3/26/2015
Conceptual
Understanding with the
Order of Operations
Connie Rivera
consultant for CREC’s Adult Training and
Development Network
CAACE
March 27, 2015
College and Career Readiness
Standards for Adult Education
Discuss
I hear these new
math standards are
more rigorous.
What comes to mind when
you hear that?
Key Instructional Shifts
Focus: Focus strongly where the
standards focus.
Standards for Mathematical
Practice
Coherence: Design learning around coherent
progressions.
Standards for Mathematical Content
Rigor: Pursue conceptual understanding,
procedural skill and fluency, and application—all
with equal intensity.
A Discovery Approach
Session Overview
Session Overview:
Making a shift to more rigorous instruction
means conceptual understanding and
application in real-world contexts have
importance equal to procedural skill and
fluency.
With a discovery
approach,
students construct
rules and discover
formulas on their
own through
carefully structured
activities.
With that in mind, we will:
• Look a sample unit that uses a discovery
approach to teaching mathematics.
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6
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3/26/2015
Lessons that Discover Rules and Formulas
Student-Centered
Scaffolded
Exercises
1+2+3+4+5+6+7+8+9=
Pursue How and Why
7
Does Order Matter?
10
10
1+2+3+4+5+6+7+8+9=
10
10
Weekend Laundry
I’m doing laundry. I discover $4 in
each of three girls’ pockets. The
next day, I find $5 in each of 2
different pockets in my husband’s
clothes.
3•4+2•5=
What is the answer to this
question?
3•4+2•5=
Pay-Day Shopping Splurge
15 + 2(7) + 10
15 + 7 + 7 + 10
Daijah-Tae went
shopping. She bought a
scarf for $15, two lip
glosses for $7 each, and a
cell phone case for $10.
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3/26/2015
Lesson 1: “Exploring Order in
Mathematics”
Lesson 1 “Exploring Order in Mathematics”
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Whyare
areMultiplication
multiplication
and
division
When
and
Division
Done
done before addition and subtraction?
before Addition and Subtraction?”
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Using context, how can this equation be
made true?
2 + 8 = 28
2_ + 8_ = 28_
2 _ + 8 _ = 28 _
2 tens + 8 ones =
How do you go about adding …
2,416 + 753 =
Why do many of us imagine
it written vertically?
2416
+ 753
28 dollars
Football Score
The football team scored 4 touchdowns, worth 6
points each; kicked 3 extra points, worth 1 point
each; and kicked 1 field goal, worth 3 points.
What was the team’s final score?
Express the relationship using words, then use that
sentence to write the situation in math language (an
expression or an equation).
Total Score is 4 touchdowns + 3 extra points + 1 field goal
2 thousands + 4 hundreds + 1 ten + 6 ones
+
7 hundreds + 5 tens + 3 ones
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3/26/2015
Basketball Score
Football Score
The football team scored 4 touchdowns, worth 6
points each; kicked 3 extra points, worth 1 point
each; and kicked 1 field goal, worth 3 points.
What was the team’s final score?
Express the relationship using words, then use that
sentence to write the situation in math language (an
expression or an equation).
Total Score is 4 touchdowns + 3 extra points + 1 field goal
T
=
4•6
+
3•1
+
1•3
Marquis’ team made
11 successful free
throws (1 point each),
25 two-point shots,
and 8 three-point
shots.
Express the relationship using words, then use
that sentence to write the situation in math
language (an expression or an equation).
Whyare
areMultiplication
multiplication
and
division
When
and
Division
Done
done before addition and subtraction?
before Addition and Subtraction?”
Basketball Score
Marquis’ team made
11 successful free
throws (1 point each),
25 two-point shots,
and 8 three-point
shots.
Express the relationship using English sentences,
then use that sentence to write the situation in
math language (an expression or an equation).
11(1) + 25(2) + 8(3)
#
Lesson
“Why
are are
Multiplication
and Division
Done
Lesson22:
“Why
Multiplication
and Division
before Addition and Subtraction?”
Done before Addition and Subtraction?”
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Number of Candies?
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3/26/2015
Exploring the Natural Order of Operations
with Bingo Chip Arrays
Exploring the Natural Order of Operations
with Bingo Chip Arrays
Exploring the Natural Order of Operations
with Bingo Chip Arrays
Exploring the Natural Order of Operations
with Bingo Chip Arrays
Lesson 3 “Exploring the Natural Order
of Operations with Bingo Chip Arrays”
Procedural Skill and Fluency
6(2 + 3) ÷ 3
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3/26/2015
6(2 + 3) ÷ 3
Conceptual Understanding
6(2 + 3) ÷ 3 6(2 + 3) ÷ 3
6(5) ÷ 3 (6∙2 + 6∙3) ÷ 3
(12 + 18) ÷ 3
30 ÷ 3
30 ÷ 3
10
10
6(2 + 3) ÷ 3
6 two
times
+
6(2 + 3) ÷ 3
6(2 + 3) ÷ 3
divided
into 3
equal
groups
6 three
times
3
5
ten
6(2 + 3) ÷ 3
two + three
two + three
two + three
two + three
6 times
ten
divided by three
Application
6(2 + 3) ÷ 3
ten
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3/26/2015
6(2 + 3) ÷ 3
Conceptual Understanding
with the Order of
Operations – Part 2
Connie Rivera
consultant for CREC’s Adult Training and
Development Network
CAACE
March 27, 2015
Procedural Fluency in Mathematics
“I know how to do
this!
… Just remind me
what that procedure
was again?”
Examining Wrong Answers
Examining Wrong Answers
A
C
63
– 44
21
B 2 45
x 34
140
13
x 12
26
+ 13
39
Examining Wrong Answers
A painter rented a wallpaper
steamer at 9 a.m. and returned it at
4 p.m. He paid a total of $28.84.
What was the rental cost per hour?
oA. $2.43
oB. $3.61
oC. $4.12
oD. $5.77
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+
·
÷
+
÷
Explain what each
student understands or
misunderstands about
evaluating expressions.
Explain what each
student understands or
misunderstands about
evaluating expressions.
Dante
Dante
Arissa
Roman
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−
43
Arissa
#
Roman
44
÷
+
Roman
Dante
Stations
Arissa
Explain what each
student understands or
misunderstands about
evaluating expressions.
45
Expressions and Array Card Match
24 Game
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3/26/2015
Prove It! True or False?
Writing Expressions for Real World Arrays
…for WRITING
EXPRESSIONS
6x2=2x6
=
PHOTOGRAPHS as
inspiration…
50
Hand-Outs Folder
•Find the Missing
Number
•Grouping Symbols
•Creating True Equations
•Properties of Numbers Challenge
Stations
•Open Middle’s Exponents and Order of
Operations
Questions, Comments, Take-Aways?
Achieve.org/EQuIP
EQuIP Exemplar
Scroll all the way down to the bottom….
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3/26/2015
Upcoming Sessions
“Using Area Models to
Teach Math Concepts”
CANMI April 10th
“Math for Science and
Social Studies”
May 1st
Thank You!
Find resources for workforce system success at:
www.workforce3one.org
Connie Rivera
CoRivera@CREC.org
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