Tip:
Highlight text to annotate it
X
The whole number parts
of mixed numbers
can be illustrated
by using the backs
of Fraction Bars.
For example,
the back of this blue bar
can be used to represent
one whole bar.
The mixed number three and three-fourths
can be represented
as shown here. Such representations
will be used in this video
to illustrate the addition
of mixed numbers.
Let's model the sum of
two mixed numbers.
Here we turned the yellow bars upside down. Each of these represents one.
We have three and a third.
Here we turned a blue bar face down.
That represents one. We have one and one-fourth.
To find the sum of these two fractions we combine the two sets of
bars.
Put the whole bars together to get four.
We can think of putting the
fraction parts together.
You can see we have a little bit more than half.
For now let's put down
one-third
plus one-fourth.
We need to get common denominators
so this is going to be four
plus one-third is going to be
four-twelfths.
One-fourth becomes three-twelfths.
So, this is equal to four
and seven-twelfths.
That's a sum of
these two mixed numbers.
To add mixed numbers,
we add the while number parts and then add the fraction parts.
As usual, addition means
put together.
Suppose it is
two and three-fifths miles from the beginning of a hiking trail from
an abandoned mine
and it is another three and three-fourths miles from the mine to the lake.
We've got
two and three-fifths
plus
three and three-fourths.
We're going to get the total distance
so we combine the
two sets of bars to get five whole bars.
Now we can think of combining the
shaded amounts of these bars. We are going to have
more than one whole bar.
Let's just write down the three-fifths. We are going to have to
do something special to get the sum of these two.
We are going to have to get common denominators.
This will be five plus,
the common denominators twenty here.
Multiply the top and bottom of this fraction by four. So we have twelve.
The top and bottom here by five to get 15.
So this is five plus
this will be twenty-seven over twenty.
That will equal...
With twenty parts to a whole bar, and we have twenty-seven parts,
so we have another whole bar, so
we get six, and
we'll have seven
parts left out of twenty.
Seven twentieths.
Seven is about a third of twenty so it's going to be approximately six and a
third miles from the beginning of the trail to the lake.
We've seen two examples now of adding mixed numbers
where we combine
sets of bars for those numbers.
Do students really need this kind of simplicity, especially for combining
whole numbers.
Here is a test results that
indicates they do.
This
question:
computer some two and two-fifths plus five;
was given to fifteen-year-old students across the United States.
43% were able to answer this correctly.
That means that
more than half of the students do not realize that
in this situation you add the whole numbers to get seven
and you keep the fraction.
several games and activities from the Fraction Bars Teacher's Guide,
involve making wholes.
That is,
finding two or more bars
who's total shaded amount
is one whole bar.
Let's look
at some examples.
In this concentration type game
for making wholes,
all 49 bars are turned face down.
In turn, each player turns over two bars and tries to make a whole.
There's a red bar with
three parts shaded. That is a half. I'm going to go for the green now,
because
they are half shaded, all shaded or a zero bar.
There is the zero.
So, at least we know where the zero is and where a half is.
The next player will probably go for the
green because one of the greens left
is a whole bar,
and the other is a half bar.
This game is called Units.
Each player is dealt a hand of six bars.
In turn they can select a seventh bar either from the playing stack or the
discard pile.
I want to make whole bars here and I have one that is half shaded, so I will
go for this green one. If this is half shaded then I'll have a whole. One chance
in three of getting a bar that is half shaded.
That is all shaded.
I have no zero bars to go with it.
So now look for combinations.
This looks promising.
Five-twelfths one-third and one-fourth
add up to a whole.
I win those bars and can take three more.
I can choose from either the discard stack, or the playing stack.
I have seven and can continue my turn.
At the end of the term each player discards one
bar to the discard pile.
If they were to make another whole, they would continue their turn.
A teacher once showed me
the following incorrect example
of how some students were using the bars to add fractions.
This is five-sixths.
This is three-sixths.
The put the bars end to end
to a show addition.
They are counting: one two three four five six seven eight - to get
eight shaded parts.
Then they are counting: one two three four five six seven eight nine ten eleven - out of 11 parts...
These mechanical kinds of things happen,
when students have not had informal experiences.
I've never seen this kind of thing happen,
when students use the games and activities for making whole bars.
For example:
Here is a whole bar,
and they have a little bit more than a whole bar.
On the small step race they can move one whole,
plus a little bit more. So five-sixths plus three-sixths is greater than one.
Eight elevenths is less than one!
We must resist the urge
to rush into a equations and abstractions
before students understand
the concept informally.
Fraction Bars Blackjack is another
game that his popular with students.
The bars are spread face down.
Each player takes one at a time adding the fractions to get as close as
possible to a sum of two, without going over.
For example,
here is a bar that is half shaded.
I have no idea what each bar will be.
That's going to be almost a whole bar.
I'll try this blue bar.
A whole bar and
four parts out of twelve.
I'll take a chance to see if I can get a little bit closer to two whole bars.
That's too big!
So i've gone over but of course, I would not
tell anyone else yet that.
I'd hold these bars till everyone
says they're "holding" and then all the players
show their bars.
If everybody is over two then no one wins.
With activities like these
students can become very good
at forming whole bars.
In one fifth grade class.
One parent complained about their child playing games in class.
At a parent teacher student conference,
the teacher asked the parent to select any five bars and compute the sum of
the fractions.
The parent
was still writing down the fractions,
when the student
gave the correct sum.
The parent
was impressed
and ended the conference by saying:
"Whatever you're doing, keep doing it."
Fraction bars Blackjack
is also on Fractionbars.com.
In this game
the player turns over two fraction playing cards.
The object is to get a sum as close as possible to two without going over.
A pencil and paper may be needed to compute the sums.
The player competes against the computer
and the greater sum that is not over two
wins the round.