Download Operations and Algebraic Thinking

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)