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Session 1 Making Meaning for Whole Number Addition and Subtraction AGENDA Welcome Sharing Student Work Samples Case Discussions Part 1 Math Activity: Modeling Story Problems Case Video Clips: Representing Subtraction on the Number Line Case Discussions Part 2

Goals Exploring mathematics for ourselves Examining print and video cases to learn how students develop ideas Investigating and sharing our own students’ work

“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one?” “I lost count.” Lewis Carroll, Through the Looking Glass “I don’t know,” said Alice.

Student Work Samples Work in pairs or triads first. Read through all the student work and analyses first. What can you learn about the mathematical ideas of these students by examining their work? What ideas are solid and what ideas appear to be missing? What are some commonalities or differences that you note across the samples? Share a potential next step with each child or questions you may have about what the nest step should be.

Case Discussion Part 1 What are the connections do you see between the children’s counting strategies and the operations of addition and subtraction? What do you notice about the operations of addition and subtraction?

Key Ideas Addition and subtraction problems can be solved by counting strategies Solving a subtraction problem by counting back requires paying attention to what you are counting, the spaces or the objects

Key Ideas The operation of addition is connected with the action of combining, subtraction is connected with the actions of taking away and comparing. Children’s counting strategies provide a way to get an answer to a particular problem as well as a way to begin to make sense of the nature of each operation; that is, the counting strategies highlight what actions are associated with addition and subtraction

Math Activity Modeling Story Problems  How did you use cubes and number lines to model the problems?  How do your number sentences match your actions?  How were the situations different?

Math Activity Use cubes, counters or tiles and a number line to perform the actions indicated in each problem. Use a post it note for each problem: Record a number sentence. Create a model that shows the action. Show the action in your model. How does it match your number sentence? Later you will decide which problem type is represented by that problem. Each table will have five post it notes!

Math Activity Part 2 Refer to the classification chart and decide how to classify each of your post its. Place your post its on the chart posters. Be sure that your table number is on each post it.

Key Ideas Focusing on the actions in a story problem highlights distinctions between problems that are represented by the same arithmetic expression Given two problems that adults would call subtraction, children may react in different ways, counting back to solve one problem and adding up to solve the other

Video Clips 1.I have 375 candy bars. I sell 90 of them. How many candy bars do I have left? 2.I am taking a trip to visit my sister in Delaware. I drive 90 miles and then stop to rest. The total distance to my sister’s house is 375 miles. How much farther do I have to go? As you watch the video, make note of the students’ solution methods and the teacher’s questions to set up the task

Case Discussion Part 2 What are the connections between the action of the story problem and the way a student represents it? How is it possible that the same problem can be solved with both an addition and a subtraction sentence? What do the connections among a story situation, a number-line model, and an arithmetic expression illustrate about the relationship between addition and subtraction?

Key Ideas The same problem may be solved by some children with addition and by others with subtraction The term difference is the name for the result of subtraction, but to find two numbers with a given difference, addition may be used Two subtraction situations that may seem the same to adults can be seen as very different by children Examining the ways in which a student uses a number line to model a problem can illuminate the way the student is thinking

Exit Question What mathematical ideas did this session highlight for you? What was the session like for you as a learner?

Lunch Assignment Consider the operations of multiplication and division as you review the cases in Chapter Two