Download What is a fraction?

Question
What is a fraction?
Answer: A topic that scares many of our
students
More seriously:
Please write down your definition of a fraction.
Then briefly discuss with a neighbor.
Dr. Richard Bisk – [email protected]
FRACTIONS
are numbers too!
Richard Bisk Ph.D.
Professor of Mathematics
Worcester State University
[email protected]
www.worcester.edu/smip
Imagine life without
fractions?
6 pies ÷ 3 people = 2 pies per person
2 pies ÷ 3 people = ? pies per person
What would we do if we didn’t have
fractions?
Skip the pie?
No way!
Cut each pie into 3 pieces
Each person now gets 2 of the 6 pieces
But what is each of the pieces?
Dr. Richard Bisk – [email protected]
Unit Fractions
Is this 1 unit shown 3 times?
Or 1 unit broken into 3 pieces?
1/3 is the number that multiplied by 3 gives 1
It takes 3 “one thirds” to make 1.
2/3 means 2 of “what it takes 3 of to make 1.”
A fraction is a point on a number line.
0
1/3
2/3
Dr. Richard Bisk – [email protected]
3/3
4/3
5/3
6/3
7/3
8/3
Common Core – Grade 3
CCSS.Math.Content.3.NF.A.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.
Dr. Richard Bisk – [email protected]
Common Core – Grade 3
more concrete version
“CCSS.Math.Content.3.NF.A.1“ Understand
a fraction 1/3 as the quantity formed by 1
part when a whole is partitioned
into 3 equal parts; understand a
fraction 2/3 as the quantity formed by 2
parts of size 1/3.
Dr. Richard Bisk – [email protected]
MIF – 2B
Dr. Richard Bisk – [email protected]
Common Core – Grade 3
CCSS.Math.Content.3.NF.A.2 Understand a
fraction as a number on the number line;
represent fractions on a number line
diagram.
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Expanding the number system
A fraction is a number; not two numbers.
They were developed to answer questions
that can’t be answered with whole
numbers.
They may look different, but they really
behave just like whole numbers.
Dr. Richard Bisk – [email protected]
Fractions are Numbers
And they have feelings too, just
like whole numbers!
Dr. Richard Bisk – [email protected]
2/3
The denominator tells us our noun - what we
have.
The numerator tells us our adjective - how
many we have.
2 dogs + 3 hats = ???
2 fifths + 3 fourths = ????
Dr. Richard Bisk – [email protected]
Addition and Subtraction
4 sevenths + 2 sevenths = 6 sevenths
7 ninths - 5 ninths = 2 ninths
If we have the same fractional unit, we
count, add and subtract fractions like
whole numbers
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Different Fractional Units
Convert to equivalent fractions with like
units.
Students must understand why this must be
done, before they learn the algorithm.
Compare to place value conversions.
Dr. Richard Bisk – [email protected]
1/2 + 1/3
Dr. Richard Bisk – [email protected]
Multiplying Fractions
5.NF.B.4 Apply and extend previous
understandings of multiplication to multiply
a fraction or whole number by a fraction.
Dr. Richard Bisk – [email protected]
Multiplying a Whole Number
and a Fraction
6 x ½ or ½ x 6
We can think of this as 6 groups of ½.
We often read ½ x 6 as ½ of 6.
Compare to: 3 x 5 means 3 groups of 5.
Dr. Richard Bisk – [email protected]
Fraction as Area
Consider 1/2 x 1/3
Start with a 1 inch square:
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Divide it into 3 equal horizontal pieces
and shade one them. This is 1/3 of the
square.
Divide the square vertically into 2
equal pieces. Shade ½ of the 1/3.
Dr. Richard Bisk – [email protected]
The blue shaded area represent ½ of 1/3 of the
square. Its area is 1/6 of the square.
The blue shaded region is a rectangle of width ½
and height 1/3.
Our area formula is length times width or
½ x 1/3 = 1/6
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Your Turn: Draw a rectangle array
model to represent
2/3 x 4/5
Dr. Richard Bisk – [email protected]
Division of Fractions
Yours is not to reason why, just invert and
multiply. (Old Proverb)
If you believe in things that you don’t
understand, then you suffer (S.W.)
Dr. Richard Bisk – [email protected]
Division of Fractions
Adult Explanation
How did we define division of
whole numbers?
6÷2=?
?x2=6
Dr. Richard Bisk – [email protected]
2 3
 ?
5 7
3
2
? 

7
5
7 3 2
 
3 7 5
???
7 3 2
?  
3 7 5
2 7 3 2
  
5 3 7 5
2 3
 ?
5 7
So
2 3 2 7
  
5 7 5 3
WOW!
DIVISION OF FRACTIONS
The Teaching Sequence
1. Review: Whole Number ÷ Whole Number
2. Teach: Fraction ÷ Whole Number
3. Teach: Whole Number ÷ Fraction
4. Teach: Fraction ÷ Fraction
Why this order?
Which is easier to model?
2/3 ÷ 4
4 ÷ 2/3
Dr. Richard Bisk – [email protected]
Why this order?
Step 2 is similar to Step 1. It can be done
using either the partitive or measurement
model.
Steps 3 and 4 often use the measurement
model
Dr. Richard Bisk – [email protected]
Example 1
Two children share 5 oranges equally.
Draw two models for 5 ÷ 2
Note relationship to multiplying by
reciprocal.
Dr. Richard Bisk – [email protected]
Example 2
Four children shared 2/3 of a pie equally.
What fraction of a pie did each get?
To draw models for 2/3 ÷ 4, we could use
a. pie
b. bar model
d. rectangle.
c. number line
Note relationship to multiplying by
reciprocal.
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Example 3
Two pies are cut into pieces that are each
¼ of a pie. How many pieces are there?
Draw models for 2 ÷ 1/4
Note relationship to multiplying by
reciprocal.
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Example 4
Find 5/7 ÷ 2/7
Draw a bar diagram.
Note relationship to multiplying by
reciprocal.
Dr. Richard Bisk – [email protected]
Dr. Richard Bisk – [email protected]
Conclusion
• Fractions are numbers too!
• Operations on fractions are extensions of
the corresponding operations on whole
numbers.
• Discrimination against fractions is a
violation of regulation bisk3.14 and is
punishable under numerous mathematical
laws.
Dr. Richard Bisk – [email protected]