# 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.

# Engaging students: Solving one-step algebra problems

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 Deetria Bowser. Her topic, from Algebra: solving one-step algebra problems.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To create a successful word problem that would both interest, and engage students, the teacher must “know his class.” Knowing one’s class involves knowing the many different students your students have. For example, if one knows that there are a lot of baseball players in the classroom, then creating word problems that involve baseball would be engaging for these students.

Additionally, to benefit all students you could do problems that involve finances. Including more “finance problems” will help students realize the importance of math, and how they can apply it in everyday life. An example of such problem would be “Damon’s earnings for four weeks from a part time are shown in the table. Assume his earnings vary directly with the number of hours worked. Damon has been offered a job that will pay him \$7.35 per hour worked. Which job is better pay (Tucker, A.)? Including word problems that students can relate to now or in the future can help students stay engaged while learning, and answer the question that is most commonly asked by students: “When will I ever use this in real life?”

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

As a teacher, creating engaging activities and/or projects can prove to be quite difficult for word problems that are one- or two-step algebra problems, due to the fact that most students completely shut down once a word problem is presented to them. To combat this I have found that making it into a cooperative game can help soothe the anxiety caused by word problems. One game that is great to play with one or two step algebra problems is called rally coach. In this game, students are paired off. Student A is expected to work on solving the problem, while Student B is expected to watch, listen, check, and praise just as a coach would. Once the students think they have the correct answer, they will raise their hand so that the teacher may check it. If they get the answer correct, then the teacher will give them another problem (this time Student A and Student B switch roles). If the answer is incorrect, they must continue working on the problem. The end goal of the game is to answer as many questions as possible before time runs out. By playing this game students are able to help each other solve one or two step word problems.

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

In future courses many problems will involve one or two step algebra problems. For instance, in science courses like chemistry and physics, one will need to know how to solve for different variables of equations. For example, if one is in a chemistry course and is given a word problem (i.e If a 3.1g ring is heated using 10.0 calories, its temperature rises 17.9°C. Calculate the specific heat capacity of the ring) that provides heat energy (Q) mass of a substance (m) and change in temperature (deltaT), but is asked to solve for the specific heat, students will need to know how to solve for the specific heat either by isolating the variable in the beginning (Cp=Q/mdeltaT) or plugging in the givens and isolating the variable (Daniell, B).

References

Daniell, B. (n.d.). Energy Slides 3 [Powerpoint that contains Specific Heat problem].

Tucker, A. (2016). Direct Variation. Retrieved September 01, 2017, from

http://www.showme.com/sh/?h=PQvPbm4

# Engaging students: Absolute value

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 Deanna Cravens. Her topic, from Pre-Algebra: absolute value.

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

A great way to teach absolute value is to do a discovery activity. A blogger and teacher, Rachel, posted on her blog, called Idea Galaxy, a great step by step on how to do a discovery activity for absolute value of integers. First the students will start out by showing the distance between two numbers on a number line, such as the distance between one and three.

They will do a few of these examples to build upon the prior knowledge of the students. Then the class will transition to another page. This one will also have number lines and will ask them problems like ‘what does negative four and four have in common?’ Some scaffolding can also be used like asking them to mark both numbers on the number line and look for similarities related to distance. After completion, students will discuss with one another about the observations they noticed. Lastly, the teacher will give them the term of absolute value and then ask students to rewrite it and put it into their own words.

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 short video YouTube video discusses absolute value and then explains one standard way that absolute value is used in real world applications. First it explains absolute value in terms of distance away from zero. It gives a few concrete examples to display, for instance -4 and 4 both have a distance from zero that is 4. So the absolute value bars will always make the number positive. Next, the video uses an example that shows a real world example. It shows a student, Lucy, who is traveling to go to a tuba lesson. She accidentally drops her sheet music and has to go back to get it. This video does a great job of showing what it would the distance would be in terms of number of blocks walked, and how far she is from where she started or her displacement. This can easily be shown at the beginning of class either as an introduction or a review. It can spark more discussion by asking for other real world examples to help show that math really is relevant and needed for every day use.

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

Absolute value can show up in many areas of future math classes. It comes up when learning about the absolute value function, working with inequalities, proofs and so much more. One specific way that absolute value is used, is in calculus. After students have learned how to take derivatives, they will learn how to take antiderivatives. If a student is given ∫1/x dx, they need to find the antiderivative. Students will know that the derivative of ln x is 1/x, however this is not the case when you take the antiderivative of 1/x. The domain of 1/x is everything except zero, so negative numbers must be taken into consideration. However, if one was to say the antiderivative is lnx, it only accounts for positive numbers. Thus, in order to make the domain match 1/x, the absolute value must be brought in. Therefore, the ∫1/x dx = ln|x|+c. Thus a very basic concept becomes for important within calculations at higher level mathematics.

# Repunit prime

In the United States, today is abbreviated 10/31. Define the $n$th repunit number as

$R_n = \frac{10^n-1}{9} = 1111\dots1$,

a base-10 number consisting of $n$ consecutive 1s. For example,

$R_1 = 1$

$R_2 = 11$

$R_3 = 111$

$R_4 = 1,111$,

and so on.

It turns out that $R_{1031}$ is the largest known prime repunit number.

# Engaging students: Independent and dependent events

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 Danielle Pope. Her topic, from Probability: independent and dependent events.

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

Students use the idea of independent and dependent events in their lives without even realizing it. Many of the word problems used to introduce probability are basic concepts that students can understand. The basic definition of an independent event is “the probability that one event occurs in no way affects the probability of the other event occurring”. Word problems can be used to demonstrate this. Asking if the probability of flipping a coin changes if you were to roll a die as well is a prime example. These two acts are something that can be easily implemented in the classroom and the technical definition can be taught. Students can then help come up with more scenarios and teach themselves the terms. Similarly this idea can be used for dependent variables with a few changes. If the “probability of one event occurring influences the likelihood of the other event” then the event is defined as dependent. Word problems could be “if you were to draw two cards from a deck of 52 cards and if on your first draw you had an ace and you put that aside, would the probability of drawing another ace change? This card questions could be more challenging by taking out more cards each time.

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

The topic of independent and dependent events can later be translated into variables when used with functions in Algebra class. Knowing and understanding the difference will help students know how to classify an event and use the correct variable and axis if asked to sketch a list of data. Just in a probability course students will learn about conditional probability, which will use the idea of dependence. Other terminology like with replacement and without replacement will be used to define a dependent event in probability. This topic can even be translated into a physics classroom when talking about time, position, velocity, acceleration etc. For example, when calculating the velocity students will either find or be given the displacement and change in time. Not knowing that the dependent event divides the independent variable or specifically with velocity, displacement divides time, If those numbers are not plugged in correctly then that will lead to the wrong answer.

How has this topic appeared in the news?

In the news, independent and dependent events show up everyday. The most common example is weather. One of the longest debates that we have been having is if global warming and climate change has influenced the world. According to an article in the Smithsonian Magazine, scientists “couldn’t prove that global warming had “caused” the heat wave of 2003, (they) did assert that warming from human emissions had doubled the risk of extreme weather events.” This observation can then be taken to a student’s science class and they can research the risk of continuing this pattern of damage to Earth. Natural disasters can also have a say in many events around the world. For example just recently “Gas prices spiked in the Baltimore-area — and nationwide — in recent days and are expected to continue to rise after a major pipeline that runs from Texas to the East Coast had to be shut down following Hurricane Harvey”. This is a prime example of a dependent event. This shortage in gas specifically in Texas then led to many people rushing to fill up their cars, resulting in gas stations running out of gas, which is just another example of dependent events.

References:

https://www.wyzant.com/resources/lessons/math/statistics_and_probability/probability/further_concepts_in_probability

http://www.smithsonianmag.com/science-nature/does-climate-change-cause-extreme-weather-events-180964506/

# Engaging students: Finding 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 comes from my former student Brittnee Lein. Her topic, from Pre-Algebra: finding prime factorizations.

• How has this topic appeared in the news?

Prime factorization is key to protecting many aspects of modern convenience. The Fundamental Theorem of Arithmetic states that every number can be broken down into a sum of two prime numbers. For relatively small numbers, this is no big deal; but for very large numbers, not even computers can easily break these down. Many online security systems rely on this principle. For example, if you shop online and enter your credit card information, websites protect that information from hackers through a process of encryption.

Something for students to think about in the classroom: Can you come up with any formula to break down numbers into their prime factors?

Answer: No! That’s why encryption is considered a secure form of cryptography. To this date, there is no confirmed algorithm for prime factorization.

Prime factorization is a classic example of a problem in the NP class. An NP class problem can be thought of as a problem whose solution is easily verified once it is found but not necessarily easily or quickly solved by either humans or computers. The P vs. NP problem is one that has perplexed computer scientists and mathematicians since it was first formulated in 1971. Most recently, a German scientist Norbert Blum has claimed to solve the P vs. NP problem in this article: https://motherboard.vice.com/en_us/article/evvp34/p-vs-np-alleged-solution-nortbert-blum

Also in recent years, A Texas student has been featured on Dallas County Community Colleges Blog for his work to find an algorithm for prime numbers: http://blog.dcccd.edu/2015/07/%E2%80%8Btexas-math-student-strives-to-solve-the-unsolvable/

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

An activity for inquiry based learning of prime numbers and prime factorization utilizes pop cubes. Students will start out with a single color-coded cube representative of the number two (the first prime), they will then move up the list of natural numbers with each prime number having its own color of cube. The composite numbers will have the same colors as their prime factors. The idea is that students will visually see that prime numbers are only divisible by themselves (each being a lone cube) and that composite numbers are simply composed of primes (multiple cubes). A good point of discussion is the meaning of the word “composite’. You could ask students what they think the word ‘composite’ means and what word it reminds them of. This leads into the idea that every composite number is composed of prime numbers. This idea comes from online vlogger Thom Gibson and the RL Moore Inquiry Based Learning Conference. Below is a picture demonstrating the cube idea:

This foundational idea can be segued into The Fundamental Theorem of Arithmetic and then into prime factorization.
One of the most practical real-world applications of prime factorization is encryption. This activity I found makes use of prime factorization in a way that is interesting and different from simply making factor trees. This worksheet would be a good assessment and challenge for students and mimics a real –world application.

https://www.tes.com/teaching-resource/prime-factors-cryptography-6145275

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

Though not actually ‘reducing’ the value of a number, prime factorization is the equivalency of numbers broken down into their smallest parts and then multiplied together. The idea of reducing numbers goes all the way back to elementary school when students are learning about fractions. Subconsciously they use a similar process to prime factorization when reducing fractions to simplest form. When reducing fractions to simplest form, the numerators and denominators themselves may not both necessarily be prime, but when put into simplest form, they are relatively prime. Being able to pick out factors of numbers –another relatively early grade school concept (going back to multiplication and division) — plays a huge deal in both fractions and prime factorization.

# Engaging students: Negative and zero 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 again comes from my former student Austin DeLoach. His topic, from Algebra: negative and zero exponents.

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

The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.

Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.

E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.