taking the pizza out of fractions
TRANSCRIPT
Taking the Pizza Out of FractionsBECKY UNKER, M.ED.
EDUCATION SPECIALIST, SPECIAL EDUCATIONUTAH OFFICE OF EDUCATION
Why Are Fractions So Important?• Proficiency with fractions is an important foundation for learning more advanced
mathematics.
• Fractions are a student’s first introduction to abstraction in mathematics and, as such, provide the best introduction to algebra in the elementary and middle school years.
• Time and emphasis are necessary for students to develop the links among fractions, decimals, and percent's and solve problems involving their use.
Francis (Skip) Fennell, NCTM President 2006-2008NCTM News Bulletin, December 2007
The Big Ideas in Fractions…
• Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as 1. On the number line, the distance from 0 to 1 is the unit.
• Fractional parts have special names that tell how many parts of that size are needed to make the whole. For example, thirds require three parts to make a whole.
• The more fractional parts used to make a whole, the smaller the parts. For example, eighths are smaller than fifths.
Big Ideas continued• The denominator of a fraction indicates by what number the whole has been divided in
order to produce the type of part under consideration. Thus, the denominator is a divisor. In practical terms, the denominator names the kind of fractional part that is under consideration. The numerator of a fraction counts or tells how many of the fractional parts (of the type indicated by the denominator) are under consideration. Therefore, the numerator is a multiplier—it indicates a multiple of the given fractional part.
• Two equivalent fractions are two ways of describing the same amount by using different-sized fractional parts. For example, in the fraction 6/8, if the eights are taken in twos, then each pair of eights is a fourth. The six-eights then can be seen to be three-fourths.
Rethinking Our Approach
• The white parts shows what part of this brownie Joe ate. How much of the brownie was eaten?
Two Students Responses: FIRST GRADER:
• A first grader studied the picture for a moment and then said the missing piece was “half of a half”.
THIRD GRADER:
• A third grader said it was an impossible amount, because the pieces were not all the same size and it therefore could not be “1 out of 3”.
• The first grader’s response suggests that children have some conceptually sound understanding of fractions, even before instruction.
• The third grader’s response suggest that children can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways.
• If models do not draw on children's formative experiences of sharing and partitioning, then they are likely to prevent teachers from cultivating the natural insights about quantities that young children have.
Sharing and the Concept of Fractional Parts• The first goal in the development of fractions should be to help children construct the
idea of fractional parts of the whole—the parts that result when the whole or unit has been partitioned into equal-sized portions or fair shares.
• Children seem to understand the idea of separating a quantity into two or more parts to be shared fairly among friends.
• They eventually make connections between the idea of fair shares and fractional parts.
• Sharing tasks are, therefore, good places to begin the development of fractions.
ActivityPlease take your bag of Cookie Crisp and act out the story as we go along…
Equal Sharing• Four children want to share 10 brownies so that
everyone gets exactly the same amount. How much brownie can each child have?
Your Task:• Create and Represent a solution to the
problem• Share and discuss your solutions with your
table group• Be ready to share with the group
Equal Sharing
• 16 kids need to share 12 pounds of clay. If they share the clay equally, how much clay would each kid get?
Your Task:• Create and Represent a solution to the
problem• Share and discuss your solutions with your
table group• Be ready to share with the group
Sharing Tasks and Fraction Language• During the discussions of student’s solutions (and discussions are essential!) is a good time to
introduce the vocabulary of fractional parts.
• This can be quite casual and, at least for younger children, should not involve any fraction symbolism.
• When a brownie or other region has been broken into equal shares, simply say, “We call these fourths. The whole is cut into four parts. All parts are the same size—fourths”.
• Children need to be aware of two aspects or components of fractional parts: (1) the number of parts and (2) the equality of the parts(in size, not necessarily in shape).
• Emphasize that the number of parts that make up a whole determines the name of the fractional parts or shares.
• They will be familiar with halves but should quickly learn to describe thirds, fourths, fifths, and so on.
Reflecting• Equal Sharing problems allow your students to learn fractions using what they already
understand as a foundation.
• If students are just beginning to learn fractions, solving Equal Sharing problems makes it possible for them to draw on what they know about partitioning and sharing to create and reflect on fractional quantities.
• If students are further along in their understanding of fractions, solving and discussing Equal Sharing problems helps them use what they know about division and benchmark fractions to reason about increasingly sophisticated relationships between fractions and whole numbers.
Models for Fractions• There is substantial evidence to suggest that the use of models in fraction tasks is
important (Cramer & Henry, 2002).
• Unfortunately, many teachers in the upper grades, where manipulative materials are not as common, fail to use models for fraction development.
• Models can help students clarify ideas that are often confused in a purely symbolic mode.
• Sometimes it is useful to do the same activity with two quite different models; from the viewpoint of the students, the activity is quite different.
• We will discuss three types of models: area or region models, length models, and set models.
Region or Area Models:• The Equal Sharing Task involved something that could be cut into smaller parts.
• The fractions are based on parts of an area or region.
• This is a good place to begin and almost essential when doing sharing tasks.
Activity:
Paper Folding
The Problem:• Zack had 2/3 of the lawn left to cut.
• After lunch, he cut ¾ of the lawn he had left.
• How much of the whole did Zack cut after lunch.
Your Task:• Fold the paper provided to solve
the problem.• Be ready to share your answer
and your experience.
Length or Measurement Models:• With measurement models, lengths are compared instead of areas.
• Either lines are drawn and subdivided, or physical materials are compared on the basis of length.
• Manipulative versions provide more opportunity for trial and error and for exploration.
• The number line is a significantly more sophisticated measurement model. From a child’s vantage point, there is a real difference between putting a number on a number line and comparing one length to another. Each number on a line denotes the distance of the labeled point from zero.
Activity:
The Magnified Inch
Activity:Please find the activity sheet, and a partner. Take a deck of playing cards for each “partnership”.
Set Models:• In set models the whole is to be understood to be a set of objects, and subsets of the
whole make up fractional parts. For example, three objects are one-fourth of a set of twelve objects. The set of 12, in this example, represents the whole or 1.
• It is the idea of referring to a collection of counters as a single entity that makes set models difficult for elementary school students.
• The set model helps establish important connections with many real-world uses of fractions and with ratio concepts.
• Counters in two colors on opposite sides are frequently used. They can easily be flipped to change their color to model various fractional parts of a whole set.
Activity: Please find the activity sheet and the two color counters on your table.
Problem Solving Strategies…
Bar Modeling Strategies
• Model drawing is a powerful problem-solving tool that opens new pathways to learning mathematics for students at every skill level.
• Model drawing is just what the name implies: drawing simple visual models to represent problems.
Let’s Try One Together:
• “ There were 80 air conditioners at a local warehouse. If 3/5 of the air conditioners were sold during one day when the heat was overwhelming, how many air conditioners were left for sale after that day?”
80
S L
L= ?
S16
S16
S16
L16
L16
5 Units = 8080 ÷ 5 = 161 Unit = 1616 x 2 = 32 There were 32 air conditioners left for sale at
the local warehouse after the hot day.
Your Turn:
• Grace has 28 marbles. Of the total marbles, 3/7 are red and the rest are blue. How many blue marbles does Grace have?
Your Task:• Use Bar Modeling to solve the word
problem.• Be ready to share with the group.
Student Commentary:• “Last year I had a teacher who was telling me what to do. He told me all these
different things to do with fractions. It took me a long, long time to learn his ways. This year my teacher allows me to try to do everything myself. She helps me, but she doesn’t show me exactly what to do. I am working hard this year figuring out how to solve problems, but I kinda like it.”
--Selma, seventh grader
Excerpt taken from: Extending Children’s Mathematics Fraction and Decimals by: Susan B. Empson, Linda Levi
Something To Ponder:“I know that many kids get to middle school without knowing procedures for fraction computation. If we show them what to do, they can get the right answer. But this doesn’t mean that they learned it. My goal isn’t to get them to be able to get the right answer for twenty problems. My goal is for them to learn mathematics.”
-Kathy Oker
sixth to eighth-grade teacher
Wingra School; Madison, Wisconsin
( excerpt taken from: Extending Children’s Mathematics Fraction and Decimals; Empson & Levi; 2011 pg. 188)
Resources:• Elementary and Middles School Mathematics Teaching Developmentally by: John A.
Van De Walle
• Extending Children’s Mathematics Fraction and Decimals by: Susan B. Empson, Linda Levi
• Bar Modeling A Problem Solving Tool –From Research to Practice, An Effective Singapore Math Strategy by Yeap Ban Har, PhD