Fraction I: Student Activity
Materials:
• Little Teddy Bears
Everyday fractions:
Almost every day we add, subtract, and reduce fractions. Many times we don’t
even realize that we are working with fractions. Can anyone think of a time when
we work with fractions but may not realize we are working with fractions? This is
an open-ended question, so give students time to respond. The following examples
may be included in the discussion.
Example 1: When we exchange five pennies for a nickel we are reducing
fractions. Each penny is 1/100th of a dollar, therefore, five pennies are 5/100 of a
dollar. When we exchange these five pennies for a nickel, we are reducing 5/100 to
1/20 since a nickel is 1/20 of a dollar. You might be asking how do I know that a
nickel is 1/20 of a dollar. Well, 20 nickels is equivalent to one dollar. In other
words, 20 nickels = 1 dollar. If we divide both sides by 20 we get 1 nickel = 1/20
dollars.
Example 2: Most conversions involve fractions. If a recipe calls for two cups of
milk, then that is equivalent to one pint of milk, since 2 cups = 1 pint. But if we
divide both sides by 2, we get 1 cup = ½ pint. Therefore, 2 cups = 2/2 pints which
is 1 pint. So when we convert 2 cups to 1 pint, we are reducing 2/2 pints down to 1
pint.
Terminology:
Fraction: A fraction is the part that is “broken off” from the whole. The whole
can be a single object such as a pie or a group of objects such as a bunch of
marbles.
Improper fraction: A fraction is improper when the numerator is larger than the
denominator. An improper fraction is greater than one.
Mixed number: A mixed number is a fraction that has both a whole number and a
fractional part. A mixed number is greater than one.
Numerator: The numerator is the top number of a fraction. It specifies the
number of parts out of the whole that one is referring to in a situation.
Denominator: The denominator is the bottom number of a fraction. It specifies
the total number of parts that are in the whole.
Fraction Bar: The fraction bar is the line between the numerator and the
denominator. The fraction bar has the equivalent mathematical operation as
division. Therefore, the fraction bar is asking us to divide the numerator by the
denominator.
Check for understanding:
In the fraction 3/4, what is the numerator?
What is the denominator?
What does the fraction bar mean in the fraction?
True or False
The fraction 5/4 is an improper fraction.
The fraction 1 ¼ is a mixed number.
Lesson:
Demonstration
Simplifying Fractions:
A fraction is simplified if the numerator and the denominator have only 1 as a
common factor. If the numerator and denominator have a common factor other
than 1, then divide both the numerator and the denominator by that number to
simplify the fraction. An easy way to visualize this is to use our little bear
manipulatives.
1. Someone from every table grab 6 blue bears and 3 red bears. This group of
bears represents the fraction 6/9 since there are 6 blue bears out of a total of
9 bears.
a. If we want to reduce the fraction 6/9, then we start by seeing how we can
evenly group the blue bears. Now the 6 blue bears can be evenly
distributed into two groups of three bears, three groups of two bears or
one group of 6 bears.
b. Then we look at how we can evenly group all the bears. The only way
that nine bears can be evenly distributed is to group them into three
groups of three bears or one group of nine bears.
c. Now we look for a common number of groups. Both the blue bears and
the total bears can be grouped into 3 groups.
d. Now we divide the blue bears into three groups of two bears. The
remaining bears are evenly distributed into these three groups. (Show
this representation on the whiteboard or overhead projector.) Therefore,
each group should have two blue bears and one red bear.
e. This group of three bears represents the fraction 2/3 since we have two
blue bears out of a total of three bears. Therefore, the fraction 6/9 is
reduced to the fraction 2/3.
Reflection:
In mathematical terms, in the above demonstration:
The number of blue bears represents the _______________.
The total number of bears represents the _______________.
The common number of groups represents the____________________.
Guided Practice
1. In order to represent the fraction 5/15, how many blue bears do we need?
Answer: 5
2. How many total bears do we need? Answer: 15
3. Gather the bears you need to represent the fraction 5/15.
4. How can you evenly distribute the blue bears?
Answer: five groups of one bear or one group of five bears
5. How can you evenly distribute all the bears?
Answer: three groups of five bears, five groups of three bears or one group
of fifteen bears.
6. What is the greatest number of groups for grouping the both the blue bears
and all of the bears.
Answer: five
7. Group the blue bears into five groups of even amounts.
8. Evenly distribute the remaining bears into these five groups.
9. What fraction does each group represent? Answer: 1/3
10. Therefore, the fraction 5/15 reduces to __________________. Answer: 1/3
Practice:
1. Using the little bears, represent the following fractions and then reduce the
fraction (if possible). Fill in the answer for the equivalent reduced fraction.
Check answers with those next to you to make sure you are getting the same
answers.
Fraction
6/10
6/8
6/15
8/12
8/16
8/20
10/15
10/25
12/27
7/22
Equivalent Reduced Fraction
3/5
¾
2/5
2/3
½
2/5
2/3
2/5
4/9
prime
Review: Rules of Divisibility
When trying to determine the number of ways to group the little bears it will be
helpful to remember the following rules of divisibility.
Divisibility by 1: All whole numbers are divisible by 1.
Divisibility by 2: A number is divisible by two if it ends in an even number.
Divisibility by 3: A number is divisible by three if you can add all the digits and
the sum that you get is divisible by three.
Divisibility by 5: A number is divisible by five if it ends in either a five or zero.
Divisibility by 10: A number is divisible by ten if it ends in a zero.
Check for Understanding
Fill in the following table, “yes” if the indicated number is divisible and “no” if it
is not divisible.
Number
45
90
Is the number divisible by
1
2
3
5
10
144
201
Demonstration
Improper Fractions / Mixed Numbers
Recall that both an improper fraction and a mixed number are greater than one. An
improper fraction is a fraction with the numerator larger than the denominator. A
mixed number is a fraction that has both a whole number and a fractional part. To
convert an improper fraction to a mixed number, divide the denominator into the
numerator and put the remainder over the divisor (numerator). Example:
1. To change 11/4 into a mixed number, divide 4 into 11.
2. Four goes into 11 two times with a remainder of 3.
3. The two is the whole number portion.
4. Put the remainder 3 over the divisor 4 to get the fractional part: ¾
5. Therefore, 11/4 is equivalent to 2 ¾ .
Practice:
Convert the following improper fractions to an equivalent mixed number.
Improper Fraction
Mixed Number
11/3
7/4
8/5
9/7
27/5
Follow the following steps to convert 2 ¾ to an improper fraction.
1. Put the whole number 2 over 1 to get 2/1.
2. Multiply both the numerator and the denominator of 2/1 by 4 (the
denominator of the fractional part: ¾) to get 8/4.
3. Add 8/4 to the fractional part ¾ to get 11/4. Remember when adding
fractions with like denominators we add the numerators and put the sum
over the common denominator.
Practice:
Convert the following mixed numbers to equivalent improper fractions.
Mixed Number
2 2/3
Improper Fraction
3 4/5
1 2/7
3 4/7
4 3/8
Demonstration
Multiplying Fractions (Multiply then reduce)
When multiplying fractions, we multiply straight across: numerator times
numerator and denominator times denominator. If possible, the resulting fraction
should be reduced. Example:
1. In order to multiply 2/3 x 3/4, we first multiply the numerators 2x3 to get 6.
2. Then we multiply the denominators 3x4 to get 12.
3. The resulting fraction is 6/12.
4. The fraction 6/12 is reduced to ½. You can reduce the fraction by using the
little bears as we did above or you can just divide both the numerator and
denominator by 6.
Check for Understanding
Answer the following questions for the problem: 3/5 x 5/6
1. What is the numerator before you simplify? Answer: 15
2. What is the denominator before you simplify? Answer: 30
3. What is the resulting fraction before you simplify? Answer: 15/30
4. What is the resulting fraction after you simplify? Answer: ½
Practice
Multiply the following fractions and fill in the answers. Check answers with those
next to you to make sure you are getting the same answer.
Multiply
2/3x6/7
3/5x1/6
5/6x3/5
3/5x5/6
3/4x2/5
Answer before Reducing
Reduced Answer
2/5x3/4
4/5x3/8
5/8x3/5
Reflection:
When filling in the above table, what pattern did you notice?
Answer: The order in which you multiply does not matter.
Demonstration
Dividing Fractions
To divide fractions, we take the reciprocal of the fraction we are dividing by and
change the operation to multiplication. Recall that to take the reciprocal of a
fraction, we just flip the fraction. For example, the reciprocal of 2/3 is 3/2. Follow
the following steps to divide 2/3 ÷ 3/4.
2
1. Change to multiplication by taking the reciprocal of ¾. Answer: 𝑥
2. Multiply straight across. Answer: 8/9
3. Can the result be reduced? Answer: No
2
3
4. Therefore, ÷ =
3
4
3
4
3
8
9
Practice
Divide the following fractions and fill in the answers. Check answers with those
next to you to make sure you are getting the same answer.
Divide
2/3÷6/7
3/5÷4/5
2/9÷1/3
3/5÷5/6
3/4÷4/5
2/5÷3/4
4/10÷6/8
5/8÷3/4
Answer before Reducing
When you are done, please fill out your evaluation form.
Reduced Answer