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Operations and Algebraic Reasoning

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Algebra… Where have you seen students use or apply algebraic reasoning? Where have you seen students struggle with algebraic ideas?

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Refreshing our memory… Glossary, Table 1 – take it out if you have it

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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 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: 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 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?

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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 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?

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Methods Direct Modeling

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Methods Counting Strategies

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Methods Derived Facts or Algorithms

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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.

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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 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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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.

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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.

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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?

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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? Where do basic facts tests fit in?

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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?

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Approaches? Solutions?

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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?”

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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?

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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”

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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?

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4,354 – 3,456 + 455 Gallery Walk Explore various strategies and explanations Any commonalities or frequently occurring ideas?

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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?

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