picks 8 red flowers and 9 yellow flowers. Maria picks 15 flowers. She picks 9 red flowers and the rest are yellow. 1. Who picks more flowers, Evie or Maria? Make a diagram and write an equation to explain how you know who has more flowers. 2. Who picks more yellow flowers, Evie or Maria? Make a diagram and write an equation to explain how you know she has more flowers. 3. Look at your diagrams and equations for Evie s flowers and Maria s flowers. What do you notice about these two situations? How are they alike? How are they different? Common Core Content Standards 2.OA.A.1 2.NBT.B.7 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting threedigit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure.
18 Tasks and Lesson Guides 17 Essential Understandings Materials Needed Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student. 1
19 18 Tasks and Lesson Guides 1 SETUP PHASE Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes. Use the counters and partpartwhole map or create a diagram to show Evie s and Maria s flowers. Write equations to tell about the flowers. (Teachers may choose to engage students in the Explore and Share, Discuss, and Analyze phase of the problem for the first problem and then follow the same cycle for the second problem.) EXPLORE PHASE Possible Student Pathways Can t get started. Uses counters and partpartwholemap to build a model. Assessing Questions What do you know about Evie s (Maria s) flowers? Tell me what you are showing here. Advancing Questions How can you show Evie s (Maria s) flowers with counters? Write an equation that describes the story problem. Has a correct sum for each collection but has not shown equations. Writes equations. Evie: = 17 Maria: = 15, possibly 15 9 = 6 Tell me how you arrived at the total number of flowers. What do the equations tell about Evie s (Maria s) flowers? Write an equation that describes Evie s (Maria s) flowers. Who has the most flowers? How do you know? Who has the most yellow flowers? How do you know?
20 Tasks and Lesson Guides 19 SHARE, DISCUSS, AND ANALYZE PHASE Evie s Flowers, solved to answer question 1. 1 EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Show us Evie s flowers on the partpartwhole map. Tell us about Evie s flowers. What do we know? (We know the 8 and the 9.) So we know that Evie has some red flowers. She has 8 red flowers and some yellow flowers, she has 9 yellow flowers. (Revoicing) What do we need to find? (We need to find out how many flowers Evie has.) Why do we need to know this? (We need to figure out who has more flowers and we don t know how many Evie has.) In Evie s situation, we know the parts, but not the whole. We have to find the whole or how many flowers she has altogether. (Marking) EU: Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. How many flowers did Evie pick? How do you know? (I counted on from 8, I said 8, 9, ) You started at 8 and counted on 9 more; 8, 9, 10 16, 17. And you got the answer of 17 flowers. (Revoicing) Okay, did anyone solve it a different way? (I moved one from 8 to the 9, then I had 7 and 10. I know 7 and 10 is 17.) Is that okay? Are we allowed to move an amount from one part to another? Why? (Challenging) Write an equation that tells about Evie s flowers. How many flowers does Evie have? How do you know? (I started at 8 and counted 9 more, 8, 9 17.) Can everyone try her method? Let s count on from 8. How will I know when to stop counting? I heard someone use 8 and 8, doubles to think about the sum of Who knows what the student was thinking? (She did is 16 and then added one more.) How do we use to think about 8 + 9? (If you add it is only 16 so you have to add one more.) Why one more? (9 is one more than 8.) We used doubles; the and we added one more. (Record ) (Marking) Who has more flowers? (Evie.) How do you know? Write this on your paper: 17 15, put in the > or < sign to tell me who has more. (Evie has 17 flowers and Maria only has 15. I wrote17 > 15, I wrote 15 < 17.) Can we record 17 > 15 or 15 < 17? Why?
21 20 Tasks and Lesson Guides 1 Maria s Flowers, solved to answer question 2. EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Show us Maria s flowers using the partpartwhole map. You showed 15 flowers. Tell us what you did next. (I moved 9 to the part.) Why did you move 9? (The 9 are the red flowers. I took those away to see how many yellow flowers were left.) In Maria s situation, we know one of the parts, 9 red flowers, and the whole, 15 flowers. We need to find the missing part, the yellow flowers. (Marking) So how many yellow flowers are there? (6 yellow flowers.) What equation did you write? (15 9 = 6) You used subtraction to solve this problem. Since we know the whole and a part, we are able to take the part away from the whole to find the missing part. (Marking) EU: Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) Did anyone write a different equation to describe Maria s flowers? (Call on a student who wrote an addition equation.) You used addition, = 15, to describe Maria s flowers. Come up and show us how you did that. (Marking) You started with the 9 and had to figure out how many more you needed to get to 15. When we are trying to find a missing part, we can choose to add or subtract. In this problem, we subtracted 15 9 to find the missing part, 6. We also solved it using addition. We started with 9 and added 6 to get to 15. If you know a part and the whole, you can find the missing part using addition or subtraction. (Recapping) Who has more yellow flowers? (Evie.) How do you know? (Evie has 9 yellow flowers and Maria only has 6.) Write this on your paper: 9 6, put in the > or < sign to tell me who has more. (Evie has 9 yellow flowers and Maria only has 6. I wrote 9 > 6, I wrote 6 < 9.) Can we record 9 > 6 or 6 < 9? Why? Application Summary Quick Write Use the subtraction problem to help solve the missing addend problem. A B C 6 +? = 13? + 14 = ? = =? =? 15 8 =? When we know the parts and have to find the whole, we can add the parts together to find the whole. When we know a part and the whole, we can use addition with a missing addend or subtraction to find the missing part. Explain in words or make diagrams to show how these equations are alike =? 2 +? = 10
22 Tasks and Lesson Guides 21 Name Fruit Pies TASK 12 Marcus makes two fruit pies. There are 20 pieces of fruit in each pie. The first pie has 12 pears and the rest are apples. The second pie has 10 pears, 2 oranges, and the rest are apples. 1. Marcus s dad claims there are more apples in the first pie. Do you agree or disagree with his dad? Explain in words and make diagrams to show why you agree or disagree with Marcus s dad. 2. What do you notice about these two situations? How are they alike? How are they different?
23 22 Tasks and Lesson Guides 12 Fruit Pies Rationale for Lesson: Students continue to develop an understanding of put together situations with an unknown addend. Students also discuss the relationship between addition with a missing addend and subtraction. The cognitive demand of the task increases, because now students are expected to grapple with situations involving three addends two addends that are known, and one unknown addend. Task 2: Fruit Pies Marcus makes two fruit pies. There are 20 pieces of fruit in each pie. The first pie has 12 pears and the rest are apples. The second pie has 10 pears, 2 oranges, and the rest are apples. 1. Marcus s dad claims there are more apples in the first pie. Do you agree or disagree with his dad? Explain in words and make diagrams to show why you agree or disagree with Marcus s dad. 2. What do you notice about these two situations? How are they alike? How are they different? Common Core Content Standards 2.OA.A.1 2.NBT.B.7 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting threedigit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Standards for Mathematical Practice Essential Understandings MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations)
24 Tasks and Lesson Guides 23 Materials Needed Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student. 12 SETUP PHASE Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes. Use the counters and your partpartwhole map or create a diagram to show each fruit pie. Write equations to tell about the fruit pies and answer the questions. EXPLORE PHASE Possible Student Pathways Can t get started. Draws a diagram. (Shows only two possible diagrams.) First pie: showing addon Assessing Questions Tell me what you know about the first fruit pie. Tell me what you are showing with your diagram. Advancing Questions Make a diagram of the first fruit pie. Write an equation that describes the story problem. Second pie: showing takeaway Arrives at the correct quantity of apples for each pie but does not show an equation. Writes equations. First pie: 12 +? = = = 8 Second pie: ? = = = = 8 Tell how you found the amounts of apples for the first pie (second pie). Tell me what your equations describe in the story problem. Write an equation that describes what you did to find the amount of apples. Was Marcus s dad right? He claims that there are more apples in the first pie. What do you think?
25 24 Tasks and Lesson Guides 12 SHARE, DISCUSS, AND ANALYZE PHASE EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Tell us what you showed. What do we know? (We know the 12 and the 20.) Say more. What do we know about the 12 and the 20? (There are 12 pears and 20 pieces of fruit.) What do we need to find? (We need to know how many apples are in the pie.) Why do we need to know this? (We are trying to figure out which pie has more apples.) For the first pie, we know a part, the 12 pears, and the whole, 20 pieces of fruit. We have to find the other part, the amount of apples. (Marking) EU: Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. How many apples are in the first pie? How do you know? (I counted on from 12. I said 12, ) Let s do this with him. He said he started at 12 and counted on until you reached 20; 12, , 20. And you got the answer of 8 apples. Okay, did anyone count a different way? (I started with 20, then I said 20, 19, ) Come up and show us on your diagram how you counted. You started counting at 20 and counted back until you got to 12; 20, 19, 18, 17, 16, 15, 14, 13, 12. You counted back 8 times, so you got the answer of 8 apples. (Revoicing) How can we count on or count back to find the number of apples in the first pie? (Challenging) Does anyone have another way for finding how many apples are in the first pie? Tell us about your equation. What do the numbers describe? EU: Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) We said that = 20 describes the apples needed in the first fruit pie. Did anyone write a different equation to describe the apples needed for the first fruit pie? (20 12 = ) You used subtraction to think about the first fruit pie. Come up and show us on the partpartwhole map how you did this. Can we write =? to think about the apples needed for the first fruit pie? How would we solve that equation? We can start with the whole amount, the 20, and then subtract the part we know, the 12 pears. (Recapping)
26 Tasks and Lesson Guides 25 Repeat this sequence of EUs and prompts for the second fruit pie. Application Write each addition problem as a subtraction problem and then solve: 9 +? = ? = ? = Summary Quick Write See recapping above What is the same between each diagram? What is different in each diagram? Diagram 1 shows 8 +? = 11 Diagram 2 shows 11 8 = Support for students who are English Learners (EL): 1. Bring in actual realworld items for students identified as English Learners so they associate the words with the items. 2. Ask students who are identified as English Learners to physically point to the counters or act out the situation as they talk through the situational problem.
27 26 Tasks and Lesson Guides TASK 31 Name What s the Unknown? Solve the story problems. Make a diagram and write an equation to describe each situation. 1. There are 7 black marbles in the bowl. There are some white marbles in the bowl. There are 19 marbles in the bowl. How many white marbles are there in the bowl? 2. There are 6 strawberries and 8 grapes in the bowl. The rest of the fruit in the bowl are bananas. There are 17 pieces of fruit in the bowl. How many bananas are there in the bowl? 3. There are 18 pencils in the box. Some of the pencils in the box are red. There are 12 blue pencils in the box. How many of the pencils in the box are red? 4. There are some books on the shelf. There are 7 books in the basket. Together there are 20 books. How many books are on the shelf?
28 Tasks and Lesson Guides 27 What s the Unknown? Rationale for Lesson: Students continue to develop their understanding of missing addend addition. Students must analyze and determine how to solve story problems using missing addend addition and subtraction. Students will use their prior knowledge of the commutative property of addition to work with missing addends in the first and second position. 13 Task 3: What s the Unknown? Solve the story problems. Make a diagram and write an equation to describe each situation. 1. There are 7 black marbles in the bowl. There are some white marbles in the bowl. There are 19 marbles in the bowl. How many white marbles are there in the bowl? 2. There are 6 strawberries and 8 grapes in the bowl. The rest of the fruit in the bowl are bananas. There are 17 pieces of fruit in the bowl. How many bananas are there in the bowl? 3. There are 18 pencils in the box. Some of the pencils in the box are red. There are 12 blue pencils in the box. How many of the pencils in the box are red? 4. There are some books on the shelf. There are 7 books in the basket. Together there are 20 books. How many books are on the shelf? Common Core Content Standards 2.OA.A.1 2.NBT.B.7 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting threedigit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure.
29 28 Tasks and Lesson Guides 13 Essential Understandings Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative Property) Materials Needed Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student.
30 Tasks and Lesson Guides 29 SETUP PHASE Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes. Use the counters and your partpartwhole map or create a diagram to show the marbles in the bowl. Write equations to tell about the marbles in the bowl and answer the questions. (The teacher may choose to review all of the questions or engage students in the Explore and the Share, Discuss, and Analyze phase question in the task.) 13 EXPLORE PHASE Possible Student Pathways Makes a diagram but does not write an equation. May show addon Assessing Questions Tell me what you have shown. How does this describe the story problem? Advancing Questions Write an equation to tell how you solved the story problem. May show takeaway Writes an equation. Has correct sum but no equation. Finishes early and provides diagrams and equations. How does your equation describe the story problem? How did you get that answer? Tell me how you thought about the task. What do you know about using subtraction to find a missing part? Write an explanation so that anyone who reads your paper will know how your equation tells the story. Write an equation to tell how you solved the story problem. Write an explanation so that anyone who reads your paper will know you understand how to use subtraction to find a missing part.
31 30 Tasks and Lesson Guides 13 SHARE, DISCUSS, AND ANALYZE PHASE Solving put together situations with a missing addend, question 1 EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Tell us what you showed for marbles in the bowl. Which strategy did you use to solve the problem? What do we know from the story problem? (We know the 7 and the 19.) Say more. What are we supposed to figure out in the story problem? We know about the 7 and the 19. (There are 7 black marbles and 19 marbles total.) What do we need to figure out? (We need to know how many white marbles are in the bowl.) We know that there are 7 black marbles and we need to find out how many white marbles are in the bowl because altogether there are 19 marbles. (Marking) EU: Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. How did you figure out how many white marbles are in the bowl? (I counted on from 7 to the 19. I said 7, 8 19.) What did you find out when you counted on? (I found out that there were 12 white marbles.) Do it with us. Okay, did anyone count a different way? (I started with 19, then I said 19, 18 7.) Come up and show us on your diagram how you counted. So you counted back from 19. How many did you count back? (12) What did you land on that told you when to stop? (7) Do it with us. Let s count back together. We can count on or count back to find the number of white marbles in the bowl; we get the answer of 12. The only difference is where we start. In one problem, we started with the 19 and stopped at the 7 when counting back; this was the subtraction problem. When counting on, we started with the 7 and stopped counting when we said 19. Tell us about your addition equation. (I wrote = 19.) When we first looked at the story problem, we only knew the 7 and the 19. What would the addition equation look like with only this information? (It wouldn t have the 12.) Come up here and show us what you mean. That s right. We could start off by writing 7 +? = 19. This is called a missing addend addition problem. We are missing one of the addends one of the parts. (Marking)
32 Tasks and Lesson Guides 31 EU: Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative Property) We just saw that we can write the equation 7 +? = 19. What would happen if we switched the 7 and the question mark? What would that missing addend equation look like? (? + 7 = 19.) Does? + 7 = 19 tell the same thing as 7 +? = 19? (Yes, both show that 7 is a part and 19.) Both of these missing addend addition equations show one part and the whole. We are allowed to write the part we know first like in 7 +? = 19, or second, like? + 7 = 19. (Marking) What will our addition equations be once we fill in the questions marks? ( = 10 and = 19.) Are we allowed to move the order of the addends? (Yes, it doesn t matter what order we put 12 and 7 together, the total is still 19.) 13 EU: Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) We said that = 19 or = 19 describe the white marbles in the bowl. Did anyone write a different equation to describe the white marbles in the bowl? (19 7 = ) You used subtraction to think about the white marbles. Show us on the partpartwhole how you did it. Can we write 19 7 =? to describe the white marbles in the bowl? How would we solve that equation? We can start with the whole, the 19 marbles, and then subtract the part we know, 7 black marbles. (Recapping) Repeat the discussion cycle above for questions 2 4. Application Summary Quick Write Solve the missing addend addition problems using subtraction. Write a subtraction equation and make a diagram to show how subtraction can be used. 7 +? = ? = 19? + 8 = 17? + 6 = 14 When we know the whole and a part, we can use subtraction to find the missing part or we can use a missing addend addition equation. Missing addend addition equations can have the unknown in the first position or second position. Solve the equations below. Then write a story problem that is described by the equations =? 4 +? = 15
33 32 Tasks and Lesson Guides TASK 41 Name Addition and Subtraction 1. Destiny and Malik solve 9 + = 16. Destiny draws the diagram below. She starts with 9 circles and draws more circles until she gets to 16. She adds on 7 circles. She writes the equation = 16. Malik solves 9 + = 16 a different way. He draws the diagram below. He starts by drawing 16 circles. Then he crosses out 9 circles. He writes the equation 16 9 = What do you notice about Destiny s diagram and Malik s diagram? How are they similar? How are they different? 2. Solve this story problem. Choose either Destiny s strategy or Malik s strategy to solve the problem. Make a diagram and write an equation to describe the situation. There are 20 dogs in the park. There are 9 black dogs and the rest are white dogs. How many are white dogs? 3. Solve this story problem. Choose the strategy that you did not use yet to solve this problem. Make a diagram and write an equation to describe the situation. There are 6 pepper plants and 9 tomato plants in the garden. The rest of the vegetable plants are carrots. There are 20 vegetable plants in the garden. How many carrot plants are there in the garden? 4. Four addition equations are shown below. Shelly claims that she can write a subtraction equation to solve each equation. Write a subtraction equation and make a diagram to show how subtraction can be used. 7 + = = = = 17
34 Tasks and Lesson Guides 33 Addition and Subtraction Rationale for Lesson: As a solidifying activity, students ground their understanding of missing addend addition. Students must analyze and determine how to solve story problems using missing addend addition and subtraction. They are expected to demonstrate that they can write subtraction equations that relate to missing addend addition equations. 14 Task 4: Addition and Subtraction 1. Destiny and Malik solve 9 + = 16. Destiny draws the diagram below. She starts with 9 circles and draws more circles until she gets to 16. She adds on 7 circles. She writes the equation = 16. Malik solves 9 + = 16 a different way. He draws the diagram below. He starts by drawing 16 circles. Then he crosses out 9 circles. He writes the equation 16 9 = What do you notice about Destiny s diagram and Malik s diagram? How are they similar? How are they different? See student paper for complete task. Common Core Content Standards 2.OA.A.1 2.NBT.B.7 Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting threedigit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.
35 34 Tasks and Lesson Guides 14 Essential Understandings Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. Materials Needed Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student.
36 Tasks and Lesson Guides 35 SETUP PHASE This task is a little different from others that we have done. The first part shows us how two students, Destiny and Malik, solve the problem 9 + = 16. Destiny and Malik make diagrams and write equations to solve the problem. Let s look at what they did together. Follow along as I read. After I am done reading, you are going to begin with part 1. (After reading about Destiny and Malik, continue.) Now put your finger on part 1. Part 1 asks for you to explain in words how Destiny s strategy and Malik s strategy are similar and different. (The teacher may choose to review all of the questions or engage students in the Explore and the Share, Discuss, and Analyze phase for each task question.) 14 EXPLORE PHASE Possible Student Pathways Can t get started. Makes a diagram but does not write an equation. Assessing Questions What do you notice about Destiny s diagram? What do you notice about Malik s diagram? Tell me what you have shown. How does this describe the story problem? Advancing Questions Write down the things that are the same about both diagrams. Then write down some ways that the diagrams are different. Write an equation to tell how you solved the story problem. Writes an equation. Has correct sum but no equation. Finishes early and provides diagrams and equations. How does your equation describe the story problem? How did you get that answer? Tell me how you thought about the task. What do you know about using subtraction to find a missing part? Write an explanation so that anyone who reads your paper will know how your equation tells the story. Write an equation to tell how you solved the story problem. Write an explanation so that anyone who reads your paper will know you understand how to use subtraction to find a missing part.
37 36 Tasks and Lesson Guides 14 SHARE, DISCUSS, AND ANALYZE PHASE Identifying similarities and differences of two diagrams, question 1. EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. What do we know about Destiny s diagram? (There are 16 circles, 9 black circles, 7 white circles.) Tell us how the equation = 16 explains the diagram. Can you use the partpartwhole and the counters to show us what this would look like? What do we notice about Malik s diagram? (There are 16 circles, 9 are crossed out, 7 not crossed out.) How does Malik s equation 16 9 = 7 describe the diagram? (16 circles, 9 are taken away, 7 are left.) How are these diagrams similar? (Both use circles, there are 16 circles, 9 circles make up one part, 7 circles make up the other part.) How are they different? (One diagram started with 9 circles and 7 more were added, the other started with 16 circles and 9 were crossed out.) How are the equations alike? (They both have 16, 9, and 7.) So they are alike because they both have the whole of 16 and the two parts 9 and 7. (Revoicing) How are the equations different? (One is adding, the parts 9 and 7 are put together to make 16. The other is a subtraction equation; it shows a part is taken away from the whole.) Why are we able to use subtraction to solve an addition problem with a missing part? (Challenging) (When we know the whole and a part, we can take away a part to find the part that is missing.) Solving put together situations with a missing addend, question 2. EU: Mapping devices and tools can help you gain a sense of the quantities involved, to notice increases and decreases, and consider the doing and undoing related to addition and subtraction. Tell us what you showed for the dog story problem. Which strategy did you use to solve the problem? What do we know from the story problem? (We know the 9 and the 20.) Say more. What are we supposed to figure out in the story problem? We know about the 9 and the 20. (There are 20 dogs in the park, 9 are black dogs.) What do we need to figure out? (We need to know how many white dogs are in the park.) We know that there are 9 black dogs and we need to find out how many white dogs are in the park because altogether there are 20 dogs in the park. (Marking)
38 Tasks and Lesson Guides 37 EU: Problems can be solved by counting all, counting on from a quantity, counting on from the largest set, or using derived facts when solving for the whole amount or the missing part of the whole. How did you figure out how many white dogs are in the park? (I counted from 9 to the 20. I said 9, ) What did you find out when you counted on? (That there were 11 white dogs in the park.) Do it with us. Anyone count a different way? (I started with 20, then I said 20, 19, 18 9.) Come up and show us on your diagram how you counted. So you counted back from 20. How many did you count back? (11) What did you land on that told you when to stop? (9) Do it with us. Let s count back together. We can count on or count back to find the number of white dogs in the park. The only difference is where we start. In one problem, we started with the 20 and stopped at the 9 when counting back; this was the subtraction problem. When counting on, we started with the 9 and stopped counting when we said 20. Both had an answer of 11. Tell us about your equation. 14 EU: Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) We said that = 20 describes the white dogs in the park. Did anyone write a different equation to describe the white dogs in the park? (20 9 = ) You used subtraction to think about the white dogs. Show us on the partpartwhole how you did it. Can we write 20 9 =? to describe the white dogs that are in the park? How would we solve that equation? We can start with the whole, the 20 dogs, and then subtract the part we know, 9 black dogs. (Recapping) Repeat the discussion cycle above for question 3. EU: Addition and subtraction are inverse operations because two or more quantities can come together and then the whole amount of objects can be taken apart, but the composition of the whole quantity remains the same. (Doing and Undoing, Inverse Operations) Show us the subtraction equation that could be used to solve for the missing part. Tell us about your equation. We just saw that the equation 20 7 = 13 can help us think about the missing part in the equation 7 + = 20. How can that be? (Challenging) Are we allowed to solve an addition problem with subtraction? Why? Repeat EUs for the remaining equations in question 4. Application Summary Quick Write No application. When we know the whole and a part, we can use subtraction to find the missing part or we can use a missing addend addition equation. No quick write for students.
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