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Fractions

Back to School Review of Equality and Inequality of Fractions

150 150 Deborah

Overview

The concept of dividing a whole into parts and then dividing the parts into smaller entities is basic to mathematics. That model of division can be illustrated by manipulatives such as fraction bars and Cuisenaire rods. Equality and inequality of fractions can be demonstrated by finding their common denominators and comparing them. By comparing unequal fractions and finding more fractions between them, students can show the density of the number line.

Dividing a Whole into Parts

A fraction is a part of a whole. Children can illustrate the concept by taking a whole circle, then cutting the circle in half, then cutting each half into halves to show fourths, then cutting each fourth into eighths, and so forth. Many different nations in ancient times, such as the Egyptians, Hindus, and Babylonians used fractions in different computations. The Arabs separated the numerator and a denominator by a bar between them.

Illustrating the Model

Manipulatives such as fraction bars and Cuisenaire rods are special types of objects that can be used to illustrate the equality or inequality of fractions. Ten white cubes equal the same length as one orange rod, and so do five red rods. It can be shown that one red rod is the same length as two white cubes, in symbol terms 2/10 = 1/5. Similarly, 3 white cubes, or 3/10, is not the same size as 1 purple rod, or 1/3.

Finding Common Denominators

The best method to test whether fractions are equal or unequal is to find a common denominator for all the fractions in the group and then compare them. Suppose the fractions are 5/8 and 3/5. The fraction 5/8 is equal to 25/40, but the fraction 3/5 is equal to 24/40. They are not equal. Suppose the fractions are 3/5 and 6/10. Since 3/5 is equal to 6/10, the fractions are equal.

Density of the Number Line

There is always another fraction that can be found between any two fractions on the number line. Take a closer look at the section between 0 and 1 on the number line. At first glance, it appears full. Halfway between 0 and 1 is ½, and halfway between that is ¼, and halfway between that is 1/8, then 1/16, then 1/32, and then there’s a point for ¾, and there’s also a point for 3/8, 5/8, and 7/8, 1/16, 3/16, 5/16, 7/16, 1/32, 3/32, and so on. Fractions are dense along the number line.

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Math Review of Complex Rational Expressions

150 150 Deborah

Overview

A complex rational expression contains a fraction within a fraction. In math language, it is a rational expression that has one or more rational expressions contained within the numerator, denominator, or both. It may or may not contain variables within it. The fractions in the numerator or denominator must be simplified first, before the other rational expression can be tackled.

Definitions

A rational expression is a ratio. Examples of simple ratios that do not contain variables are 2/3, 6/7, and 7/8. Rational expressions can also include variables, such as (3x)/7, (9z + 2)/4, or 3/(y + 1). They can be simplified in one or two steps. Complex rational expressions (also called complex fractional expressions) are ratios that contain fractions within fractions. Simplifying them often takes more than one or two steps.

Figure 1: A complex fraction in symbol form.

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Simplifying the Numerator

Suppose a rational expression contains the terms (2/3 + 1/4) in the numerator and 5 in the denominator. In one method, the first step in simplifying the rational expression is to find the LCM for the fractions in the numerator, changing each fraction to the equivalent common denominator, and performing the operation. In this example 2/3 becomes 8/12 and ¼ becomes 3/12. Adding 8/12 and 3/12 equals 11/12. If one or more variables in the rational expression is in the numerator, the process is similar. Suppose another complex rational expression contains a variable in the numerator, so that the terms are ([4x]/7 + x/9)/2. In that case, the first step would be similar: The LCM of 7 and 9 is 63, so the numerator becomes 36x/63 + 7x/63 or 43x/63. The denominator is still 2.

Figure 2: Simplifying the ratio when a fraction is in the numerator.

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Simplifying the Denominator

If the rational expression is in the denominator, the process is similar to solving the rational expression in the denominator. However, if there is a variable in the denominator, the denominator cannot equal zero. That would be the same thing as dividing by zero, which is undefined by definition. Suppose the numerator is 3 and the denominator is 11/12 + 1/20. Using the same method, the LCM is 60, so the fractions in the denominator become 55/60 + 3/60 or 58/60. (In turn, 58/60 can be simplified to 29/30.) The rational expression becomes 3/(29/30).

Figure 3: Simplifying the ratio when a fraction is in the denominator.

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Simplifying the Entire Ratio

Whether the expression is (11/12)/5, (43x/63)/2, 3/(29/30), 12/(3x + 1)/2, or even (3/(2x + 1)/1/2), solving the ratio in the numerator or denominator is only part of the process. The ratio is not fully simplified until the fraction within the fraction is done. Suppose the expression is (11/12)/5. That is equal to 11/12 ∙ 1/5 = 11/60. Similarly, (43x/63)2 means the same thing as 43x/63 ∙ ½ or 43x/126.

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Math Review of Least Common Multiples and Denominators in Rational Expressions

150 150 Deborah

Overview

Rational expressions are like fractions, and must have common denominators in order to be added or subtracted.  In order to find the smallest denominator in common, the least common multiple should be found between the denominators.  The least common multiple can also be calculated when the denominators are algebraic expressions.

Least Common Multiple

The least common multiple for rational expressions is found in two steps.  Factor the denominators in the rational expressions to be added or subtracted.  The least common multiple will be the factors that both denominators have in common times the factors that are not in common.  Suppose the rational expressions to be added are x/36 and x/48.  The constant 36 can be factored as 2∙2∙3∙3.  The constant 48 can be factored as 2∙2∙2∙2∙3.  The factors that 36 and 48 have in common are 2∙2∙3.  The factors that they do not have in common are an additional 2.2.3.  The least common multiple is 2∙2∙2∙2∙3∙3, or 144.  To check, 144/36 is 4 and 144/48 is 3.

Figure 1:  The least common multiple (LCM) contains the factors in common and the factors not in common.

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Finding the Numerator

Before the expressions can be added, both rational expressions must be changed to their equivalents with the common denominator as the least common multiple.  Using the example, x/36 is multiplied by 4/4 as 4x/144 and x/48 is multiplied by 3/3 as 3x/144.  Then they can be added as rational expressions with common denominators.

Adding the Expressions

Still using the example, 4x/144 + 3x/144 equals 7x/144.  The expression is already in simplest terms, because neither 7 nor x are factors of 144.  Suppose x equaled 2.  Then, the first fraction would be 8/144 and the second fraction would be 6/144, or 14/144.  The fraction 14/144 can be simplified to 7/72.  Suppose x equaled 5.  Then the first fraction would be 20/144 and the second fraction would be 15/144.  Adding them together would be 35/144, which would be in simplest terms.

Figure 2:  To add rational expressions, each expression must have the same denominator.

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LCM of Algebraic Expressions

Finding the least common multiple when algebraic expressions are in the denominators of rational expressions is a similar process to finding the least common multiple for fractions with a constant in the denominator.  Suppose that the denominator of one expression is (x2 – 49) and the denominator of another expression is (x2 + 14x + 49).  The expression (x2 – 49) can be factored as (x + 7)(x – 7).  The expression (x2 + 14x + 49) can be factored as (x + 7)(x + 7).  The factor each expression has in common is (x + 7).  The factors they do not have in common are (x + 7)(x – 7).  The least common denominator would be (x + 7)(x + 7)(x – 7).

Figure 3:  To calculate the LCM, factor algebraic expressions for the common factors and the factors not in common.

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Math Review of Multiplying Fractions and Rates

150 150 Deborah

Overview

Multiplication of fractions and rates with variables is similar to multiplying regular fractions and rates. It is important to cancel out common factors and units.

The General Pattern

The general pattern for multiplying fractions is to multiply the numerators and multiply the denominators to create a new fraction. In symbol form, the rule is for real numbers a, b, c, and d, where b and d are not equal to zero, a/b ∙c/d is equal to ac/bd. Following the pattern, 2/3 ∙7x/4 can be solved by (2∙7x)/(3∙4), or 14x/12. In simplest terms, the new fraction is 7x/6, as both the numerator and denominator can be divided by their common factor 2/2.

Figure 1: The rule in symbol form for multiplying fractions.

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Factoring Before Multiplying

Another way to solve a problem where there are common factors is to take out the common factors before multiplying. A problem such as (2∙ 7x)/(3∙4) has common factors that can be cancelled out before it is multiplied. Using the Commutative Property, the problem becomes (2/4)(7x/3). Using Equal Fractions, 2/4 can be simplified to ½ and then multiplied as 7x/(2∙3), or 7x/6. Suppose the problem is 9/(3y) ∙6y. Another way to simplify the problem is to set it up as multiplication of the fractions 9/(3y) ∙6y/1, so it looks like (9∙6y)/(3y). That simplifies even further, because 6y is evenly divisible by 3y, as 2.

Figure 2: The process of factoring before multiplying.

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Rates

Rates can be multiplied by other quantities. Suppose a car is traveling at a rate of 70 miles per hour. In 2 hours, it will travel 140 miles. The hours in the miles per hour and the time traveled (2 hours) cancel each other out. Similarly, suppose a typist types 70 words per minute. In 10 minutes, the typist can type a 700-word document.

Multiplying Rates

Rates can be multiplied similar to fractions. Suppose a commuter drives 30 minutes per day to work, 5 days per week. That commuter drives 30 min/day times 5 days/week. How many minutes per week does she drive? The days cancel out, so the equation becomes 30 minutes times 5 or 150 minutes/week. Also, 150 minutes/week can be simplified to 2 hours and 30 minutes, by dividing by 60 minutes/ hour.

Figure 3: Rates can be multiplied. For example, commuting time is equal to time per day multiplied by days per week.

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Math Review of Fractions, Decimals, and Percents

150 150 Deborah

Overview:

Fractions can be converted to corresponding decimals, and both fractions and decimals can also be converted to percents.  As with many equivalent expressions in mathematics, fractions, decimals, and percents can also be converted back into their original forms.

What Are Fractions?

A fraction is a part of the whole so that it is in the form a/b, where a is the numerator and b is the denominator.  A simple fraction has an integer in its numerator and a nonzero integer in its denominator.  Fractions are useful in equations.  Some fractions, such as 3/8, are in simplest terms, while others, such as 6/8, can be simplified to an equivalent fraction ¾ in simplest terms. Fractions can be used to perform all operations, such as addition, subtraction, multiplication, and division.

How can Fractions Be Changed to Decimals?

Although fractions are useful, they can be harder to use in operations than integers and decimals.  Fortunately, they are easily changed to decimal form and back again.  One of the meanings of a simple fraction such as ¾ is 3 divided by 4 or 0.75.  Similarly, the fraction 5/8 is equal to 5 divided by 8 or 0.625.  The decimal can easily be changed back into a fraction by remembering that a decimal such as 0.75 means 75/100.  By dividing 75/100 by 25/25 changes it to the equivalent simplest form, or ¾.  The decimal 0.625 can be changed to its fractional form 625/1000 and then divided by 125/125 to its equivalent simplest form 5/8.  Fractions such as 1/3 can be changed to decimal forms that have repeating forms, such as 0.333….  The repeating forms can then be changed into their fractional forms.

How can Fractions Be Changed to Percents?

Fractions can also be changed into percents, because percents are a particular type of fraction.  A percentage means “times 1/100”, so that a fraction such as ½ is 50% or ¾ is 75%.  They can be changed similarly as decimals to fractions, so that 50% means 50/100 or ½, and so on.

How Can Decimals Be Changed to Percents?

Decimals can be changed easily to percents, because the percent is merely the same thing as hundredths, or two decimal places to the right.  Therefore, 0.625 (or 5/8) is the same thing as saying 62.5% and 0.75 is the same thing as saying 75%.  To convert percents to decimals , the decimal places are moved to the left, so that 49% is the same thing as 0.49 and 55.12% is the same thing as 0.5512.

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Math Review of Adding and Subtracting Fractions

150 150 Deborah

Overview:

Adding and subtracting fractions is similar to adding and subtracting whole numbers, with one important difference.  The denominators must be the same, so that if the fractions do not have the same denominator, they must be changed into equivalent fractions with the same denominator.  Then the general pattern of adding or subtracting fractions can be followed.

What Is the Putting-Together Model for Addition?

By the putting-together model for addition, similar units can be combined.  Suppose there are x CD’s in one box and y CD’s in another box.  The total number of CD’s in both boxes is x + y.  Similarly, if there are t tulip bulbs planted in one flowerbed and 6 more tulip bulbs planted in another flowerbed, there are t + 6 total bulbs planted all together.  In order for elements to be combined, they must be the same type of unit.  Also, there must not be an overlap in the things to be added together.  None of the elements in the x set can be also in the y set for the resulting sum to be x + y.

How Can Fractions Be Added?

Adding fractions is a type of putting- together.  If the fractions have the same denominators, they are the same type of unit, so that 1/8 and 2/8 can be added together as 3/8.  Similarly, 4/16 and 5/16 can be added together as 9/16.  Generally, the rule using variables is that a/c + b/c, when c is not equal to 0 is the same thing as adding (a +b)/c.

What about Common Denominators?

In order for fractions to be the same type of unit, they must have common denominators.  Suppose the fractions were 1/3 and 1/4.  In order for them to be added, 1/3 would have to be changed to its equivalent 4/12 and ¼ would have to be changed to its equivalent 3/12.  Then they could be added together as 7/12.

How Can Fractions Be Subtracted?

Subtraction is adding the inverse of the number to be subtracted, so that a – b means the same thing as a + -b.  Similarly subtracting a/c- b/c means the same thing as a/c + –b/c.  In order for fractions to be subtracted, they must also have a common denominator.  Suppose the expression were 5/6 – ½ .  The equivalent of ½ is 3/6, so that 5/6 – 3/6 is 2/6.  In simplest terms, 2/6 is 1/3.

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Math Review: Equality and Inequality of Fractions

150 150 Deborah

Overview: What Are Fractions?
Fractions consist of a numerator, a, and a denominator, b, so that the fraction takes the form a/b.  The denominator is the number of equal parts in a whole, and the numerator is the number of parts considered.  For example, if a pizza is divided into 8 equal pieces, and 3 pieces are taken, the fraction to describe the relationship is 3/8.  There is a close relationship between fractions and division, so that for any numbers a and b, and b is not equal to zero, a/b means that a is divided by b.

What Are Mixed Numbers and Improper Fractions?
In many fractions, the numerator is smaller than the denominator, so the fraction is less than 1.  For example, 1/8 is less than 1.  Using the division model, where a is divided by b to describe a fraction, the numerator can be larger than the denominator, such as 8/5.  The number 8/5 can be represented 2 different ways, either as a mixed number 1 3/5, or an improper fraction 8/5.  Think, 5/5 is equal to 1, and there are 3/5 left over.  Even though the mixed number and the improper fraction mean the same thing, they are used in different circumstances.

What Are Common Denominators?
In order to compare fractions with each other, both must have the same denominator.  In order to do that,  find the least common multiple of both denominators.  What is the common denominator for 1/6 and 1/4?  The least common multiple is 12,  so 1/6 x 2/2 would equal 2/12, and 1/4 x 3/3 would equal 3/12.

What Are Equal Fractions?
Fractions are equal when one fraction can be obtained from the other.  The fractions 1/6 and 2/12 are equal because 2/12 means 1 X 2/6 X 2.  Similarly, the fractions 1/4 and 3/12 are equal because  3/12 means 1 X 3/4 X 3.  This can also be shown graphically with fraction bars, as 1/2 can be represented by 2/4, 4/8, 8/16 and so on.  There are an infinite number of equal pairs of fractions.

What Are Unequal Fractions?
Although 1/6 and 1/4 can be transformed into equivalent fractions with the same common denominator, so that 1/6 is equal to 2/12 and 1/4 is equal to 3/12, 1/6 is not equal to 1/4.  That can be shown clearly because 2/12 and 3/12 are not equal.  The properties of equal fractions and unequal fractions together also illustrate the density of the number line, because there are an infinite number of fractions on the number line.

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Fractions to Decimals and Decimals to Fractions

150 150 Deborah

Fractions can be changed to decimals and decimals to fractions in a few simple steps. The rule is that both fractions and decimals are different ways to express the same relationship between numbers, a ratio, using division and multiplication. All fractions can be written as a/b, where the two numbers have a relationship with one another.

The top number(the numerator) is part of the bottom number (the denominator). The words numerator and denominator are just math language for part and whole, or top number and bottom number. To turn the fraction into a decimal, just divide the numerator by the denominator. Thus, 2/5 is the same thing as “2 divided by 5” or 0.4, 1/2 is 1 divided by 2, or 0.5, and 3/4 is 3 divided by 4 or .75.

Some numbers don’t divide as nicely, and they have an interesting pattern of repeating decimals. Therefore , 1/3 is 1 divided by 3, or 0.33, 2/3 is 2 divided by 3 or 0.66, 1/6 is 1 divided by 6 or 0.166, and 5/6 is 5 divided by 6 or 0.833. (Notice I didn’t use 2/6, because that’s the same as 1/3, 3/6 is the same as 1/2, and 4/6 is the same as 2/3.)

Decimals fall into two different groups, terminating and repeating, and there are slightly different steps for turning each type from decimal into fraction. It’s easier to see with an example, using long division. With the fraction 3/8, 3 is divided by 8, or .375, and every remainder gets smaller until the remainder is zero. At that point, the decimal stops repeating. If that last remainder is not zero (as in 5/6), it will eventually repeat itself as in 0.833.

When the decimal doesn’t repeat, it can be expanded and solved into a fraction that way. For example, 0.4 can be expressed as 4/10 divided by 2/2 (to put it in simplified terms ) or 2/5. When the decimal does repeat itself, turn the statement into an equation in order to get rid of the repeating part.

If X = .2222, then 10X =2.222. Then you can subtract the two equations: 10x-x = 2.22-.22, 9x = 2, or x = 2/9. When 2 is divided by 9, the answer is 0.2222; full circle either way.

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Cognitive Friday: Maximizing Learning Styles

150 150 Suzanne

A learning style inventory is an assessment that shows people the modes in which they learn best. It is not an assessment in the common high-stakes sense of the word, but instead a tool to provide information to teachers and learners.  Students learn using one of three dominants modes – their visual brain, their auditory brain or their kinesthetic brain; the Learning Style Inventory can help you recognize which is your style, and which is not. The latter point may be particularly important for some students who do not do well learning in the way that their teacher wants them to learn.

Imagine a teacher in the front of the room reading from a list of 50 common words – Cold, Juice, Level, Hard, Cat and so on. As the teacher reads, students check a box in one of four columns:
1) I see the word,
2) I see an image of the word,
3) I know the sound of the word,
4) I have a physical feeling when I hear the word.  For each word, the child marks which of these four events occur for them. The first two are most common for visual learners, the third is a technique used by auditory learners, and the fourth is the natural response of someone who is kinesthetic.

Many teachers take the time at the beginning of the year to use this assessment with their whole class as a way to plan instructional approaches. If a third of the class is kinesthetic learners, the teacher knows that hands-on learning experiences will be important to incorporate into teaching.  However, the dominant learning style is visual learning, and classrooms are generally set up for that modality. This can cause real problems for students who are not primarily visual learners.

This information helps to understand what to do and what not to do while teaching children. For instance, a kinesthetic learner may have difficulties learning in an environment too full of visual stimuli. Endless posters in a classroom may provoke a range of feelings for a student, which could be highly distracting. A kinesthetic learner needs to know how to channel distractions and maximize the visual and auditory clues they receive. Likewise, a visual learner may be uncomfortable with hands-on learning and may shy away from those learning opportunities.

Even if a teacher does not do a formal learning style inventory with their students, there are ways that a parent can discern this through simple conversation. Just asking what happens in their child’s mind when they hear the word “cat” can start a conversation in which learning style – and learning obstacles – can become much better understood. Taking the time to understand how a child responds to stimuli and cues will go a long way in developing stronger study skills among all students.

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Priming and Decision-Making

150 150 Suzanne

The psychological phenomenon of priming affects a number of mental processes. Priming is like an advanced notification system, however a person is receiving notifications that affect what they think and how they feel. Priming is done through the words we hear and see, and the messages we are surrounded by. It is ubiquitous and natural and students can benefit from the messages they are primed with.

Priming can impact decision-making, and so at the points in which students are asked to make decisions, the messages they receive are critically important. When students choose electives or extracurricular activities, the messages they receive about trying something different may occasion a learning opportunity otherwise missed. When thinking about colleges, the many message students receives provide a positive influence on their college-oriented thinking.

Another way that this phenomenon can work is through creating experiences for students which will mirror future experiences that might anticipate. Such role-playing opportunities allow students a chance to make good and bad decisions and reflect on them. They will be better prepared for situations in the future insofar as understanding the implications of the decisions that they make.

One potential pitfall with this aspect of unconscious messages is the ways in which social issues and current events are treated. As students become more aware of the world they live in, the tone taken in an examination of a current event can help to shape the way in which a student will see the world. Political and social values will be created in classroom and learning experiences, and the way these ideas are framed will have an impact.

As students work their way through the transitions that school calls for – moving from elementary to middle to high school – offers many opportunities for students to make decisions that will impact their future. At each point, it is important to remember that the people and environment guiding those decisions will have a strong impact.

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