Content Deepening
th
6 Grade Math
January 24, 2014
Jeanne Simpson
AMSTI Math Specialist
Welcome
Name
School
What
are you hoping to learn
today?
2
He who dares to teach
must never cease to
learn.
John Cotton Dana
3
Goals for Today
 Implementation
of the Standards of
Mathematical Practices in daily lessons
 Understanding
of what the CCRS expect
students to learn blended with how they
expect students to learn.
 Student-engaged
learning around highcognitive-demand tasks used in every
classroom.
Agenda

Statistics and Probability

Progression

Standards Analysis

Resources

High-Cognitive Demand Tasks

Expressions and Equations

Inequalities

Resources

Standards of Mathematical Practice

Fractions
acos2010.wikispaces.com

Electronic version of handouts

Links to web resources
Statistics and Probability
THE STRUCTURE IS THE STANDARDS
The natural distribution of prior knowledge in classrooms should not
prompt abandoning instruction in grade level content, but should
prompt explicit attention to connecting grade level content to content
from prior learning. To do this, instruction should reflect the
progressions on which the CCSSM are built. For example, the
development of fluency with division using the standard algorithm in
grade 6 is the occasion to surface and deal with unfinished learning
with respect to place value. Much unfinished learning from earlier
grades can be managed best inside grade level work when the
progressions are used to understand student thinking.
 http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422
KNOWLEDGE GAPS
This is a basic condition of teaching and should not be ignored in the
name of standards. Nearly every student has more to learn about the
mathematics referenced by standards from earlier grades. Indeed, it is
the nature of mathematics that much new learning is about extending
knowledge from prior learning to new situations. For this reason,
teachers need to understand the progressions in the standards so
they can see where individual students and groups of students are
coming from, and where they are heading. But progressions
disappear when standards are torn out of context and taught as
isolated events.
 http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422
Learning Progression Jigsaw



Read your assigned section

K-5 Data, pages 1-5

6-8 Overview, pages 6-7

Grade 6, pages 8-10

Grade 7, pages 11-14
Chart paper

Summarize what needs to be learned.

How can this document help you in your classroom?
Be prepared to share
10
Analysis Tool
6.SP.1 Recognize a statistical question as one that
anticipates variability in the data related to the question
and accounts for it in the answers.
Content
Standard
Cluster
Develop
understanding
of statistical
variability.
Which
Standards in
the Cluster
Are Familiar?
What’s New
or
Challenging in
These
Standards?
Which
Standards in
the cluster
Need
Unpacking or
Emphasizing?
How Is This
Cluster
Connected to
the Other 6-8
Domains and
Mathematical
Practice?
What resources do you already have for
teaching statistics?
How do these match the standards?
SP Resources
 Raisin
Activity
 MARS
– Mean, Median, Mode
 CMP
 Data
About Us
 Common
 Lessons
 How
Core Investigations
for Learning
MAD are You?
 Shakespeare
vs. Rowling
Raisin Activity

Count the number of raisins in your box.

Make a box plot for the number of raisins in each
brand’s box.

Find the median, range, and interquartile range
for each brand.

Make a dot plot of the data. Find the mean and
the mean absolute deviation.
Mean, Median, Mode, and Range
Computer Games: Ratings
Imagine rating a popular computer game.
You can give the game a score of between 1 and 6.
P-16
Computer Games: Ratings
Rate the game Candy Crush with a score between
1 and 6.
P-17
Bar Chart from a Frequency Table
Score
1
2
3
4
5
6
Frequency
Mean score
Median score
Mode score
Range of scores
P-18
Matching Cards
1.
Each time you match a pair of cards, explain
your thinking clearly and carefully.
2.
Partners should either agree with the
explanation or challenge it if it is unclear or
incomplete.
3.
Once agreed stick the cards onto the poster
and write a justification next to the cards.
4.
Some of the statistics tables have gaps in
them and one of the bar charts is blank. You
will need to complete these cards.
P-19
Sharing Posters
1.
One person from each group visit a different
group and look carefully at their matched
cards.
2.
Check the cards and point out any cards you
think are incorrect. You must give a reason why
you think the card is incorrectly matched or
completed, but do not make changes to the
card.
3.
Return to your original group, review your own
matches and make any necessary changes using
arrows to show if card needs to move.
P-20
AMSTI Connected Math Unit
How MAD are You?
(Mean Absolute Deviation)
 Fist
to Five…How much do you know about
Mean Absolute Deviation?
0
= No Knowledge
5
= Master Knowledge
Create a distribution of nine
data points on your number
line that would yield a
mean of 5.
Card Sort
 Which
data set seems to differ the least
from the mean?
 Which
data set seems to differ the most
from the mean?
 Put
all of the data sets in order from
“Differs Least” from the mean to “Differs
Most” from the mean.
The mean in each set equals 5.
Find the distance (deviation) of each point from the
mean.
Use the absolute value of each distance.
3
3
3
2
1
1
3
4
Find the mean of the absolute deviations.
6
 How
could we arrange the nine points
in our data to decrease the MAD?
 How could we arrange the nine points
in our data to increase the MAD?
 How MAD are you?
Shakespeare vs. Rowling
An effective mathematical
task is needed to challenge
and engage students
intellectually.
Comparing Two
Mathematical Tasks
Solve Two Tasks:
• Martha’s Carpeting Task
• The Fencing Task
Comparing Two
Mathematical Tasks
How are Martha’s Carpeting Task
and the Fencing Task the same
and how are they different?
Similarities and Differences
Similarities
Differences
• Both are “area”
problems
• The amount of thinking
and reasoning required
• Both require prior
knowledge of area
• The number of ways the
problem can be solved
• Way in which the area
formula is used
• The need to generalize
• The range of ways to
enter the problem
Comparing Two
Mathematical Tasks
Do the differences between the
Fencing Task and Martha’s
Carpeting Task matter?
Why or Why not?
Criteria for low cognitive demand tasks
• Recall
• Memorization
• Low on Bloom’s Taxonomy
Criteria for high cognitive demand tasks
• Requires generalizations
• Requires creativity
• Requires multiple representations
• Requires explanations (must be “worth
explaining”)
Patterns of Set up, Implementation,
and Student Learning
Task Set Up
Task Implementation
Student Learning
A.
High
High
High
B.
Low
Low
Low
C.
High
Low
Moderate
Stein & Lane, 2012
•
•
•
•
•
•
Factors Associated with the Maintenance and
Decline of High-Level Cognitive Demands
Routinizing problematic aspects of the task
Shifting the emphasis from meaning, concepts, or
understanding to the correctness or completeness of the
answer
Providing insufficient time to wrestle with the demanding
aspects of the task or so much time that students drift into offtask behavior
Engaging in high-level cognitive activities is prevented due to
classroom management problems
Selecting a task that is inappropriate for a given group of
students
Failing to hold students accountable for high-level products or
processes
(Stein, Grover & Henningsen, 2012)
Factors Associated with the Maintenance and
Decline of High-Level Cognitive Demands
• Scaffolding of student thinking and reasoning
• Providing a means by which students can monitor their own
progress
• Modeling of high-level performance by teacher or capable
students
• Pressing for justifications, explanations, and/or meaning
through questioning, comments, and/or feedback
• Selecting tasks that build on students’ prior knowledge
• Drawing frequent conceptual connections
• Providing sufficient time to explore
(Stein, Grover & Henningsen, 2012)
“Not all tasks are created equal, and
different tasks will provoke different
levels and kinds of student
thinking.”
(Stein, Smith, Henningsen, & Silver, 2011)
“The level and kind of thinking in
which students engage determines
what they will learn.”
(Hiebert et al., 2011)
Expressions and Equations
Activities



Inequalities

Look at standards. What is required?

Illustrative Mathematics tasks
MARS – Laws of Arithmetic lesson

Arithmetic with whole-number exponents

Order of operations

Finding area of compound rectangles by evaluating expressions
Math-Magic

Using variables
What are students asked to do with
inequalities?

6.EE.5 – Understand solving an equation or inequality as a process of answering a
question: which values from a specified set, if any, make the equation or
inequality true? Use substitution to determine whether a given number in a
specified set makes an equation or inequality true.

6.EE.6 – Use variables to represent numbers and write expressions when solving a
real-world or mathematical problem; understand that a variable can represent an
unknown number, or, depending on the purpose at hand, any number in a specified
set.

6.EE.7 – Solve real-world and mathematical problems by writing and solving
equations of the form x + p = q and px = q for cases in which p, q and x are all
nonnegative rational numbers.

6.EE.8 – Write an inequality of the form x > c or x < c to represent a constraint or
condition in a real-world or mathematical problem. Recognize that inequalities of
the form x > c or x < c have infinitely many solutions; represent solutions of such
inequalities on number line diagrams.
Log Ride
(6.EE.5)
A theme park has a log ride that can hold 12 people. They also have a weight
limit of 1500 lbs per log for safety reasons. If the average adult weights 100 lbs
and the log itself weights 200, the ride can operate safely if the inequality
150A + 100C + 200 < 1500
is satisfied (A is the number of adults and C is the number of children in the log
ride together). There are several groups of children of differing numbers waiting
to ride. If 4 adults are already seated in the log, which groups of children can
safely ride with them?
Group 1: 4 children
Group 2: 3 children
Group 3: 9 children
Group 4: 6 children
Group 5: 5 children
Fishing Adventures (6.EE.8)
Fishing Adventures rents small fishing boats to tourists for
day-long fishing trips. Each boat can hold at most eight
people. Additionally, each boat can only carry 900 pounds of
weight for safety reasons.
a.
Let p represent the total number of people. Write an
inequality to describe the number of people that a boat
can hold. Draw a number line diagram that shows all
possible solutions.
b.
Let w represent the total weight of a group of people
wishing to rent a boat. Write an inequality that describes
all total weights allowed in a boat. Draw a number line
diagram that shows all possible solutions.
Laws of Arithmetic
Projector Resources
Laws of Arithmetic
Pre-Assessment Task
Projector Resources
Laws of Arithmetic
Pre-Assessment Task
Projector Resources
Laws of Arithmetic
Pre-Assessment Task
Projector Resources
Laws of Arithmetic
Teacher Notes
Projector Resources
Laws of Arithmetic
Writing Expressions
4
5
3
Write an expression to represent the
total area of this diagram
Projector Resources
Laws of Arithmetic
P-58
Compound Area Diagrams
4
Area
A
1
2
4
Area B
5
5
1
Which compound
area diagram
represents the
expression:
2
5 + 4 x 2?
5
Area C
Projector Resources
2
4
1
Laws of Arithmetic
P-59
Matching Cards
1. Take turns at matching pairs of cards that you think belong
together. For each Area card there are at least two
Expressions cards.
2. Each time you do this, explain your thinking clearly and
carefully. Your partner should either explain that reasoning
again in his/her own words or challenge the reasons you gave.
3. If you think there is no suitable card that matches, write one of
your own on a blank card.
4. Once agreed, stick the matched cards onto the poster paper
writing any relevant calculations and explanations next to the
cards.
You both need to be able to agree on and to be able to
explain the placement of every card.
Projector Resources
Laws of Arithmetic
P-60
Sharing Work
1. If you are staying at your desk, be ready to explain the
reasons for your group’s matches.
2. If you are visiting another group:
– Copy your matches onto your paper.
– Go to another group’s desk and check to see
which matches are different from your own.
3. If there are differences, ask for an explanation.
If you still don’t agree, explain your own thinking.
4. Return to your original group, review your own matches
and make any necessary changes using arrows to
show that a card needs to move.
Projector Resources
Laws of Arithmetic
P-61
Variables and
Expressions
6.EE.2 – Write, read, and evaluate expressions in which letters stand for
numbers
6.EE.2a – Write expressions that record operations with numbers and with
letters standing for numbers
6.NS.6 – Understand a rational number as a point on a number line.
Extend number line diagrams and coordinate axes familiar from previous
grades to represent points on the line and in the plane with negative
number coordinates.
Math Magic – Trick #1
Pick a number… any number! (keep it a secret
though)
 Add 1 to that number
 Multiply by 3
 Subtract your ‘secret’ number
 Add 5
 Divide by 2
 Subtract your secret number

Math Magic – Trick #1
Now
map the digit you got to a letter in
the alphabet.
For example:
A =1
Pick
B=2
C=3
D=4
a name of a country in Europe that
starts with that letter.
Countries in Europe












Albania
Andorra
Austria
Belarus
Belgium
Bosnia and
Herzegovina
Bulgaria
Croatia
Cyprus
Czech Republic
Denmark
Estonia













Finland
France
Germany
Greece
Hungary
Iceland
Ireland
Italy
Latvia
Liechtenstein
Lithuania
Luxembourg
Macedonia











Malta
Moldova
Monaco
Netherlands
Norway
Poland
Portugal
Romania
Russia
San Marino
Serbia and
Montenegro









Slovakia
(Slovak
Republic)
Slovenia
Spain
Sweden
Switzerland
Turkey
Ukraine
United
Kingdom
Vatican City
Math Magic – Trick #1

Take the second letter in the country's name, and pick and
animal that starts with that letter.

Think of the color of that animal.
Directions
a. Think of a number
b. Add 5
c. Multiply by 3
d. Subtract 3
e. Divide by 3
f. Subtract your original number
How The
Numbers Change
2
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
d. Subtract 3
e. Divide by 3
f. Subtract your original number
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
21
d. Subtract 3
e. Divide by 3
f. Subtract your original number
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
21
d. Subtract 3
18
e. Divide by 3
f. Subtract your original number
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
21
d. Subtract 3
18
e. Divide by 3
6
f. Subtract your original number
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
21
d. Subtract 3
18
e. Divide by 3
6
f. Subtract your original number
4
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
b. Add 5
7
c. Multiply by 3
21
d. Subtract 3
18
e. Divide by 3
6
f. Subtract your original number
4
7
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
7
b. Add 5
7
12
c. Multiply by 3
21
36
d. Subtract 3
18
33
e. Divide by 3
6
11
f. Subtract your original number
4
4
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
7
b. Add 5
7
12
c. Multiply by 3
21
36
d. Subtract 3
18
33
e. Divide by 3
6
11
f. Subtract your original number
4
4
15
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
7
15
b. Add 5
7
12
20
c. Multiply by 3
21
36
60
d. Subtract 3
18
33
57
e. Divide by 3
6
11
19
f. Subtract your original number
4
4
4
Algebraic
Expressions
Directions
How The
Numbers Change
a. Think of a number
2
7
15
b. Add 5
7
12
20
c. Multiply by 3
21
36
60
d. Subtract 3
18
33
57
e. Divide by 3
6
11
19
f. Subtract your original number
4
4
4
Algebraic
Expressions
n
Directions
How The
Numbers Change
Algebraic
Expressions
a. Think of a number
2
7
15
n
b. Add 5
7
12
20
n+5
c. Multiply by 3
21
36
60
d. Subtract 3
18
33
57
e. Divide by 3
6
11
19
f. Subtract your original number
4
4
4
Directions
How The
Numbers Change
Algebraic
Expressions
a. Think of a number
2
7
15
n
b. Add 5
7
12
20
n+5
c. Multiply by 3
21
36
60
3n + 15
d. Subtract 3
18
33
57
e. Divide by 3
6
11
19
f. Subtract your original number
4
4
4
Directions
How The
Numbers Change
Algebraic
Expressions
a. Think of a number
2
7
15
n
b. Add 5
7
12
20
n+5
c. Multiply by 3
21
36
60
3n + 15
d. Subtract 3
18
33
57
3n + 12
e. Divide by 3
6
11
19
f. Subtract your original number
4
4
4
Directions
How The
Numbers Change
Algebraic
Expressions
a. Think of a number
2
7
15
n
b. Add 5
7
12
20
n+5
c. Multiply by 3
21
36
60
3n + 15
d. Subtract 3
18
33
57
3n + 12
e. Divide by 3
6
11
19
n+4
f. Subtract your original number
4
4
4
Directions
How The
Numbers Change
Algebraic
Expressions
a. Think of a number
2
7
15
n
b. Add 5
7
12
20
n+5
c. Multiply by 3
21
36
60
3n + 15
d. Subtract 3
18
33
57
3n + 12
e. Divide by 3
6
11
19
n+4
f. Subtract your original number
4
4
4
4
General MacArthur's # Game

Write down the number of the month you were born in

Double it

Add 5

Multiply by 50

Add in your age

Subtract 365

What is your number?
Your Game!

Create your own magic with math to share with
your friends and family.

Complete the worksheet with directions you make
up.

Try three different numbers.

Use algebraic expressions to show why the magic
works.
Exit Question

Ricardo has 8 pet mice. He keeps them in two cages that
are connected so that the mice can go back and forth
between the cages. One of the cages is blue, and the
other is green. Show all the ways that 8 mice can be in
two cages.
Standards of Mathematical Practice
SMP Proficiency Matrix
Students:
Make sense of
problems
(I) Initial
Explain their thought processes in solving a
problem one way.
(IN) Intermediate
Explain their thought processes in solving a problem
and representing it in several ways.
(A) Advanced
Discuss, explain, and demonstrate solving a
problem with multiple representations and in
multiple ways.
1b
Persevere in
solving them
Stay with a challenging problem for more
than one attempt.
Try several approaches in finding a solution, and
only seek hints if stuck.
Struggle with various attempts over time, and learn
from previous solution attempts.
2
Reason
abstractly and
quantitatively
Reason with models or pictorial
representations to solve problems.
Are able to translate situations into symbols for
solving problems.
Convert situations into symbols to appropriately
solve problems as well as convert symbols into
meaningful situations.
3a
Construct viable Explain their thinking for the solution they
arguments
found.
Explain their own thinking and thinking of others with Justify and explain, with accurate language and
accurate vocabulary.
vocabulary, why their solution is correct.
3b
Critique the
reasoning of
others.
Model with
Mathematics
Explain other students’ solutions and identify
strengths and weaknesses of the solution.
1a
4
5
6
7
8
Understand and discuss other ideas and
approaches.
Use models to represent and solve a problem, Use models and symbols to represent and solve a
and translate the solution to mathematical
problem, and accurately explain the solution
symbols.
representation.
Use appropriate Use the appropriate tool to find a solution.
tools
strategically
Attend to
Communicate their reasoning and solution to
precision
others.
Select from a variety of tools the ones that can be
used to solve a problem, and explain their
reasoning for the selection.
Incorporate appropriate vocabulary and symbols
when communicating with others.
Compare and contrast various solution strategies
and explain the reasoning of others.
Use a variety of models, symbolic representations,
and technology tools to demonstrate a solution to
a problem.
Combine various tools, including technology,
explore and solve a problem as well as justify their
tool selection and problem solution.
Use appropriate symbols, vocabulary, and labeling
to effectively communicate and exchange ideas.
Look for and
make use
of structure
Look for structure within mathematics to help Compose and decompose number situations and
them solve problems efficiently (such as 2 x 7 x 5 has relationships through observed patterns in order to
the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which simplify solutions.
See complex and complicated mathematical
expressions as component parts.
Look for and
express
regularity in
repeated
reasoning
Look for obvious patterns, and use if/ then
reasoning strategies for obvious patterns.
Discover deep, underlying relationships, i.e.
uncover a model or equation that unifies the
various aspects of a problem such as
discovering an underlying function.
is (2 x 7) x 5, the student can mentally calculate 10 x 7.
Find and explain subtle patterns.
SMP Instructional Implementation
Sequence
1.
Think-Pair-Share (1, 3)
2.
Showing thinking in classrooms (3, 6)
3.
Questioning and wait time (1, 3)
4.
Grouping and engaging problems (1, 2, 3, 4, 5, 8)
5.
Using questions and prompts with groups (4, 7)
6.
Allowing students to struggle (1, 4, 5, 6, 7, 8)
7.
Encouraging reasoning (2, 6, 7, 8)
SMP Proficiency Matrix
Students:
Make sense of
problems
(I) Initial
Explain their thought processes in solving a
problem one way.
(IN) Intermediate
Explain their thought processes in solving a problem
and representing it in several ways.
(A) Advanced
Discuss, explain, and demonstrate solving a
problem with multiple representations and in
Grouping/Engaging Problems
multiple ways.
1b
Persevere in
solving them
Stay with a challenging problem for more
than one attempt.
Try several approaches in finding a solution, and
only seek hints if stuck.
Struggle with various attempts over time, and learn
from previous solution attempts.
2
Reason
abstractly and
quantitatively
Reason with models or pictorial
representations to solve problems.
Grouping/Engaging
Problems
Explain their thinking for the solution they
found. Showing Thinking
Are able to translate situations into symbols for
solving problems.
Convert situations into symbols to appropriately
solve problems as well as convert symbols into
meaningful situations.
Understand and discuss other ideas and
approaches.
Explain other students’ solutions and identify
strengths and weaknesses of the solution.
1a
3a
Construct viable
arguments
3b
Critique the
reasoning of
others.
Model with
Mathematics
4
5
6
7
8
Pair-Share
Questioning/Wait Time
Pair-Share
Grouping/Engaging
Problems
Questioning/Wait Time
Grouping/Engaging Problems
Grouping/Engaging Problems
Showing Thinking
Encourage Reasoning
Explain their own thinking and thinking of others with Justify and explain, with accurate language and
accurate vocabulary.
vocabulary, why their solution is correct.
Questioning/Wait Time
Grouping/Engaging Problems
Questioning/Wait Time
Compare and contrast various solution strategies
and explain the reasoning of others.
Questions/Prompts for
Groups
Use a variety of models, symbolic representations,
and technology tools to demonstrate a solution to
a problem.
Use models to represent and solve a problem, Use models and symbols to represent and solve a
and translate the solution to mathematical
problem, and accurately explain the solution
symbols.
representation.
Use appropriate Use the appropriate tool to find a solution.
tools
Grouping/Engaging Problems
strategically
Attend to
Communicate their reasoning and solution to
precision
others.
Showing Thinking
Select from a variety of tools the ones that can be
used to solve a problem, and explain their
reasoning for the selection.
Incorporate appropriate vocabulary and symbols
when communicating with others.
Grouping/Engaging Problems
Allowing Struggle
Grouping/Engaging Problems
Showing Thinking
Combine various tools, including technology,
explore and solve a problem as well as justify their
tool selection and problem solution.
Use appropriate symbols, vocabulary, and labeling
to effectively communicate and exchange ideas.
Showing Thinking
Encourage Reasoning
Look for and
make use
of structure
Look for structure within mathematics to help Compose and decompose number situations and
them solve problems efficiently (such as 2 x 7 x 5 has relationships through observed patterns in order to
the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which simplify solutions.
See complex and complicated mathematical
expressions as component parts.
Look for and
express
regularity in
repeated
reasoning
Look for obvious patterns, and use if/ then
reasoning strategies for obvious patterns.
Discover deep, underlying relationships, i.e.
uncover a model or equation that unifies the
various aspects of a problem such as
discovering an underlying function.
Questions/Prompts for
Groups
is (2 x 7) x 5, the student can mentally calculate 10 x 7.
Grouping/Engaging
Problems
Allowing Struggle
Find and explain subtle patterns.
Allowing Struggle
Encourage Reasoning
Encourage Reasoning
Fractions
FINDING THE MISSING PIECES
Middle Grades Fractions
Jeanne Simpson
NCTM 2014
Difficulty with learning fractions is pervasive and is
an obstacle to further progress in mathematics and
other domains dependent on mathematics,
including algebra. It has also been linked to
difficulties in adulthood, such as failure to
understand medication regimens.
National Mathematics Panel Report, 2008
WHY ARE FRACTIONS SO DIFFICULT?
There are many meanings of fractions (part-whole,
measurement, division, operator, ratio).
Fractions are written in an unusual way.
Instruction does not focus on a conceptual
understanding of fractions.
Students overgeneralize their whole-number
knowledge. (McNamara & Shaughnessy, 2010)
Van de Walle, Karp, & Bay-Williams, 2013
KEY IDEAS NEEDED FOR CONCEPTUAL
UNDERSTANDING
 The meaning of fractions
 Partitioning
 Unit fractions
 Models
 Number lines
 Equivalent fractions
 Comparing fractions
PARTITIONING
PARTITIONING IS KEY TO UNDERSTANDING AND
GENERALIZING CONCEPTS RELATED TO FRACTIONS
SUCH AS:
 Identifying “fair shares”
 Identifying fractional parts of an object
 Identifying fractional parts of sets of objects
 Comparing and ordering fractions
 Locating fractions on number lines
 Understanding the density of rational numbers
 Evaluating whether two fractions are equivalent or finding equivalent fractions
 Operating with fractions
 Measuring
STAGES OF PARTITIONING
 Sharing – two equal parts
 Algorithmic halving – equal parts that are powers of two
 Evenness – even numbers that have odd factors
 Oddness – partitioning into an odd number of equal
parts involves thinking about the relative size of each
part to the whole before partitioning
 Composition – using rows and columns (multiplicative)
DO I TEACH THESE STRATEGIES? NO!
 Teachers should make intentional choices about which fractions they
use to teach, reinforce, and strengthen concepts that can be built on
understanding the impact of partitioning.
 Provide students with a variety of models
 Students should partition the models into a variety of fractional
parts, starting with powers of two
 Have students share their strategies so that all students are
exposed to a variety of ways of thinking.
 Over time, students will take on other strategies as they are ready.
UNIT FRACTIONS
CCSS 3.NF.A.1
Understand a fraction 1/b as the quantity formed
by 1 part when a whole is partitioned into b equal
parts; understand a fraction a/b as the quantity
formed by a parts of size 1/b.
 (Grade 3 expectations in this domain are limited to
fractions with denominators 2, 3, 4, 6, 8.)
MODELS FOR FRACTIONS
Area
Whole = area of defined region
Parts = equal areas
Fraction = part of area covered
Length
Whole = Unit of distance or length
Parts = equal distances or lengths
Fraction = location in relation to other
points on the line
Set
Whole = one set
Parts = equal number of objects
Fraction = the count of objects in the
subset
THREE TYPES OF MODELS
 Area models (regions, part
to whole relationships)
 Set models (fractional part
of a set of objects)
 Se
•Number lines
traveled
(distance
or location)
MODELS
Learning is facilitated when students interact with
multiple models that differ in perceptual features
causing students to continuously rethink and
ultimately generalize the concept.
Rectangles are better than circles. They are easier
to partition equally, and they allow multiplicative
reasoning.
MODELS DIFFER IN CHALLENGES
 How the whole is defined
 How “equal parts” are defined
 What the fraction indicates
Model
Whole
Equal Parts
Fraction Indicates
Area
Defined region
Equal area
The part covered of whole
unit of area
Set
What is in the
set
Equal
number of
objects
The count of objects in the
subset of the defined set of
objects
Number
line
Unit of
distance or
length
(continuous)
Equal
distance
The location of a point in
relation to the distance from
zero with regard to the
defined unit
TO PREVENT OVER-RELIANCE…
Let students become comfortable with
model.
Then give them a problem where the model
is cumbersome.
Vary the model so students do not overgeneralize.
The ultimate goal is a mental model.
RESEARCHERS SAY….
Over time, students should move from the need
always to construct or use physical models to
carrying the mental image of the model, while
still being able to make a model as they learn new
concepts or encounter a difficult problem.
Petit, Laird, Marsden (2010)
“Students who are asked to practice the
algorithm over and over…stop thinking. They
sacrifice the relationships in order to treat the
numbers simply as digits.”
Imm, Fosnot, Uittenbogaard (2012)
Contact Information
Jeanne Simpson
UAHuntsville AMSTI
[email protected]
[email protected]
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