In an NAEP study only 35 % of thirteen- year-olds chose the correct answer to

3/4 + 1/2.

When adding or subtracting fractions, children may have ideas about whole numbers that conflict with their ideas about fractions.

Estimate the answer to 12/13 + 7/8.

A, 1

B 2

C. 19

D. 21 E. Do not know.

Only 24 percent of the thirteen-year-olds chose the correct answer, 55 percent selected 19 or 21-they added either the numerators or the denominators.

Poor Understanding

1. Not viewing fractions as numbers but meaningless symbols to manipulate

2. Focus on numerators and denominators as separate numbers not 1 number.

3. Adding or subtracting fractions conflicts ideas about whole numbers

4. Insufficient time spent on concept of fraction, order and equivalence

5. Manipulatives often are not used enough or abandoned too quickly.

6. Many teachers do not have a good understanding of fractions

Why are fractions so difficult FOR CHILDREN

To master higher level math concepts

Why are fractions important

Unfortunately high quality research on fractions is scarce (IES) and evolving.

The following recommendations based on comprehensive review of research in 2013.

Developmental Sequence

K.5

The student will identify the parts of a set and/or region that represent fractions for halves and fourths.

Learning Standard

Essential Strategy 1

.

Provide opportunities for students to work with irregular partitioned, and unpartitioned areas, lengths, and number lines.

Strategies for instruction

Reference

**Fraction**

Math Curse

by Jon Scienska and Lane Smith

LUNCH is pizza and apple pie

Each pizza is cut into 8 equal slices

Each pie is cut into 6 equal slices

And you know what that means

If I want 2 slices of pizza should I ask for

A. 1/8

B. 2/8

C. 2 slices of pizza

What is another way to say ½ of an apple pie?

A.2/6

B.3/6

C.La moitie d’one tarte aux pommes

Which tastes greater?

A.½ a pizza

B.½ apple pie

We haven’t studied fractions yet, so

I take 12 carrot sticks 3 at a time and eat them 2 at a time

1.Parts of a whole: when an object is equally divided into parts, then a/b denotes a of those b parts.

2. The size of a portion when an object of size a is divided into b equal portions.

3. The quotient of the integer a divided by b.

4. The ratio of a to b. For every b quantity there is a available.

5. An operator: an instruction that carries out a process, such as “4/5 of”.

Numerator & Denominator

A fraction is made up of 2 numbers. The top number is called the NUMERATOR and the bottom number is called the DENOMINATOR. In the fraction the 3 is the numerator and the 4 is the denominator.

DENOMINATOR

This number shows how many equal 'pieces' something has been divided into. In the fraction, the denominator is 4 which means there are 4 equal pieces that make up the whole

NUMERATOR

This shows how many of those pieces there are. In the fraction there are 3 out of the total of 4 pieces.

Study indicates understanding of fractions in 5th grade uniquely predicts success in algebra

To make sense of every day use

Measurement of time (age, I’m 4 and a half)

Sharing and partitioning (cookies or pizza)

Measurement of quantity (cooking)

Different types of fractions

It does not matter how many equal pieces a whole is split into, if all the pieces are then taken, we have

the whole ,

For example,

We have some more mathematical names to describe some fractions. If the numerator is smaller than the denominator, the value of the fraction is less than 1 and it is called a

proper fraction

. For example

If the numerator is larger than the denominator and hence the value of the fraction is greater than 1, then it is called an

improper fraction.

For example

Improper fractions arise where more than one whole has been split up, and they can also be written as a mixture of whole numbers and fractions. For example, if we have 3/2 then we can think of this as 2/2 plus another 1/2 , and the 2/2 form a whole. So

Similarly with, say, 8/3. Every 3 lots of 1/3 makes a whole one, so we have 2 whole ones and 2 left over. In other words, we calculate 8 ÷ 3: 3 goes in to 8 twice remainder 2, so

These are referred to as

mixed fractions.

Now let us look at turning mixed fractions into improper fractions. Suppose we start with . We want it written in quarters. Now 3 wholes, divided into quarters, give us 12 quarters. And we also have another quarter. In total we have 13 quarters, so as an improper fraction In effect we have multiplied each whole number by 4, then added on the one quarter. So, to convert from mixed fractions to improper fractions you multiply the whole number by the denominator then add the numerator before writing it all over the denominator.

Fractions may appear as proper fractions, improper fractions or mixed fractions. They may also appear in many equivalent forms.

Equivalent fractions

are different fractions that name the same number, and they have the same value, even though they may look different.

Mistakes with fractions

1.Believing that fractions’ numerators and denominators can be treated as separate whole numbers.

(e.g., 2/4 + 5/4 = 7/8 or 3/5 – 1/2 = 2/3)

2. Failing to find a common denominator when adding or subtracting fractions with unlike denominators.

(e.g., 4/5 + 4/10=8/10).

3. Believing that only whole numbers need to be manipulated in computations with fractions greater than one.

(e.g., 53/5 – 21/7 = 3).

4. Leaving the denominator unchanged in fraction addition and multiplication problems.

(e.g., 2/3 × 1/3 = 2/3).

5. Failing to understand the invert-and-multiply procedure for solving fraction division problems.

(e.g., 2/3 ÷ 4/5 = 8/15)

Why Are Fractions so Important?

Lisa Hannich

For comparing fractions

For adding fractions

For subtracting fractions

For resolving proportion problems

For scaling problems

For calculus and beyond

Why are fractions so difficult to learn?

http://www.homeschoolmath.net/teaching/teaching-fractions.php

Click here, you also can check and

video

of manipulatives and the use of

visual models

The article--Teaching Fractions: What, When, How?

Nadine Bezuk

Kathleen Cramer

http://www.cehd.umn.edu/ci/rationalnumberproject/89_1.html

Pre K – 2. Foundational Knowledge:

Build on students informal understanding of sharing and proportionality.

1. introduce concepts of fractions by sharing a set of objects and partitioning a single object.

2.Extend to concepts of ordering and equivalence by partitioning the number of objects or number of sharers

Grade 2 – 3 Ensure students recognize that fractions are numbers

Use Number Lines as Central Representational Tool

1.Use measurement activities and number lines to help students understand that fractions are numbers,

2. Provide opportunities for students to locate and compare fractions on number lines.

3. Use number lines to improve students’ understanding of fraction equivalence, fraction density

Developmental sequence (2)

Mathematics Standards of Learning for Virginia Public Schools

Kindergarten

K.7

The student will recognize a penny, nickel, dime, and quarter and will determine the value of a collection of pennies and/or nickels whose total value is 10 cents or less.

Grade One

1.3

The student will identify the parts of a set and/or region that represent fractions for halves, thirds, and fourths and write the fractions.

1.7

The student will

a) identify the number of pennies equivalent to a nickel, a dime, and a quarter; and

b) determine the value of a collection of pennies, nickels, and dimes whose total value is 100 cents or less.

1.8

The student will tell time to the half-hour, using analog and digital clocks.

Learning Standard (2)

Mathematics Standards of Learning for Virginia Public Schools

Grade Two

2.3

The student will

a) identify the parts of a set and/or region that represent fractions for halves, thirds, fourths,

sixths, eighths, and tenths;

b) write the fractions; and

c) compare the unit fractions for halves, thirds, fourths, sixths, eighths, and tenths.

2.10

The student will

a) count and compare a collection of pennies, nickels, dimes, and quarters whose total value is $2.00 or less;

Grade Three

3.3

The student will

a) name and write fractions (including mixed numbers) represented by a model;

b) model fractions (including mixed numbers) and write the fractions’ names; and

c) compare fractions having like and unlike denominators, using words and symbols (>, <, or =).

Grade Four

4.2

The student will

a) compare and order fractions and mixed numbers;

b) represent equivalent fractions; and

c) identify the division statement that represents a fraction.

4.3 The student will

a) read, write, represent, and identify decimals expressed through thousandths;

b) round decimals to the nearest whole number, tenth, and hundredth;

c) compare and order decimals; and

d) given a model, write the decimal and fraction equivalents.

Learning Standard (3)

Grade Four (2)

4.5

The student will

a) determine common multiples and factors, including least common multiple and greatest common factor;

b) add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5,

6, 8, 10, and 12, and simplify the resulting fractions, using common multiples and factors;

c) add and subtract with decimals; and

d) solve single-step and multistep practical problems involving addition and subtraction with

fractions and with decimals.

Grade Five

5.1

The student, given a decimal through thousandths, will round to the nearest whole number, tenth, or hundredth.

5.2

The student will

a) recognize and name fractions in their equivalent decimal form and vice versa; and

b) compare and order fractions and decimals in a given set from least to greatest and greatest to least.

5.6

The student will solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form.

Grade Six

6.1

The student will describe and compare data, using ratios, and will use appropriate notations, s u c h a s ab , a t o b , a n d a : b .

6.2

The student will

a) investigate and describe fractions, decimals, and percents as ratios;

b) identify a given fraction, decimal, or percent from a representation;

c) demonstrate equivalent relationships among fractions, decimals, and percents; and

d) compare and order fractions, decimals, and percents.

6.4

The student will demonstrate multiple representations of multiplication and division of fractions.

6.6

The student will

a) multiply and divide fractions and mixed numbers; and

b) estimate solutions and then solve single-step and multistep practical problems involving

addition, subtraction, multiplication, and division of fractions.

NCTM Standards

Mathematics Standards of Learning for Virginia Public Schools

Pre-K–2 Expectations:

1. understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

2.understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Grades 3–5 Expectations:

1.use models, benchmarks, and equivalent forms to judge the size of fractions;

2.recognize and generate equivalent forms of commonly used fractions, decimals, and percents;

3.develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience;

4.use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals;

Grades 6–8 Expectations

1.work flexibly with fractions, decimals, and percents to solve problems;

2.compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;

3.understand and use ratios and proportions to represent quantitative relationships;

4.understand the meaning and effects of arithmetic operations with fractions, decimals, and integers

5.use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;

6.select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;

7.develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;

Essential Strategy 2.

Provide opportunities for students to investigate, assess, and refine mathematical “rules” and generalizations.

Essential Strategy 3

. Provide opportunities for students to recognize equivalent fractions as different ways to name the same quantity.

Essential Strategy 4.

Provide opportunities for students to work with changing units.

Essential Strategy 5

. Provide opportunities for students to develop their understanding of the importance of context in fraction comparison tasks.

Essential Strategy 6

. Provide meaningful opportunities for students to translate between fraction and decimal notation.

Essential Strategy 7.

Provide students with multiple strategies for comparing and reasoning about fractions.

Strategy 8.

Provide opportunities for students to engage in mathematical discourse and share and discuss their mathematical ideas, even those that may not be fully formed or completely accurate.

Strategy 9.

Provide opportunities for students to build on their reasoning and sense-making skills about fractions by working with a variety of manipulatives and tools.

Tips for teaching fractions

Engage your students’ interest in fractions.

Stress the importance of fractions in the world around them and in successful careers.

Emphasize that fractions are used in a variety of ways.

Practice understanding of fractions by using math manipulatives.

Practice basic words or phrases by giving students a problem and a list of relevant terms, e.g., "numerator," "denominator,“

Practice fractions by having students observe their surroundings, e.g., what fraction of classmates have black hair, have brown eyes.

Practice fraction problems by having students write their own fractions based on their own experiences.

Practice fraction problems by having students work in small groups to create their own surveys around fractions based on classmates' preferences

**Teng Sun**

**Reid C Taylor**