3rd Grade Standards for Fractions and Decimals: 4.N.3 Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on the number line. 4.N.4 Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 1-1/2), find equivalent fractions, mixed numbers, and decimals, and order fractions. 4.N.5 Identify and generate equivalent forms of common decimals and fractions less than one whole (halves, quarters, fifths, and tenths). 4.N.6 Exhibit an understanding of the base ten number system by reading, naming, and writing decimals between 0 and 1 up to the hundredths. Taken from 2000 Massachusetts Curriculum Frameworks
What is a fraction? What is a fraction? A fraction is part of an entire object or unit whole. One-fourth of the whole is shaded in. Two-fourths of the whole is shaded in. Three-fourths of the whole is shaded in. Here the whole is divided into four equal parts called fourths. Four-fourths are shaded in here.
A fraction is part of a collection of objects. A fraction is part of a collection of objects. This collection has 12 pieces, and four-twelfths of the pieces are yellow.
A fraction is a place on the number line. 12 A fraction is a place on the number line. 0 1 Here, there is a point at one-half.
How do fractions get their names? The bottom number of a fraction is the denominator. It tells us how many equal pieces the whole has been divided into, how many pieces are in the collection, or how many equal pieces the number line has been divided into between 0 and 1. The denominator is usually named after the number in the denominator: four becomes fourths, 12 becomes twelfths, etc. Here the numerator is 1, and the denominator is 2. When we say ½, we mean that we have one piece out of two pieces. 1 2 3 4 4 12 Here the numerator is 3, and the denominator is 4. When we say ¾, we mean that we have three pieces out of four pieces. Here the numerator is 4, and the denominator is 12. When we say 4/12, we mean that we have four pieces out of twelve pieces. The top number of a fraction is the numerator. It tells us how many pieces of that size we have.
Comparing Fractions Just like whole numbers, we can compare the size of different fractions to put them in order. Take a look at this chart. Which fractions are bigger than other fractions? Which fractions seem to line up?
Comparing Fractions with the Same Denominator When you are trying to compare fractions with the same denominator, just compare the numerator. (Remember that the numerator tells us how many pieces there are. How would you tell which one is bigger?) Whichever fraction has the larger numerator is the larger fraction. ¼ < 2/4 2/3>1/3 3/5>1/5
What happens when the fractions have different denominators? If they have the same numerator, sometimes you can just compare them based on their denominator. The larger the denominator, the smaller the pieces will be because the whole has to be cut into more pieces. ¾___3/5 Three big pieces is larger than 3 small pieces, so ¾>3/5.
Finding a Common Denominator Often times, you will have two fractions that have different numerators and different denominators. In these cases, you need to find a common denominator. You are trying to get the whole divided into the same number of pieces, so you can compare the number of pieces.
Which one is smaller? 1/2 2/5 • First list multiples of 2 and 5: • 2: 2, 4, 6, 8, 10, 12, 14… • 5: 5, 10, 15, 20… • Circle the one they have in common. • Multiply each fraction by a fraction equal to 1: 1/1, 2/2, 3/3, etc., to get the common multiple in the denominator. This is the same as multiplying the numerator and the denominator by the same number. • 1 x 5 5 2 x 2 4 • = = • 2 x 5 10 5 x 2 10 • Then compare the “new” fractions. 5/10 > 4/10 So 2/5 < 1/2. Fractions that are equal to each other are called equivalentfractions.
+ 1/3 Adding and Subtracting with Fractions Before adding and subtracting fractions, the fractions must have the same denominator. What does it mean to add fractions? You are trying to add to pieces of a whole to see how much you have. What happens if the pieces are different sizes? 1/2 These pieces don’t line up very well, so we will need to find pieces that line up with both ½ and 1/3.
Make sure your fractions have a common denominator. If they don’t have one, find one. 1/5 + 2/5—These fractions are ready to add because they have the same denominator. ½ +2/3—You need to find a common denominator before you can add these. • Then add (or subtract) across the numerators and keep the same denominator. 1+2 3 5 5 = So 1/5+2/5=3/5.
What happens when the numerator is bigger than the denominator? 3/2 A fraction with a numerator that is larger than the denominator is called an improper fraction. These are fractions that are larger than one. When might you get a fraction larger than one?
Moving from Improper Fractions to Mixed Numbers Often, the improper fraction will be easier to think about if you change it to a mixed number. A mixed number is a combination of a whole number and a proper fraction: 1 1/3, 2 ½, etc. • Subtract a fraction equal to 1 from the improper fraction: • 3/2-2/2=1/2 • Sometimes you will have to do this more than once. Keep track of how many times you subtract one because this will become your whole number in the mixed number. • Then rewrite the fraction as the whole number you subtracted with the remaining fraction beside it. • 3/2= 1 ½ 3/2=?
What is a Decimal? A decimal is like a fraction because it is a number between 0 and 1. 0.1 0.25 0.5 0.95 Decimals are often added to whole numbers by joining them to the whole number with a decimal point. 1.1 2.25 4.5 6.95
Decimals are fractions. Decimals are fractions with denominators of 10, 100, 1000, and other powers of 10. 0.1=1/10 0.25= 25/100 0.5= 5/10 0.95=95/100 To write a decimal as a fraction find the place value of the last digit of the decimal. That number will be the denominator. Then write the digits in the decimal over the denominator.
Place Value with Decimals This number would be read: One thousand two hundred thirty four and five hundred sixty seven thousandths.
Rules for Reading Decimals • Always use “and” between the whole number and the decimal to show where the decimal point is. (Never say “and” when you are reading a number without a decimal.) • The decimal always has the name of the last digit’s place value even if there are non-zero digits in the other places. For example, if the last digit is in the tenths place, the decimal is in tenths. If the last digit is in the thousandths place, the decimal is in thousandths.
Every Fraction Can Be Written As a Decimal • How do we write ½ as a decimal? • List multiples of 2 and find one that is a power of 10. • 2, 4, 6, 8, 10 • Multiply the numerator and denominator of ½ each by 5 to get the fraction in the new denominator. • 1 x 5 5 • 2 x 5 10 • Then write the numerator of the fraction in the appropriate place values in the decimal. • 5/10=0.5 Do you remember what we did to find common denominators of fractions? To write a fraction as a decimal, most of the time you can find a common denominator that is a power of 10 (10, 100, 1000, 10,000, etc.) and multiply each fraction by a fraction equivalent to 1 to get the new fraction with the new denominator. Then write the new fraction as a decimal. =
Writing Fractions as Decimals • 0.333 • 1.000 • -9 • 10 • -9 • 10 • -9 • 1 Sometimes, the denominator of a fraction will never have a multiple that is also a power of 10. This happens with 1/3. In these cases, you have to divide the numerator by the denominator to find the decimal. Writing zeros after the decimal point doesn’t, change the number, but it makes it so that we can divide 3 into what looks like 10. Act as if the numbers are whole numbers and complete the division, but remember to raise the decimal point to the answer line.
Repeating Decimals As we saw when we tried to write 1/3 as a decimal, some decimals keep repeating forever. These decimals are called repeating decimals. Because we can’t keep writing the pattern forever, we write a bar over the part of the decimal that repeats to show that it is a repeating decimal. 1/3=0.33
Summary • A fraction is part of a whole, part of a collection, and a point on a number line. • Before adding, subtracting, or comparing fractions, you must have a common denominator. • Decimals are fractions with denominators that are powers of 10: 10, 100, 1000, etc. • All fractions can be written as decimals.