Mathematics Task Arcs


 Caitlin White
 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 1 Adding to Situational Tasks: Solving for Unknowns in All Positions 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 Lesson Progression Chart Tasks and Lesson Guides TASK 1: Three Collections Lesson Guide TASK 2: Buying Fruit Lesson Guide TASK 3: Order of Groups Lesson Guide TASK 4: Picking Apples Lesson Guide TASK 5: Collecting Stickers Lesson Guide TASK 6: Bags of Candy Lesson Guide TASK 7: Boxes of Pencils Lesson Guide TASK 8: Addition or Subtraction Lesson Guide... 55
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6 mathematics Grade 1 Introduction Adding to Situational Tasks: Solving for Unknowns in All Positions A SET OF RELATED S
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8 Introduction 7 Overview This set of related lessons provides a study of adding to story situations that explore solving for unknowns in all positions (whole, change, start). The related tasks are aligned to 1.OA.A.12 and 1.OA.B.34 Content Standards of the CCSSM. Tasks 1 and 2 have add to situations with unknown sums and explore the commutative property of addition. This is followed by a solidifying task, Task 3, for the commutative property. Tasks 4 and 5 then return to add to situations with increased cognitive demand. The tasks now present situations that have an unknown as the sum and situations that have the unknown as the change. Tasks 6 and 7 continue with add to situations and again increase in cognitive demand with both the unknown as the change and this time as the start. Subtraction is introduced as a means for solving missing addend problems in Task 4 and is reinforced in Tasks 5, 6, and 7. Students gain an understanding of subtraction as an unknownaddend problem. This is solidified in Task 8. 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 the format of addition equations and of partpartwhole relationships and the partpartwhole mapping device. Through engaging in the lessons in this set of related tasks, students will: solve add to situational tasks using a variety of strategies and partpartwhole maps; determine if the unknown in a situational task represents a part of the situation or the whole amount in a situation; apply and discuss the commutative property of addition; utilize subtraction to solve missingaddend problems; and use the partpartwhole map to respond to situational problems. By the end of these lessons, students will be able to answer the following overarching questions: Why can we add two sets in any order? What do we know about adding two sets in any order? What strategies can be used to solve story problems with an unknown addend? 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. 1.OA.A.1 1.OA.A.2 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 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. 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. A quantity in a set can be moved to the other set and the sets can be combined, but the whole amount will remain the same because no additional items were added or taken away (e.g = ). 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. 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. A quantity in a set can be moved to the other set and the sets can be combined, but the whole amount will remain the same because no additional items were added or taken away (e.g = ).
10 Introduction 9 CCSS for Mathematical Content: Operations and Algebraic Thinking Essential Understandings Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.B.3 1.OA.B.4 Apply properties of operations as strategies to add and subtract. Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition) Understand subtraction as an unknownaddend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) 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; = 9, 9 5 = 4) 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
11 10 Introduction Tasks CCSSM Alignment Task 1.OA.A.1 1.OA.A.2 1.OA.B.3 1.OA.B.4 Task 1 Three Collections Developing Understanding Task 2 Buying Fruit Developing Understanding Task 3 Order of Groups Solidifying Understanding Task 4 Picking Apples Developing Understanding Task 5 Collecting Stickers Developing Understanding Task 6 Bags of Candy Developing Understanding Task 7 Boxes of Pencils Developing Understanding Task 8 Addition or Subtraction Solidifying Understanding
12 Introduction 11 Task MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8 Task 1 Three Collections Developing Understanding Task 2 Buying Fruit Developing Understanding Task 3 Order of Groups Solidifying Understanding Task 4 Picking Apples Developing Understanding Task 5 Collecting Stickers Developing Understanding Task 6 Bags of Candy Developing Understanding Task 7 Boxes of Pencils Developing Understanding Task 8 Addition or Subtraction Solidifying Understanding
13 12 Introduction Lesson Progression Chart Overarching Questions Why can we add two sets in any order? What do we know about adding two sets in any order? What strategies can be used to solve story problems with an unknown addend? TASK 1 Three Collections Developing Understanding TASK 2 Buying Fruit Developing Understanding TASK 3 Order of Groups Solidifying Understanding TASK 4 Picking Apples Developing Understanding Content Solves add to situational problems, whole unknown. Solves situational problems with two and three addends. Explores commutative property of addition. Solves add to situational problems, whole unknown. Solves situational problems with three addends. Establishes understanding of the commutative property of addition. Solidifies understanding of the commutative property of addition using 2 and 3 addends. Compares add to situational problems, whole unknown and change unknown. Explores unknown addend situations as subtraction. Strategy Counting: all on Known facts: doubles doubles +1 doubles 1 Counting: all on Known facts: doubles doubles +1 doubles 1 Counting: all on Known facts: doubles doubles +1 doubles 1 Counting: all on back Representations Starts with context and asks students to construct a model or use the partpartwhole map to show the partpartwhole relationship and write equations. Starts with context and asks students to construct a model or use the partpartwhole map to show the partpartwhole relationship and write equations. Asks students to solve sets of addition equations with two or three addends, make noticings about the sets, and agree or disagree with a claim about the commutative property. Starts with context and asks students to construct a model, use the partpartwhole map to show the partpartwhole relationship, and write addition and subtraction equations.
14 Introduction 13 TASK 5 Collecting Stickers Developing Understanding TASK 6 Bags of Candy Developing Understanding TASK 7 Boxes of Pencils Developing Understanding TASK 8 Addition or Subtraction Solidifying Understanding Content Compares add to situational problems, whole unknown and change unknown. Explores unknown addend situations as subtraction. Compares add to situational problems, change unknown and start unknown. Establishes unknown addend situations as subtraction. Compares add to situational problems, change unknown and start unknown. Establishes unknown addend situations as subtraction. Solidifies understanding of subtraction as an unknown addend. Strategy Counting: all on back Counting: on back Counting: on back Counting: on back Representations Starts with context and asks students to construct a model, 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, 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, use the partpartwhole map to show the partpartwhole relationship, and write addition and subtraction equations. Asks students to solve sets of addition equations, make noticings about things that are the same in the sets, and then analyze examples and nonexamples of the relationship between missing addend equations and subtraction.
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16 mathematics Grade 1 Tasks and Lesson Guides Adding to Situational Tasks: Solving for Unknowns in All Positions A SET OF RELATED S
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18 Tasks and Lesson Guides 17 Name Three Collections TASK 1 Three friends have collections. Use counters or create a picture to show each collection. Try to write an equation that shows each friend s collection. Juan collects marbles. He has 6 marbles and he gets 5 more marbles. Martha collects blocks. She has 5 blocks and she gets 6 more blocks. Jasmine collects books. She has 5 books and she gets 3 more books. Then later in the day, she gets 3 more books. Which friend has the most items in their collection? Be prepared to explain to the class how you know who has the most in their collection.
19 18 Tasks and Lesson Guides 1 Three Collections Rationale for Lesson: Students explore ways to solve add to situational problems with two and three addends with unknown sums using a variety of strategies and discuss the reason why two addends can be added in any order (the commutative property of addition). Task 1: Three Collections Three friends have collections. Use counters or create a picture to show each collection. Try to write an equation that shows each friend s collection.* Juan collects marbles. He has 6 marbles and he gets 5 more marbles. Martha collects blocks. She has 5 blocks and she gets 6 more blocks. Jasmine collects books. She has 5 books and she gets 3 more books. Then later in the day, she gets 3 more books. Which friend has the most items in their collection? Be prepared to explain to the class how you know who has the most in their collection. *Since students develop at different rates, some students will be able to show each collection with counters and others will make a picture. Some students will be able to write equations, while others will only be able to show and explain their thinking in words. Common Core Content Standards 1.OA.A.1 1.OA.A.2 1.OA.B.3 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Apply properties of operations as strategies to add and subtract. Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition) Standards for Mathematical Practice 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.
20 Tasks and Lesson Guides 19 Essential Understandings Materials Needed 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. 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. Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student. 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 your partpartwhole map or create a picture to show the collections. Write equations to tell about the collections. EXPLORE PHASE Possible Student Pathways Can t get started. Uses partpartwhole map to build a model. Assessing Questions What do you know about Juan s marbles? Tell me what you are showing here. Advancing Questions How can you show Juan s marbles with counters? Write an equation that describes the story problem. Has a correct sum for each collection. Writes equations. Juan: = 11 Martha: = 11 Jasmine: = 11 Tell me how you arrived at the total number of marbles (blocks, books) in the collection. What do the equations tell about the collections? Write an equation that describes each students collection. Who has the most in their collection? How do you know?
21 20 Tasks and Lesson Guides 1 SHARE, DISCUSS, AND ANALYZE PHASE Juan s marbles (Repeat this sequence of EUs and prompts for Martha s blocks.) 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 Juan s marbles on the partpartwhole map. Tell us about Juan s marbles. What do we know? (We know the 6 and the 5.) So we know that Juan has 6 marbles and gets 5 more. (Revoicing) What do we need to find? (We need to find out how many marbles Juan has.) In this situation, we know the parts, not the whole. We have to find the whole. (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. Write an equation that tells about Juan s collection. How many marbles does Juan have? How do you know? (I counted all of the marbles. I said 6, 7, 8 11.) You started with the 6 and counted 5 more. We call this way counting on. (Marking) I heard someone use 5 and 5 to think about the sum of Who knows what the student was thinking? Can we use to think about 5 + 6? What would we do? (Challenging) Sometimes we can use doubles to solve problems that aren t doubles, but if we do this, then we may have to add on or take one away. (Marking) Jasmine s books 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. Tell me about Jasmine s books. There are three parts? Are we allowed to have three parts? You can show two parts in the one part on the partpartwhole map. What will that look like? Someone show us her books using the partpartwhole map. (Challenging) How many books does Jasmine have? If we start at 5, then what will we do to find out how many books Jasmine has? (5 6, 7, 8, 9, 10, 11.) Did someone use a different way to figure out how many books Jasmine has? What equation can we write to tell about Jasmine s books?
22 Tasks and Lesson Guides 21 EU: Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) Who has the most items in their collection? Martha and Juan only had two parts in their collections and Jasmine has three parts? So Jasmine has the most, right? (Challenging) How can Juan and Martha have the same amount in their collections if Juan starts with 6 and Martha starts with 5? (Challenging) So if = 11 and = 11, can we write = 5 + 6? Is this okay? Why or why not? When we are adding, we are allowed to switch the order of the numbers because the sum remains the same. (Recapping) 1 Application What is = What is = What is = What is = What is = What is = Summary Quick Write When we add two addends, we get a sum. We can switch the order of the addends and we will still get the same sum because we add the same amounts. We did not take any away from the collection. Write an equation to describe each partpartwhole map and explain how you know they will have the same sum.
23 22 Tasks and Lesson Guides TASK 2 Name Buying Fruit Morgan and Marshall go to the store to get some fruit. They each have a shopping basket. Create pictures or use counters to show the fruit in Morgan s basket and the fruit in Marshall s basket. Try to write an equation for Morgan s fruit and an equation for Marshall s fruit. Morgan has 5 oranges in her shopping basket. She adds 3 cherries to the shopping basket and then adds 2 bananas. Marshall has 2 bananas in his shopping basket. He adds 3 cherries to the shopping basket and then adds 5 bananas. Marshall says that he has more fruit than Morgan. Be prepared to explain to the class why you agree or disagree with Marshall.
24 Tasks and Lesson Guides 23 Buying Fruit Rationale for Lesson: Students explore add to situational problems with two and three addends with unknown sums and discuss the reason why two addends can be added in any order. (Commutative property of addition) 2 Task 2: Buying Fruit Morgan and Marshall go to the store to get some fruit. They each have a shopping basket. Create pictures or use counters to show the fruit in Morgan s basket and the fruit in Marshall s basket. Try to write an equation for Morgan s fruit and an equation for Marshall s fruit.* Morgan has 5 oranges in her shopping basket. She adds 3 cherries to the shopping basket and then adds 2 bananas. Marshall has 2 bananas in his shopping basket. He adds 3 cherries to the shopping basket and then adds 5 bananas. Marshall says that he has more fruit than Morgan. Be prepared to explain to the class why you agree or disagree with Marshall. *Since students develop at different rates, some students will be able to show the fruit with counters and others will make a picture. Some students will be able to write equations, while others will only be able to show and explain their thinking in words. Common Core Content Standards 1.OA.A.1 1.OA.A.2 1.OA.B.3 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Apply properties of operations as strategies to add and subtract. Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition) Standards for Mathematical Practice 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.
25 24 Tasks and Lesson Guides 2 Essential Understandings Materials Needed 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. 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. Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student. 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 picture to show the fruit the students bought. Write equations to tell about the fruit and then answer the question. EXPLORE PHASE Possible Student Pathways Can t get started. Uses partpartwhole map to build a model. Assessing Questions Tell me what you know about how much fruit Morgan puts in her basket at the store. Tell me what you are showing with your partpartwhole map. Advancing Questions Show Morgan s fruit using the partpartwhole map. Write an equation that describes the story problem. Has a correct sum for each collection. Writes equations. Morgan: = 10 Marshall: = 10 Tell me how you found the amounts of fruit. Tell me what your equations describe in the story. Write an equation that describes what you did to find the amounts of fruit. Who bought the least amount of fruit and how do you know?
26 Tasks and Lesson Guides 25 SHARE, DISCUSS, AND ANALYZE PHASE Morgan s fruit 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 about Morgan s fruit. Who can show us Morgan s fruit on the partpartwhole map? What do we know about Morgan s fruit? Yes, there are three parts. Are we allowed to have three parts? (Challenging) If we know the parts, what are we trying to find? 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 much fruit did she buy? Tell us how you found that answer. (I count all of them. I said 1, 2 10/I said 5, 6, 7 10.) Student responds, I counted the 5, then the 3, then the 2. You started with the biggest number 5, and then counted 6, 7, 8 and then 9, 10. (Revoicing) We call this counting on. (Marking) Someone said they used doubles to figure out how much fruit Morgan has. Who understands what this person might have done? Teacher writes 5 + (3 + 2). You put together the and got 5. Are we allowed to do this? Why? Repeat the previous sequence of EUs and prompts for Marshall s fruit. EU: Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) Do we agree with Marshall, does he have more fruit? Marshall has 10 and Morgan has 10, so they have the same amount? How can that be; Marshall started with 5 and Morgan started with 2? We wrote 5 + (3 + 2) and (2 + 3) + 5. What do you notice about the amount of fruit they have? When we are adding, we are allowed to switch the order of the numbers because the sum remains the same. (Recapping) Application Solve these problems. Create a picture or use counters to show the equations = = What do you notice about these two problems? Summary We can add three addends to arrive at a sum. We can switch the order of the addends and we will still get the same sum because we added the same amounts.
27 26 Tasks and Lesson Guides 2 Quick Write Explain how you know these two equations will have the same sum = = Supports 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.
28 Tasks and Lesson Guides 27 Name Order of Groups 1. Solve each set of equations TASK 3 A = = B = = C = = D = = 2. Look at each set of equations. What do you notice about the sets? 3. Omar claims that you can add the numbers in an addition equation in any order. Do you agree or disagree with Omar s claim? Make a picture and be prepared to explain to the class why you agree or disagree with Omar.
29 28 Tasks and Lesson Guides 3 Order of Groups Rationale for Lesson: Students solidify understanding of the commutative property of addition by working with two and three addends. Task 3: Order of Groups 1. Solve each set of equations. A = = B = = C = = D = = 2. Look at each set of equations. What do you notice about the sets? 3. Omar claims that you can add the numbers in an addition equation in any order. Do you agree or disagree with Omar s claim? Make a picture and be prepared to explain to the class why you agree or disagree with Omar. Common Core Content Standards 1.OA.B.3 Apply properties of operations as strategies to add and subtract. Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition) Standards for Mathematical Practice Essential Understandings Materials Needed 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. 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. 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. Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) 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 This task is a little different from others that we have done. Put your finger on the first problem. It shows and You have to show a picture for and then write the sum. Then you have to show a picture for and find the sum. When you are done with the picture and you have found the sum, then you answer the question. Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes and then work with a partner. Use counters and the partpartwhole map and create pictures to explain your thinking. 3 EXPLORE PHASE Possible Student Pathways Can t get started. Uses the partpartwhole map. Assessing Questions What do you know about the first set of problems? Tell me what you have shown. What does this tell about the problems in the set? Advancing Questions Draw a picture to show the problems in the first set. What is the same and what is different about the sets? Has correct sum. Finishes early. How did you get the answers for the equations in the first set? Tell me how you thought about the sets. What do you think about Omar s claim? Write an explanation that tells why and have the same sum. Do you think Omar s claim will always work when we re adding? Why or why not?
31 30 Tasks and Lesson Guides 3 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. Show us your picture for the first two expressions. Tell us about the two expressions. What do we know about =? What do we need to find? Yes, 8 and 9 are both parts and we need to find the whole, the sum. (Revoicing) What do we know about =? What do we need to find? Yes, 9 and 8 are both parts and we need to find the whole, the sum. (Revoicing) So in both equations in the first set, we know the parts and have to find the whole, the sum. (Recapping) What do you notice about the parts? The parts are the same, but in a different order. (Marking) What might this mean for the whole? (Challenging) 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 the sums? (Call on students who count all and count on.) You counted all and you counted on from the 8. You both got the sum of 17. How did both of you get the same sum? (Challenging) Did anyone use doubles to solve 8 + 9? Tell us about the doubles you used. Can you use either of the doubles, or 9 + 9, to figure out the sum? If you do 8 + 8, then what do you have to do to 16 to get the sum of 17? (Add one to 16 to get 17.) If you use 9 + 9, then what do you have to do to the sum of 18 to get the 17? (Take one away from 18 to get 17.) Why do we add 1 when we use to solve and subtract 1 when we use to solve 8 + 9? Repeat the previous sequence of EUs and prompts for each of the equations. EU: Two quantities can be combined in any order and the whole quantity will remain the same. (Commutative property; = 5 + 4) What do you notice about the sets of equations? Each set has the same sum. (Marking) What else do you notice? Student responds that some have the same parts. That s right, Sets 1, 2, and 4 have the same parts the same addends but they are in a different order. (Revoicing) What have we learned about adding the same parts in a different order? We know that the order in which we add the parts does not matter; the sum will be the same. (Recapping)
32 Tasks and Lesson Guides 31 Application No application. Summary So what do we notice about all the sets of expressions? What s the pattern?  They are all addition equations.  The equations in each set have the same sum.  The equations in each set have the same numbers, but in different orders. Are we permitted to switch the numbers? What will always be true when we switch the numbers of two addition problems? This is called the commutative property. It says that numbers can be added in any order and it won t change the sum. 3 Quick Write Write an equation for each partpartwhole map. Which two partpartwhole maps show that parts can be added in any order without changing the sum? Be prepared to tell how you know.
33 32 Tasks and Lesson Guides TASK 4 Name Picking Apples Javon and Kya are picking apples at the orchard. Javon picks 9 apples in the morning and 6 apples in the afternoon. How many apples did Javon pick at that orchard? Kya picked 7 apples in the morning. She picked more apples in the afternoon. Now she has 15 apples. How many apples did Kya pick in the afternoon? Javon says he has more apples than Kya. Is he right? Explain your thinking using counters, pictures, or words. Try to write equations to show Javon s apples and Kya s apples.
34 Tasks and Lesson Guides 33 Picking Apples Rationale for Lesson: Students explore the relationship between addition with an unknown addend and subtraction by solving add to situational problems with an unknown whole and then an unknown change (addend). 4 Task 4: Picking Apples Javon and Kya are picking apples at the orchard. Javon picks 9 apples in the morning and 6 apples in the afternoon. How many apples did Javon pick at that orchard? Kya picked 7 apples in the morning. She picked more apples in the afternoon. Now she has 15 apples. How many apples did Kya pick in the afternoon? Javon says he has more apples than Kya. Is he right? Explain your thinking using counters, pictures, or words. Try to write equations to show Javon s apples and Kya s apples. *Since students develop at different rates, some students will be able to show the apples with counters and others will make a picture. Some students will be able to write equations, while others will only be able to show and explain their thinking in words. Common Core Content Standards 1.OA.A.1 1.OA.B.4 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Understand subtraction as an unknownaddend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Standards for Mathematical Practice Essential Understandings Materials Needed 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. 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. 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; = 9, 9 5 = 4) Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student.
35 34 Tasks and Lesson Guides 4 SETUP PHASE Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes. Use counters and a partpartwhole map or create pictures to show Javon s apples and Kya s apples. Write an equation to tell about the apples. EXPLORE PHASE Possible Student Pathways Can t get started. Some students may have trouble figuring out how to start thinking about the task. Other students may only have an issue when they get to Kya s unknown change situation. Uses partpartwhole map to build a model. Javon Assessing Questions What do you know about Javon s apples? What do you know about Kya s apples? Tell us about your partpartwhole map. Advancing Questions Show Javon s apples with the counters. Draw a picture to show Kya s apples. Write an equation that describes the situation. Kya Writes an equation. Javon: = 15 Kya: = 15 OR 15 7 = 8 Arrives at the solution. Determines that Kya picked 22 apples (Misconception, adds ) Tell me about your equations. Show us how you found your answers. How did you find that Kya picked 22 apples in the afternoon? Who has the most apples? How do you know? Write an equation that describes the situation. Use the partpartwhole map to show the apples Kya picked. What are the parts? What is the whole?
36 Tasks and Lesson Guides 35 SHARE, DISCUSS, AND ANALYZE PHASE Javon s apples 4 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 Javon s apples on the partpartwhole map. What do we know? What do we need to find? We know the parts, and have to find the whole. (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 does Javon have? How do you know? Show us how you figured it out. You counted all by putting them all together and counting from 1 1, 2, 3 14,15. (Revoicing) You started at 9 and counted on 6 more, 9, 10, 11, 12, 13, 14, 15. (Revoicing) Who has an equation that tells about Javon s apples? Tell us about your equation. Kya s apples 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. Now let s see Kya s apples on the partpartwhole map. What parts of Kya s apples do we know? What are we trying to find out? We know one of the parts and the whole. We need to find the other part the number of apples she picked in the afternoon. We call this the change. (Marking) How is Kya s situation similar to Javon s? How is it different? 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 did Kya pick in the afternoon? How do you know? (I counted all of them. I said 1, 2, I counted on from 7. I said 7, 8 15.) You started at 7 and counted on till you reached 15; 7, 8, 9, 10, 11, 12, 13, 14, 15. And you got the answer of 8 apples. (Revoicing) Okay, did anyone count a different way? (I started with 15, then I said 15, 14, 13 8.) You started counting at 15 and counted back 7; 15, 14, 13, 12, 11, 10, 9, 8. And you got the answer of 8 apples. (Revoicing) Does that mean we can count on or count back to find the number of apples Kya picked in the afternoon? How is that possible? (Challenging) Does anyone have another way for finding how many apples Kya picked in the afternoon? Tell us about your equation. What do the numbers describe?
37 36 Tasks and Lesson Guides 4 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; = 9, 9 5 = 4) We said that = 15 describes Kya s apples. Did anyone write a different equation to describe Kya s apples? (Call on a student who wrote a subtraction equation.) You used subtraction to think about Kya s apples. Come up and show us how you did that. (Marking) Can we write 15 7 =? to think about Kya s apples? How would we solve that equation? When we know the whole and one part, we can find the unknown part by subtracting. (Recapping) Application Solve for the missing number in each problem. What do you notice about these problems? = 3 + = = Summary When we know the whole and a part, we can subtract the part from the whole to find out what changed. Addition is when groups come together and subtraction is when groups are taken apart; they do the opposite things. Quick Write Solve each equation and draw a picture. How are the pictures similar? 4 + = =
38 Tasks and Lesson Guides 37 Name Collecting Stickers TASK 5 Aaron likes to collect stickers. He has animal stickers and sports stickers. Aaron wants to know if he has more animal stickers or more sports stickers. Aaron starts with 13 animal stickers. He gets 7 more animal stickers. Aaron starts with 7 sports stickers. Then he gets some more sports stickers. He now has 19 sports stickers. Does Aaron have more animal stickers or more sports stickers? Be prepared to share your thinking with the class. You can draw pictures to show and try to write an equation to describe each type of sticker.
39 38 Tasks and Lesson Guides 5 Collecting Stickers Rationale for Lesson: Students explore the relationship between addition with an unknown addend and subtraction by solving add to situational problems with an unknown whole and then an unknown change (addend). Task 5: Collecting Stickers Aaron likes to collect stickers. He has animal stickers and sports stickers. Aaron wants to know if he has more animal stickers or more sports stickers. Aaron starts off with 13 animal stickers. He gets 7 more animal stickers. Aaron starts with 7 sports stickers. Then he gets some more sports stickers. He now has 19 sports stickers. Does Aaron have more animal stickers or more sports stickers? Be prepared to share your thinking with the class. You can draw pictures to show and try to write an equation to describe each type of sticker.* *Since students develop at different rates, some students will be able to show each collection of stickers with counters and others will make a picture. Some students will be able to write equations, while others will only be able to show and explain their thinking in words. Common Core Content Standards 1.OA.A.1 1.OA.B.4 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Understand subtraction as an unknownaddend problem. For example, subtract 10 8 by finding the number that makes 10 when added to 8. Standards for Mathematical Practice Essential Understandings Materials Needed 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. 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; = 9, 9 5 = 4) Counters or other appropriate manipulative, 25 per student. Partpartwhole map, 1 per student. Student task reproducible, 1 per student.
40 Tasks and Lesson Guides 39 SETUP PHASE Listen as I read the task. Follow along on your paper. Work by yourself for a few minutes. Use counters and a partpartwhole map, or create pictures to show Aaron s stickers. Write an equation to tell about his animal stickers and an equation to tell about his sports stickers. 5 EXPLORE PHASE Possible Student Pathways Uses a partpartwhole map, creates a picture, or builds a model. Animal Assessing Questions Tell us about what you are showing with your partpartwhole map. Advancing Questions Write an equation that describes the situation. Sports Arrives at the solution. Writes an equation. Animals: = 19 Sports: = 19 OR 19 7 = 12 Determines that there are 26 sports stickers. (misconception, adds ) Finishes early. Show us how you found your answers. Tell me about your equations. How did you find that there were 26 sports stickers? Tell me how you thought about this problem. Write an equation that describes the situation. Are there more animal stickers or sports stickers? How do you know? Use the partpartwhole map to show the sports stickers. What are the parts? What is the whole? Write about how you can solve addition situations with subtraction.
41 40 Tasks and Lesson Guides 5 SHARE, DISCUSS, AND ANALYZE PHASE Animal stickers 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. Can someone share their partpartwhole map that shows the animal stickers? What do we know from the problem? What do we need to find? That s right, we know the parts, the addends, but we don t know the whole, the sum. (Revoicing) How many animal stickers are there? How do you know? Show us how you figured it out. (Call on a student who counted on, 13, 14, 15, 16, 17, 18, 19, 20.) Does anyone have another way of finding the total number of animal stickers? Tell us about your equation. What do these numbers describe? Sports stickers 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. Now let s see a picture of the sports stickers. What did we know about the sports stickers? What did we need to find out? So in this situation, we know one of the parts, the number of sports stickers he had at the start. We also know how many he had at the end. We needed to find out how many stickers were added. We learned yesterday that this is called the change. (Marking) How many sports stickers were added? How do you know? Show us how you figured it out. (Call on a student who counted on, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19.) How else can we count the sports stickers? (Call on a student who counted back, 19, 18, 17, 16, 15, 14, 13, 12.) Does anyone have another way for finding how many sports stickers were added? Tell us about your equation. What do these 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; = 9, 9 5 = 4) We know that 7 +? = 19 tells about the sports stickers. Is there another equation we could use to think about this problem? (Call on a student who wrote a subtraction equation.) Oh, so you used subtraction to solve the sports stickers situation. Come up and write that equation for us on the board. Can someone show us what 19 7 would look like with counters on the overhead? Are we allowed to use 7 +? = 19 and 19 7 =? to think about the sports stickers? Why? (Challenging) When we have a missing part, in an addition problem, we are allowed to use subtraction or addition to solve for it. In this case, the change was unknown. We can think about this situation as (teacher points to the equations) either 7 plus the change equals 19 or 19 minus the 7 equals the change. (Recapping)
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