01.02 Working with Fractions
- Fractions: Conceptual vs. Procedural Knowledge
- Fractions Are: Numbers
- Fractions Are Not: 2 Numbers
- Pro Tips for Problem Solving with Fractions
- Equivalent Fractions
- Identify Equivalent Fractions
- Make Fractions Equivalent
- Comparing Fraction Size
- Must be Equivalent
- Ordering Fractions
- Unit Fractions and Whole Numbers
- Lowest (Simplest) Form
- Fractions: Same Number; Different Way
- Equivalent Fractions
- Relationship of Fractions to Decimals
- Change fractions to decimals
- Change decimals to fractions
- Solve Problems with Fractions
- Adding and Subtracting – Same or Different Unit?
- Add fractions – Same Unit
- Subtract Fractions – Same Unit
- Subtract Fractions – Regrouping
- Multiplying and Dividing – Unit Not Important
- Multiply Fractions
- Divide Fractions
- Adding and Subtracting – Same or Different Unit?
- Seeing Fractions Everywhere
Cornell Note-Taking System Instructions:
- Record: During the lecture, use the note-taking column to record the lecture using telegraphic sentences.
- Questions: As soon after class as possible, formulate questions based onthe notes in the right-hand column. Writing questions helps to clarifymeanings, reveal relationships, establish continuity, and strengthenmemory. Also, the writing of questions sets up a perfect stage for exam-studying later.
- Recite: Cover the note-taking column with a sheet of paper. Then, looking at the questions or cue-words in the question and cue column only, say aloud, in your own words, the answers to the questions, facts, or ideas indicated by the cue-words.
- Reflect: Reflect on the material by asking yourself questions, for example: “What’s the significance of these facts? What principle are they based on? How can I apply them? How do they fit in with what I already know? What’s beyond them?
- Review: Spend at least ten minutes every week reviewing all your previous notes. If you do, you’ll retain a great deal for current use, as well as, for the exam.
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Hey, welcome to the course working with fractions, where I’ll clarify everything you need to know to gain confidence with fractions.
In this course, we’ll prioritize concepts over procedures. So I’ll give you a very clear definition of fractions, a ton of pro tips about fractions, and then we’ll solve example problems. It’ll be fun.
Let’s start at the beginning and understand that a fraction is a number, and that a fraction is NOT 2 numbers. So while you see 2 digits in the fraction ½, it represents 1 amount, just like 0 and 1 do. Thinking of fractions as numbers on a number line will help you to conceptualize a fractional amount. Think about where 1/2 belongs relative to 0 and 1. These are also numbers: 1/16 and 15/16, and also represent a partial amount relative to the whole.
Understand that fractions can represent the same amount, or equivalent, even when the digits in them are different. See here how each smaller line represents 1/16, then we have eight units to the midpoint. and where 1/2 is considered one step away from zero, 8/16 is considered eight steps. but they represent the same amount – so 1/2 and 8/16 are equivalent fractions. If you have a strong knowledge of multiplication tables, you ‘ll quickly see the relationship numerically between these two fractions. Change 8/16 to ½ by dividing the numerator and the denominator by 8, or change ½ to 8/16 by multiplying both by 8.
It’s important to be comfortable with this concept so fractions become second nature.
Another important tip to remember is that all fractions will be easier to manipulate if you work with the simplest form. You probably learned the word “reduce” when changing fractions to their simplest form. Why would a recipe say that I have to use 8/16 of a cup of milk when it can say I need ½, the simplest form of 8/16? You have to understand equivalent fractions in order to be able to work with the simplest form , so think of these together.
You have to understand how to compare fractions by size. When the denominator in the fractions you’re comparing is different, relative size can be misleading. It’s like comparing apples to oranges. So to order fractions correctly, you have to create equivalent fractions by giving them the same denominator. Here again, knowing your multiplication tables inside out helps make this an automatic task. When I see ½, ¾, and 5/12, I immediately think to make the common denominator 12, because 12 is divisible by 2, 4, and 12. Now I can make an equivalent fraction for each – like this (6/12, 9/12, and 5/12 remains the same.). Now you can compare the original fractions like Apples to Apples, and see that 5/12 is the smallest, 1/2 is bigger, and 3/4 is the largest.
Understanding the relative size of fractions helps you do a lot more mental math during problem solving. A unit fraction is any fraction that has 1 as a numerator, so represents the smallest unit of that fractional amount. 1/16 is a unit fraction on a measuring tape, where you will typically an inch divided into 16 equal parts. In contrast, 16/16, which would be here on the number line, represents a whole number. In this case, the whole inch. You can improve your conceptual understanding of relative fractions size by comparing the numerator to the denominator and imagining where it would go on a number line. The closer to 1 the numerator is, the smaller that number. As the numerator approaches the denominator, you can see you’re getting closer to one. In any fraction in which the numerator is exactly half the denominator, that’s always the numerical middle point. Now you can estimate with fractions by saying about half almost all or nearly none.
This tip spans two slides. The numbers you see here are not all equivalent, but they are three examples of different ways fractions can be written. You’ll want to get comfortable with each form and what they mean relative to a number line value of 1. A proper fraction is simply a number that is less than 1 – recognized because the numerator is less than the denominator. An improper fraction is a number that’s greater than 1 because you have more parts, represented by the numerator, then what constitutes one. And a mixed number just represents a whole number and a fractional part of that whole number. So 1 and ⅓ is a bit more than 1 on the number line, but still less than 1 and a half. Can you see that? Changing from an improper fraction to a mixed fraction is simply dividing the numerator by the denominator, like this. Now let’s look at another way fractional amounts can be represented.
Surprise! It’s a decimal. Did you know that fractions and decimals represent the same amount but in a different form? Many problems on tests require you to change a fraction to a decimal. To do that, you have to know the place value names that are less than one, or to the right of the decimal. Maybe you learned to say your decimals like this . . “point 5”, but I beg you to get into the habit of using the place value name of the digit on the right because that will immediately tell you the value of that number. Isn’t 5 tenths easier to visualize as 1/2 than point 5? So now, start writing every decimal amount as a fraction just by making the place value name the denominator. So 15 hundredths is this decimal – and this fraction. (15/100 = 3/20). Practice this so you start to relate relative size of decimals to fractions – allowing you to change decimals into fractions like a pro.
Now let’s take a look at problem solving using fractions. When adding and subtracting fractions, the first thing you want to notice is whether the units, or the denominators, are the same. If they are, adding or subtracting is just a matter of counting up or down the number line like with whole numbers. Adding 2/16 + 3/16 + 4/16 gives you 9/16. It’s as simple as adding the numerators and keeping the denominator. Subtracting is the same, so think about “ 10/16 – 8/16?” Right! 2/16. Typically your final answer should be stated using an equivalent fraction in its simplest form. You remember how to do that? Divide the numerator and denominator of 2/16 by 2 and you’ll get the equivalent fraction of 1/8 . Simple huh?
Don’t get all nervous about adding and subtracting fractional problems that begin with different units! Remember earlier when we created equivalent fractions? Just do that first and then perform the operation on those equivalent fractions. So for 1/2 + 3/4 find a denominator that’s divisible by both 2 and 4. Use 4. And just like you did when you ordered the fractions according to their size, now that you have 1 denominator for both you can make equivalent fractions. 1/2 becomes 2/4 and 3/4 Remains the same. Watch me do a think aloud while I subtract 1 fraction from another that represents a different unit. Remember: create equivalent fractions that share a common unit, and then think of moving up or down the number line. You’ll do great.
Just like when you have to regroup, or borrow, when subtracting whole numbers, you might have to when subtracting fractions. But there’s nothing mysterious about it if you see the fractions as numbers relative to a number line. In this problem here I have mixed fractions. I have no problem subtracting the 4 from the 6 but I can already see that 2/3 is larger than ⅓. When subtracting you cannot change the order of the numbers, so you’ll have to borrow from the whole number.. My first recommendation is to line the fractions up vertically like this. Now borrow. Making it five and take one. Since you can’t subtract numbers that are in different forms, just write an equivalent number so the forms match. Change the one to an equivalent fraction, which in this case is 3/3 . No ,choosing 3 as the denominator is not random, I chose it because the other fractions have 3 as their denominator .Now that I have borrowed 3/3 ,or 1, from the 6, I have to add it to the fraction that’s currently there. This gives me 5 + 4/3 -4 + ⅔. Now you can easily perform subtraction after regrouping.
The cool thing about multiplying fractions is that you don’t have to consider the unit or the denominator. Simply multiply across the numerators, then across the denominators, and you have your product. From 6/12 you should automatically see that 1/2 is the same value but the simplest form. Most tests will want your answer in this format. Here’s another multiplication problem, but this one makes you choose when to create the fractions in the simplest form. Something important about multiplying fractions – your product is SMALLER than the original numbers, unlike when multiplying with whole numbers. Be aware of this when checking word problems.
Dividing by fractions requires a special understanding. Typically folks teach only the shortcut to the answer, while never explaining why it actually works. So I’ll start with the concept behind the shortcut so you can see how it works, then teach you the shortcut. The rule to understand what is happening is simply 2 steps: 1) get the common denominators , then 2) divide the numerators. When you establish that the common denominator is 12, written out longhand, the problem looks like this: (3*3)/4*3) ÷ (2*4)/(3*4).You have completed step 1. When you divide across, the denominators = 1, which leaves you to finish with step 2, which here is 9 ÷8 which equals 1 and ⅛. But that’s the long way around. The shortcut, which results in the same answer, is to 1) change the sign to multiplication, then 2) invert the divisor like this. This gives you a shorter equation to solve, but results in ¾ * 3/2 = 9/8 = 1 and ⅛. The shortcut simply turn the division problem into a multiplication problem, and it’s quicker. Just be sure you understand the reason behind the shortcut to understand why it works .
I hope you feel more confident about fractions. Key points to take away: fractions are numbers (not two numbers) that are easily recognized on a number line. Being able to quickly change fractions to . equivalent forms, from improper to mixed fractions, and from decimals will make your life so much easier. Practice regrouping when subtracting fractions. Build it into your muscle memory. and finally, play with math in your head in your daily life. See fractions everywhere. That’s what’ll really turn you into a math person. So that’s it for working with fractions and as always…