Elementary Mathematics

Week 1: Base Systems

Multiplying Fractions

Pattern Block Model

Abstractly:
When we multiply fractions abstractly,
we just multiply straight across and simplify
when needed.

Ex. 2/3 x 1/4
Multiplying straight across, we get
2/12. This fraction can be simplified by
dividing both the denominator and numerator
by 2 which would give us a total of 1/6

Ex. 3/5 x 2/3
Multiplying these two fractions,
we get 6/15. This can't be simplified
anymore, so this our final answer.

Pictorially:
When multiplying fractions pictorially,
since we are using the pattern block model
we would draw out certain pattern blocks depending
on the fractions given.

Ex. 2/3 x 1/4
If two hexagons made a whole:
The pattern block that represents
1/4 in this case is the trapezoid. So,
we would draw one trapezoid in
one of the hexagons. Now, we have to
find out what 2/3rds of 1/4th is. In this case,
2 triangles fill in 2/3rds of the trapezoid, and
since the triangle represents 1/12 when two
hexagons are a whole, then the answer is 2/12.
Simplified, we get 2/3 x 1/4 = 1/6

Ex. 4/12 x 1/2
First, I always find it easier to solve
the problem abstractly before I draw it out.
So, 4/12 x 1/2 = 4/24. Simplified, that's 1/6.

Ex. 4/12 x 1/2 (continued)
If four hexagons made a whole:
The pattern block that represents
1/12 is the rhombus, and since the
fraction in the equation calls for 4/12,
then we would draw in 4 rhombi in the
4 hexagons. Now, we have to figure out
what's half of 4/12. To do so, we'll draw in
1 triangle in each of the rhombi which would
give us 4 triangles. Finally, because the fraction
that resembles triangles, if four hexagons make a
whole, is 1/24, then the total fraction would be 4/24.
Simplified, that's 1/6. So, 4/12 x 1/2 = 1/6

Multiplying Fractions with
Pattern Blocks Video:

Fraction Bar Model

Abstractly:
This is used the same way
as when we add and subtract
fractions, and multiplying fractions
with pattern blocks.

Ex. 3/4 x 1/2 = 3/8
This is the simplified fraction

Ex. 3/8 x 1/3 = 3/24.
Simplified, that's 1/8

Pictorially:
When using the fraction bar model
pictorially, we would draw out one box,
divide it into a certain number of parts,
and shade some parts in depending on
the fractions given.

Ex. 3/4 x 1/2
Drawing one box,
we would first take the 1/2 fraction,
divide the box into two parts horizontally, then
shade in 1 of them. Next, we take the 4 from 3/4,
then divide it into four columns. We now have a total
of 8 parts in the box with 4 shaded, but since we only
need 3, in order to solve the problem correctly, we
cross out one part. This will leave us with 8 parts with
only 3 shaded. Therefore, 3/4 x 1/2 = 3/8

Ex. 3/8 x 1/3
Drawing one box, we would
first take the 3 from 1/3 and
divide the box into 3 rows
shading in 1 row. Taking the
8 from 3/8, we would draw out 8
columns. There are now 24 total parts
with 8 parts shaded. Since we only need 3 parts
in order for the answer to make sense, we would
cross out 5 of those shaded parts, leaving us with
the fraction 3/24. Simplified, gives us 1/8. Therefore,
3/8 x 1/3 = 1/8

Dividing Fractions

Pattern Blocks

Abstractly:
We do things a little
differently when divide fractions
abstractly.

Ex. 1/3 divided by 1/2
To solve this, we would
turn the division sign into
a multiplication sign and
put in the reciprocal of 1/2
which is 2/1. Multiplying
1/3 x 2/1, we get 2/3

Ex. 1 1/2 divided by 1/6
First changing the mixed number
into an improper fraction, we get
3/2 x 6/1 = 18/2 which equals 9.

Pictorially:
This can be shown in a similar way
from when we multiply fractions.

Ex. 1/3 dividing by 1/2
To figure this out pictorially,
we first have to ask ourselves,
in this case, how many 1/2s fit in
or cover 1/3s?
If one hexagon made a whole:
The rhombus makes 1/3, and the
trapezoid equals 1/2. So, we need to find
out how much of the trapezoid covers the
rhombus. As we first draw the rhombus,
then draw a triangle at the end to make a
trapezoid, we see that the trapezoid covers
2/3 of the rhombus. So, 1/3 x 2/1 = 2/3

Ex. 1 1/2 divided by 1/6
How many 1/6 fit in or cover
1 1/2?
If two hexagons equal a whole:
The rhombus equals 1/6, and since
two hexagons won't make it to 1 1/2,
we add another hexagon and draw in
three rhombi in each hexagon. With 3
rhombi in each hexagon, we end up with
a total of 9 rhombi. So, 1 1/2 or 3/2 x 6/1 = 9

Dividing Fractions with
Pattern Blocks Video:

Fraction Bars:

Ex. 5/6 divided by 2/3
First, solving this abstractly,
we get 5/6 x 3/2 = 15/12 = 5/4 = 1 1/4.
Pictorially, we draw out two big rectangles
on either side of each other drawing the 5/6
fraction in the first rectangle in rows. In the second
rectangle, we draw out the 2/3 fraction in columns.
Next, we add 3 columns in the 5/6 rectangle and 6 rows
in the 2/3 fraction. We now have 15 parts shaded in the 5/6 rectangle with one row left unshaded, and 12 parts shaded in the 2/3 rectangle with one column left unshaded.

Ex. 5/6 divided by 2/3 (continued)
Then, we would cross out the number
of parts in the 5/6 rectangle so that it matches
the number of parts in the 2/3 rectangle. In this
case, we would cross out 12 parts out of 15 from the
5/6 rectangle which gives us 1 group of 12 parts which
equals 1. Now, we have 3 remaining parts shaded, so, we
would have the fraction 3/12. Simplified, that's 1/4. Bringing
the 1 group of 12 to the 3/12 parts remaining, we now have
a mixed number of 1 1/4.

Ex. 6/8 divided by 2/4
Abstractly, we get:
6/8 x 4/2 = 24/16 = 6/4 = 3/2 = 1 1/2
Having a rectangle with a 6/8 fraction and
another rectangle with the 2/4 fraction, we
add 4 columns to the 6/8 rectangle and 8 rows
in the 2/4 rectangle, leaving us with two rows and
two columns unshaded. We now have 24 shaded parts
in the 6/8 rectangle, and 16 shaded parts in the 2/4 rectangle.

Ex. 6/8 divided by 2/4 (continued)
Crossing out 16 parts in the 6/8
rectangle, we have 1 group of 16 parts
which is 1. There are now 8 remaining parts,
which then, leaves us with the fraction of 8/16
or 1/2. Bringing the 1 group of 16 and the 8/16
remaining parts together, we get a mixed number
of 1 1/2

Why We Invert and Multiply:

We invert and multiply fractions instead of
just dividing them because it's easier to find
answer when we change one of the fractions
into its reciprocal and multiply rather then
dividing them straightforward.

If we were to just divide fractions,
we would end up solving it like, for example:
6/8
___
2/4
which can get pretty confusing. We would end
up getting the same answer if we solved it this way,
but multiplying those fractions with the second fraction
being its reciprocal is the best way to solve this equation.

To solve this:
6/8
___
2/4,
we would take the 8
from 6/8 and multiply
with the fraction 8/8 or 1.
The denominator of 6/8, and the numerator of
8/8 cancels out, leaving us to multiply the numerator
of 2/4 with the denominator of 8/8. This will give us
6/16/4. Next, we would take the 4 and multiply 6/16/4
by 4/4. The two denominators of 4 cancel each other out,
so, when we multiply the numerators 6 and 4, we end up
with the fraction 24/16.