Operations and Algebraic Reasoning

Algebra…

• Where have you seen students use or apply algebraic reasoning?

• Where have you seen students struggle with algebraic ideas?

Refreshing our memory…

• Glossary, Table 1 – take it out if you have it

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

• 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: Agree or Disagree

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

Problem Types and their history

• Cognitively Guided Instruction – Problem Types (Types of tasks)

• Is that all there is to CGI ??????

• Does it matter how students solve these problems? Why or why not?

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• Counting Strategies• Algorithms or Derived Facts

• There were 8 seals playing. 3 seals swam away. How many seals were still playing?

• Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan?

Methods

• Direct Modeling

Methods

• Counting Strategies

Methods

• Derived Facts or Algorithms

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.

Direct modeling, counted or invented strategy?

• There were 8 seals playing. 3 seals swam away. How many seals were still playing?

• 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.

Direct modeling, counted or invented strategy?

• Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan?

• A student draws 5 tallies and circles them. They then draw another 5 tallies and circle them and then count their 10 tallies. They do this one more time and count 15 tallies.

Direct modeling, counted or invented strategy?

• Susan is 5 years old. Mark is 15 years old. How many times older is Mark than Susan?

• A student writes the equation 5x3 = 15 and also the equation 15 divided by 5 = 3.

How students solve problems

• Does it matter what strategy students use? Why?

• What does it look like for students to be proficient with a problem type? Does the strategy that they use indicate they are proficient?

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? • Where do basic facts tests fit in?

Factors and Multiples

• Three cruise ships are in port today. They arrive back to port and leave the same day. The Allure of the Seas arrives every 3 days. The Oasis of the Seas arrives every 4 days. The Quantum of the Seas arrives every 6 days.

• Over the next 200 days, on what days will 2 of the ships be in port at the same time? When will 3 of the ships be in port at the same time?

Approaches? Solutions?

Factors and Multiples

• Where is the algebra with this type of work?

• In the following case-– Where is there “algebraic reasoning”?– How does the teacher promote “algebraic

reasoning?”

Task Modification

• Investigations Unit– examine a number sense unit

• Look for “opportunities” to modify tasks to match “more difficult” task types

• Modify/write tasks– What is an appropriate size of numbers? – What are the task types? – How would you assess?

Teaching experiment…

• 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”

Working with Large Numbers

• On your own solve 4,354 – 3,456 + 455 in three different ways

• Write a story problem to match this problem.

• Pick one of your strategies… how did algebraic reasoning help you complete the task?

4,354 – 3,456 + 455

• Gallery Walk

• Explore various strategies and explanations

• Any commonalities or frequently occurring ideas?

4,354 – 3,456 + 455

• Sharing out strategies

• How can estimation help us BEFORE we start?

• Rounding…. Rounding to which place helps us get the best estimate? – What is the point of rounding?