## Fractions Introduced Conceptually

Introduction: Fractions are used in many ways. Before rushing into how to do arithmetic with fractions, students should experience what fractions mean. Why do we really use fractions? How do fractions express real ideas that we use or experience? Only after the concept makes sense does the arithmetic make sense.

Purpose: To help students develop an understanding of the meaning and use of fractions from everyday experiences. Some of these lessons develop thematically with other units, some are designed to be reinforced daily.

Background: Some of these lessons were used in a 1st-2nd grade multi-age setting. The rest can be used in third and fourth grades, as well as, reinforced at higher grades.

Reformatted: January 2010

How Are Fractions Used? This series of sample exercises demonstrate the many common uses of fractions. They demonstrate what the fractions mean in the discussion. Use exercises like these, until the students can express the concept of fractions. Then move onto arithmetic. These exercises are designed to be used daily to fill in small periods of time, say, as a warm-up activity.

1. Parts of a Whole - sharing, splitting equally
 Money Activities: What part of a dollar is a penny, a dime, a quarter, 50¢, 66¢. How did the quarter get it's name? What do we call a 50¢ piece? What could we call the dime, nickel or cent? Pizza Activities: Have students discuss, and demonstrate by drawing, whether they want 1/6, 1/8, 1/12, or 1/16 of the pizza. Demonstrate by cutting the pizza and handing out the pieces.
2. Comparing Parts of a Group - "what part of...," "... out of ...."
 Calendar Activity: Each day when the date is written have a student write out what fraction of the month, week, year, or school year has passed. Advanced students may reduce the fractions or discuss why the fraction can not be reduced. Reading Activity: Have students count the letters in a sentence, and determine what fraction of the letters are e, t, or a. What fraction of the words have more than one syllable? Students in Class Activity: Each day pick a characteristic such as wearing red, bringing lunch, playing ball, girls, done their assignments, etc. Determine what fraction of the class is in this group. Money in your Pocket: Have students determine what fraction of their money is nickels. There may be two correct answers, first: the fraction of the counted number of coins, second: the fraction of the value of the coins.
3. Property of Measurement - part of inches, feet, hours, days, ....
 Clocks Activities - Hours: Have students determine what part of an hour is 1, 6, 15, 30, or 60 minutes. Point out what we call "half past," or "quarter till." Have them draw a clock and shade the area from the hour until quarter after. Clocks - Minutes: Have students determine what part of one minute is: 1, 15, 60, or 90 seconds. Compare these answers to similar question for hours. Clocks - Days: Have students determine what part of a day is: 1 hour, 2 hours, 8 hours, and 12 hours. Have students estimate what part of a day they spend in school, sleeping, eating, watching TV. Clocks - School: Have students estimate what part of the school day is spent in lunch, recess, math, or gym. Which of these is the largest? Ruler Activities: Have students draw lines that are 1/4", 1/2," 3/4", 5/8", 11/16." Have them determine which line is longer? Have students practice mixed numbers by drawing lines of length 1 1/2", 2 3/4, 5 1/8, etc. Have students draw lines 2 1/2" and 5/2" and compare their lengths. Have students explain how they did their comparisons. Have students measure items around the room. Liquid Measurement Activities: Have student use a measuring cup or a graduated cylinder. They should actually have a chance to look at the markings on the cup. What fraction of a cup is 4 oz. or 8 oz? How many cups (accurate to 1/8th) does it take to fill another container, such as a mug?
4. Comparisons in Measurement - ratio, twice as much, half as long, unit weight, speed
 Liquid Measurement: Have students determine what fraction of a quart jar is held by a drinking glass. What part of a gallon is a quart? Why do we call that part of a gallon a "quart?" Distance Measurement: Have students determine the ratio of the width of the room to the length of the room, or the width of their desk to the length of their desk, or the size of their desk to the teacher's desk. Weight: Have students determine the ratio of the weight of two different objects that they might want to carry. Have them discuss what this ratio says about strength, or workload. Weight per item: Have students determine the weight of a piece of candy by weighing a box of candy and dividing by the number of pieces. Compare this to the weight of one piece. Speed: Measure out 50ft and use a stop watch to determine how much time it takes a student to run that distance. Show them how the ratio of distance / time is their speed. Let them discuss whether a faster student will have a longer or shorter time.
5. Probability and Statistics -expected value, predict how often, is it frequent or rare
 Dice Activity: Have students roll a die many times and count how many times they rolled a 3, and how many times they rolled total. Have them guess how many 3s they will get with a different number of rolls. Have them compare their results to the ratio of the number of 3s on a die to the number of faces (1/6.) Color chips Activity: Place chips of different colors in a container. have students pick out a chip, record the color, and replace the chip many times. Have them compare the ratio of times they picked a red to the total number of picks. Have them compare this to the ratio of reds to the total number of chips in the bag.

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### Repeating many activities, like the ones above, will help students understand the uses of fractions. It will help prepare them for their higher math classes in higher grades. Activities like this will help them reason about the answers they get to arithmetic problems. By thinking about what fractions mean, they can look at their arithmetic and ask, "Does this result make sense?" "Can I take 1/2 a pizza plus 1/3 a pizza to get a total of 1/6 a pizza? No. The result is too small!"

Encourage students to notice where they observe fractions in everyday life. Encourage them to notice how fractions can be used to create ideas such as speed, unit weight, or probability. Read package labels to promote the idea of unit weight or grams per serving. Have students notice similarities in speech such as "Half as much," and "Twice as much." If half is a fraction then maybe twice (2/1) is a fraction also.

### Extra: Promoting Algebra Concepts with Fractions:

Once students have got the idea of 1 being anything divided by itself, extend the pattern to include variables and arithmetic operations. Fill in the numerator or denominator to make each fraction equal 1: With this exercise you can motivate the idea that anything, even if we don't know what it is (a variable), fits the pattern.
 1 2 3 4 345 2+2 83+12 pi b+b -- -- -- -- -- -- --- --- ---- --- ---- -- ---- ---- 1 2 5 10 1,083 x y 