This lesson will help students understand what the division of fractions truly means.
Main Curriculum Tie:
Mathematics Grade 6
Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 1.
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
- fraction pies
- pencil and paper
Background For Teachers:
When dividing fractions, students will have a better understanding if
they think of how many times the divisor will “fit” into the dividend. The
reason we use reciprocals when dividing fractions is to make the divisor
Intended Learning Outcomes:
Represent mathematical situations.
Invitation to Learn:
Ask the question, “What does it mean to divide 8 by 4?” Have
students draw a representation of what it means. Encourage the students
to share their representation. Ask students to then draw a model to show
what 3/4 divided by 1/4 means.
The students will typically look at the above division problem as
dividing 8 into 4 parts. Help them to see that you can look at the
problem in another way; how many fours are in 8. This works the same
for 3/4 divided by 1/4. How many 1/4's are in 3/4?
- Give each student a fraction pie. Instruct the students to layout
before them the disk equivalent to 1/2. Ask the students to
demonstrate what 1/2 divided by 1/4 would be. (How many 1/4
are in 1/2). Have the students do five or six similar problems in
which the first number is bigger. Be sure that they are very
comfortable with this way of thinking before moving on to the
next step. They should always use the language of how many
dividends are in the divisor.
- Next have the students lay out a disk equivalent to 1/4. Ask them
to demonstrate 1/4 divided by 1/8. Then ask them what they think
1/4 divided by 1/2 would be? Help them to see that there is only
1/2 of the 1/2 in 1/4. 1/4 divided by 1/2 = 1/2. Do similar
problems until the students are comfortable with the concept.
Help them to see that when dividing fractions, sometimes the
quotients are larger than the dividend or the divisors.
- Next give each student a ruler. Present to the students 5 1/2
divided by 1/2. Ask the students to demonstrate the answer using
their rules. (How many 1/2 inches are in 5 1/2 inches). Instruct
them to estimate, then find the exact answer. Guide them
through several examples of dividing mixed numbers.
- Give the students several written problems and have them
estimate what they think the answers would be. Working in
groups, and then having the groups report on their procedures
would develop opportunities for more math talk.
- Ask the students to develop story problems for several division of
fraction expressions. Encourage the students to share their
- Ask the students if they would like to learn a short cut for
dividing the fractions. Demonstrate the following two ways of
representing 1/4 divided by 1/2:
- Explain, “It is difficult to think about dividing by 1/2. But, it is
easy to divide by 1. Is there a way that we can change the 1/2 to a
1?” Guide the students to the understanding that the divisor can be
multiplied by 2/1, its reciprocal. Help them to see that they must
multiply the dividend and the divisor by 2/1 to get the correct
The shortcut is to multiply the dividend by the reciprocal of the
divisor. Have the students record in their journal the meaning and
methods of fraction division.
This perspective of division works very well with decimals also.
The following are several real world situations in which division of
decimals is used. Have the students explain to one another how they
would solve these situations. Then have them write equations for their
You need $69.99 to purchase a new video game. You decided to sell
boxes of chocolates to kids at your school to earn the money. You buy 15
1/2 lbs of chocolate for $ 35.00. The clerk suggested that you package the
chocolates in either 1/4, 1/3, 1/2 or 3/4 lb boxes. Determine how many
boxes of chocolates you could fill for each of the fractions. Select which
size you think would sell the best. How much would you charge for each
box of chocolates? Be sure that you make enough to pay for the
chocolate and your game. Do you think students would buy the
chocolates for the price you are asking?
You decide to make sock bags for your friends for Christmas. You
find in the closet 5 1/4 yards of cloth. You need 3/4 of a yard to make one
bag. How many bags could you make?
You get a job installing tires on cars in an assembly line. You get paid
$5.00 for each set of tires you install. It takes you 1/4 of an hour to put
on a set of tires. If you work for 8 1/2 hours, how much would you expect to make?
Homework & Family Connections:
Have the students use manipulatives to explain to their parents or
siblings what division of fractions means.
Assign students to measure the amount of food in a box (cereal, rice,
noodles etc.). Have them determine how many batches they could make if
a recipe calls for 1/4, 1/2, 1/3, 2/3, or 3/4 cups of their selected food.
Have them write a math sentence using division of fractions to represent
what they did.
Assign students to measure the length of their bedroom. Then have
them determine how many 3/4 foot tiles they would need to lie tiles
across the length of the room. Have them write a math sentence using
division of fractions to represent what they did. They could also find how
many 1 1/2 foot dressers they could line up, or how many 1/3 foot candy
bars, or 3 1/2 ft wide beds, etc.
As students respond and explain their response to you and fellow
students, assess their understanding. Their journal entry would be
another opportunity for assessment.
Created Date :
Jul 22 2003 08:06 AM