Transcript appleton

Meaningful Mathematics
Fractions
and
Common Core Standards for State
Mathematics
joe georgeson
retired
UWM
Brookhill/WSMI
agenda:
•introduction Common Core
•learning intentions/success criteria
•third grade expectations
•unit fractions
•number line and area models
•importance of the whole
•break
•operations and algorithms
•closing-questions/comments
A brief introduction
Focus
do less but do it better
Coherent
progression from grade to grade
Rigor
develop skills
built on understanding
in the context of applications
K-8 Domains and HS Conceptual Categories
William McCallum, The University of Arizona
Common Core Teaching Practice
Standards
•Solve
Problems and Persevere
•Reason abstractly and quantitatively
•Model with Mathematics
•Attend to Precision (language)
“You have just purchased an expensive Grecian urn and asked the
dealer to ship it to your house.
He picks up a hammer, shatters it into pieces, and explains
that he will send one piece a day in an envelope for the next
year.
You object; he says “don’t worry, I’ll make sure that you
get every single piece, and the markings are clear, so you’ll be able to
glue them all back together. I’ve got it covered.”
Absurd, no? But this is the way many school systems
require teachers to deliver mathematics to their students; one
piece (i.e. one standard) at a time.
They promise their customers (the taxpayers) that by the end of
the year they will have “covered” the standards.”
some of my beliefs
Mathematics should have meaning, for
teachers and students.
Algorithms should be taught - based on
good and deep understanding.
Teaching is not telling.
Students Construct Knowledge.
Testing Gets in the way of Learning
Learning Intentions......
Deepen conceptual understanding of
fractions and operations with
fractions;
Understand and appreciate the need
for context, models, and meaning
when teaching about fractions.
Success Criteria......
Model problems with fractions using
number lines and areas;
Use reasoning and models to
demonstrate understanding of
fractions and operations with
fractions.
Words from CCSSM
Number Operations and Fractions
Fractions are Numbers
Geometry Standards
First Grade: partition squares, circles,
rectangles into halves, fourths using
vocabulary “half of,” or “fourth of.”
Second Grade: recognize that equal
shares may not have the same shape
Third Grade: Express and Describe unit
fractions as part of the whole. Partition a
shape into 4 equal parts and describe each
part as “1/4 of the whole.”
Third Grade
Number Operations and
Fractions Standards
unit fractions
fractions on a number line
equivalent fractions
1/5 is the amount of one part
when a whole has been
partitioned into 5 equal parts
area model
1
5
one part when a whole has
been partitioned into 5
equal parts
linear model
1
5
one part when a whole has
been partitioned into 5
equal parts
discrete model
1
5
one part when a whole has
been partitioned into 5
equal parts
1/5 also marks the
coordinate on a number line
of the point 1/5:
area model
4/5 is the amount in 4 parts
when a whole has been
partitioned into 5 equal parts
linear model
4 parts when a whole has been
partitioned into 5 equal parts
8
represents 8 parts of a whole
3
that was partitioned into 3 equal parts
8
1
 8 groups of
3
3
1 1 1 1 1 1 1 1
      
3 3 3 3 3 3 3 3
1
1
1
3 groups of  3 groups of  2 groups of
3
3
3
2
2
1 and 1 and  2
3
3
This is the whole, partitioned into 5 equal parts:
1
7
Then, would require 7 parts each of which is of the
5
5
original whole.
2
1 whole and of another whole.
5
If the whole is the distance from 0 to 1
on a number line:
7
would be the distance on that number line,
Then,
5
counting by 1/5 until you got to 7/5.
All fractions are composed of
unit fractions.
5/8 is 5 groups of 1/8
12/5 is 12 groups of 1/5
9/9 is 9 groups of 1/9
Describe each fraction using Common Core
vocabulary. Practice. Be Precise.
5
6
5 parts of a whole that was
partitioned into 6 equal parts
6
5
6 parts of a whole that was
partitioned into 5 equal parts
3
2
4
2 wholes and 3 parts of a whole that
was partitioned into 4 equal parts
3
convert 4
to a fraction
5
traditionally, converting a mixed
number to a fraction:
multiply 4 times 5 then add 3
23/5
a more meaningful
strategy
Since 4 is 4 groups
of 1, and each whole
is 5/5, the mixed
number can be
decomposed and
recomposed as
shown:
3
4
5
3
1 1 1 1
5
5 5 5 5 3
   
5 5 5 5 5
23
5
number
how
many
the unit
result
5
6
5
sixths
5 groups of 1/6
10 feet
10
feet
10 groups of 1 foot
8
8
ones
8 groups of 1
12
5
12
fifths
12 groups of 1/5
6 cents
6
cents
6 groups of 1 cent
9x
9
x
9 groups of x
Note:
For all of these examples, it is
important to also show an area
or linear model that accurately
represents each.
5
8
a whole is
partitioned into 5
equal parts and I
have 5 of them
What does “equal parts” mean?
Which of these shows
3
?
4
Unit FractionsArea Model
Start with a square.
Fold along every line of
symmetry.
Now fold each corner in to
the center and unfold.
How many unit fractions can you find?
Explain how a unit fraction is shown for
each.
Shade in 1/2 of the square.
Be creative.
How many ways could
you show 1/2?
12,870
Unit Fractions Linear Model
Fraction Strips
This strip is the unit-the whole.
Fold your other strips to show
halves, fourths, eighths
thirds, sixths, ninths
twelfths
Standard 3.NF.3a
Two fractions are equivalent if they
are the same size or name the same
point on a number line.
Use your fraction strips to find
fractions equivalent to 1/3 and 1/4.
Use your fraction strips to draw a
number line going from 0 to 1/2.
Mark off other points on that
segment using each of your fraction
strips.
Find as many equivalent fractions as
you can on that number line. Explain
why they are equivalent.
Fold your “fourths” fraction
strip to show a length of 3/4.
How do you see this fraction
composed of unit fractions?
Fraction Strips-What do you
notice?
Four children want to share
seven candy bars. How much
would each child get?
Take turns explaining your
thinking to each other.
area model
everyone gets 7 parts of a
whole that was partitioned
into 4 equal parts, or 7/4
linear model
everyone gets 7 parts of a
whole that was partitioned
into 4 equal parts, or 7/4
Locate the number 5/4 on this
number line. Describe your
thinking.
A student did it this way. Is he
right? If not, why not?
Which is larger?
1/8 or 1/9?
Explain how you know
using a linear and an
area model.
Knowing the Whole
Which is greater:
1
1
or
2
3
Talk to each otherexplain how you know.
Which is greater?
1
3
1
of this class
2
1
1
foot or mile
2
3
1
2
1
of Lambeau
3
1
If $1.00 is the whole, 50 cents is .
2
1
But, if $2.00 is the whole, 50 cents is .
4
Is this true?
1 1

2 2
John looked at this picture and said
the yellow was 2/6 of the rectangle.
Is he right? Why?
Jim looked at this picture and said
the yellow represented 2/3. Is he
right? Why?
Pat said the yellow was just 2. How
could she be right?
This rectangle is 4/5 of some whole.
Sketch a rectangle like this one and then
show what the whole would look like.
You need to have 5 equal parts. The red
rectangle is 4 of those parts.
Here is a shape. It is 2/3 of some whole.
What could that whole look like?
You would need 3 equal parts
to make a whole partitioned
into thirds.
The blue hexagon is two of
them.
Here is a shape. It is 3/2 of some whole.
What could that whole look like?
Since the shape contains 3
one-halves, any two of them
would make a whole.
These are the 7 pieces-called tangrams.
Cut them out.
What number
would represent
each piece if the
whole square is 1.
If the red triangle is 1, what number
would represent each of the other pieces?
If the blue square is
1, what number would
represent the other
pieces?
Pattern Blocks
Generally, the hexagon is defined to
be the whole or the unit. What
happens if this is no longer true?
triangle
rhombus
trapezoid
hexagon
1
4
4
1
1/2
1 1/2
1/4
3/4
1/2
triangle
rhombus
trapezoid
hexagon
1
2
3
6
2
4
6
12
2/3
4/3
2
4
1/3
2/3
1
2
1/4
1/2
3/4
6/4
1/4
1/2
3/4
1 1/2
1/4
1/2
3/4
6/4
1/4
1/2
3/4
6/4
1/12
1/6
1/4
1/2
Number Lines
What are the missing numbers?
How do you know?
What are the missing numbers?
How do you know?
What are the missing numbers?
How do you know?
Grade 3 NF Standards
Number & Operations—Fractions¹ 3.NF
Develop understanding of fractions as numbers.
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.
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a.Represent a fraction 1/b on a number line diagram by defining the
interval from 0 to 1 as the whole and partitioning it into b equal parts.
Recognize that each part has size 1/b and that the endpoint of the
part based at 0 locates the number 1/b on the number line.
b.Represent a fraction a/b on a number line diagram by marking off a
lengths 1/b from 0. Recognize that the resulting interval has size a/b
and that its endpoint locates the number a/b on the number line.
3. Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
a.Understand two fractions as equivalent (equal) if they are the same
size, or the same point on a number line.
•Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4,
4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a
visual fraction model.
•Express whole numbers as fractions, and recognize fractions that are
equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1;
recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number
line diagram.
3. Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
d. Compare two fractions with the same numerator or the same
denominator by reasoning about their size. Recognize that comparisons
are valid only when the two fractions refer to the same whole. Record
the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
Back in 10 minutes
4th grade:
Use a visual fraction model to show that 4/5 is 4
groups of 1/5 and that 10x3/4 is 10 groups of
3/4 which is 10 groups of 3 groups of 1/4
4th grade standard: 4.NF.4
Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent
5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction
by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5),
recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
Example:
There are 10 containers and each one
contains 4/5 of a pound of salt. How much
salt do all these containers hold?
4
10 
5
4
10 groups of
5
1
10 groups of 4 groups of
5
1
40
40 groups of or
5
5
4
10 
5
10 4 40
 
1 5 5
5th grade:
Divide a whole number by a unit fraction
Divide a unit fraction by a whole number
5th grade standard: 5.NF.7a and b
Interpret division of a unit fraction by a non-zero whole number, and compute such
quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction
model to show the quotient. Use the relationship between multiplication and division
to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
Interpret division of a whole number by a unit fraction, and compute such quotients.
For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to
show the quotient. Use the relationship between multiplication and division to explain
that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
10 ÷ 1/3 = ???
either
??? X 1/3 = 10
How many groups of 1/3 make 10?
or
1/3 X ??? = 10
1/3 of a group of some size = 10
10 ÷ 1/3 = ???
either
??? X 1/3 = 10
How many groups of 1/3 make 10?
or
1/3 X ??? = 10
1/3 of a group of some size = 10
1/3 ÷ 10 = ???
either
??? X 10 = 1/3
How many groups of 10 make 1/3?
or
10 X ??? = 1/3
10 groups of some size = 1/3
1/3 ÷ 10 = ???
either
??? X 10 = 1/3
How many groups of 10 make 1/3?
or
10 X ??? = 1/3
10 groups of some size = 1/3
Make a meaningful story for this
division:
10 ÷ 1/3
and
1/3 ÷ 10
The “capstone”
in 6th grade:
6.NS.1 Apply and extend previous understandings of multiplication and division to
divide fractions by fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving
division of fractions by fractions, e.g., by using visual fraction models and equations to
represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use
a visual fraction model to show the quotient; use the relationship between
multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is
2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3
people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a
cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2
square mi?
Is 4 x 6 the same as 6 x 4?
4 groups of 6
6 groups of 4
What is the difference between
2
2
 10 and 10 
3
3
Numerically they are the same. In meaning
they are very different:
John ran 2/3 miles every day for 10 days.
How far did he run?
I make $10 per hour. I can only keep 2/3 of
it. How much can I keep?
Write a story for each of these:
3
3
 24 and 24 
4
4
Listen to each others stories.
Discuss how they show the multiplication.
How would you represent them with a linear
or area model?
There are 24 students. Three-fourths of
them play a sport. How many play a sport?
There are 24 people at a party. Everyone
got three fourths of a small pizza. How
many pizzas were needed?
My gas tank was empty. I put 10 gallons in
and it is now 2/3 full. How much does the
whole tank hold?
Is this division or multiplication?
Write an expression for the problem.
divide: 10 ÷ 2/3
multiply: 2/3 X ??? = 10
John has 10 candy bars. He is going to give
each of his friends 2/3 of a candy bar. How
many friends could he give this candy to?
Is this division or multiplication?
Write an expression for the problem.
divide: 10 ÷ 2/3
multiply: ??? X 2/3 = 10
The algorithm, multiply by the reciprocal, is not done until
6th grade.
That is the capstone. It only comes after students
understand multiplication and division in a meaningful way.
Could you explain, without using mathematical jargon, why
division of fractions is done this way?
Could you explain it in a meaningful context?
to a student?
I have 5/6 of a cake left after a party. My
freezer containers each hold 1/3 of the cake.
How many containers can I fill? Can I fill part of
another container? If so, how full will it be?
Solve this and represent it with a tape diagram
or number line, or other way to show it.
How many groups of 1/3
can I fit into 5/6?
I have 3/4 oz. of gold. I want to make rings
that have 1/8 of an ounce. How many rings
could I make?
Show this with a linear model (tape diagram
or number line).
Write a series of equations that reflect your
reasoning.
I have 10 feet of rope and I want to make smaller
pieces, each 2/3 of a foot in length. How many of
these smaller-sized pieces could I make?
How would you figure this out without using any
algorithm? Pretend you are looking at this without
learning how to divide fractions.
Explain your reasoning to the person next to you.
The algorithm says to multiply by the reciprocal.
2 10  3
10  
3
2
What does the 10 times 3 represent in the context
about the rope?
What does the divide by 2 represent?
How has the meaning and role of the “3” and the “2”
changed as you solved the problem?
The “common denominator”
algorithm:
2 30 2
10  

3 3 3
Does this work all the time?
Does it make sense?
John has 2/3 of a gallon of lemonade. How many water
bottles can he fill if each bottle holds 1/12 of a gallon?
Draw a visual model of the problem.
Solve using both the invert and multiply algorithm and
the common denominator algorithm.
Explain the meaning of each step in the algorithm as
represented in the diagram and context of the actions
with the lemonade.
Interpreting 7/8:
There was a pizza. It was partitioned into 8
equal parts and I ate 7 of them. I ate 7/8 of
the pizza.
There were 7 pizzas. I have 8 in my family,
including me. How much pizza do we each get?
(Share 7 pizzas with 8 people)
Learning Intentions......
Deepen conceptual understanding of
fractions and operations with
fractions;
Understand and appreciate the need
for context, models, and meaning
when teaching about fractions.
Success Criteria......
Model problems with fractions using
number lines and areas;
Use reasoning and models to
demonstrate understanding of
fractions and operations with
fractions.
My final thoughts:
“It’s the story that is important
in learning mathematics.”
“Your Professional Development is
a forever process.”
Resources:
Common Core Standards for School Mathematics
Progressions Documents for the Common Core:
http://ime.math.arizona.edu/progressions/
http://isupportthecommoncore.net
http://www.illustrativemathematics.org/illustrations/1189
http://commoncoretools.me
Fordham Institute Interview with Jason Zimba
http://edexcellence.net/events/common-core-curriculum-controversies