Teaching Middle School
Mathematics
Fractions, decimals and percentages
Ratios, rates and proportions
Work out the problem on your card, then find 3
other people who have the same number as you do.
Sit with them to work a collaborative problem.
Goals for the four sessions

Identify tasks and activities that are effective for
teaching the content of each session

Use visual representations and the concreterepresentational-abstract approach to enhance
students’ understanding of the content of each
session

Identify mathematical language and develop its
use in students

Use mathematical discourse to promote
engagement and deep processing

Provide tasks that promote student engagement
and mathematical reasoning within the content
of each session

Assess students’ understanding and proficiency
in order to provide useful feedback and make
needed changes to instruction
Common Core for fractions

Read through each standard, marking
“solve word problems” and
“use visual fraction models”

Match each problem to a standard

Anything surprise you about the CCSS?
Common Core Standards
Components of Mathematical Proficiency
1. Conceptual understanding
2. Procedural skill and fluency
3. Application
Adding or subtracting fractions
Conceptual understanding
Apples
and Oranges
(same size pieces of the whole)
Procedural Skill and Fluency
1
2
+
3
1
4
=
1
3
1
+
2
=
Application
A pitcher contains
5
8
3
2
4
pints of orange juice.
After you pour of a pint into a glass, how
much is left in the pitcher?
5.4 – 3.25 =
Learning Progression within
Conceptual Understanding
Concrete - Representational - Abstract
Objects
1
2
1
Pictures
Draw the essential
insight that allows
Use fraction circles, bars students to add
or pattern blocks.
fractions of different
size pieces.
+3
Symbols
1
2
+ 3 = ? becomes
1
3
6
+6=6
2
5
Read “The Role of Representations…” p. 494. Find a “Golden Sentence”.
Multiplying fractions
1.
2.
3.
Conceptual Understanding
Procedural Skill
Application
Conceptual
Understanding
Procedural Skill
Application
Learning Progression within
Conceptual Understanding
Concrete - Representational - Abstract
Objects
2
4x5
Make 4 equal groups,
add them.
Pictures
Draw this on a number
line.
Symbols
Notice that the size of
the fraction pieces stay
the same.We’re only
multiplying the
numerator times the
number of groups:
4 x 2 = 8.
4 x 2/5 = 8/5
Learning Progression within
Conceptual Understanding
Concrete - Representational - Abstract
Objects
1
2
2
x5
Make ½ of a group.
Read the multiplication
1
2
as of .
2
Pictures
Bar model:
Area model:
5
NLVM.usu.edu
Symbols
The area model shows
that we made pieces of
a new size – tenths. The
multiplication results in
2 tenths. Notice that
this is the result of
multiplying the
numerators (as in the
earlier example) and the
denominators.
See pp. 23-25
Learning Progression within
Procedural Skill
Acquisition - Fluency - Generalization
C-R-A
Practice*
Extensions
* Guided practice with feedback is critical
One extension
Generalize this process to multiplication
involving mixed numbers.
1
1
×1
2
3
Draw a visual fraction model.
Connect to the procedure.
Fun problems: If the rectangle has a value of
1
, show 1.
3
Learning Progression within
Application
Near Transfer – obvious connection to
previous problems to establish the “type”
Far Transfer – problem-solving skills are
required:
1. What “type” of problem is this?
2. What do I know that I can use? (KWL)
3. Is there a drawing or chart that will
help?
4. Other problem-solving strategies


3/4 of a pan of brownies was sitting on
the counter.You decided to eat 1/3 of the
brownies in the pan. How much of the
whole pan of brownies did you eat?
A cake mix uses
1
1
2
1
2
2
cups of flour. You
want to make recipes of this cake.
How much flour do you need?
Learning Progressions
The Common Core gradually increases
complication of working with fractions.
The Operations with Fractions packet steps
students through these learning
progressions carefully and systematically.
Interlude… Alternate Algorithms
6 )234
-120
114
-60
54
-30
24
-24
0
20
10
5
4
39
Fluently divide
multi-digit
numbers using
the standard
algorithm.
6.NS.2
This type of division is called
repeated subtraction
“When I reflect on this past unit, I think
that learning the alternate algorithms was
extremely helpful for me. I chose the
scaffolding algorithm as my algorithm of
choice for a good reason. Growing up
through elementary school, middle school,
and high school, I always struggled with long
division. I never really got the grasp of an
algorithm that made sense to me.”
“This scaffolding method has made long
division unbelievably easier for me. I finally
understand how to solve those problems and
can do them on my own now. I originally was
taught how to carry the one and cross out
certain numbers. But really I had no idea what
my teacher was talking about. This scaffolding
method not only helps me with my long
division, but it also helps me with my
multiplication tables, as well as adding. This
scaffolding method will stay with me forever,
and I truly do believe I will use this for the rest
of my life.”
This type of division is called
fair shares, or partitioning
Partial Products Algorithm
How would this work for 2.3 x 1.8?
So what about fraction division?

One serving (1/2 cup) of broccoli
contains 47 mg of calcium. Kids ages 9-18
need to get 1300 mg of calcium daily to
build strong bones. How many cups of
broccoli would this be?

See the Acquisition-Fluency-Generalization
scheme for division
Two types of division

Partitive (fair shares)

We want to share 12 cookies equally
among 4 kids. How many cookies does
each kid get?

How would you solve this with objects?

The number of groups is known; the
number in each group is unknown.

Measurement (repeated subtraction)

For our bake sale, we have 12 cookies and
want to make bags with 2 cookies in each
bag. How many bags can we make?

How would you solve this with objects?

The number in each group is known; the
number of groups is unknown.
Why is this important?

A box of Cheerios contains
3
4
1
12
2
cups.
Each serving is cups. How many servings
are in a box of Cheerios? How much is
left over?

Partitive or Measurement division?

Write 3 additional problems like these.
Dividing a fraction by a whole
number

We have ½ of a pizza and want to share it
equally among 4 people. How much pizza
does each person get?

1/2 ÷ 4

Try this with fraction manipulatives.

8/3 ÷ 4

What’s a procedure?
Dividing a whole number by a
fraction

We have a dozen large cookies and want
to give ½ cookie to each child. How many
children can we serve?
1
÷
2

12

(How many times does ½ go into 12?)

What’s a procedure?
Dividing a fraction by a fraction

A serving size is ¼ cup. How many
servings are in 5/4 cup?

Solve it. Write it as an equation.
5

4

÷
1
4
How many ¼’s are in 5/4?
This is the same as asking “How many 1’s
are in 5?” Procedure: Get common
denominators.
Dividing a fraction by a fraction

Cheerios problem – solved by mental
math or a drawing (C-R-A)
1
3
12 ÷
2
4

Translated to symbols

Procedure? Find common denominators...
50
3

÷
4
4

then ask “how many 3’s in 50?”
Write two more similar problems to
solve with common denominators.
What about “invert and multiply?”

See pp. 29-34 in Operations with
Fractions for an instructional approach.

Why “invert and multiply” works:
1
 12
2

3
4
÷ =
25
3
÷
2
4
=
25
4
×
2
3
÷
3
4
×
4
3
The last step is justified by recognizing
that if we multiply both numbers in a
division problem (i.e. in a fraction) by a
constant, we get an equivalent problem.
Decimals
2.3 x 1.8
A generalization from multiplying
fractions: Multiplying decimals
Understanding-Skill-Application
 C-R-A
 A-F-G

Think-Pair-Share  Poster
 2.3 x 1.8

See the decimals and percents assessment
Why do we “count decimal places?”
2.3
x 1.8
.2 4
1.6
1.3
2
5.1 4
Yellow 2 x 1
Orange 0.3 x 1
Blue 2 x 0.8
Green 0.3 x 0.8
2.3
1.8
Decimals Forever!