Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

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.

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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.

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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.

green lineE1.  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.

MonkeyFraction1

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: MonkeyFraction2

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.BalloonFraction

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/

Engaging students: Expressing a rate of change as a percentage

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 Samantha Smith. Her topic, from Pre-Algebra: expressing a rate of change as a percentage.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

The TLC show Extreme Cheapskates follows the lives of Americans who are very frugal with their money. In this clip, a man takes his wife to the movies and does everything he can to save money. Expressing a rate of change as a percentage is most commonly associated with spending money, such as a sign in a store saying “50% off all merchandise.” Using this clip as an introduction, I can have my students practice calculating how much money they are saving on buying certain items. I can bring in a catalog and coupons and have my students “buy” 3 items and calculate how much they saved. This is a real world application that students will use for the rest of their lives. Looking back on the video, students may notice that the man had a rate of change of 100%. Instead of paying full price for the drink and popcorn, he saved 100% of his money (or paid 0%). Even though his wallet was happy, I’m sure his wife wasn’t after seeing this on TV.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Facebook is the largest social networking site on the internet. There are many high school students that constantly check their Facebook and most of them post to get attention from their followers. The article link below gives 7 pieces of advice on how to get more attention on Facebook. For example, number 1 says “Photo posts get 39% more interaction.” As I am introducing the topic of changing rates as a percentage, I can have my students try to analyze what these numbers really mean. The important thing to stress about this article is not the just the numbers themselves, but the verbs attached to the percentages such as “increasing.” This shows the rate is changing. Combining this topic and a website the students use every day is sure to grab their attention.

http://blog.bufferapp.com/7-facebook-stats-you-should-know-for-a-more-engaging-page

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C3. How has this topic appeared in the news?

I know, for myself, that I love eating fast food, and I’m sure I am not the only one. However, after New Year’s Resolutions are made, many people choose to give up the glorious taste and convenience of fast food for options that are healthier. This trend causes many fast food chains, such as McDonalds, to lose customers. As mentioned in the article below, McDonald’s guest counts have fallen 16% in the U.S. in 2013. This causes the company to make changes to attract more customers. Rates of change expressed as percentages are very common in the analysis of businesses. Students will perk up when they hear this topic because it is interesting to see how their personal diet choices effect major restaurants.

http://abcnews.go.com/Business/wireStory/mcdonalds-profit-fewer-customers-21634926

Engaging students: Probability and odds

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 Michelle Nguyen. Her topic, from Pre-Algebra: probability and odds.

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

As a teacher, I would place 100 red marbles and 25 blue marbles in a bag and have each group of students draw a marble each time from a bag for five times. After drawing a marble, the student would put the marble back and then redraw. After five times, the class would come together and the students would compare how many red marbles to how many blue marbles they have. The students will compare the ratios and guess if there are more red marbles or blue marbles in the bag given. By doing this, the students will see whether there is a big chance of drawing a red or blue marble. After doing the activities, I would ask questions that will scaffold the students into saying that there is a higher probability in picking a red marble than a blue marble because the red marbles are picked more often when compared to the blue marbles that got picked.

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

With the popularity of gambling rising in the French society, mathematical methods were needed for computing chances. A popular gambler named De Mere talked to Pascal about questions about chances. Therefore, Pascal talked to his friend Fermat and they began the study of probability. The created the method called classical approach which is the probability fractions we use today. In order to verify the results of the classical approach, Fermat and Pascal used the frequency method. During this method, one would repeat a game a large number of times with the same conditions. Bernoulli wrote a book named Ars Conjectandi in 1973 to prove the classical approach and the frequency method are consistent with another one. Later on Abraham De Moive wrote a book to provide different examples of how the classical methods can be used. As time passed by, probability moved from games of chance to scientific problems. Laplace wrote a book about the theory of probability but he only considered the classical method. After the publication of this book, many mathematicians found that the classical method was unrealistic for general use and they attempted to redefine probability in terms of the frequency method. Later on, Kolmogorov developed the first rigorous approach to probability in 1933. There are still researches going on about probability in the mathematical field of measure theory.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

In the movie “21” there is math problem that is similar to the popular Monty Hall problem. In the movie, a kid is given the chance to pick one out of three doors with a car in it in order to win. Once a door is chosen, the announcer will open a door without a car. Therefore, the start off is 33% of a car existing and 66% with an empty door. Since a door was open, the chance of switching your choices gives you a higher winning percentage because the one you chose at the beginning will still be 33% while switching will change your chances to 66%. This youtube video is a clip from the movie:

References:

http://www.math.wichita.edu/history/activities/prob-act.html#prob1

http://staff.ustc.edu.cn/~zwp/teach/Prob-Stat/A%20short%20history%20of%20probability.pdf

http://www.examiner.com/article/21-and-the-monty-hall-problem

 

 

Engaging students: Prime Factorizations

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 Michael Dixon. His topic, from Pre-Algebra: prime factorizations.

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A1. What word problems can your students do now?

One word problem that is easily relatable would be something involving food!

For instance: “Don loves peanut butter and jelly sandwiches. One day he noticed a jumbo jar of peanut butter has 72 servings and a jar of jam only has 40 servings. If he opened the [first] jars on the same day and used exactly one serving each day, how many days until he emptied a peanut butter jar and a jam jar on the same day? Use prime factorization to solve.”

Obviously, this involves finding the least common multiple of 72 and 40. I would introduce this problem at the beginning of class, after my students have already been introduced to the idea of prime factorizations. I do not expect that my students would know how to calculate the lcm using prime factorizations, rather I would want to strike up a class discussion asking students to explore what they know about factorizations and see if they can find any patterns that would lead to the solution. I want to lead them to the idea that prime factorizations make finding the lcm far easier than listing the multiples of each number, especially when large numbers are involved.

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B1. Future Curriculum

As mentioned in the previous paragraph, students can learn to use prime factorizations to calculate the greatest common factor or the least common multiple of numbers easily. To take this quite a bit further, we can introduce students to the idea of using factorizations, gcd, and lcm in formal abstract proofs. We would ask them to actually prove anything, just think about the ideas. Ask students how they know that the math that they use everyday actually works. Why does every number have a unique factorization? Why can I calculate the gcd and lcm of any two numbers, and know that that answer is the only answer? Then explain that later on, in higher level math classes, we actually flawlessly prove why our number system works, and how and why primes are important, such as in the Euler Phi function. Without prime factorizations, we would be unable to prove quite a lot of the math that we take for granted.

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E1. How can technology be used to engage students?

After your students have been working with prime factorizations for a while and they are getting more proficient, what’s an obvious escalation? Make the numbers larger! Ask your students to factor numbers like 198 and 456. See how long it takes them to work through these. Then, ask them how long it would take to factor numbers like 2756 or even 12857. How could they do these? Is it even reasonable to try? What about 51,234,587 (this is actually prime)?

Here we can introduce using a computer, and using a computer to do the calculations for us. Just a simple website is adequate to show them just how useful computers are when doing large calculations. A website such as Math is Fun is an excellent tool to demonstrate the magnitude of some prime numbers and composite numbers, and show that even as numbers get very, very large, they are not divisible by any numbers other than themselves and one.

References

www.mathsisfun.com/numbers/prime-factorization-tool.html

http://tulyn.com/wordproblems/prime_factorization-word_problem-7928.html

Engaging students: Solving one-step linear equations

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 Jessica Trevizo. Her topic, from Pre-Algebra: solving one-step linear equations.

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

            Many students have played “Around the World” at one point in their elementary childhood, or have at least heard of the game. Around the World is an activity that is commonly used by elementary school teachers when they are teaching multiplication. Students are supposed to sit in the form of a circle. One person is chosen to attempt to go around the world. He/she will stand behind a student and will compete against the student that is sitting down. Once both students are ready the teacher holds up a multiplication card. The student who responds with the correct answer first gets the chance to move on to the next person. If the student who is standing up loses then he/she gets to sit down while the other student who obtained the correct answer advances. Every person has to attempt the problem on a sheet of paper, but they are not allowed to call out the answer. The student who “goes around the world” first is the winner. If a student is not able to complete the entire circle then the student who advanced the farthest is the winner. The same idea will be used after the students have learned how to solve one step linear equations.  After having a deep conceptual understanding of the topic it is very important for the students to keep practicing problems.  Around the World allows the students to keep practicing in an entertaining way. The students should be able to solve the equations within 30 seconds since it only requires one step to solve. The ability to use calculators with this activity will vary depending on the level of difficulty of the problems as well as the teacher.

 

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B1. How can this topic be used in your students’ future courses in mathematics or science?

            Being able to solve one step linear equations is an important skill that every student should acquire. After the students learn how to solve one step linear equations they are expected to be able to solve multi-step equations, solve absolute value equations, solve inequalities, finding the side lengths of a shape given a certain area in geometry, etc. If the students are not able to master solving one step linear equations then they will have a very difficult time in other math courses.

In geometry the Pythagorean Theorem requires the skill to solve one step equations. Students are expected to solve for the missing variable in order to find the missing side length of a right triangle. In Algebra II the students are required to manipulate equations in order to solve systems of linear equations through substitution. Also this basic skill is necessary when finding the inverse of a function. This topic is also used in physics. For example, if the student is asked to find the acceleration of an object given only the force and the mass, then it involves using Newton’s second law which states Force=mass*acceleration.

 

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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This website is an amazing tool that allows the students to visualize how to solve linear equations using algebra tiles. If the teacher decides to teach this lesson using algebra tiles in the classroom, then this website will allow the students to continue to practice at home. Also, the website automatically lets the student know if he/she responded correctly. Obtaining quick results allows the student to know whether or not they truly understand how to solve the equations as opposed to having a worksheet with 50 problems for homework and not knowing if the same mistake was repeated.  Also, by using the online algebra tiles the students are able to understand the zero pair concept and see how it is being applied. This website can also be used for other algebra topics such as factoring, the distributive property, and substitution.

http://illuminations.nctm.org/Activity.aspx?id=3482