operations and algebraic thinking: addition and subtraction
TRANSCRIPT
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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
• 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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