Fractions, Ratios, Money, Decimals and Percent

Lesson Four

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 rock-scissors-paper 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 fifty-four cookies and fifty-four 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 half-gallon 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 one-third off sale. How could we tell the price
of something that is now one-third 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 one-third 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.