Conceptual Understanding with the Order of Operations...

3/26/2015

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Conceptual Understanding with the

Order of OperationsConnie Rivera

consultant for CREC’s Adult Training and Development Network

CAACE

March 27, 2015

Discuss

I hear these new math standards are

more rigorous.

What comes to mind when

you hear that?

College and Career Readiness College and Career Readiness College and Career Readiness College and Career Readiness Standards for Adult EducationStandards for Adult EducationStandards for Adult EducationStandards for Adult Education

Standards for Mathematical Content

Standards for Mathematical

Practice

Key Instructional ShiftsKey Instructional ShiftsKey Instructional ShiftsKey Instructional Shifts

Focus: Focus strongly where the

standards focus.

Coherence: Design learning around coherent

progressions.

Rigor: Pursue conceptual understanding,

procedural skill and fluency, and application—all

with equal intensity.

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 that in mind, we will:

• Look a sample unit that uses a discovery

approach to teaching mathematics.# 5

Session OverviewSession OverviewSession OverviewSession Overview

6

With a discovery

approach,

students construct

rules and discover

formulas on their

own through

carefully structured

activities.

A Discovery Approach

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7

Student-Centered

Pursue How and Why

ScaffoldedExercises

Lessons that Discover Rules and Formulas

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 =

10

10

10

10What is the answer to this

question?

3 • 4 + 2 • 5 =

Does Order Matter?

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 =

Weekend Laundry Pay-Day Shopping Splurge

Daijah-Tae went

shopping. She bought a

scarf for $15, two lip

glosses for $7 each, and a

cell phone case for $10.

15 + 2(7) + 1015 + 7 + 7 + 10

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Lesson 1 “Exploring Order in Mathematics”

13#

Lesson 1: “Exploring Order in Lesson 1: “Exploring Order in Lesson 1: “Exploring Order in Lesson 1: “Exploring Order in

Mathematics”Mathematics”Mathematics”Mathematics”

# 14

Why are multiplication and division done before addition and subtraction?

When are Multiplication and Division Done When are Multiplication and Division Done When are Multiplication and Division Done When are Multiplication and Division Done

before Addition and Subtraction?”before Addition and Subtraction?”before Addition and Subtraction?”before Addition and Subtraction?”

2 + 8 = 28

Using context, how can this equation be

made true?

2 _ + 8 _ = 28 _2 tens + 8 ones = 28 dollars

2_ + 8_ = 28_

How do you go about adding …

2,416 + 753 =

Why do many of us imagine

it written vertically?

2416

+ 753

2 thousands + 4 hundreds + 1 ten + 6 ones

+ 7 hundreds + 5 tens + 3 ones

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).

Football Score

Total Score is 4 touchdowns + 3 extra points + 1 field goal

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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).

T = 4 • 6 + 3 • 1 + 1 • 3

Football Score

Total Score is 4 touchdowns + 3 extra points + 1 field goal

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 words, then use

that sentence to write the situation in math

language (an expression or an equation).

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)

# 22

Why are multiplication and division done before addition and subtraction?

When are Multiplication and Division Done When are Multiplication and Division Done When are Multiplication and Division Done When are Multiplication and Division Done

before Addition and Subtraction?”before Addition and Subtraction?”before Addition and Subtraction?”before Addition and Subtraction?”

Lesson 2 “Why are Multiplication and Division Done before Addition and Subtraction?”

23#

Lesson Lesson Lesson Lesson 2: “Why are Multiplication and Division 2: “Why are Multiplication and Division 2: “Why are Multiplication and Division 2: “Why are Multiplication and Division

Done before Addition and Subtraction?”Done before Addition and Subtraction?”Done before Addition and Subtraction?”Done before Addition and Subtraction?” Number of Candies?

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Exploring the Natural Order of Operations with Bingo Chip Arrays




29

Lesson 3 “Exploring the Natural Order of Operations with Bingo Chip Arrays” Procedural Skill and Fluency

6(2 + 3) ÷ 3

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6(2 + 3) ÷ 3

6(2 + 3) ÷ 3

6(2 + 3) ÷ 3

6(5) ÷ 3 (6∙2 + 6∙3) ÷ 3

30 ÷ 3

10

(12 + 18) ÷ 3

30 ÷ 3

10

Conceptual Understanding

6(2 + 3) ÷ 3

6(2 + 3) ÷ 3

6 two

times

6 three

times+

divided

into 3

equal

groups

ten

6(2 + 3) ÷ 3

3

5 6 times

divided by three

ten

6(2 + 3) ÷ 3two + three

two + three

two + three

two + three

ten

Application

6(2 + 3) ÷ 3

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6(2 + 3) ÷ 3Conceptual 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

13

x 12

26

+ 13

39

63

– 44

21

AC

45

x 34

140

B 2

Examining Wrong AnswersA 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

Examining Wrong Answers

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� � + �� ÷ �

43#

Explain what each

student understands or

misunderstands about

evaluating expressions.

Dante

ArissaRoman

� · � + �� ÷ �

44#Arissa Roman

Explain what each




Dante

�� − �� ÷ �

� + �

45

Arissa

RomanDante

Explain what each




StationsStationsStationsStations

Expressions and Array Card Match 24 Game

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Prove It! True or False?

6 x 2 = 2 x 6

=50

…for WRITING

EXPRESSIONS

PHOTOGRAPHS as inspiration…

Writing Expressions for Real World Arrays

Hand-Outs Folder

•Find the Missing

Number

•Grouping Symbols

•Creating True Equations

•Properties of Numbers Challenge

•Open Middle’s Exponents and Order of

Operations

StationsStationsStationsStations

Questions, Comments, Take-Aways? Achieve.org/EQuIP

Scroll all the way down to the bottom….

EQuIP Exemplar

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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

56#

Connie [email protected]

Conceptual Understanding with the Order of Operations...

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