ELEMENTARY FALS FORMATIVE ASSESSMENT LESSONS

Transcription

1 ELEMENTARY FALS FORMATIVE ASSESSMENT LESSONS

2 Formative Assessment Lesson Alpha Counting Dots in Various Arrangements Mathematical goals This lesson unit is intended to help you assess how well students are able to count objects up to 20 no matter how they are arranged and also how well they are able to represent their counts with written numerals. It will help you to identify students who have the following difficulties: Not being able to mark where they start/stop counting Arranging objects in order to be counted One-to-one correspondence Common Core State Standards This lesson involves mathematical content in the standards from across the grade, with emphasis on: Know number names and the count sequence. Count to tell the number of objects. (K.CC.5) Count to answer how many? questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects. (K.CC.3)Write numbers from Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). This lesson involves a range of mathematical practices, with emphasis on: 2. Reason abstractly and quantitatively. 7. Look at and make use of structure. Introduction This lesson is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. Students work in small groups on collaborative discussion tasks, to match cards with dots in regular patterns, structured 10 frame patterns, unstructured arrangements, and numerals. As they do this, they interpret the cards meanings and begin to link them together. Throughout their work, students justify and explain their decisions to their peers. Students return to their original assessment tasks, and try to improve their own responses. Materials required Each individual student will need: Two copies of the assessment task Counting Dots. A sheet of sticky dots. 1

3 Formative Assessment Lesson Alpha A mini-whiteboard, marker, and eraser. Each small group of students will need the following resources: Card sets: A, B, C, and D. All cards should be cut up before the lesson and it would be helpful if each set were a different color. There is also a projector resource to help with the whole class discussion. Time needed Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact timings will depend on the needs of the class. Before the Lesson Assessment task: Counting Dots (15 minutes) Have students do this task individually in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or in small groups (they should still work individually. Give each student a copy of the assessment task Counting Dots. Count the dots in each space and write the numeral that tells how many dots on the line. Put stickers in the space to match the numeral. It is important that the students are allowed to answer the questions without your assistance, as far as possible. Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal. Assessing students responses Collect students responses to the task. Make some notes about what their work reveals about their current levels of understanding, and their different problem solving approaches. We suggest that you do not score student s work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. 2

4 Formative Assessment Lesson Alpha We suggest that you write a list of your own questions, based on your students work, using the ideas that follow. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson. The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help students to find a variety of ways that work in different situations and make sense to them. Common Issues: Student writes numbers that are more than the number of dots. Student writes numbers that are less than the number of dots. Suggested questions and prompts: Did you count each dot only once? Is there a way to know if you have already counted a dot? How do you know where you started counting and where you stopped? Think about how you could remember where you started. How can you be sure that you counted every dot? Suggested lesson outline Collaborative Activity: matching Card Sets A, B, C, and D (30 min.) Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Card Sets A(numerals) and B (structured 10-frames). Introduce the lesson carefully: I want you to work as a team. Take turns placing a numeral card with a dot card that has the same number of dots. Each time you do this, explain your thinking clearly to your partner. If your partner disagrees with your match then challenge him or her to explain why. It is important that you both understand why each card is matched with another one. There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving. Make a note of student approaches to the task You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may consistently loose count or forget where they started and re-count dots. 3

5 Formative Assessment Lesson Alpha Support student problem solving Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. If one student has placed a particular card to make a match, challenge their partner to provide an explanation. If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. The projector resources may be useful when doing this. Placing Card Set C (regular patterns) As students finish with matching card sets A and B, hand out card set C. These provide students with a different way of interpreting the numbers. Do not collect card set B. an important part of this task is for students to make connections between different representations of numbers. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. The following patterns may be noticed: Placing Card Set D (unstructured) As students finish matching card set C, hand out card set D. Continue to monitor work and listen to discussion. Taking two class periods to complete all activities If you have to divide the lesson into two class periods, you may want to have a way for students to save the work they have done with matching the card sets. You may give each group a chart paper or poster board and have them tape the cards down with their matches. You may chose to have them do this even if you are not dividing up the class period just to use as a visual during the class discussion. Sharing Work (10 minutes) When students get as far as they can with matching the card sets, ask one student from each group to visit another group s work. Students remaining at their desk should explain their reasoning for the matched cards on their own desk. If you are staying at your desk, be ready to explain the reasons for your group s matches. If you are visiting another group, check to see which matches are different from your own. (they might need some kind of representation of their own matches to take with them perhaps a cell phone picture?) If there are differences, ask for an explanation. If you still don t agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. 4

6 Formative Assessment Lesson Alpha Extension activities Ask students who finish quickly to make another set of cards that shows each number in a different way. Plenary whole-class discussion (10 minutes) Give each student a mini-whiteboard, pen, and eraser. Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending what has been learned to new examples, and then examining some of the conclusions listed above. Show students the examples of number configurations on the slide resources and have them write the matching numeral on their white board. Then, showing them the slide with all three representations, discuss which was easiest to count and why. Have students explain their process for counting the dots on other slides, always asking, Is there another way? Have students draw a configuration on their whiteboards for numerals on the slides and discuss their approaches and why they chose them. You might even group students by the configuration they drew (scattered, regular, 10-frame). Improving individual solutions to the assessment task (10 minutes) Return to the students their original assessment, Counting Dots, as well as a second blank copy of the task (and more sticky dots if needed). Look at your original responses and think about what you have learned during this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. (with kindergarten, you may only focus on the questions students really seemed to need and state these out loud and ask them again as you move around the room or work with a small group at a time.) 5

7 Formative Assessment Lesson Alpha Counting Dots Count the dots in each picture. Write the numeral that tells how many dots in the box beside the picture. 6

8 Formative Assessment Lesson Alpha Count out sticker dots that match the number in the box. Place them in the box beside of the number

9 Formative Assessment Lesson Alpha Card Set A: Numeral Cards

10 Formative Assessment Lesson Alpha Card Set B: Structured Ten Frames 9

11 Formative Assessment Lesson Alpha Card Set C: Regular Patterns 10

12 Formative Assessment Lesson Alpha Card Set D: Unstructured 11

13 Math Formative Assessment Lesson Pieces of the Hundred Chart Grade 1 Mathematical goals This lesson unit is intended to help you assess how well students are able to count to count from any number and recognize patterns on the 100 chart that will help them to count on or back. It will also help you to assess how well they write numbers between 1 and 100. It will help you to identify students who have the following difficulties: Recognizing patterns on the hundred chart. Counting on or back from a given number Common Core State Standards This lesson involves mathematical content in the standards from across the grade, with emphasis on: Extend Counting Sequence Understand Place Value (1.NBT.1) Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. (1.NBT.2) Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). This lesson involves a range of mathematical practices, with emphasis on: 2. Reason abstractly and quantitatively. 7. Look at and make use of structure. Introduction This lesson is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. Students work in small groups on collaborative discussion tasks, to place cards with numerals onto a section taken from a hundred chart. Throughout their work, students justify and explain their decisions to their peers. Students return to their original assessment tasks, and try to improve their own responses. Materials required Each individual student will need: Two copies of the assessment task Torn Hundred Chart. A mini-whiteboard, marker, and eraser. Each small group of students will need the following resources: Two place cards with squares, some of which are marked with numerals in the pattern of the hundreds chart. Two sets of colored cards that will fill in the blanks on the two charts along with a few numbers that will not fit either chart. Hundred Charts on hand for groups that need more support 1

14 Math Formative Assessment Lesson Time needed Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact timings will depend on the needs of the class. Before the Lesson Assessment task: Torn Hundred Chart (15 minutes) Have students do this task individually in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or in small groups (they should still work individually.) Give each student a copy of the assessment task Torn Hundred Chart Fill in the missing numbers in the white spaces of the torn hundred charts. It is important that the students are allowed to answer the questions without your assistance, as far as possible. Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal. Assessing students responses Collect students responses to the task. Make some notes about what their work reveals about their current levels of understanding, and their different problem solving approaches. We suggest that you do not score student s work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students work, using the ideas that follow. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson. The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help students to find a variety of ways that work in different situations and make sense to them. 2

15 Math Formative Assessment Lesson Common Issues: Students are placing cards in incorrect spot. When going down to the next line, students write the number that comes next Suggested questions and prompts: Why did you put that card here? Think about our hundreds chart, how is each row organized? --You may even want to carry a hundred chart with you to allow students a resource when questioning. What would go into the spaces between the number that is already there and the one that you wrote? What are some of the patterns you notice in a hundred chart? Why did you put that card here? ----You may even want to carry a hundred chart with you to allow students a resource when questioning. Suggested lesson outline Collaborative Activity: Placing Card set onto torn hundreds charts Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Set A with two place cards and a set of numeral cards. Introduce the lesson carefully: I want you to work as a team. Take turns placing a numeral card onto the piece of the hundred chart. Each time you do this, explain your thinking clearly to your partner. If your partner disagrees with your placement then challenge him or her to explain why. It is important that you both understand why each card is placed where it is. There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving. Make a note of student approaches to the task You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may consistently try to place numbers in counting order without noticing that some of the numbers will not show on their section of the chart. Support student problem solving Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. If one student has placed a particular card on the chart, challenge their partner to provide an explanation. If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. The projector resources may be useful when doing this. 3

16 Math Formative Assessment Lesson Place Card Set B As students finish with matching Place card set A, hand out card set B. These provide students with a bit less structure in placing the numerals. Do not collect Place card set A. An important part of this task is for students to make connections between different arrangements of numerals. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. The following patterns may be noticed: Students observe that the numbers in vertical lines have the same digit in the ones place and one higher in the tens place. Students observe that numbers in horizontal lines have the same digit in the tens place and one higher in the ones place. Students observe that diagonally the digits in each place either rise or fall depending on the direction. Place Card Set C When students move to Place Card Set C have them fill out there puzzles by writing in their responses. When they have completed this, give them Card Set C to see if their responses match. Place Card Set D When students move to Place Card Set D have them fill out there puzzles by writing in their responses. When they have completed this, give them Card Set D to see if their responses match. Taking two class periods to complete all activities If you have to divide the lesson into two class periods, you may want to have a way for students to save the work they have done with the Place card sets. You may have each group tape the cards down with on their place cards. You may chose to have them do this even if you are not dividing up the class period just to use as a visual during the class discussion. Sharing Work (10 minutes) When students get as far as they can with placing the card sets, ask one student from each group to visit another group s work. Students remaining at their desk should explain their reasoning for the placement of the cards on their own desk. If you are staying at your desk, be ready to explain the reasons for your group s placements. If you are visiting another group, check to see which placements are different from your own. (They might need some kind of representation of their own placement to take with them perhaps a cell phone picture?) If there are differences, ask for an explanation. If you still don t agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Extension activities Give students who finish quickly a mini set of place cards and cards to make their own puzzle. Plenary whole-class discussion (10 minutes) Give each student a mini-whiteboard, marker, and eraser. 4

17 Math Formative Assessment Lesson Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending what has been learned to new examples, and then examining some of the conclusions listed above. Show students the examples of place cards on the slide resources and have them write the missing numeral on their white board. Discuss why they think that number goes there. Have students explain their process for deciding which number goes in the blank and have them describe patterns that they see. Improving individual solutions to the assessment task (10 minutes) Return to the students their original assessment, Torn Hundreds Chart, as well as a second blank copy of the task. Look at your original responses and think about what you have learned during this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. (With first graders, you may only focus on the questions students really seemed to need and state these out loud and ask them again as you move around the room or work with a small group at a time.) 5

18 Math Formative Assessment Lesson Torn Hundreds Chart These are chunks taken from a hundred chart. Fill in the empty boxes with the correct numbers

19 Math Formative Assessment Lesson Place Card Set A

20 Math Formative Assessment Lesson Place Card Set A

21 Math Formative Assessment Lesson Place Card Set B

22 Math Formative Assessment Lesson Place Card Set B

23 Math Formative Assessment Lesson Place Card Set C

24 Math Formative Assessment Lesson Place Card Set C 43 12

25 Math Formative Assessment Lesson Place Card Set D Place Card Set D 13

26 Math Formative Assessment Lesson 36 Card Set A 14

27 Math Formative Assessment Lesson

28 Math Formative Assessment Lesson Card Set A

29 Math Formative Assessment Lesson Card Set B

30 Math Formative Assessment Lesson Card Set B

31 Math Formative Assessment Lesson Card Set C

32 Math Formative Assessment Lesson Card Set C

33 Math Formative Assessment Lesson Card Set D

34 Math Formative Assessment Lesson Card Set D

35 Math Formative Assessment Lesson Extension 23

36 Math Formative Assessment Lesson Extension 24

37 Math Formative Assessment Lesson Extension 25

38 Mathematics Formative Assessment Lesson What is the Value of the Place? Mathematical goals This lesson unit is intended to help you assess how well students are able to recognize the value of the place in a number. It will help you to identify students who have the following difficulties: Recognizing place value of ones, tens, hundreds. Use correct mathematical vocabulary to explain place value to a partner. Make connections of place value to previously taught lessons. Common Core State Standards This lesson involves mathematical content in the standards from across the grades, with emphasis on: 2.NBT: Understand Place Value (2.NBT.1AB) Understand that the three digits of a three digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following special cases: a. 100 can be thought of as a bundle of ten tens called a hundred. b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). (2.NBT.3) Read and write numbers to 1000 using base ten numerals, number names, and expanded form. This lesson involves a range of mathematical practices, with emphasis on: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. Introduction This lesson is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. Students work in small groups on collaborative discussion tasks to represent 3 digit numbers in multiple ways. Throughout their work, students justify and explain their decisions to their peers. Students return to their original assessment tasks, and try to improve their own responses. Materials required Each individual student will need: Two copies of the assessment task What s the Value? A mini whiteboard, marker, and eraser. Each small group of students will need the following resources: A set of arrow cards. A set of numeral cards. There is also a projector resource to help with the whole class discussion. 1

39 Mathematics Formative Assessment Lesson Time needed Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes for a follow up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact timings will depend on the needs of the class. Before the Lesson Assessment task: What s the Value (15 minutes) Have students do this task individually in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow up lesson. Depending on your class you can have them do it all at once or in small groups (they should still work individually.) Give each student a copy of the assessment task What s the Value. Circle all the statements that are true about the number, and explain your thinking. It is important that the students are allowed to answer the questions without your assistance, as far as possible. Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal. Assessing students responses Collect students responses to the task. Make some notes about what their work reveals about their current levels of understanding, and their different problem solving approaches. We suggest that you do not score student s work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students work, using the ideas that follow. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson. The solution to all these difficulties is not to teach one particular way of counting one to one matchingbut to help students to find a variety of ways that work in different situations and make sense to them. 2

40 Mathematics Formative Assessment Lesson Common Issues: Students group cards based on hundred place only. Suggested questions and prompts: Why do you think those are a match? I see that number is 362 and there are 3 hundred flats represented, but what about the rest of the blocks, did you count all those too? How are you tracking that you have counted all the blocks? Is there a way we can make sure we have counted everything? Students are selecting the wrong arrow cards. For example, if the number is 243, the students just may get out the 2 4 and 3 rather than Did you line your arrows up? What does it mean when we line our arrows up and the numbers cover each other up? Look at you 2, what is the value of the two in you number and what is the value on your arrow card? Drawings are not represented correctly in set C. How are you tracking that you have counted all the blocks? Is there a way we can make sure we have counted everything? Suggested lesson outline Collaborative Activity: Match Base Ten Representations and Build Numbers with Arrow Cards Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Set A of numeral cards, arrow cards, and base ten cards Introduce the lesson carefully: I want you to work as a team. Match each of the base ten cards to the correct numeral card. Take turns building the numeral using the arrow cards. Each time you do this, explain your thinking clearly to your partner. If your partner disagrees with your placement then challenge him or her to explain why. It is important that you both understand why each number is built the 3

41 Mathematics Formative Assessment Lesson way it is. There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving. Make a note of student approaches to the task You can then use this information to focus a whole class discussion towards the end of the lesson. In particular, notice any common mistakes. Partners should be engaged in checking their partner, asking for clarification, and taking turns. When calling on students make sure you allow the struggling groups to share first. Support student problem solving Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. If one student has built a number in a particular way, challenge their partner to provide an explanation. If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. The projector resources may be useful when doing this. Place Card Set B As students finish with matching Place card set A, hand out card set B. These are developed to be more difficult. There are three representations of each number using base ten cards which manipulate the tens and ones. Do not collect Place card set A. An important part of this task is for students to make connections between different ways of building the numerals. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. Place Card Set C As students finish with Card Set B, hand out Card Set C. Students will now be challenged to draw two different representations of each number. Do not take up the previous sets of cards. Students may use these for guidance in making their decisions. Sharing Work (10 minutes) When students get as far as they can with the card sets, ask one student from each group to visit another group s work. Students remaining at their desk should explain their reasoning for the placement of the cards on their own desk. 4

42 Mathematics Formative Assessment Lesson If you are staying at your desk, be ready to explain the reasons for your group s work. If you are visiting another group, check to see which matches are different from your own. (They might need some kind of representation of their own placement to take with them perhaps a cell phone picture?) If there are differences, ask for an explanation. If you still don t agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Extension activities Extension 1: Challenge those students who complete card set C to go back and add a third representation of the number. Extension 2: Provide the students with Card Set D and let them explore representations in the thousands. Plenary whole class discussion (10 minutes) Give each student a mini whiteboard, marker, and eraser. Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending what has been learned to new examples, and then examining some of the conclusions listed above. Show students the examples of number arrows on the slide resources and have them write the numeral they represent on their white board. Discuss why they think that number is built. Have students explain their process for deciding which number goes in which place. Improving individual solutions to the assessment task (10 minutes) Return to the students their original assessment, What s the Value?, as well as a second blank copy of the task. Look at your original responses and think about what you have learned during this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. (With second graders, you may only focus on the questions students really seemed to need and state these out loud and ask them again as you move around the room or work with a small group at a time.) Resources: Arrow Cards thous.pdf (Adapted From: Uncovering Student Thinking in Mathematics Grades K 5: 25 Formative Assessment Probes for the Elementary Classroom, Tobey and Minton, Corwin Press, 2011) 5

43 Mathematics Formative Assessment Lesson What s the Value? 243 Circle all the true statements about this number, and explain your thinking. A) 2 tens and 43 ones B) 243 ones C) 2 hundreds 403 ones D) 24 tens and 3 ones E) 2 hundreds and 43 ones F) 1 hundred and 143 ones 6

44 Mathematics Formative Assessment Lesson Card Set A Card Set B

45 Mathematics Formative Assessment Lesson Card Set A 8

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51 Mathematics Formative Assessment Lesson Card Set B 14

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57 Mathematics Formative Assessment Lesson Card Set C

58 Mathematics Formative Assessment Lesson

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60 Mathematics Formative Assessment Lesson

61 Mathematics Formative Assessment Lesson Extension 1 Students will return to card set C and will be asked to draw each number a third way. Extension

62 Math Formative Assessment Lesson Mathematical goals What Subtraction Strategy Did You Use? Grade 3 This lesson unit is intended to help you assess how well students are able to apply and understand a variety of different strategies to subtraction. It will help you to identify students who have the following difficulties: Misunderstandings regarding methods of subtracting lack of conceptual understanding of the properties of numbers misconception that there is only one correct algorithm for each operation misconception that there is no need to understand other methods of subtracting Common Core State Standards This lesson involves mathematical content in the standards from across the grades, with emphasis on: Use place value understanding and properties of operations to perform multi-digit arithmetic. (3.NBT.2) Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. This lesson involves a range of mathematical practices, with emphasis on: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others Introduction This lesson is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. Students work in small groups on collaborative discussion tasks, match student work with student explanation and solve subtraction problems using various strategies. Throughout their work, students justify and explain their decisions to their peers. Students return to their original assessment tasks, and try to improve their own responses. Materials required Each individual student will need: Two copies of the assessment task What Subtraction Strategy Did You Use? Each small group of students will need the following resources: Card Set A Have Activity B ready, but do not pass out Have Activity C ready, but do not pass out 1

63 Math Formative Assessment Lesson Time needed Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact timings will depend on the needs of the class. Before the Lesson Assessment task: What Subtraction Strategy Did You Use? (15 minutes) Students should have been exposed to lessons that promote each of the strategies used in this activity. (think addition, compensation, and split/jump, standard algorithm). Have students do this task individually in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or in small groups (they should still work individually.) Give each student a copy of the assessment task What Subtraction Strategy Did You Use? Circle the method that most closely matches how you solved the problem. Explain whether each method makes sense mathematically. It is important that the students are allowed to answer the questions without your assistance, as far as possible. Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal. Assessing students responses Collect students responses to the task. Make some notes about what their work reveals about their current levels of understanding, and their different problem solving approaches. We suggest that you do not score student s work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students work, using the ideas that follow. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson. The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help students to find a variety of ways that work in different situations and make sense to them. 2

64 Math Formative Assessment Lesson Common Issues: Students always subtract the small number from the larger number rather than regrouping. 46 students subtracted Students do not demonstrate place value understanding. They struggle with breaking two-digit numbers into tens and ones. Students do not think about decomposing numbers into tens and ones for easier adding and subtracting. Suggested questions and prompts: Which numbers did you subtract? Does it matter which order numbers are in when you subtract? What do the digits in 37 stand for? What is their value? How does the position of a number affect its value? How would the value of the 3 and 7 change if we swapped their places? Can you write 37 in expanded form? How could you make this number easier to work with? Suggested lesson outline Collaborative Activity: Subtraction Card Sort and Use of Multiple Strategies Organize the class into groups of three or four students. With larger groups, some students may not fully engage in the task. You may want to consider heterogeneous grouping kids who anchor to different strategies. Give each group a copy of the problem or show the problem on the board or slide. Introduce the lesson carefully: I want you to work as a team. You will have to work together and match students work with the matching explanation. Explain your thinking clearly to your partner. If your partner disagrees with your thinking then challenge him or her to explain why. It is important that you both understand how each problem was solved. There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving. Make a note of student approaches to the task You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may not consider or understand the thought process behind the algorithm of subtraction. They may try to borrow without knowing the foundation math behind the algorithm. Therefore they do not recognize when they conclude with an incorrect answer. Students may become confused with the problems that show multiple ways to subtract if they have not been exposed to decomposing numbers. Support student problem solving 3

65 Math Formative Assessment Lesson Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. When a student creates a match, challenge their partner to provide an explanation. If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. The projector resources may be useful when doing this. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. Activity B As students finish with Card Set A and are able to explain their reasoning give them Card Set B. Card Set B will consist of a subtraction problem for kids to solve using 4 different strategies (standard algorithm, think addition, compensation, and split/jump). Activity C As students finish with Activity B and are able to explain their reasoning give them Activity C. Activity C will consist of a subtraction problem for kids to solve using 4 different strategies (standard algorithm, think addition, compensation, and split/jump). Sharing Work (10 minutes) When students get as far as they can with working the problems, ask one student from each group to visit another group s work. Students remaining at their desk should explain their reasoning for the way they worked the problem at their own desk. If you are staying at your desk, be ready to explain the reasons for your group s work. If you are visiting another group, check to see which answers or explanations are different from your own. (They might need a copy of their own groups work to take with them. If there are differences, ask for an explanation. If you still don t agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Extension activities Students who finish quickly may work another problem or may try to find a different strategy or strategies for working the original problem. Plenary whole-class discussion (10 minutes) Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending what has been learned to new examples, and then examining some of the conclusions listed above. Choose students to share how their group used the different strategies to work the problem. Discuss which strategy they liked best and why. Improving individual solutions to the assessment task (10 minutes) Return to the students their original assessment, Which Subtraction Strategy Did You Use? as well as a second blank copy of the task. 4

66 Math Formative Assessment Lesson Look at your original responses and think about what you have learned during this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. (With first graders, you may only focus on the questions students really seemed to need and state these out loud and ask them again as you move around the room or work with a small group at a time.) 5

67 Math Formative Assessment Lesson Which Subtraction Strategy Did You Use? Katrina, Jenny, Debbie, and Seth each subtracted 28 from Circle the method that most closely matches how you solved the problem. 2. Explain whether each method makes sense mathematically. A) Katrina s Method B) Jenny s Method = =37 C) Debbie s Method =37 D) Seth s Method (Adapted From: Uncovering Student Thinking in Mathematics Grades K-5: 25 Formative Assessment Probes for the Elementary Classroom, Tobey and Minton, Corwin Press, 2011) 6

68 Math Formative Assessment Lesson A) Jessica s Method Card Set A B) Charlie s Method = = =17 C) Renee s Method = 17 D) Teresa s Method

69 Math Formative Assessment Lesson 8

70 Math Formative Assessment Lesson Card Set A 9

71 Math Formative Assessment Lesson 10

72 Math Formative Assessment Lesson Activity B

73 Math Formative Assessment Lesson Activity C

74 Formative Assessment Lesson Materials A Snail in the Well Mathematical goals This lesson unit is intended to help you assess how well students are able to use addition and subtraction in a problem solving situation. In particular, this lesson aims to identify and help students who have difficulties with: Choosing an appropriate, systematic way to collect and organize data. Examining the data and looking for patterns Describing and explaining findings clearly and effectively. Common Core State Standards This lesson involves a range of mathematical practices from the standards, with emphasis on: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 8. Look for and make use of repeated reasoning. This lesson asks students to select and apply mathematical content from across the grades, including the content standards: K-OA: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. 1-OA: Represent and solve problems involving addition and subtraction. 2-OA: Represent and solve problems involving addition and subtraction. 3-OA: Solve problems involving the four operations, and identify and explain patterns in arithmetic. Introduction This lesson unit is structured in the following way: Before the lesson, students attempt the task individually. You then review their work and formulate questions for students to answer in order for them to improve their work. At the start of the lesson, students work individually to answer your questions. Next, they work collaboratively, in small groups, to produce a better collective solution than those they produced individually. Throughout their work, they justify and explain their decisions to peers. In the same small groups, students critique examples of other students work. In a whole-class discussion, students explain and compare the alternative approaches they have seen and used. Finally, students work alone again to improve their individual solutions. Materials required Each individual student will need two copies of the worksheet A Snail in the Well. Each small group of students will need d a copy of Sample Responses to Discuss and whichever samples of student work chosen.

75 Time needed Approximately fifteen minutes before the lesson, a one-hour lesson, and ten minutes in a followup lesson. All times are approximate. Exact timings will depend on the needs of the class. Before the lesson Assessment task: Have the students do this task in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson. Give each student a copy of A Snail in the Well. Introduce the task briefly and help the class to understand the problem and its context. Spend fifteen minutes on your own, answering this question. Show your work. Don t worry if you are not sure if your solution is correct. There will be a lesson on this material [tomorrow] that will help you improve your work. Your goal is to be able to answer this question with confidence by the end of that lesson. It is important that students answer the question without assistance, as far as possible. Students who sit together often produce similar answers, and then, when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual places. Experience has shown that this produces more profitable discussions. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully. We suggest that you do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores, and distracts their attention from how they may improve their mathematics. Instead, help students to make further progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write your own lists of questions, based on your own students work, using the ideas below. You may choose to write questions on each student s work. If you do not have

76 time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson. Common issues - Suggested questions and prompts: Common Issues Student forgets to consider what the snail does each day and each night. Suggested questions and prompts How could you simplify this into an easier task? What sort of diagram might be helpful? How can you show the path the snail follows until he gets out of the well? Student work is unsystematic. What pattern do you notice? What is the same and what is different about the how the snail moves during the day and at night? How can you organize your work? Student assumes that the initial pattern continues indefinitely and over-generalizes. What assumptions can you make about how far the snail travels each day? Does the snail always fall back? Student writes answer without explanation. How could you explain/show how you reached your conclusions so that someone in another class understands? How can you use numbers, words, or diagrams to describe the path of the snail? Student correctly identifies when the snail gets out of the well. Think of another way of solving the problem. Is this method better or worse than your original one? Explain your answer. Can you make a new problem with a different size well and/or a snail that travels different amounts each day and night? Suggested lesson outline Improve individual solutions to the assessment task (10 minutes) Return your students work on the A Snail in the Well problem. Ask students to re-read both the A Snail in the Well problem and their solutions. If you have not added questions to students work, write a short list of your most common questions on the board. Students can then select a few questions appropriate to their own work and begin answering them. Recall what we were working on previously. What was the task?

77 Draw students attention to the questions you have written. I have read your solutions and I have some questions about your work. I would like you to work on your own to answer my questions for ten minutes. Collaborative activity 1: Organize the students into small groups of two or three. In trials, teachers found keeping groups small helped more students play an active role. Students should now work together to produce a joint solution. Put your solutions aside until later in the lesson. I want you to work in groups now. Your task is to work together to produce a solution that is better than your individual solutions. You have two tasks during small-group work, to note different student approaches to the task, and to support student problem solving. Note different student approaches to the task Notice how students work on finding the path of the snail. Notice what strategies they use and how they organize their data. Note the representations they use, including incorrect versions, for use in whole-class discussion. You can use this information to focus the whole-class plenary discussion towards the end of the lesson. Support student problem solving Try not to make suggestions that move students towards a particular approach to this task. Instead, ask questions to help students clarify their thinking. If several students in the class are struggling with the same issue, you could write a relevant question on the board. You might also ask a student who has performed well on one part of the task to help a student struggling with that part of the task. The following questions and prompts would be helpful: What information have you been given? What do you need to find out? What changes in the diagram? What stays the same? How will you write down your pattern? Why do you think your conjecture might be true? Teachers who have used this activity noted that the students sometimes lose sight of both color patterns and how the patterns are related. If this issue arises in your class, help the student to focus his or her attention on both the white and black bead patterns and the connections between the patterns. How are the white beads changing? How are the black beads changing? What effect does the change of the white bead have on the change in the black beads? You may find that some students do not work systematically when organizing their data. What can you do to organize your data to show both patterns?

78 If students have found function rules or equations, focus their attention on improving explanations, or exploring alternative methods. How can you be sure your explanation works in all cases? Ask another group if your argument makes sense. Show me that this equation works. Some stronger explanations are shown in the Sample Responses to Discuss. Make a note of student approaches to the task Give each small group of students a copy of the Sample Responses to Discuss. Choose the samples of student work that match your students level of understanding. Display the following questions on the board or OHP using the provided sheet: Analyzing Student responses to discuss. Describe the problem solving approach the student used. You might, for example: Describe the way the student has organized the data. Describe what the student did to calculate the day the snail gets out of the well. Explain what the student could do to make his or her solution correct or clearer if they calculated correctly. This analysis task will give students an opportunity to evaluate a variety of alternative approaches to the task, without providing a complete solution strategy. During small-group work, support student thinking as before. Also, check to see which of the explanations students find more difficult to understand. Identify one or two of these approaches to discuss in the plenary discussion. Note similarities and differences between the sample approaches and those the students took in small-group work. Plenary whole-class discussion comparing different approaches (20 minutes) Organize a whole-class discussion to consider different approaches to the task. The intention is for you to focus on getting students to understand the methods of working out the answers, rather than just numerical solutions. Focus your discussion on parts of the two small-group tasks students found difficult. Let s stop and talk about different approaches. Ask the students to compare the different solution methods. Read through your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work. Which approach did you like best? Why? Which approach did you find it most difficult to understand? Sami, your group used that method. Can you explain that for us?

79 Improving individual solutions to the assessment task (10 minutes) If you are running out of time, you could schedule this activity for the next lesson or for homework. Make sure students have their original individual work on the A Snail in the Well task on hand. Give them a fresh, blank copy of the A Snail in the Well task sheet. If a student is satisfied with his or her solution, ask the student to try a different approach to the problem and to compare the approach already used. Solution The snail will reach the top of the well on the 8 th day. He does gain one foot per day, so that at the end of the 7 th day he is 7 feet from the bottom of the well. When he travels up 3 ft on the 7 th day he does not slide back down because he reaches the top at 10ft and is out of the well. Analysis of Student Responses to Discuss Will s Method Will noticed that up 3 then down 2 always results in a gain of 1 ft. He overgeneralizes this fact and concludes in will take 10 days for the snail to get out of the well. Timothy s strategy of noticing a gain of 1 foot per day is correct but he did not systematically organize the data to show the complete path of the snail. Will s Solution

80 Whitney s Method Whitney made a list of each day and the amount the snail gained. She overgeneralizes this fact and concludes it will take 10 days for the snail to get out of the well. Whitney s strategy of noticing a gain of 1 foot per day is correct but she did not show the complete path of the snail. Whitney s Solution Chuck s Method Chuck correctly shows subtraction and addition calculations organized by day so that he arrives at the correct solution to the problem. He could enhance his solution by labeling each day and night clearly and/or showing an additional representation (picture, words, chart, etc.) to validate the solution.

81 Chuck s Solution Tim s Method Tim draws a diagram and attempts to show the path of the snail including how the snail moves up the well during each day and slides back each night. However, his diagram is not clear and consistent so he cannot accurately tell how many total feet the snail has traveled. Tim s diagram could be enhanced with additional labels showing feet, day, night, etc.

82 Tim s Solution Denise s Method Denise clearly and accurately shows the path of the snail, labeling feet, day and night. Her distinction in bold versus slashed lines makes the distinction between night and day easy to see. To enhance her solution, Denise could verify her solution with another representation (chart, words, etc.) to show how they are related. Denise s Solution

83 Bill s Method Bill made a chart to organize the data in this problem. He makes columns to identify the overall day in the well, the distance the snail is from the top of the well each day and each night and the total feet the snail is from the top at the end of the day. On the 8 th day Bill shows there are 0 feet left for the snail to travel. His work is organized and labeled so that he correctly arrives at an accurate solution. Bill s labels may be a bit confusing for someone else reading his work. He could enhance his solution by clearly labeling the columns so it easy to tell the difference between day and d, and N and total. He could also further verify the solution with alternative representation. Bill s Solution

84 A Snail in the Well Student Materials

85 A Snail in the Well Student Materials Sample Responses to Discuss Here is some work on A Snail in the Well from students in another class. For each piece of work: 1. Write the name of the student whose solution you are analyzing. 2. Describe the problem solving approach the student used. For example, you might: Describe the way the student has organized the data. Describe what the student did to calculate the day the snail reaches the top of the well. 3. Explain what the student needs to do to complete or correct his or her solution. s_ Solution

86 A Snail in the Well Student Materials s_ Solution s_ Solution

87 Will s Solution

88 Whitney s Solution Chuck s Solution

89 Tim s Solution Denise s Solution

90 Bill s Solution

91 Formative Assessment Lesson Materials Multiplication, Division, and Interpreting Remainders Mathematical goals This lesson unit is intended to help you assess how well students are able to use a variety of strategies to multiply and divide. In particular, this unit aims to identify and help students who have difficulties with: The traditional multiplication algorithm. The traditional division algorithm. Interpreting remainders. Common Core State Standards This lesson involves a range of mathematical practices from the standards, with emphasis on: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. 8. Look for and make use of repeated reasoning. This lesson asks students to select and apply mathematical content from across the grades, including the content standards: 3-OA: Represent and solve problems involving multiplication and division. 3-OA: Understand properties of multiplication and the relationship between multiplication and division. 4-OA: Use the four operations with whole numbers to solve problems. 5-OA: Operations and Algebraic Thinking 3-NBT: Use place value understanding and properties of operations to perform multi-digit arithmetic. 4-NBT: Use place value understanding and properties of operations to perform multi-digit arithmetic. 5-NBT: Perform operations with multi-digit whole numbers. Introduction This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their work, and formulate questions for students to answer, to help them improve their solutions. During the lesson, students work in pairs and threes to match the word problem, model, and multiple strategies of the same multiplication or division problem. In a whole-class discussion, explain their answers. Finally, students return to their original assessment task, and try to improve their own responses. Materials required Each individual student will need two copies of the worksheet Multiplication, Division, and Interpreting Remainders. Each small group of students will need a packet of Card Set A and Card Set B copied in color. The card sets should be cut up before the lesson.

92 Formative Assessment Lesson Materials Time needed Approximately fifteen minutes for the assessment task, a one-hour lesson, and 15 minutes for the students to review their work for changes. All timings are approximate. Exact timings will depend on the needs of the class. Before the lesson Assessment task: Have the students do this task in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson. Give each student a copy of Multiplication, Division, and Interpreting Remainders. Introduce the task briefly and help the class to understand the problem and its context. Spend fifteen minutes on your own, answering these questions. Don t worry if you can t figure it out. There will be a lesson on this material [tomorrow] that will help you improve your work. Your goal is to be able to answer these questions with confidence by the end of that lesson. It is important that students answer the question without assistance, as far as possible. If students are struggling to get started, ask them questions that help them understand what is required, but do not do the task for them. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully. We suggest that you do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores, and distracts their attention from how they may improve their mathematics. Instead, help students to make further progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on the next page. These have been drawn from common difficulties anticipated. We suggest that you write your own lists of questions, based on your own students work, using the ideas below. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson.

93 Formative Assessment Lesson Materials Common issues: Suggested questions and prompts: Common Issues Student doesn t match the cards correctly because he or she doesn t have a conceptual understanding of multiplication. Student doesn t understand Distributive Property. Student doesn t understand the area model for multiplication. Students don t know what to do with the remainder. Suggested questions and prompts If you are multiplying 27x4, what does the 2 represent? the 7? What would happen if you multiplied 20 x 4 and 7 x 4? Could you take those answers and do a calculation to get the answer to 27x4? How can these number(s) we are multiplying be broken apart? What could you do with those numbers to solve this problem? In the problem 27x14 let s look at the number 27. How many 10s are in 27? How many ones? How could you model 27? Now let s look at 14? How many tens? ones? How could you model 14? Is there a way to take those two models and fit them on a rectangle to discover 27x14 without doing any calculations? What does this remainder represent? Suggested lesson outline If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, then we suggest you end the lesson after the first collaborative activity and continue in a second lesson. Collaborative activity 1: matching Card Sets Models 1A, 1B, 1C, 1D, 1E, and 1F (30 minutes) Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Card Sets 1A, 1B, and 1C.

94 Formative Assessment Lesson Materials Introduce the lesson carefully: I want you to work as a team. Take it in turns to match a Model card with either a Lattice card or a Distributive Property card. Each time you do this, explain your thinking clearly and carefully. If your partner disagrees with the placement of a card, then challenge him/her. It is important that you both understand the math for all the placements. There is a lot of work to do today, and it doesn't matter if you don't all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may know Lattice multiplication and the Distributive Property but may not understand the model. Make a note of student approaches to the task Try not to make suggestions that move students towards a particular approach to this task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. Students will correct their own errors once the Partial Product and Traditional Algorithm cards are added. For students struggling to get started: There is more than one way to tackle this task. Can you think what one of them might be? [Working out the answer in either the Distributive Property problem or the Lattice multiplication model and matching that card to the model or finding the answer from the model and matching it to either the Lattice or Distribution Property card.] How can you calculate products with the Distributive Property? with the Lattice model? This Distributive Property card shows (20 + 2) x (10+5)? What would the original multiplication problem be for this model? Does that multiplication problem match any of the other cards on the table?

95 Formative Assessment Lesson Materials If one student has placed a particular card, challenge their partner to provide an explanation. Maria placed this Lattice card with this Model. Martin, why has Maria placed it here? If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. Placing Card Set 1D and 1E: Partial Products and Traditional Algorithm As students finish placing the Model, Distributive Property, and Lattice cards, hand out Card Set 1D and 1E: Partial Products and Traditional Algorithm. These provide students with a different way of interpreting the situation. Do not collect the card sets they have been using. An important part of this task is for students to make connections between different representations of multiplication problems. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. Extension activity Hand out Card Set 1F: Word Problems to students who finish. These are word problems that will match each set of representations. Wrap up the lesson when students are completely satisfied with their own work by handing out the poster template Representations of Multiplication. Students should use it to record the position of their cards. The poster template allows students to record their finished work. It should not replace the cards during the main activities of this lesson as students can more easily make changes when working with the cards, and they encourage collaboration. Sharing work (10 minutes) When students get as far as they can with matching cards, ask one student from each group to visit another group's work. Students remaining at their desk should explain their reasoning for the matched cards on their own desk. If you are staying at your desk, be ready to explain the reasons for your group's matches. If you are visiting another group, write your card placements on a piece of paper. Go to another group's desk and check to see which matches are different from your own. If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.

96 Formative Assessment Lesson Materials When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Taking two lessons to complete second card sort The second card sort, which deals with division, will be completed after the Representation of Multiplication card sort. Repeat the process used in the previous card sort when using the Division Card Sort. Hand out Card Sets 2A and 2B. Once students have matched these sets correctly, give them sets 2C and 2D. Use Card Set 2E for extensions. Improve individual solutions to the assessment task (10 minutes) Return to the students their original assessment, Multiplication, Division, and Interpreting Remainders, as well as a second blank copy of the task. Look at your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. If you find you are running out of time, then you could set this task in the next lesson, or for homework. Solutions Assessment Task: Multiplication, Division, & Interpreting Remainders For questions 1 and 2 student responses will vary depending on how they solve multiplication problems. Be sure that the responses have correct interpretations of each model. Question 3 a) $ = $6 R 2 (or $6.25) Report remainder as a decimal because it is money. b) 25 7 = 3 R 4 Round the remainder up because the 4 people will need a bench to sit on. c) $60.00 $8.00 = 7 R 4 Ignore the remainder because you don t have enough left over to buy a whole pizza. These materials were adapted from Everyday Mathematics, Uncovering Student Misconceptions in Mathematics, and the National Library of Virtual Manipulatives. When teaching these multiplication and division strategies, Teaching Student Centered Mathematics by Van de Walle will be a useful resource.

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104 CARD SET 1 REPRESENTATIONS OF MULTIPLICATION Model Lattice Distributive Property Partial Products Traditional Problem Card Set 1A Card Set 1B Card Set 1C Card Set 1D Card Set 1E Card Set 1F 15 X X Each pack of baseball cards has 15 cards. How many cards are in 22 packs? 12 X X How many eggs are in 12 dozen? 11 X X The boy scout troop eats about 23 grapes each on their campout. How many total grapes did the troop of 11 boys eat?

105 26 X X The deck Scott is building needs 26 boards and each board needs 17 nails. How many nails does Scott need to buy? 14 X X An opossum sleeps an average of 19 hours per day. How many hours does it sleep in a 2-week time period? 13 X X Cam bought 13 different colored folders and each had 28 dots. How many total dots are on her folders?

106 21 X X Bags of reese s cups have 21 individually wrapped peanut butter cups. How many pieces are in 25 bags? 15 X X The zoo has 15 monkeys who eat 15 bananas each day. How many bananas does the need each day for the monkeys?

107 CARD SET 2 - DIVISON Model Groupings Remainder Interpreting Problem Remainders CARD SET 2A CARD SET 2B CARD SET 2C CARD SET 2D CARD SET 2E 2 7/18 Write the remainder as a fraction. The 18 cheerleaders each want a piece of pink ribbon to wear for the breast cancer march. There is 43? inces of ribbon. How much ribbon should each girl get?? 7 Round up the remainder. You are organizing a trolley ride for 95 total students, teachers & parents. If each trolley can seat 15 people, how many trolleys do you need?

108 ? 5 There is no remainder. Mr. Jones bought 95 new pencils to give his class of 19 students. How many pencils will each student get?? 3.67 Write the remainder as a decimal. The soccer team bought their coach a $55.00 sweatshirt. The 15 players split the bill evenly. How much did each pay?? 6 Ignore the remainder. Compact discs are on sale for $13.00 including tax. How many can you buy with $84.00?

109 ? 14 There is no remainder. There are 84 girls in a basketball league and 6 girls on each team. How many teams are there?? 7 Ignore the remainder. Sally scooped out 87 pieces of candy to buy at the store. She wants to divide the candy evenly among 12 people. How many pieces of candy will each person get?

110 Formative Assessment Lesson Arrays, Number Puzzles, and Factor Trees Mathematical goals This lesson unit is intended to help you assess how well students are able to use clues about numbers including: factors, multiples, prime, composite, square, even, odd, etc. In particular, this unit aims to identify and help students who have difficulties with: Understanding the difference between primes and composites, or factors and multiples. Common Core State Standards This lesson involves a range of mathematical practices from the standards, with emphasis on: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. This lesson asks students to select and apply mathematical content from across the grades, including the content standards: 4-OA: Use the four operations with whole numbers to solve problems. 4-OA: Gain familiarity with factors and multiples. 5-NBT: Understand the place value system. 6-NS: Compute fluently with multi-digit numbers and find common factors and multiples. Introduction This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task that is designed to reveal their current understanding and difficulties. You then review their work, and formulate questions for students to answer, to help them improve their solutions. During the lesson, students work in pairs and threes to match the array cards, number clues and factor trees. In a whole-class discussion, explain their answers. Finally, students return to their original assessment task, and try to improve their own responses. Materials required Each individual student will need two copies of the worksheet Arrays, Number Puzzles, and Factor Trees. Each small group of students will need a packet of Card Set 1: Arrays A-J, Number Puzzles 1-8, & Factor Trees a-g; and Card Set 2: Arrays K-T, Number Puzzles 9-16, & Factor Trees h-n copied and cut up before the lesson.

111 Formative Assessment Lesson Time needed Approximately fifteen minutes for the assessment task, a one-hour lesson, and 15 minutes for the students to review their work for changes. All timings are approximate. Exact timings will depend on the needs of the class. Before the lesson Assessment task: Have the students do this task in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson. Give each student a copy of Arrays, Number Puzzles & Factor Trees. Introduce the task briefly and help the class to understand the problem and its context. Spend fifteen minutes on your own, answering these questions. Don t worry if you can t figure it out. There will be a lesson on this material [tomorrow] that will help you improve your work. Your goal is to be able to answer these questions with confidence by the end of that lesson. It is important that students answer the question without assistance, as far as possible. If students are struggling to get started, ask them questions that help them understand what is required, but do not do the task for them. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. The purpose of this is to forewarn you of the issues that will arise during the lesson, so that you may prepare carefully. We suggest that you do not score students work. The research shows that this is counterproductive, as it encourages students to compare scores, and distracts their attention from how they may improve their mathematics.

112 Formative Assessment Lesson Instead, help students to make further progress by asking questions that focus attention on aspects of their work. Some suggestions for these are given on below.. These have been drawn from common difficulties anticipated. We suggest that you write your own lists of questions, based on your own students work, using the ideas below. You may choose to write questions on each student s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson. Common issues - Suggested questions and prompts: Common Issues Student doesn t find all factor pairs for a given number. Conceptual understanding of prime numbers. Student doesn t realize that a prime number will have only one factor pair, only one rectangular array, and it will not have a factor tree if it is prime. Student should also recognize the only EVEN prime number is 2. One is neither prime nor composite. Students do not understand that you do not use one as a factor in factor trees. Not understanding that if there are an odd number of factors for a number, the number is a square number. Suggested questions and prompts How can you make sure you haven t left out any factor pairs for a number? How can you use rectangular arrays to decide if you have all the factor pairs? How can you use a factor tree to decide if you have all the factor pairs? How could you make a chart to decide if you have all the factor pairs? How many other rectangular arrays can you find for each number? What do you notice about the number of rectangular arrays possible for each number? Which number(s) do not have factor tree? Why not? Why are more of the numbers even than odd? Why don t any of the factor trees include the number 1 as a factor? Which rectangular arrays are actually squares? How many total factors do each of your numbers have? How can you tell if a number will have a square as one of its arrays? Suggested lesson outline If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, then we suggest you end the lesson after the first collaborative activity and continue in a second lesson. Collaborative activity 1- matching Card Set 1: Arrays A-J, Number Puzzles 1-8, & Factor Trees a-g (30 minutes)

113 Formative Assessment Lesson Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Card Set 1: Arrays A-J, Number Puzzles 1-8, and Factor Trees a-g. Introduce the lesson carefully: I want you to work as a team. Take it in turns to match am Array with either a Number Puzzle or a Factor Tree. Each time you do this, explain your thinking clearly and carefully. If your partner disagrees with the placement of a card, then challenge him/her. It is important that you both understand the math for all the placements. There is a lot of work to do today, and it doesn't matter if you don't all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice any common mistakes. For example, students may know how to find the number represented by each array or factor tree, but have difficulty finding the answer for each number puzzle.

114 Formative Assessment Lesson Formative Assessment Lesson Make a note of student approaches to the task Try not to make suggestions that move students towards a particular approach to this task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. Students will correct their own errors as they find matches for all the cards. For students struggling to get started: There is more than one way to tackle this task. Can you think what one of them might be? [Finding the number represented by the rectangular arrays is the most efficient way to find the numbers that will match the number puzzle and factor trees.] How can you narrow down which numbers might be the solution to each number puzzle?[it may be better to give students who were unable to complete much of the pre-assessment task, only the arrays & factor trees to begin with, then give them the set of number puzzles once they have the list of possible numbers]. The factor trees are missing numbers, how can you figure out which numbers go in the blanks? The use of a hundred s board may be beneficial to some students to mark off numbers that do not fit the criteria for the number puzzles. All of the solutions are for numbers of 100 or less. If one student has placed a particular card, challenge their partner to provide an explanation. Maria placed this Array card with this Number Puzzle. Martin, why has Maria placed it here? If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. Collaborative activity 2 - matching Card Set 2: Arrays K-T, Number Puzzles 9-16, & Factor Trees h-n (30 minutes)

115 Formative Assessment Lesson Formative Assessment Lesson Card Set 2 can be used in the exact same way as card set one. Set 2 does include clues using the exponents which are not included in set 1. You may have students work on set one the first day and set two a second day, or have all groups of students use card set one and only have card 2 as an extension activity for who finish the first card set quickly. If you have all students complete both sets, an Extension Activity on either day would be to have students write their own number puzzles and make corresponding arrays and factor trees to match the puzzles they have made. Groups who finish making their own number puzzles can switch creations with other groups to test out the new puzzles. Wrap up the lesson when students are completely satisfied with their own work by them glue or tape their cards together in solution sets including arrays, puzzles & factor trees that match. They should have eight matching solution sets for Card Set 1 and another eight for Card Set 2. Sharing work (10 minutes) When students get as far as they can with matching cards, ask one student from each group to visit another group's work. Students remaining at their desk should explain their reasoning for the matched cards on their own desk. If you are staying at your desk, be ready to explain the reasons for your group's matches. If you are visiting another group, write your card placements on a piece of paper. Go to another group's desk and check to see which matches are different from your own. If there are differences, ask for an explanation. If you still don't agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Taking two lessons to complete second card sort The second card set, will be completed after Card Set 1 has been taped or glued on a large sheet of paper. Have students make up their own number puzzles as an extension n either day as needed for an extension. Improve individual solutions to the assessment task (10 minutes) Return to the students their original assessment, Arrays, Number Puzzles & Factor Trees, as well as a second blank copy of the task. Look at your original responses and think about what you have learned this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. If you find you are running out of time, then you could set this task in the next lesson, or for homework.

116 Formative Assessment Lesson Solutions Assessment Task: Arrays, Number Puzzles & Factor Trees For questions 1 and 2 student responses will vary depending on how they solve multiplication problems. Be sure that the responses have correct interpretations of each model. Question 1: # of shifts # of hours

117 Question 2: The dimensions of the sides of the rectangles should be the factors for 32. Students should drawn rectangles with the following dimensions: 1x32, 2x16, 4x8 (or the reverse 32x1, 16x2, 8x4) Question 3: Numbers with exactly three factors would be square numbers. Square numbers between 12 and 40 are: 16, 25, & 36. Only 25 has only three factors. Question 4: only one because prime numbers only have two factors one and itself. For example 7 is prime and has only factors 1 and 7 so the only array would be 1x7 (or 7x1). Question 5: Rosa is correct. Strings of factors at the bottom of a factor tree do not include 1. Question 6: All prime numbers are odd, except for the number 2 so ODDS have more prime numbers. These materials were adapted from Connected Mathematics: Prime Time, and Investigations in Number Data & Space: Mathematical Thinking.

118 Student Materials Arrays, Number Puzzles & Factor Trees 1.) A restaurant is open 24 hours a day. The manager wants to divide the day into workshifts of equal length. Show the different ways this can be done. The shifts should not overlap, and all shifts should be a whole number of hours long. 2.) Sammie s Game Store wants to rent a space of 32 square units. Find all the possible ways Sammie can arrange the squares. How are the rectangles you found related to the factors of 32? 3.) Lewis has chosen a mystery number. His number is larger than 12 and smaller than 40 and it has exactly three factors. What could his number be? 4.) How many rectangles can you build with a prime number of square tiles? 5.) Rosa claims the longest string of factors at the bottom of a factor tree for 30 is 2x3x5. Lon claims there is a longer string at the bottom of the factor tree: 1x2x1x3x1x5. Who is correct? Why? 6.) Which group of numbers evens or odds- has more prime numbers? Why?

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