Engaging students: Determining the largest fraction

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Shama Surani. Her topic, from Pre-Algebra: determining which of two fractions is largest if the denominators are unequal.

A2. How could you as a teacher create an activity or project that involves your topic?

An activity that involves students to determine which of two fractions is greatest is called Compare Fractions, which is a two-player math game found at the website http://www.education.com.  The objective of the game is to work together to determine who has created the largest fraction.  The materials necessary is a deck of cards with the face cards removed, notebook paper, and a pencil.  Below are the directions of this game:

1. Review the concepts of numerator and denominator.
2. Decide on a dealer and have him/her shuffle the cards.
3. Divide the deck evenly among the players.
4. Have the players place their cards face down in a pile in front of them.
5. To begin playing, have players turn over two cards from their respective decks and place them in front of themselves.
6. Players can then decide which card they want to be in the numerator and which card they want to be the denominator.
7. Now the players have to calculate who has the largest fraction.  There are a variety ways this can be done.  Encourage different methods in determining which fraction is larger.  One way is to multiply the numerator and denominator of each fraction by the denominator of the other fraction. For example, with the fractions 5/6 and 4/7, compute 5/6 x 7/7 = 35/42 and 4/7 x 6/6 =24/42.  The largest fraction is 35/42 so 5/6 must be greater than 4/7.
8. The player who has the largest fraction wins all of the cards played in the round.  For the instance of a tie (when the both students have equivalent fractions), split the cards evenly among the players.
9. The game is over when the players have accumulated all of the cards.
10. Have the players count their cards.  Whoever has the most cards, wins.

I believe this activity will be fun for the students because they are creating their own fractions with the cards.  Once the students are comfortable with determining which of the two fractions is greatest, the teacher can start timing the students if he/she wants to.

B2.  How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have learned how to draw a number line and determine where on the number line two natural numbers are located.  They would have known how to compare numbers or order the numbers from least to greatest or greatest to least.  Then, the students were exposed to fractions as being a part of whole, and being called rational numbers. This concept is then extended to ordering fractions with equal denominators with visual diagrams and using the number line.  In a visual illustration, the students can be exposed to two circles of the same size but divided into the same amount of sections.  For example, both circles can be divided into four equal sections, but one can have two sections filled in while the other has three sections filled in.  Students then can determine which circle is larger.  In this case, the circle with three sections filled in is larger.  Then this concept is extended to be written in fraction format where the first circle is 2/4 and the other circle is 3/4. When the students have fractions with equal denominators, they look at the numerator to see which fraction is larger or smaller.  Determining which of two fractions is greatest if the denominators are not equal extends off this previous concept.  The best way is to show the students visually how different shapes such as a square or a circle can be divided equally into different sections. For example, the first circle might be divided into four sections with three sections shaded while the other circle can be divided into eight sections with seven shaded. In fraction form, the first circle is 3/4 while the other circle is 7/8.  Here the students will notice that the denominators are different but by looking at the shaded circles, they can see that 7/8 is larger than 3/4.

E1.  How can technology be used to effectively engage students in this topic?

Technology is increasing day by day, and in many respects, technology can be the tool for aiding learning in the classrooms.   One way that technology can be used to effectively engage students in determining which of two fractions is greatest when the denominators are unequal by playing simple online games.  Since several schools are distributing i-Pads to their students, I have found an i-Pad application called “Fraction Monkeys” that the students can download for free for this lesson.  This application is a wonderful tool in demonstrating how fractions with same or different denominators are located on the number line.  The objective of this game is that a monkey with a fraction will appear on the screen.  The student will have to place the monkey correctly on the number line.  Sometimes the card the monkey holds up is in reduced form, so the student will have to think about how that reduced form relates to the number line.

For example, below is a picture of a number line with the denominator being 16.   When the student is finished placing the monkeys on the correct location, they will notice that the monkeys were placed differently depending on what fraction they received.

By providing the students with a guided worksheet, the students will be able to compare which fractions are greater and which fractions is less than the other by viewing the number line.  For example:

$\displaystyle \frac{7}{8} ~~ ? ~~ \frac{3}{4}$

The student will answer that 7/8 is greater than 3/4 since 3/4 comes before 7/8 on the number line.  I believe this activity will help the students conceptualize how to compare fractions.  In addition, in case when the student incorrectly places a monkey on the number line, a hint with little squares pops up where the student can visually see how their fraction relates to the number line.  Below is a picture demonstrating this:

Another computer game that involves comparing fractions is named “Balloon Pop Math.”  This is also a good resource to use because it shows balloons with fractions with a visual of a circle divided in equal sections.  The idea of this game is to pop the balloon with the smallest fraction to the largest fraction with different denominators.   Below is a picture from the game demonstrating the fractions 7/8 and 4/5.  The students will be able to see that 4/5 is less than 7/8 by looking at the circle so the student will pop the balloon that contains 4/5.  This game is also wonderful to use because it contains three levels.  The first level allows the students to compare two fractions.  The next one allows the student to compare three fractions, and the last level allows the students to compare four fractions.  This will be a good engagement activity to allow the students to do before teaching about how to compare which two fractions is greater than the other.

References:

http://www.fractionmonkeys.co.uk/activity/

http://www.sheppardsoftware.com/mathgames/fractions/Balloons_fractions1.htm

http://www.education.com/activity/article/capture-that-fraction/