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Operations and Algebraic Thinking: Addition and Subtraction

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What’s the Sum Complete the task with people around you - sheet sheet Find the sum of all the numbers in the rectangle Look for strategies or patterns that support your exploration of the task

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What’s the Sum Gallery walk What strategies do you see people use? What representations do you see?

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Sharing Strategies

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Modifications of this task What grade is this appropriate for? How would you modify this to: – Decrease difficulty? – Increase the rigor?

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Algebra? Say what? Where is the algebra in What’s the Sum? Patterns? Equations? Generalizations?

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Algebra and Addition/Subtraction Starting with the familiar problem types – Glossary, Table 1 chart also in the DPI Unpacking document Take a few minutes – Come up with a “progression” from easy to hard for these problem types? – Construct a viable argument about your progression and why certain things come before or after others

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Problem Types: Agree or Disagree The problem types are research-based and come from research with young children doing these tasks.

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Problem Types: Agree or Disagree This idea of problem types are all over Investigations curriculum in various grades

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Problem Types: Agree or Disagree Writing tasks to fit a specific problem type is a tasks that my teachers can do.

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Problem Types: Agree or Disagree When we think about problem types with addition and subtraction it does not matter at all about how students “solve” tasks (e.g., manipulatives, drawing, counting, number lines).

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Problem Types and their history Cognitively Guided Instruction – Problem Types (Types of tasks) – Methods in which students solve tasks – Decisions that teachers go through to formatively assess students AND then pose follow-up tasks

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Methods Direct Modeling Separate (Result Unknown) There were 8 seals playing. 3 seals swam away. How many seals were still playing? A student would….. A set of 8 objects is constructed. 3 objects are removed. The answer is the number of remaining objects.

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Methods Counting Strategies Separate (Result Unknown) There were 8 seals playing. 3 seals swam away. How many seals were still playing? A student would….. Start at 8 and count backwards 3 numbers. The number they end on would be their answer.

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Methods Invented algorithms /derived strategies Separate (Result Unknown) There were 8 seals playing. 3 seals swam away. How many seals were still playing? What would students do? “4 plus 4 is 8, so 8 minus 4 is 4. But I am only taking away 3 so there should be 5 seals playing.”

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Direct modeling, counted or invented strategy? There were 8 seals playing. 3 seals swam away. How many seals were still playing? The student starts at 8 on a number line and count backwards 3 numbers. The number they land on is their answer. The student puts 3 counters out and adds counters until they get to 8. The number of counters added is their answer. The student draws 8 tallies and crosses out 3. The number left is their answer. The student starts at 3 and counts up until they get to 8. As the student counts they put a finger up (1 finger up as they say 4, 5, 6, 7, 8). The number of fingers up is their answer.

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Problem Types and Strategies What does it look like for students to be proficient with a problem type?

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Common Core Connection “Fluently add and subtract” – What do we mean when students are fluent? Fluently (Susan Jo Russell, Investigations author) – Accurate, Efficient, Flexible What do these mean?

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Taking this back to our schools What does this have to do with teachers in various grades? Pick two grade levels that you work with in your school. Write 3 tasks using the various problem types (involving addition and subtraction). Describe how the three strategies might look with students in that grade level.

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For next time…. Select students who are struggling Pose a few problems for a problem type Observe and question Pose a follow-up task that “meets them where they are”

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Sums of numbers Find 2 4-digit numbers that will add up to 9,999. Do not use a 0 in any of your numbers. Find at least 4 possible answers. Find 2 4-digit numbers that add up to 10,101. Do not use a 0 in any of your numbers. Find at least 4 possible answers.

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Sums of Numbers With people around you discuss: – What were your initial strategies? – How was the first task different from the second task? – What makes this kind of task challenging for students?

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Sums of Numbers As a whole group How would you differentiate this for students? Where is the algebraic reasoning?

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