Operations and Algebraic Thinking

Quick write (sentence each)

• What is algebraic thinking?

• How is algebraic thinking connected to operations?

• Why do students struggle so much with algebra?

Finding a rule…Choose four consecutive two-digit odd counting numbers (e.g., 21, 23, 25, 27). Take the product of the middle two numbers and subtract the product of the first number and the last number. Try a few samples and formulate a rule. Can you write an equation to match this? Explain why the rule works.

What would happen if you used four consecutive counting numbers in the above problem? Would your rule change? If so, find a new rule, and explain why this new rule works and why it is different from the original rule. What about the equation? If the rule does not change, does your explanation change?

Unpacking

• Strategies and Approaches• Math topics that emerged • Grade level connections

Strategies and Approaches

What kind of math “emerged” ?

Grade level connections for this task?

Evens and Odds

• Where does the concept of even and odd numbers first get introduced?

• For younger grades before the terms get introduced, what “building blocks” lead to an understanding of even/odd?

• For older grades after the terms get introduced, what concepts are supported by even/odd numbers?

Common Core Connections • 2.OA.C.3 Determine whether a group of

objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

• What are we expecting students to do with this Standard? Come up with a task that 2nd grade students would do.

Evens and Odds

• Opportunities for proof in mathematics

Proofs? Like that nonsense in high school geometry?

• An example of proof and/or mathematical reasoning

• What was the example?• What did your students do?

• C+M+L = 190• C = M• M+L = 119• 119 + C = 190

Examples

• Importance of question posing to help students have opportunities to “prove”

• Continuing ideas across days• How big is a foot book, measurement using

different sized units• Measurement with kids’ feet and different

sized units get different measurements

What if….

• In a second grade classroom, they have been doing a good bit of work on skip counting.

• After a few days, a second grade student says:• “Why is it that we can count to 10 by 2s, and we can

count to 10 by 5s, but we can’t count to 5 by 2s?”

• Is the student correct in their thinking? Mathematically explain why or why not using a picture.

• What experience/task could they do in order to explore this concept?

Case Analysis: Laying the Groundwork

• Objects of inquiry not objects of exemplars– What’s the difference?

• As you read:– What understandings and misconceptions are

evident among the students?– What “teacher moves” support the students’

learning?– What tasks would you want students to explore next

to deepen their understanding?

Case Analysis

• Find someone else in the room and discuss:– Quick summary of the case– Teacher moves– Student understanding/misconceptions

Common Core Connections • 3.OA.D.9 Identify arithmetic patterns (including

patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

HELP!!!!

What are the CCSSM Authors after with 3.OA.D.9?

Even and odd

• What constitutes “proof” of even/odd numbers in various grade levels?

• What would tasks look like in Grades 2, 3, 4, or 5 that would align with grade level CCSSM and address the idea of even and odd numbers?

• Take some time and create some tasks.

Sharing out

Carrying it forward…

• What opportunities do you have to work with teachers on “algebraic ideas” ?

• What resources do you have access to?

• Design a multi-step task that address even/odd ideas targeted at one of your grade levels

Division connection

• When I take half of a number and it is a whole number the original number is divisible by ___ ?

• When I take half of a half of a number and I get a whole number the original number is always divisible ___?

• When I take half of a half of a half of a number and I get a whole number the original number is divisible by ____ ?

• Prove with pictures and equations!

Exit ticket

• One big take away• One looming question• Rank order the following topics (1 is top):

– Algebra in fractions/decimals– Algebra in multiplication/division– Algebra in addition/subtraction– Algebra and patterns (arithmetic and geometric)