Math 103 Lecture 5 notes page 1
Fraction derived from Latin fractus meaning "to break"
Numerator comes from Latin word meaning "numberer," hence numerator tells how many equal-sized parts there are.
Denominator comes from Latin word meaning "namer," hence denomiator tells what kind of parts there are.
Note: Cathcart text states there’s no reason for children to learn the terms “numerator” and
“denominator,” just say “top number” and “bottom number .” note: Warren Buck (TA) and I wrote a paper on this topic; you can read it on the course website.
The early Egyptian number system had symbols with numerators of 1. Most fractions with numerators other than 1 were expressed as a sum of different fractions with numerators of 1.
Ex: 7/12 = 1/3 + 1/4
Fractions with denominators 60 or powers of 60 were common in ancient Babylon about 2000 B.C., where 12,35 meant 12 + 35/60 . (recall that the Babylonian system is base 60)
The same method was used in Islamic and European countries and is presently used in the measurements of angles, where 13°19'47" means 13 + 19/60 + 47/60 2 .
The modern notation for fractions is Hindu origin, and came into use in Europe, 16th-century books.
The 6 th National Assessment of Educational Progress (NAEP) noted that most fourth, eighth, and twelfth-grade children tested were able to correctly choose pictorial representations for simple fractions, but they had more difficulty representing equivalent fractions. NAEP results showed that children had difficulty with items such as:
Jose ate 1/2 of a pizza. Ella ate 1/2 of a pizza.
Jose said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that Jose could be right.
Over half of the fourth graders did not answer the item correctly, and only about one-fourth of the children gave satisfactory responses to this question.
The NCTM Standards for grades 3-5 state:
"All students should develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers; use models, benchmarks, and equivalent forms to judge the size of fractions."
"All students should recognize and generate equivalent forms of commonly used fractions, decimals and percents."
Cognitive Issues with Fractions:
1. model –> verbal
2.
3. verbal –> model verbal/model –> symbolic
Suggested approach to teaching Fractions:
1. discrete before continuous
2.
3. relationship with part to whole relationship with so many equal parts
4.
5. equivalent fractions fractions do not have value by themselves

Math 103 Lecture 5 notes page 2
6.
fraction magnitude
Discrete quantities : collections consisting of distinct items. Because collectionsconsists of separate, whole items, they have no “in-betweens.” Such quantities are counted.
Continuous quantities : involve non-discrete quantities that include parts of a unit, or “oinbetweens.” Such quantities must be measured, not counted.
Which of the following are continuous and which are discrete?
Time elevation above sea level
Height
Population of the US size of a car inventory size of the national debt
Air pressure level of math anxiety
Examples of continuous and discrete fraction models: figure 10-2 in Cathcart, & UpClose notes
Part to Whole
Train yourself and your students to define the whole explicitly.
Two Pizza questions: May and June ordered two pizzas. Both pizzas were the same size and each was divided into four equal pieces. The shaded area in the diagram below represent the pieces that were eaten.
1. How many pizzas did May and June eat?
2. What fraction of the pizza pieces did May and June eat?
3. May and June ate completely what fraction of the pizzas?
4. How much pizza did May and June eat?
Equal Parts
When a child is first introduced to fractional parts, they may not understand that the parts must be equal. examples
Equivalent Fractions examples
Fractions do not ave value by themselves.
Joe ate ½ of a pizza. Ellen ate ½ of a pizza. Did they eat the same amount?
Ellen’s mother gave her 1/3 of a cookie. Joe’s mother gave him ½ of a cookie. Did they get the same amount?
Fraction Magnitude
Students begin to think about fractions as numbers on a number line, and are able to compare 2/3 and 4/5 and determine which has the greater value.
3.
4.
Fraction Error Analysis:
1. Color in one-fourth (1/4) of the circles:
2. Write a fraction to show what part of the pizza was eaten.
Write a fraction to show what part of the pie was eaten.
The fraction
2/8
¼ is equal to which of the following?
2/5 2/4 3/12

Math 103 Lecture 5 notes page 3
5. Circle the larger fraction: 1/4 1/3
4 Meanings for Common Fractions : (see figure 10-12 in Cathcart)
1.
Part of a whole
2.
3.
Quotient
Ratio
4.
Operator
Examples from UpClose Notes
ISAT examples
“Take Note” handout
Books:
Fraction Action
Give Me Half
Gator Pie by Loreen Leedy by Stuart J. Murphy by Louise Mathews
Fraction Fun
More For Me!
M&M Counting Book by David A. Adler by Sydnie Meltzer Kleinhenz
The Hershey’s Milk Chocolate Fractions Book by Jerry Pallotta and Rob Bolster
The Rainbow Fish by Maras Pfister by Barbara McGrath
Fraction computation is the math topic with which most adults have had the least success and the most bad memories! The traditional curricular emphasis on mastering computation with fraction symbols without first developing understanding of fractions and related concepts lead to this frustration and lack of achievement. The temptation to have children progress quickly to working symbolically with fraction computation may arise from the thinking that because children already know how to operate with whole numbers, they are ready to use the operation with fractions.
However, most children have difficulty relating operations on whole numbers with operations on fractions.
Divided People
“Congress tells us that just 2% of America’s elementary school teachers have science degrees and only 1% hold math degrees. Which explains why only 11 out of 21 elementary teachers can divide
13/4 by 1/2 and come up with the correct answer. Even more embarrassing, every single one of a similar group of 72 Chinese teachers got it right. The answer, according to our college intern at this newspaper (The C-U News Gazette) assures us, is 3 ½ .” (Is this a typo?)
Four goals of instruction regarding fraction computation:
1.
Children need to recognize situations that involve operations on fractions.
2.
3.
Children need to find the answer to fraction computation problems by using models.
Children need to estimate the answer and understand the reasonableness of results to
4.
fraction computation.
Children need to find an exact answer to fraction computation problems.
Notice the finding an exact answer is only goal of instruction. The other goals are equally important.
Estimate 12/13 + 7/8

Math 103 Lecture 5 notes page 4
When asked this problem, only 24% of 13-year-old students in a national assessment said the answer was close to 2. Most said it was close to 1, 19, 21, all of which reflect common computational errors in adding fractions and suggest a lack of understanding of the operation being carried out and a lack of number sense.
Fraction Addition: 2/3 + 1/2
The inclination to merely add across numerators and denominators should be offset by having children verbalize and formulate computational rules. For Ex:
“I have 2 thirds and 1 half, so I change the fractions so they are the same-sized parts, then
I add the number of parts in each group.”
As children work through introductory activities, they progress from thinking of fractions as “part of things” to operating with them as numbers, each with a precise location of the number line.
Introductory activities should have children work with fractions in different contexts because a narrow view of thinking of fractions only as parts of a whole is thought to be a reason for the unsatisfactory performance in fraction work. The part-whole, measurement, and part-of-a-set interpretations should be embodied in different problem-solving situations. In time, the quotient, operator, and ratio interpretations will also be explored. Allow adequate time for computation to emerge from children’s active explorations with concrete and pictorial representations.
Two common misunderstandings of multiplication and division with fractions are that “multiplication makes bigger” and “division makes smaller,” which is true for whole numbers. For example, many children expect that 6 ÷ ½ is 3, because they believe the answer to a division problem is smaller than the numbers in the problem. Instruction must lead children to recognize that these generalizations are incorrect and help them make sense of answers to fractions problems.
An assessment question might be:
Which quotient is greatest?
8 divided by 1 8 divided by ½ 8 divided by ½
When you see 6÷2, the question is: “How many 2’s does it take to make 6?”
When you see 4÷ ½ the question is: “How many ½ ’s does it take to make 4?”
If you give the student the problem 4÷ ½ , you probably will get a wrong answer. Instead, tell him you have candy bars scores in ½ ’s, and you want to know how many ½ ‘s would be in 4 candy bars, or, ask how many students would each get a ½ piece . . . you will get a right answer!
Models for fraction computation:
½ of ¾
¾ ÷ 2
¾
¾
÷
÷
½
1/8
Also see Cathcart figure 11-3, 11-7, 11-8
Three ways of posing a fraction problem:
Missing fraction: Jerry got 3 of his sister's 12 candies.
What fraction of the candies did Jerry get?

Math 103 Lecture 5 notes page 5
Missing outcome:
Missing start:
1/4 of 12 candies is ? is 1/4 of what?
Fraction Terminology:
Proper fraction = a/b where 0 < |a| ≤ |b|
Improper fraction = a/b where 0 < |b| ≤ |a|
Reduced fraction = a/b is in simplest form if a and be have no common factor greater than 1, i.e., a and b are relatively prime.
Equal fractions = represent same amount
Fundamental Property of fractions: two fractions a/b and c/d are equal if ad = bc.
Fraction, Rational Number, and Ratio comparison
Fraction = a/b where b ≠ 0
Ratio = a comparison of any two numbers
Rational number = a/b where a and b are integers and b ≠ 0. (The fact that b ≠ 0 is necessary because division by 0 is impossible.)
Ex: 1/ √ 2 is a fraction and a ratio but not a rational number
Ex: 1/0 is a ratio but not a fraction or rational number.
Fundamental law of fractions:
a/b = an/bn a, b, n can be any number but b ≠ 0,n ≠ 0
What is a ratio?
•
A fixed relationship: ex. the length of a square’s diagonal is always √ 2 of the length of its side
•
A multiplicative relationship:
•
An ordered relationship: ex. 3 is three-fifths of 5
25 mpg is vastly different from 25 gpm!
Ratio – a division of two numbers: a/b
Unlike rational numbers, the numbers in a ratio can be nonrational, or zero! Ex: 1: √ 2 or 2:0
A ratio can compare: a) 2 measures involving different units (rates) ex: cell phone minutes cost 35¢/min. ex: electricity costs 8.5¢/kwh b) part to its whole (part:whole) ex: of 5 students in the group, 2 are boys
(this is the same meaning as common fractions) c) part of a whole to another part (part:part) ex: the group has 2 boys and 3 girls
(this is a different meaning than common fractions)

Math 103 Lecture 5 notes page 6
When the measuring units describing two quantities being compared are different, the ratio is a called a rate . ex: Two candy bars for 97¢
The map scale is 1:100
Sales tax is 7%
When the second term is 1, the rate is referred to as the unit rate.
The speed limit is 55 miles per hour.
I can type 40 words per minute.
The two numbers in a ratio are referred to as the first term and the second term.
Ratios are introduced formally in intermediate grades, but informal exploration should begin in the primary grades. Here are some natural illustrations of ratios. Notice how doubling a ratio can be linked to children’s informal knowledge.
•
Ex: 1 nickel makes 5 pennies, 2 nickels makes 10 pennies
•
Ex: 1 musical note is 2 beats; 2 notes is 4 beats
•
Ex: 1 cup of flour makes 4 servings; 2 cups makes 8 servings
•
Ex: 1 yard is 3 feet; 2 yards is 6 feet
•
Ex: 1 dozen is 12 eggs; 2 dozen is 24 eggs
•
Ex: a person has 2 arms; a person has 10 fingers
Students often do not clearly distinguish between part to part and part to whole and hence confuse them.
Miss Brill gave her class the following word problems:
5.
Two boys in five in Ms. Pink’s chorus could not sing. She wisely assigned the nonsigners the role of stagehands and gave them the task of desingning the sets. What is the ratio of
6.
male stagehands to male singers in Ms. Pink’s chorus?
There are four boys for every five girls in Ms. Pink’s chours. What part of the chorus is boys.
For A, Darlene and Ellen answered 2:5; Fredricka and Georgia answered 2:3. For B, Darlene and
Fredricka answered 4:5; Ellen and Georgia answered 4:9.
Data Given
Problem A
2 nonsigners: 5 total
Problem B
4 boys:5 girls
(part:part)
Correct Answer
Darlene’s Answer
(part:whole)
2:5
(part:whole)
4:5
(part:part)
Ellen’s Answer
Fredricka’s Answer
Georgia’s Answer
Five boys in Ms. Pink’s chorus were tenors and three were basses. What is the ratio of tenors to basses in the chorus?

Math 103 Lecture 5 notes page 7
Children should understand that combining ratios is not the same as adding fractions.
To illustrate, suppose there are 2 boys and 3 girls in one group and 3 boys and 4 girls in another group.
If the two groups joined, the ratio of boys to girls would be 5 to 7. However, 2/3 + 3/4 is definitely not 5/7.
Similarly, while the ratio of boys to group is 5/12, it would be incorrect to write 2/5 + 3/7 = 5/12
The comparison of two fractions such as 1/2 and 1/3 is not possible unless it can assumed that both fractions have the same whole.
The same is not true of ratios. To compare two ratios, the whole need not be the same for each. Ex: if 1 of every 2 boys buys a pennant from the Booster Club and 1 of every 3 girls does, the ratio of boys buying a pennant is more than the ratio of girls buying a pennant. This is true regardless of the number of boys and girls there are.
ISAT examples
Cathcart Figure 10-18
Ratio denoted by a/b or a:b or “a to b” where a and b do not have to be integers!
Proportion: a statement that two ratios are equivalent
Property: If a, b, c, and d are real numbers and b ≠ 0 and d ≠ 0, then a/b = c/d if and only if ad = bc; i.e, “the product of the means equals the product of the extremes.”
An important aspect of proportional reasoning is the ability to discriminate between proportional and non-proportional reasoning. For example:
Sue and Julie were running equally fast around a track. Julie started first. When she had run 9 laps, Sue had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?
Research in 1993 found that most pre-service teachers in a math methods course wrote a proportion to solve the problem.
Qualitative prediction or comparison situations such as the following can be used to assess proportional reasoning.
If today Eric ran fewer laps in more time than he did yesterday, was his speed faster, slower, the same, or can’t you say for sure?
If Lisa ran more laps than Craig, and she ran for less time, who was the faster runner – Lisa,
Craig, or they ran the same speed, or can’t you say for sure?
Direct Variation : two variables that are in the same ratio regardless of their values
Ex: If a store charges $1.19 for a marker, then the cost “c” for “m” markers is:
C = 1.19 m and can also be looked at as: c/m = 1.19
Inverse Variation : when the product of two variables remains constant, they are inversely proportional or they vary inversely
Ex: see-saw
Division with Zero
The definition of division states that a ÷ b = c if and only if a = b x c, where c is a unique number.

Math 103 Lecture 5 notes page 8
Division problems have only one answer.
0 ÷ 3 = c means 0 = 3 x c
The only number that can be substituted for c to make this true is zero. Therefore, 0 ÷ 3 = 0.
5 ÷ 0 = c means 5 = 0 x c
No number can be substituted for c to make this true because anything times zero will always be zero. Therefore, 5 ÷ 0 is undefined.
0 ÷ 0 = c means 0 = 0 x c
Any number can be substituted for c to make this a true statement. Since the definition of division says that c must be only one (unique) number, we say that the answer is indeterminate. Therefore,
0 ÷ 0 is indeterminate.