Operations and Algebraic Thinking: Addition and Subtraction

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

What’s the Sum

• Gallery walk

• What strategies do you see people use?

• What representations do you see?

Sharing Strategies

Modifications of this task

• What grade is this appropriate for?

• How would you modify this to:– Decrease difficulty? – Increase the rigor?

Algebra? Say what?

• Where is the algebra in What’s the Sum?

• Patterns?• Equations?• Generalizations?

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

Problem Types: Agree or Disagree

• The problem types are research-based and come from research with young children doing these tasks.

Problem Types: Agree or Disagree

• This idea of problem types are all over Investigations curriculum in various grades

Problem Types: Agree or Disagree

• Writing tasks to fit a specific problem type is a tasks that my teachers can do.

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).

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

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.

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.

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.”

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.

Problem Types and Strategies

• What does it look like for students to be proficient with a problem type?

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?

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.

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”

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.

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?

Sums of Numbers

• As a whole group

• How would you differentiate this for students?

• Where is the algebraic reasoning?