Modeling
Math for
Mastery
Libby Serkies, MA Ed, PhD Candidate 
Central Illinois Adult Education Service Center
L-Serkies@wiu.edu
Number Sense is…
…an awareness and understanding about what
numbers are, their relationships, their
magnitude, the relative effect of operating on
numbers, including the use of mental
mathematics and estimation.
Fennel & Landis, 1994
According to the experts…
It is imperative to teach students the meaning of what
they are doing when they manipulate numbers during
arithmetic computations. Meaning not only increases
the chances that information will be stored in longterm memory, but also gives the learner the
opportunity to change procedures as the nature of the
problem changes. Without meaning, students
memorize procedures without understanding how and
why they work. As a result, they end up confused
about when to use which procedure.
- Dr. David Sousa
Number Sense
Adding Fractions
Adding fractions
Fractions with like denominators:
3 1
+ =
5 5
Adding fractions
Fractions with like denominators:
3 1
+ =
5 5
Adding fractions
Fractions with like denominators:
4
3 1
+ =
5
5 5
Adding Fractions
Adding fractions
Fractions with unlike denominators:
2 1
+ =
5 4
How do we combine these two fractions
when they don’t have the same terms?
Step 1:
Create 2 fraction bars of the same size, and divide one
with horizontal lines to denote the fraction we are
using. Divide the other with vertical lines to denote the
fraction we are using.
Step 2:
Color in the fractions to represent the numerators of
both fractions:
Step 3:
Transfer the marks (lines) from the right
fraction to the left:
Step 3 - part II:
Then transfer the marks from the left fraction to the right:
Now we have common terms! Each box in both fractions is the
same size.
Step 4:
Next count & move all the colored boxes
from one fraction to the other.
Step 5:
Next, count the number of colored boxes.
This is your new numerator.
1
5
2
3
6
7
4
8
9
13
10
11
12
2
5
+
1
4
=
13
Step 6:
Next, count the total number of boxes. This is
your new denominator.
1
5
2
3
6
7
4
8
9
13
10
11
12
14
15
16
17
18
19
20
2
5
+
1
4
=
13
20
Extension
Ask your students:
 What can you do to 5 to get 20?
(Multiply by 4)
 What can you do to 4 to get 20?
(Multiply by 5)
2 ×𝟒
+
5×𝟒
1×𝟓
=
4×𝟓
13
20
Subtracting Fractions
Subtracting Fractions
Begins the same way adding fractions does look for or create common terms:
2
2
−
=
3
5
Step 1:
Create models to show fractions:
2
3
−
2
5
=
Step 2:
Color in the representations of each
2
fraction:
3
−
2
5
=
Step 3:
Construct common terms - transfer marks:
2
3
−
2
5
=
Step 4:
Remove the smaller number of colored
boxes from the larger:
2
3
−
2
5
=
Step 5:
Count the # of originally colored boxes left
from the first fraction. This is your new
2
numerator:
3
1
2
3
−
4
2
5
=
4
Step 6:
Count the # of total boxes in the first fraction.
This is your new denominator:
2
3
1
6
2
7
3
8
−
4 5
9 10
11 12 13 14 15
2
5
=
4
15
Step 6:
Solve:
2
3
−
2
5
=
4
15
Simplify if necessary
Remember to ask your students what
they NOW know about subtracting
fractions with unlike denominators!
Multiplying Fractions
Multiplying fractions (modeling)
2 1
x =
3 2
Step 1:
Draw representations of the fractions we
2
1
are using: x =
3
2
Step 2:
Transfer marks and portions of one fraction
2
1
to the other:
x =
3
2
Step 3:
The denominator of the product is the
total number of pieces our diagram has:
2
3
x
1
2
=
6
1
2
3
4
5
6
Step 4:
The numerator of the product is the
number of pieces that are shaded twice:
2
1
x =2
3
2
6
1
2
3
4
5
6
Step 5:
Finish the problem:
1 2
2
x =
2 3
6
or
1
3
By extension…
ASK: What do you now know
about multiplying fractions?
Dividing Fractions
Dividing fractions (modeling)
1
2
÷
4
3
DIVIDEND
DIVISOR
Step 1:
Draw a representation of the dividend -- 1/4 and
the divisor -- 2/3.
Step 2:
Draw a representation of the dividend -- 1/4 and
the divisor -- 2/3.
Step 3:
Transfer the markings from one fraction to the
other:
Step 4:
Number the originally shaded tiles - this is your
new numerator:
1
2
3
𝟏
𝟒
÷
𝟐
𝟑
=
3
Step 5:
Number the tiles shaded by the divisor - this is your
new denominator:
1
2
3
4
5
6
7
8
𝟏
𝟒
÷
𝟐
𝟑
=
3
8
Step 6:
Combine 1
2
3
÷
1
2
3
4
5
6
7
8
𝟏
𝟐
𝟑
÷
=
𝟒
𝟑
𝟖
Finally:
Check your answer with “keep-change-flip”
1
4
Keep
1
4
÷
Change
x
2
3
3
2
=
FLIP
=
3
8
Algebra
Tiles
Box Multiplication!
(𝟓 𝒙𝟒 − 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟑) (𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓)
𝟓𝒙𝟒
𝟐𝒙𝟑
+ 𝟒𝒙𝟐
+(−𝟑𝒙)
+𝟓
+ 𝟎𝒙𝟑
+ (−𝟑𝒙𝟐 )
+ 𝟐𝒙
+𝟑
Box Multiplication!
(𝟓 𝒙𝟒 − 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟑) (𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓)
𝟓𝒙𝟒
+ 𝟎𝒙𝟑
+ (−𝟑𝒙𝟐 )
+ 𝟐𝒙
+𝟑
𝟐𝒙𝟑
10𝑥 7
0𝑥 6
−6𝑥 5
4𝑥 4
6𝑥 3
+ 𝟒𝒙𝟐
20𝑥 6
0𝑥 5
−12𝑥 4
8𝑥 3
12𝑥 2
−15𝑥 5
0𝑥 4
9𝑥 3
−6𝑥 2
−9𝑥
25𝑥 4
0𝑥 3
−15𝑥 2
10𝑥
15
+(−𝟑𝒙)
+𝟓
Box Multiplication!
(𝟓 𝒙𝟒 − 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟑) (𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓)
𝟓𝒙𝟒
+ 𝟎𝒙𝟑
+ (−𝟑𝒙𝟐 )
+ 𝟐𝒙
+𝟑
𝟐𝒙𝟑
10𝑥 7
0𝑥 6
−6𝑥 5
4𝑥 4
6𝑥 3
+ 𝟒𝒙𝟐
20𝑥 6
0𝑥 5
−12𝑥 4
8𝑥 3
12𝑥 2
−15𝑥 5
0𝑥 4
9𝑥 3
−6𝑥 2
−9𝑥
25𝑥 4
0𝑥 3
−15𝑥 2
10𝑥
15
+(−𝟑𝒙)
+𝟓
Box Multiplication!
(𝟓 𝒙𝟒 − 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟑) (𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓)
10𝑥 7 + 20 𝑥 6 − 15𝑥 5 − 6𝑥 5 + 25𝑥 4 − 12𝑥 4 + 4𝑥 4 + 9𝑥 3 + 8𝑥 3 + 6𝑥 3 − 15𝑥 2 − 6𝑥 2 + 12𝑥 2 + 10𝑥 − 9𝑥 + 15
𝟏𝟎𝒙𝟕 + 𝟐𝟎 𝒙𝟔 − 𝟐𝟏𝒙𝟓 + 𝟏𝟕 𝒙𝟒 + 𝟐𝟑𝒙𝟑 − 𝟗𝒙𝟐 + 𝒙 + 𝟏𝟓
Meaning Making:
The single greatest tool/instructional
method we can use in our classrooms.
Encouraging students to “discover”
mathematical principles through guided
delivery and explicit questioning will
establish the foundation they need to
succeed in higher level math.
My Contact Info:
Libby Serkies
CIAESC (Central IL Adult Ed Service Center)
Western Illinois University
Horrabin Hall 5
Macomb, IL 61455
L-Serkies@wiu.edu
309-830-2029