Fractions Mortensen Math http://crewtonramoneshouseofmath.blogspot.com/2014/07/base-ten-blocks-for-fractions-success.html When working with fractions, start with small denominators-keep the denominators < 10. Fraction - from Latin fractus - a breaking, We mean breaking something into equal parts. Numerator - how many parts we are talking about, how many do we count Denominator - how many parts each thing is broken into - tells us what we are counting. Numbers have two parts the "how many" part and the "what kind" part. With fractions the numerator tells how many and the denominator (de nome of it) tells what kind. - Crewton Ramone When the numerator and denominator are the same, the fraction represents ONE whole thing. We can only add, subtract, or divide fractions when they have the same denominator. (Concept #2 - SAME) The concept of the whole or one thing can be confusing. The whole can be one pizza, one bag of candy, or six boxes of cookies. A fraction can represent parts of any whole thing. Fractions are taught as portions of a whole, but are also division and ratios. All fractions represent division ( ) problems. The numerator can be divided by the denominator. Proper Fraction - the numerator is less than, or equal to, the denominator Improper Fraction - the numerator is greater than the denominator. Represents a number greater than one. Mixed Number - a whole number alongside a fraction (3½) Equivalent Fraction - fractions are said to be equivalent if the ratios are equal, Fractions that represent the same amount and have the same value Reduced Fraction - a fraction written in its lowest terms (simplest ratio) - The numerator and denominator have be divided by their highest common factor. Reciprocal - the number we multiply by to get a product of 1, - In fractions, the reciprocal is the inverse of the original fraction. The reciprocal of 3/4 is 4/3. The reciprocal of ½ is 2. (Concept #5) Common - when discussing fractions take a moment to define common as shared In order to add, subtract, or divide fractions they have to have a common denominator. (Some say same. ) Least Common Denominator (LCD) - The smallest shared multiple of each of the denominators. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 1
Which is Bigger? Play the Ratio Game before discussing fractions. This can be used to work on skip counting, multiplication, division, area, perimeter, and probably other things I haven t thought of. Adding Fractions with Common Denominators If the denominators are the same, all the math occurs above the fraction line. The denominator tells you what you are counting. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 2
Adding & Subtracting Fractions with Different Denominators Photo from Crewton Ramone s House of Math We can only add or subtract fractions when they have the same denominator. (Concept #2 - SAME) Use the Ratio Game to make Equivalent Fractions with Common Denominators. Add blocks in the same ratios until the denominator is the same. (The name of the game is to make the bottoms the same!) Whatever you add to the bottom, add to the top. (Another way of looking at this step is to think about multiplying by different ones in sequence. 2/2, 3/3, 4/4, etc.) Always add to the side with the smaller denominator. It really works! kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 3
Photo from Crewton Ramone s House of Math Now put the number of times you moved over above the numerator and below the denominator. -Side note: You have just multiplied by 3/3 and 4/4, or both fractions by 1. - Photo from Crewton Ramone s House of Math Put the total number of blocks into the square and add the numerators. Once we have the same denominator, the denominator is just along for the ride. It tells us what we are counting, not how many. Photo from Crewton Ramone s House of Math kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 4
Multiplying Fractions Remember multiplication is across, then up. ½ X ¼ is Across one of two, and Up one of four. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 5
The shortcut to remember when multiplying fractions is to just multiply the numbers that are across from each other - numerator times numerator and denominator times denominator. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 6
Dividing Fractions Start with using bars. Photo from Yes, But Why? By Ed Southall This doesn t always work out evenly. To avoid partial fractions, find common denominators before dividing. Short video showing fraction division - https://youtu.be/7twklvxfb8s The shortcut when dividing fractions is to Flip, then Multiply. This means invert the second fraction, then multiply the two fractions normally. The reason this works is division of fractions is a complex fraction, one fraction over another. If you multiply both the numerator and the denominator by the reciprocal of the denominator (the denominator flipped), you end up with a denominator of one. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 7
Decimals I t is helpful to review place value, multiplying by 10, and dividing by 10 before introducing decimals. Decimal comes from the Latin, decma meaning a tenth part. We use a decimal point to show where whole numbers end and fractional parts begin. Decimals show numbers between integers with decimal places. (Concept #1) Photo from mathsisfun.com It is helpful to point out the Ones place is the fulcrum, not the decimal point. The decimal point is just telling us everything after it is less than one. Students have to use their imagination to see the red 100 square as one unit. Use tens to demonstrate dividing one unit into ten equal parts. (1/10=.1) Then, use ones to divide each 1/10 into ten equal parts. (1/100=.01) Tenths Hundredths The last place tells you how to say the value. 0.3 = three tenths 4.72 = four & 72 onehundreths 54.295 = 54 & 295 onethousandths kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 8
It might be helpful to use the blocks and symbols to show that decimals, fractions, and percents are just different ways of naming and writing the same value. The symbols for each name look and sound different but are for the most part the same. 35/100=.35=35%. * Photo from private Facebook group Mortensen Math sells blocks to demonstrate decimals in the Basic Operations Kit. Percentages Review the roots - per and cent. Start with a hundred square and different values. Write these as fractions and %s. Review dividing by 100 and converting percentages to decimals. Use a hundred square to set up problems. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 9
Here is one way to think about it. Percent = % = out of a hundred. If we have a fraction like 3/4 and we want to convert it to percent we need to change the fraction of the denominator to "out of one hundred". ¾ =?/100. To make these two fractions equal we need to find what we multiply 4 by to get to 100 then multiply 3 by this number. Or we could say 3(100/4). We multiply 3 by 25 to get 75. Now we have 75 out of 100 = 75/100 = 75%. There is a small leap from there to decimals because 75/100 is the same as 7 tenths and five hundredths or 75 hundredths because 100 is our 1 so we can easily convert that to.75 in decimal form *. Solving for numbers that are not multiples of ten requires four steps. * Text within quotes on pages 7 & 9 are by Anna Tarnowski, taken from a private Facebook group. Anna is a Mortensen Math tutor whose web page is Anna s Math Page. All photos without a citation are personal photos from Kirk s Tutoring. kirkstutoring@gmail.com (404) 435-2493 Kirkstutoring.com 10