what subtraction strategy did you use? grade 3 · math formative assessment lesson alpha version 1...

Math Formative Assessment Lesson Alpha Version

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What Subtraction Strategy Did You Use? Grade 3 Mathematical goals 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


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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.


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Common Issues: Suggested questions and prompts: Students always subtract the small number from the larger number rather than regrouping. 46 students subtracted 8-6. -28 22

Which numbers did you subtract? Does it matter which order numbers are in when you subtract?

Students do not demonstrate place value understanding. They struggle with breaking two-digit numbers into tens and ones.

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?

Students do not think about decomposing numbers into tens and ones for easier adding and subtracting.

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


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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.


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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.)


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Which Subtraction Strategy Did You Use?

Circle Explain

65 -28 ______ 37

(Adapted From: Uncovering Student Thinking in Mathematics Grades K-5: 25 Formative Assessment Probes for the Elementary Classroom, Tobey and Minton, Corwin Press, 2011)

+2 30 +30

+5 60


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Card Set A

56 -39 ______ 17

+1 40

+10

+6 50


8


9

Card Set A


10


11

Activity B

42-28


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Activity C

81-56

what subtraction strategy did you use? grade 3 · math formative assessment lesson alpha version 1...

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