Engaging students: Expressing probability as a fraction and 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 Jenna Sieling. Her topic, from probability: expressing a probability as a fraction and as a percentage.

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

This topic is something that can really be applied in many places. Especially in sports, weather, and economics, probabilities as fractions and percentages are used daily. This can become very relatable to high school students no matter what they are interested in or plan to study in college. An activity that can be used in the classroom is starting a fake fantasy football league. Although I have never played in a fantasy football league, I know that to win in your group you need to look at the statistics of each player doing well. Given a class of hopefully around 30 students, we can start a week long activity of our own fantasy football league in the classroom and the students can be given different statistics each day to calculate the probability of their players being a good advantage for their team. This is just one activity that could catch the interest of students who may not usually be interested in probabilities.

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

One of the most popular majors for young students to fall into is business and probabilities become an important concept to understand if you plan to work in the business world. By making this point to a class, I feel the students will take the importance of this subject to heart. Business is not the only future path that would be using probabilities in the form of fractions or percentages. Fields like meteorology, economics, and even education majors would use the concept of probabilities to help teach elementary school students the basics to help them further on. If a student goes on to study history, at one point he or she will have to look at the economic history and understand the probability of these events happening and the probability of them happening again. The student would need to know how to multiply integers by fractions or percentages to gain conceptual knowledge of probability and its use.

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

I googled different online games to use for probability games and the most useful games, I found from Mathwire.com. Most games on this website were dice-based probability games but I think these are fun, easy games that could be assigned as homework. One game on the website was a game named SKUNK. The aim of the game is to guess the probability that a pair a dice will give you the highest amount of points. Each letter in the name SKUNK counts as one round and at the end of all the rounds, the person with the highest amount of points wins. Each player has to roll the dice once within one round and calculate the probability of getting the highest amount on each round. After looking at this game and others on this website, I realized that I could also explain the probability you need to understand to play poker if it was a popular game between friends and family. I could easily find a website to create a mock poker game and show students the idea of probability within poker.

Engaging students: Order of operations

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 comes from my former student Megan Termini. Her topic, from Pre-Algebra: order of operations.

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

The order of operations appears in pop culture in many different ways. An example is the song “Cupid Shuffle” by Cupid. There are certain steps that you do in a specific order. If you do not follow the order, then it is no longer the cupid shuffle. An activity would be incorporating the order of operations into the “Cupid Shuffle”. For example, the chorus is,

“Parentheses, Parentheses, Parentheses, Parentheses,

Exponents, Exponents, Exponents, Exponents,

Now Mult. or Div., Now Mult. or Div.

There are certain dance moves to go along with each step in the song. Here is a video of some students doing the song and dance (Reference A). This is a very effective way of teaching the students the order of operations(PEMDAS) because many students love music and dancing, and they are more likely to remember the song and dance moves than just memorizing the order itself.

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

There are tons of activities that you could do that involve the order of operations. As the teacher, you would want to create an activity that is fun and engaging for the students. Something that involves everyone in the class and not just a few students. One activity that would-be fun is Order of Operations War. Many students love playing the card game war. Now it is the same game just involving the order of operations. Each student will get a deck of cards and evenly deal them. Then they will get note cards with each of the operations on it. They will each flip 3 cards, arrange them with the operations and try to get as close to the target number as they can. The person who gets the closest is the winner of the round. This game would be a great way of getting all the students involved and a good way of learning the order of operations. (Reference B)

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

Learning the order of operations is very important for the students to learn, especially for their future courses in mathematics or science. The order of operations is used is almost every mathematics course from then on and most of the science courses. That is why is it very important to understand how it works. You know that you will use them in math and science course, but also you will use the idea of order of operations in computer sciences courses. When programming, the code has to be in a specific order to work. Just like a math problem, if you don’t apply the operations in the correct order, then you won’t get the correct answer.

References:

A. (2014, March 11). Retrieved September 01, 2017, from https://www.youtube.com/watch?v=EfgtWthLvk4

B. Order of Operations War With Just A Deck of Cards. (n.d.). Retrieved September 01, 2017, from http://us9.campaign-archive2.com/?u=3c5f5b9960a466398eccb35f8&id=cf58289e69&e=c87fd3cb28

Engaging students: Laws of Exponents

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 comes from my former student Lyndi Mays. Her topic, from Pre-Algebra: the Laws of Exponents (with integer exponents) While thinking about different activities that I could do with Laws of Exponents I decided to try making a bingo card. I like this idea because it’s a way for students practice on different problems while playing a game. The way I have it set up to use in a classroom, I have questions that I would ask. One example is . I would put this up on the board and the student has to solve it and see if they have the answer on their card. I would tell the students what the answers were until after we were done with the activity so that they’re not just waiting to hear the answer instead of doing the work. If a student got a “bingo” then I would check their answers and if they got them all right then I would have an incentive like 5 extra points on a homework assignment of their choice or something along those lines.

So, if I wrote on the board the equations $x^4(x)$, $x^0 y^5$, $(2x^2-3y^5)^0$, and $x^5 y^{-2}$ . If a student received this card, then on these questions they would get a “bingo” on the descending diagonal from left to right. You’ll also notice that I included some wrong answers in a few of the spots. Hopefully the students would notice they were not all the way simplified and would know they couldn’t use those.

Students can use Laws of Exponents to help them understand Laws of Logarithms. They will use the Laws of Exponents throughout Calculus courses when taking the derivatives or integrals of different problems. It’s important for students to understand these laws so that they can simplify problems and use them to their advantage. One example is when the student is asked to solve $\int x^{-4} \, dx$. If the student has a good understanding of the Laws of Exponents, then their first reaction will be to change it to $\int dx/x^4 = -1/3 x^3 + C$. Having this understanding is necessary for this problem and helps when students already know the Laws of Exponents so that they’re not having to learn extra material basically.

Archimedes is the one that discovered the Laws of Exponents. He did this by breaking everything down as much as possible. To show an example,

$3^4 \times 3^2$ = (3×3×3×3) (3×3)  We can do this just by know the definition of exponents

= 3×3×3×3×3×3     Once we remove the parentheses we see we’re just multiplying 3 together 6 times.

= $3^6$                         This is just the definition of exponents again

Teaching the students the Laws of Exponents this way can show them how a mathematician discovers all these rules that we follow and gives them a better understanding of the laws. Opening up this interest might help the students become more interested in math. Another example that I would show students would be $y^5/y^3$. From here I would show the students that we could break it down to $(y \times y \times y \times y \times y)/(y \times y \times y)$. Hopefully, then the students would see that you could divide and get rid of the denominator, $y×y=y^2$, and this is why it is ok to subtract when a term with an exponent is being divided by something with the same base. This is also a really good way to show students why they can NOT use these laws when they’re working with terms with different bases.

References:

Exponentiation. (2017, September 1). In Wikipedia, The Free Encyclopedia. Retrieved

23:05, September 1, 2017, from https://en.wikipedia.org/w/index.php?title=Exponentiation&oldid=798388543

When the Professor Won’t Bump Up Your 39% to an A

I don’t know about you, but this happens in my office every semester:

Engaging students: Dividing fractions

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 comes from my former student Kerryana Medlin. Her topic, from Pre-Algebra: dividing fractions.

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

One of the more practical uses of dividing fractions is cooking. Anybody who has baked in the past will know that sometimes one does not possess the proper measuring cup for the job and that they have to crunch some numbers. (This happens a lot when in college.)

The basic idea behind the activity is to ask the students to follow a recipe using a 1/3 cup measuring cup and a teaspoon. This will also allow them to practice dividing whole numbers by fractions, which strengthens to concept as well. They will be reminded that a whole number can be expressed as the number over one.

The ingredient list would be as follows:

Treats:

5-6 cups of rice cereal

1 cup of marshmallow fluff

1/3 cup of sprinkles

Buttercream:

½ cup unsalted butter

1 ½ cups powdered sugar

1 ½ teaspoons of vanilla extract

1-3 teaspoons of milk

They would be asked to figure out how many 1/3 cups each component would take. This would also help the students to use the skill of adding fractions (1 and ½ being 3/2) before dividing. The recipe would ultimately make rice cereal treats with icing on top (enough for the entire class). This is envisioned as an activity in which the students work either individually or in small groups to do the calculations and then come together as a class to provide answers and give me the proper amount of ingredients to put into the recipe.

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

Dividing fractions involves prior knowledge from fractions, generally. If dividing by flipping the dividend and then multiplying the resulting two fractions, the student must use their knowledge of multiplication of fractions and inverses, assuming that they have learned anything about inverses at this point. If the student is taught to find the greatest common denominator first, then they will use their knowledge of greatest common denominators and basic division to find the quotient. They will also be reminded of the concept of whole numbers being expressed as fractions in this topic.

How did people’s conception of this topic change over time?

Originally, division of fractions would have been thought of in terms of practical use only and was likely conceptual since the symbolism of fractions was not the clearest. An example of fraction systems that were more difficult to comprehend, would be the Egyptian system, since they would add together unit fractions to represent non-unit fractions, unless it was fraction that had a repeating unit fraction, such as 2/7 = 1/7 + 1/7 (Weisstein). When symbols became clear, the division was done by taking the fractions, finding their common denominator, then dividing the numerators and denominators, leaving the quotient. The Babylonians mostly used the method of taking the inverse of the divisor and then multiplying by the dividend (O’Connor and Robertson, 2000). This is still a popular method. Today we can do either, but some believe that doing this operation algebraically might be better for students because thinking about division of fractions in only a practical sense will stifle their imagination (Ahia and Fredua-Kwarteng, 2006).

References:

Ahia, Francis and Fredua-Kwarteng, E.. (2006) Understanding Division of Fractions: An Alternative View.

O’Connor, J. and Robertson E.. (2000). An overview of Babylonian mathematics. Retrieved from

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html

http://mathworld.wolfram.com/EgyptianFraction.html

Pi vs. Pie

Courtesy Bedtime Math:

Engaging students: Ratios and rates of change

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 Kelly Bui. Her topic, from Algebra: ratios and rates of change.

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

The activity I created would involve having the entire class make Rice Krispies treats as either groups or partners. The recipe I linked below calls for 6 cups of Rice Krispies, but for the sake of the activity, each table will receive 1 ½ cups of the cereal. Every table will receive the original recipe and determine how many large marshmallows they will need and how many tablespoons of butter they will need for the recipe to be modified to using only 1 ½ cups of cereal. This activity will allow students to use the ratios to convert measurements, such as 40 marshmallows / 6 cups of Rice Krispies. After a group finishes their calculations and finds the ratio of each ingredient in respect to the amount of cereal, they can begin making their Rice Krispies treats. To extend this to a project, the Rice Krispies Treats activity can be done in class and students will be assigned to find a recipe which involves either using the ratio to create a smaller or larger serving of the recipe.

How can this topic be used in your students’ future courses in mathematics or sciences?

Ratios are introduced in middle school when we compare a part to another other part or part to whole. Students also begin to grasp that 12 inches / 1 foot is a relationship between two quantities because there are 12 inches per foot. We also see the use of ratios in high school chemistry when converting units. A simple ratio we first learn is that density is the ratio of mass to volume. This can then be extended, for example, when students begin to solve for the number of moles of an element given its mass in grams. Before teaching a chemistry class that 1 mole = 6.022×10^23, instructors could begin with simple conversions of the length of a state in miles and converting that length into inches. Once students understand the process and the concept that we are taking one unit and converting it to another unit, it will be easier to apply it to more complex situations in chemistry.

As a class, to get into the process of using ratios to convert units, the students can make their own conversion ratios with different objects to model this relationship. For instance, 4 fire extinguishers are the length 1 lab table and 8 lab tables are the length 1 school bus, and based on these ratios, students must find the length of a school bus in terms of fire extinguishers. This activity will allow the students to use objects they see every day and create a relationship among them.

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

Shark Tank is a show which involves 5 or 6 sharks (investors) and entrepreneurs that go into the “shark tank” to pitch their ideas seeking one or more partners who will invest in their business. Most entrepreneurs seek a money amount for an amount of stake in their company or business. If the entrepreneurs are lucky, they will get a deal with one or more of the sharks. In the video below, Aaron Krause pitches his product, the “Scrub Daddy” in which he asks for a $100,000 investment for a 10% equity in his company. We see the topic of ratios appear in this business-related show because 10% equity of$100,000 means he values his company at $1 million, in other words our ratio is 10% /$100,000. This ratio can be used to find the value of the company at 100%. In addition, the sharks also like to know the breakdown of the cost per unit. In this video, Mr. Krause states that it takes $1.00 to create a scrub daddy and he sells it for$2.80 wholesale. This gives the sharks the knowledge of how much they would earn for 1 Scrub Daddy. Given the sharks are willing to negotiate, like in the video, Lori gets 20% equity of the company. For each $2.80 / 1 Scrub Daddy, she will earn$0.56.

Engaging students: Adding a mixture of positive and negative numbers

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 comes from my former student Rachel Delflache. Her topic, from Pre-Algebra: adding a mixture of positive and negative numbers.

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

An activity that could be helpful for engaging students in engaging in addition and subtraction would be Snakes and Humans. The activity is done using red and black counting tiles—the red representing humans (positive integers) and the black representing snakes (negative Integers). The activity begins by letting the students know that whenever one snake meets one human they leave together (i.e. cancel each other out). After the introduction is given, a series of addition problems can be given. After the student are comfortable with the addition problems, more challenging problems can be given such as 5- (-3) or 5 humans minus 3 snakes. From this point, the students have to figure out how to take away three snakes when they are only given 5 humans to begin with. The trick is that they have to add three human/snake pairs to the original group of humans before they can take away the three snakes, which results in them ending with 8 humans. This activity is beneficial in engaging students because it allows them to explore addition and subtraction of negative and positive integers without the anxiety that seeing traditional math problems may cause students.

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

This topic builds on students prior understanding of addition and subtraction of positive integers. Adding a negative integer can be introduced as subtracting a positive integer, which is something students should already be comfortable with. By equating it to something the students already know, it allows the students to have more confidence in their abilities going into the lesson After the students have mastered adding a negative number, the lesson would be able to move onto subtracting a negative number, a more unfamiliar topic to the students. For this part of the lesson, an activity like the one above could be use to allow the students to discover that subtracting a negative integer is the same as adding a positive integer and why. The benefit to building on a procedure that the students are already comfortable with is that it allows the students to be more comfortable going into the lesson.

How can technology be used to engage students with this topic?

One website that can be used to help engage students is http://www.coolmath-games.com/. While this website does not have instructional aspects, it does have games that are centered around math. One such game was Sum Points, in which the player tries to make the total points on the board equal to zero by adding and subtracting different numbers. The benefit of this website is that it allows students to sharpen their abilities in adding and subtracting integers without feeling like they are doing math. Students tend to enjoy using computers, and playing games on the computer tends to be a favorite for students. This tool gives them the pleasure of playing on the internet, while also allowing them to stay on task with learning.