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Fractions, Ratios, Money, Decimals and Percent
Lesson Four
Purpose  Learn to be aware of fractions in life . 
Summary  We ask our students to think about the sharing,
cutting and dividing fractions in their lives. 
Materials  None. 
Topic  Teacher lead discussion on the sharing, cutting
and dividing that lead to fractions in our lives. 
Homework  Students bring examples from home to school to share. 
four marbles...
54 ÷ 4 is a division problem. The answer that a calculator gives is 13.5. But
mathematics is more than
numbers on a page or the answer that a calculator finds.
Teacher: Four children have 54 marbles to share among themselves. How many
marbles will each
child receive? Please be prepared to explain your answer to the class.
Patrick: The answer is 14 because when I do it on my calculator, I get 13.5, but
we are supposed
to round up when we get a .5, so its 14.
Calculators give us answers for the arithmetic that we do , but are calculators
any use in giving students
an awareness of what the numbers mean ? Patrick has learned to round up when he
sees .5, but has he
learned to think about what the numbers mean?
Teacher: Patrick, what does your calculator say fourteen times four is?
Patrick: 56.
Teacher: Then if I said to you that four children each have fourteen marbles,
how many marbles
would they have altogether?
Patrick: 56.
Teacher: But the first problem I gave you said the four children have only 54
marbles, not 56.
You would need to have started with 56 to give each child fourteen.
Rounding is a useful skill to know. Knowing when to round and when to not is
another useful skill.
Jesse: You don't round. The answer is just 13.5. Everybody gets 13 and a
half.
Teacher: How do you propose to give anyone half a marble?
Jesse: Hit the extra marbles with a hammer?
Teacher: Is that really how you would share any extra marbles that you had?
Jesse: No.
Aaron and Danielle: Each child gets thirteen marbles.
Then they play rockscissorspaper to see
who gets the two left over. So three children get thirteen marbles and one gets
fifteen.
Teacher: That way seems like it would work. Who has another way that the
children might
share?
The teacher questions each student's explanation to see if the students
understand the numbers. Then
the teacher uses the numbers 54 ÷ 4 to ask a different question .
Teacher: Four children have 54 cookies to share among themselves. How many
cookies will each
child receive? Please be prepared to explain your answer.
What is the difference between fiftyfour cookies and fiftyfour marbles?
Individual marbles cannot be
broken up and shared. Individual cookies can. Mathematics is more than numbers
on a page.
Dividing, cutting, sharing...
Teacher: What are examples of sharing in your life?
Sharing food and drinks.
Sharing toys.
Sharing colored pencils or crayons.
Sharing math materials in class.
Sharing clothes.
Sharing the swing at recess.
Sharing secrets.
Sharing money.
Sharing comic books.
Sharing combs and brushes.
Sharing lipstick.
Sharing rides.
Sharing answers for homework.
Sharing answers secretly for a test.
Sharing ideas.
Sharing prizes won.
Sharing with the class.
Teacher: We know that candy bars can be shared fractionally. Can comic books?
Can pencils?
Which things do we divide using fractions and which do we divide another way?
Can you explain your answer?
Teacher: What are examples of things you can cut?
Cutting cakes and pies.
Cutting firewood.
Cutting class.
Cutting up in class.
Cutting glass.
Cutting grass or hay.
Cutting someone down.
Cutting hair.
Cutting fingernails.
Cutting out paper dolls.
Cutting out the pattern for a shirt or dress.
Cutting with a knife.
Cutting down a tree.
Cutting back on spending.
Cutting through the water.
Cutting pictures from a magazine.
Cutting folded paper into snowflake designs.
Cutting diamonds.
Cutting cattle with a horse.
Teacher: We know that cut up cakes and pies can be described fractionally.
Can cutting hair?
Which of the things that we can cut can be described with fractions and which
cannot?
Explain your answers.
Teacher: What are examples of when you might use your skills of division?
Dividing the arithmetic problems on the workbook page.
Dividing to see if the multiplication answer was right.
Dividing handsful of squares into groups.
Dividing portions in the cafeteria.
Dividing into teams.
Dividing to find averages.
Dividing up the chores.
Dividing up the Sunday paper.
Dividing miles into gallons to see the mileage for the car.
Dividing up the land.
Dividing up the loot.
Dividing up the time.
Dividing up the day.
Dividing to find the cost of one.
Dividing to find your share.
Teacher: We know that we can divide squares into groups and if the groups don't
divide evenly,
the remainder is a fraction. Would dividing people into teams ever leave a
fraction?
Which of the things that we can divide might have fractions in their answers at
least
sometimes? What would the fractions be? Which things never have fractions in
their
answers?
Explain your answers.
Fractions everywhere...
We ask our students to think about fractions in school. How do we help our
students to think about
fractions at home?
Teacher: Your homework assignment for tonight is to find examples of fractions
and bring them
with you tomorrow when you come to school.
You may bring in your examples in any way you can. Cut them out (ask permission
first!), write
them up, draw them, or bring them in your head.
Where do you think you might look to find examples? Where are fractions used?
If our students do not know where to look, we provide some clues. Examples come
from the sharing,
cutting and dividing discussions we have had in class. Where else might our
students look at home?
In newspapers, magazines and cookbooks. On packages for foods and drinks.
Looking means
listening, too. What fractions are used on the radio, on TV, or in conversations
with relatives?
Examples can come from places outside the home. Store advertising signs , sizes
for shoes and clothes,
product names , street addresses, distances to freeway exits. Every situation
that our students find can
be the inspiration for a hundred situations more.
The awareness we create...
The object of this lesson is not the answers that we find, it is the awareness
we create. Fractions are
everywhere around.
We have three halfgallon bottles of soda to share for our party this afternoon.
We also have paper
cups for everyone. Can we figure out in advance if three bottles hold enough
soda to give
everybody in class at least one full cup? If any soda were left, how much would
each child get
for seconds? If three bottles are not enough, how many more will we need?
How should we cut Aaron and Kyle's birthday cake so that everyone gets a piece?
What fraction of
the cake will each piece be? Should we cut it differently, so that there will be
cake left over for
seconds? What fraction would these pieces be?
How shall we divide the class to have four teams for P. E. today? Will the teams
come out evenly? If
four teams does not give each team the same number of players and we want equal
teams, how
many teams should we have?
Julie found the fraction 1/3 in the newspaper for a onethird off sale. How
could we tell the price
of something that is now onethird less than its original price?
We ask questions for awareness. We create problems that are real. We take the
risk of not knowing
where the lesson may be going and let the lesson take us where it leads. Some
problems may lead to
work with fractions. Some may not. All involve thinking about how to use
mathematics. All involve
connecting math to life.
Teachers ask questions for different reasons in the United
States and in Japan. In the
United States, the purpose of the question is to get an answer. In Japan,
teachers pose
questions to stimulate thought. A Japanese teacher considers a question to be a
poor
one if it elicits an immediate answer, for this indicates that students were not
challenged to think.
J. W. Stigler & H. W. Stevenson, How Asian Teachers Polish Each Lesson to
Perfection,
American Educator, Volume 15, Number 1, Spring 1991.
Is there a particular method for calculating the answers that we should teach
our students? Can our
students figure out if three bottles of soda are enough to share? Can they
decide how to cut a cake so
that everybody gets a piece? Can the children in our room divide themselves into
teams? Can they
show us what it means to take onethird off?
Answers are not the goal. Thinking is. Fractions are something that we think
about.
The assessment...
If we do not have soda bottles, or cakes, our students show us how they would
divide, cut, or share
with manipulatives. Unifix Cubes , Power Blocks and water from the fountain are
available for our
students to act out how they might divide, cut, or share. The assessment for
each problem that our
students do is in the proofs they offer for the answers that they find.
What is the assessment for a lesson that may lead us in directions that we did
not plan?
Teacher: Write down everything you have learned about fractions. If drawings
help you show
what you have learned, add drawings to the writing that you do. You may use your
spelling
notebooks if you are not sure how to spell a word.
Assessments do not have to involve special problems that we create. Assessments
can be as simple as
asking our students to tell us what they think they know. As we read their
statements, we can see what
has been learned and what we should teach next.
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