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, 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, Gallery Walk Explore various strategies and explanations Any commonalities or frequently occurring ideas?

4,354 – 3, 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?