March 2004 Tip
FRACTIONS! Oh, say the word and strong men weep! Why are we so hung up with fractions? After all, who among us did not have to share a favorite snack “equally” with a sibling or friend? We KNOW fractions, we just get a little confused with all those numbers over numbers that we use to represent fractions. Teaching fractions to your students will be matter of fact common sense, IF we start with things to divide, not numerical representations.
Primary Grades: From PK through 3rd grade, students need stuff and sets to divide into fractional parts, and not just “pies” or “pizzas” either. Be sure to include squares, other rectangles, hexagons, octagons, even an isosceles or equilateral triangle can be shared easily into halves. Use marshmallows, crackers, any food items for parts of a set. Younger students need to fold, cut, and compare sizes. And students of ALL ages should be asked constantly what fractional parts they have and how they can prove they have that fractional part.
Here’s a typical primary lesson (or any level really that is having difficulty understanding “fraction-ness”).
Distribute four 6” x 6” squares (or any shape easily divided into the fraction you are studying) to each student. Instruct the students to fold as you are folding.
Hold up one of the squares and tell the students that this square is going to be the size and shape of the whole you’ll be working with today so you are not going to fold or cut it. Label this square “Whole.”
Hold up a second square. Fold the square over once and ask, “When I open the square, how many pieces will I have?” (2) Have the students open
their squares to verify.
Cut on the fold, and ask, “What fractional part is this piece?” _ “How do you know?” We cut the square into 2 equal pieces, and this is 1 of
the pieces. Have the students cut on the fold. If you are working with 2nd graders and above, you’ll want to have them label each half.
“We’re going to label each fractional piece so that we remember later what it is. This is 1 (write the 1 numerator) out of (draw the fraction
line) 2 equal pieces (write the 2 denominator.) “Label each of your halves with the numbers that tell us what fractional part each is.” Be sure
to ask students as they are labeling, why they are using the numbers 1 and 2, and what that line means. DON’T use the terms numerator and
denominator unless your state standards specifically require younger students to use those terms. Let’s keep this simple. Notice also, however,
that I did NOT tell them “the number on the top” and “the number on the bottom.” We’re simply labeling how many pieces we are talking about (1)
out of the total number of equal pieces (2).
Ask the students what _ and _ make? (1 whole) So you are saying that 2 halves make 1 whole. (For older students, show how that is written _ + _ = whole. And we could write the whole another way as 2/2 or 2 halves .)
Hold up a third square, and fold over once. Ask students how many equal pieces there will be (2). And what fractional part of the whole will each piece be ( _ ). Then fold the folded square over again. Ask how many equal pieces there will be now? (4) What will each fractional part be named? ( _ ). And how do they know that each will be _ of the whole? (There are 4 equal pieces, and that will be 1 of them).
Cut and label your pieces. As you are cutting, ask students to tell you what fractional piece you are cutting off ( _ ), and how many parts are still to be cut ( _ ). How do you know? (I cut off 1 of the fourths, and there are 3 of the fourths left.) So, 4/4 _ = _ . (Repeat this type of thinking for the rest of the cuts. Also, add back fourths so that students see the sense of adding and subtracting like denominators.
Continue in this fashion for eighths and, if you wish, sixteenths.
You can use this manipulative kit to compare and order fractions, to add and subtract like denominators, and to find equivalent fractions. Just keep it simple! Marilyn Burns has a wonderful Fraction Kit set of activities including games. Check out her website for more http://www.mathsolutions.com/.
Intermediate Grades: Once students understand “fraction-ness” and you’ve given them an intuitive sense of what it means to compare, order, find equivalence and combine/separate, THEN it’s time to move into symbol manipulations. But if your students do not have the prerequisite understanding of what fractions are all about, you are not only “spitting in the wind,” you are also contributing to the misunderstanding of fractions and their relatives, decimals, ratios and percents. PLEASE take the time to teach with stuff, not just pie pictures or worse, symbols.
OK, assuming students understand the concept of fractions, then have them draw pictures of fractions (and remember that fractions can be part of a SET as well!).
Remember, though, these are pictures that STUDENTS create - not pictures in a book or on a worksheet. The students must construct their own understanding. And while we’re at it, try to frame your questions in problem solving situations.
Marty ate _ of a pizza for dinner and Carolyn ate _ of the same-sized pizza. Prove who ate more.
Alex wanted _ of the salami log, but the log was cut into 16 equal pieces. How many pieces should Alex take to equal _ of the log?
Middle Grades: Students in the middle grades must add and subtract unlike denominators and ultimately multiply and divide fractions. Once again, it is VERY important that students manipulate fraction pieces or at the least, draw their own pictures of what is happening. And, of course, this age student needs a thorough understand of fractions to grasp ratio and percents - BIG IDEAS in the middle grades!
I know what you are saying about teaching multiplication of fractions, “Just teach them to multiply straight across - how simple!” or for division, “Ours is not to reason why. Just invert and multiply!” Of course! But now really, does multiplication make any sense when you teach those short cuts. When you multiply whole numbers, the product is larger than either of the 2 factors, but when you multiply fractions, the product is smaller. Hmm, wonder why that is? Or Division - the quotient is always smaller than the dividend in whole numbers; but for fractions, the quotient is larger! What a mystery!
Using grids can be very helpful for multiplication. Let’s take a look.
Martha wanted _ of the piece of cake that was left on the cake plate. If the cake had been cut originally into 8 pieces, what fractional part of the cake will she be taking?
This is a multiplication problem: _ of 1/8 or _ X 1/8.
Let’s make a grid that shows 8 pieces to represent the cake. Color in 1 piece, because that is what was left on the cake plate.
Then take _ of that piece. Of course, if you take _ of that piece, you need to take _ of ALL of the pieces of the whole to see what the fractional part would be.
So _ x 1/8 = 1/16
By doing enough of these problems, students see the pattern of multiplying the numerators over the denominators.
When you introduce division of fractions in word problems, it’s easy to see why the quotient is larger than the dividend.
Martha needed 1/8 yard pieces of fabric. If she has a _ yard piece of fabric, how many eighth yard pieces can she cut?
In division, whether you are dividing whole number of fractions, you are trying to see how many of the divisors there are in the dividend. 8 ÷ 2 really means how many 2s are there in 8. There are 4 twos in 8.
In the problem above, you are trying to see how many yard pieces there are in 1/8 _ yard. Use a picture. First, divide the rectangle into half and mark the _ yard.
Next divide the yard into eighths. How many eighths are in the half yard? (4)
So, _ ÷ 1/8 = 4. There are 4 eighths in that half. It takes many, many examples of working with word problems and models to see the pattern of division of fractions; but it’s worth the effort. Students see WHY the quotient is larger than the dividend - it makes sense.
Just like our initial experiences with primary fractions, every new step in fraction concepts needs to be built using manipulatives, models or student-drawn pictures.
Wishing you and your students loads of fun with fractions!
Ms Fritzie
