Engaging students: Graphing Square Root Functions

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 student Alexandria Johnson. Her topic, from Algebra II: graphing square root functions.

An interesting word problem that students should be able to solve after completing a lesson on graphing square root functions would be: “The Chandra satellite detected x-rays coming from the region of the sky containing the galaxy cluster JKS041. The electrons in the gas are emitting the X-rays, and colliding at high speed with the protons in the gas. The energy of the x-rays at the time they were emitted by the hot gas was 21,400 electron Volts (eV). This energy is shared equally between the electrons and protons. The speed of a proton is related to its kinetic energy by E = 1/2mV^2 where E is the energy in Joules, V is the proton speed in meters/sec, and m is the mass of a proton (m = 1.7 x 10-27 kg). About how fast are the protons moving? (Note: 1 eV = 1.6 x 10^-19 Joules)”. Students can arrange the problem into a square root function to solve for velocity: V=sqrt(2E/m). Using the information provided students can convert eV to E and solve for m. Once this information is found, students can plug in the numbers to solve for V. Note: this question is difficult and some students may struggle with the calculations. A simpler question about the relationship between kinetic energy and velocity could be used in place of this one. Question provided by https://spacemath.gsfc.nasa.gov/weekly/6Page70.pdf.

In Physics, students will be able to use square root functions to describe the relationship between different variables. Having the knowledge of graphing square root functions will allow students to represent these relationships graphically. For example, to find kinetic energy, students use the formula E=(1/2)*m*v^2, where m=mass and v=velocity. Students can manipulate the equation to find v which would be v=sqrt(2E/m). Given m, students should be able to graph the relationship between v and E. When solving for volume, students can rearrange the equation into the form y=a*sqrt(x-h)+k, where h=0, k=0 y=v, x=E, and a=sqrt(2/m). knowing how to graph a square root function, students can graph this equation.

A useful resource when creating a lesson about graphing square root functions is https://teacher.desmos.com/. This website provides teachers with existing activities that the students can complete. Also, it allows the teacher to create activities for the student. An activity that is already created for teacher use is called Polygraph: Square root functions. In this activity, students play a game similar to the board game Guess Who. Students pair up and are given a set of graphs of square root functions. Partner 1 chooses a graph. Then, Partner 2 asks questions about the graphs to try to find the graph that Partner 1 chose. Students compare various graphs and communicate these differences. Though the website doesn’t offer any other premade activities at this time, teachers can use the activity type “marble slides” to create an activity that shows how a, h and k affect the parent function of square roots.
Work cited

“Chandra Spies the Most Distant Cluster in the Universe.” Space Math, NASA, Chandra Spies the Most Distant Cluster in the Universe. Accessed 15 Sept. 2017.
“Square Root Functions.” Desmos Classroom Activities, teacher.desmos.com/polygraph/custom/560ad29158fd074d156300b6. Accessed 15 Sept. 2017

Fun with pancakes

I’m not sure who had the idea to cook pancakes like these, but I’m sure glad he/she did.

Happy Phi Day!

In the United States, today is abbreviated 1/6/18, matching the first four significant digits of the irrational number $\phi = (1 + \sqrt{5})/2$, otherwise known as the golden ratio. For a good introduction to $\phi$, see http://www.foxnews.com/story/2007/06/18/happy-phi-day-perfect-time-for-some-phinancial-fun.html.

For today, I’ll give a fun fact that I learned last year; for its national flag, the country of Togo chose a rectangle whose dimensions matches the golden ratio.

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 Trent Pope. His topic, from Pre-Algebra: probability and odds.

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

This website contains problems that would be great for odds. On the worksheet it has you solving problems about the chances of getting different gumballs from a gumball machine and chances of winning gift cards in a drawing. These worksheets would be great because there are real life applications with these examples. On the worksheet students are to solve what color gumballs they could draw from the machine. This will give them a visual representation of their odds. In order to find their odds they must know all the required information such as the number of total gumballs and the number of each color. Then the instructor can ask the students any question about what they can draw. The other problem is that there are gift cards, coupons, and free admission to a theme park that a student draws from a hat. This would be another great example of how students can find the odds of what they can draw.

http://www.algebra-class.com/odds-and-probability.html

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

This project idea comes from the game show Deal or No Deal. The purpose of the project would be for students to see what the odds are of winning more money than the amount offered from the Banker. For instance, the banker will offer you $100,000 to leave the show without seeing what is in your briefcase. The contestant would then look to see how many briefcases are left that could contain an amount greater than$100,000. If there are five chances out of the twenty remaining briefcases, the student would have a 5/20 chance, or 25% chance, to win more money. So, the contestant might want to say no deal because there is a higher chance of winning more money should he/she stay in the game. Students could go multiple rounds of this and see if their chances increase as the game goes on. This would engage students and they would look forward to winning the game show.

http://www.teachforever.com/2008/02/lesson-idea-probability-using-deal-or.html

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

This topic has appeared in many examples through pop culture. One is in the movie 21. Also, I have found a YouTube video demonstrating probability and odds. The video gives an example of how a game show changes your odds of winning a brand new car. There are three doors and the host asks you which door you think has the new car. When you do this you have a 33% chance of selecting the right door. After you have made a selection, the host goes and selects one of the remaining doors to open. Remember that the host knows which door the car is behind. He opens the door to show you that it does not contain the car. Then, the host asks you if you would like to change your door or keep it. Because of variable change you are more likely to pick the car if you change your decision. This increases your chances to 66% of choosing the right door. I thought this was a great way to engage students about probability and odds because it is all about your chance of selecting the correct door. You have one chance to pick the right door, but three doors to pick from. This is all about odds. It increases after the host opens a door because you have a second chance to select the correct door. This would apply to all game shows, and people would be able to make personal connections.

Engaging students: Multiplying 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 again comes from my former student Saundra Francis. Her topic, from Pre-Algebra: multiplying fractions.

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

This multiplying fractions project found on Teachers Pay Teachers by Mix and Match gets students interested in the topic through creating a dog house. For the project students have to use the size of their chosen dog to discover the dimensions of the doghouse. The students will then scale down their doghouse by multiplying fractions to create a model doghouse. Once the students have discovered the dimensions of the model they can build the model doghouse. There are worksheets provided on the website that will guide the students through this process, the also have word problems related to the doghouse for extra multiplying fractions practice. This project would engage students because they will be able to create their own doghouse and they will be given an opportunity to build it. It also will help students understand how to multiply fractions through working out how it relates to scaling items.

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

In previous math course students should have learned how to multiply, reduce fractions to the simplest form, and how to covert mixed numbers to improper fractions. Using these concepts students will be able to multiply fractions based on previous understanding of fractions and multiplication. A YouTube video titled Review of Fraction Concepts created by mathtutordvd (https://www.youtube.com/watch?v=7Wrde6iFVcA) reminds students what a fraction is and what it represents. It also reviews term such as numerator and denominator, which are important terms for students to know when they learn how to multiply fractions. This will engage students’ prior knowledge by giving them a refresher and will prepare them for learning how to multiply fractions. It also might help students that were previously having a hard time understanding the concept of fractions once they watch the video. It also discusses the important of fractions, which will help students realize how it can apply to their daily lives.

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

The Multiplying Fractions Song by NUMBERROCK (https://www.youtube.com/watch?v=CcDGRLosAf0) is an excellent video to engage students and help them understand how to multiply fractions.. This video goes through two examples of multiplying fractions while rapping. The examples used about finding treasure and digging for dinosaur bones will catch students’ attention. The video not only gives student procedural knowledge, the steps to multiply fractions, but explains why we are able to multiply fractions through the images. They sing “multiply the numerator, then multiply the denominator” which students can repeat when they are working on problems later in the lesson. In the video, models are displayed that show students how to multiply using a model, which is part of the TEKS. The diagrams also show students why multiplication of fractions works and gives them a better understanding of the concept. The rap song and cartoon visuals draws students attention and help them remember the topic being learned.

References:

https://www.teacherspayteachers.com/Product/Multiplying-Fractions-Project-2351217

https://www.youtube.com/watch?v=7Wrde6iFVcA

https://www.youtube.com/watch?v=CcDGRLosAf0

Engaging students: Permutations

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 Sarah McCall. Her topic, from probability: permutations.

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

In high school math, word problems are essentially unavoidable. They can be a pain, but they do help students to be able to see applications of what they are learning as well as good problem solving skills. So, if we must make use of word problems, we might as well make them as engaging/fun as possible. Some examples of ones that I found and would use in my classroom:

1. Permutation Peter went to the grocery store yesterday and met a super cute girl. He was able to get her phone number (written on the back of his receipt), but today when he went to call her he couldn’t find it anywhere! He knows that it consisted of 7 digits between 0 and 9. Help Permutation Peter by figuring out how many combinations of phone numbers there are.
2. Every McDonald’s Big Mac consists of 10 layers: 2 patties, 3 buns, lettuce, cheese, onions, special sauce, and pickles. How many different ways are there to arrange a Big Mac?

How has this topic appeared in pop culture?

Many students are easily confused when they first learn the difference between permutations and combinations, because for most permutations is an unfamiliar concept. One way to show students that they have actually seen permutations before in everyday life is with a Rubik’s cube. To use this in class, I would have students pass around a Rubik’s cube, while I explained that each of the possible arrangements of the Rubik’s cube is a permutation. I would also present to them (and explain) the equation that allows you to find the total number of possibilities (linked below) which yields approximately 43 quintillion permutations. This means it would be virtually impossible for someone to solve it just by randomly turning the faces. Who says you won’t use math in the real world!

How can technology be used to effectively engage students with this topic?
In a day and age where a majority of our population is absorbed in technology, I believe that one of the most effective ways to reach high school students is to encourage the constructive use of technology in the classroom instead of fighting it. Khan academy is one of the best resources out there for confusing mathematics topics, because it engages students in a format that is familiar to them (YouTube); not to mention it may be effective for students’ learning to hear a different voice explaining topics other than their normal teacher. In my classroom, I would have my students use their phones, laptops, or tablets to work through khan academy’s permutation videos, examples, and practice problems (link listed below).

References

https://www.quora.com/How-are-permutations-applied-in-real-life

https://prezi.com/q3aaem0k2xie/permutations-in-the-real-world

https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group

https://www.khanacademy.org/math/precalculus/prob-comb/combinatorics-precalc/v/permutation-formula

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 Peter Buhler. His topic, from Pre-Algebra: expressing a rate of change as a percentage.

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

As a teacher, one activity that could be used to engage students would be to use a real world application. This topic is unique, as it can be applied directly to shopping at a store. This activity could include having students bring in a catalog of a sale (either from a grocery store or department store) to the classroom to use. Then students would be encouraged to calculate percent discounts based on markdowns, or they could use a fixed percent discount (ex: 30% off everything) and calculate the new prices of various items from the store.

This activity is not only effective for teaching the topic, but also engages students since this is a topic that everyone deals with on a regular basis. Also, allowing students to bring in catalogs gives the students the freedom to operate within the classroom, as opposed to being given a generic worksheet and asked to solve those problems. An extension of this could be to introduce exponential growth (which is still rate of change and uses percentages) and can be applied to banking, credit, mortgages, and other applications that students may know little about.

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

Although the rate of change and percentages may be introduced at the junior high level, students will continue to use various aspects of these topics even into college level math courses. Derivatives are a huge part of calculus, and it is a known fact that derivatives are simply the rate of change of the original function. On the other hand, percentages can also lead to discussions around probability, chemical compositions within a compound, or even calculating grades for a certain class. All of these deal with using rate of change or percentages in classes outside of pre-algebra.

One application of this could be to introduce derivatives in a class outside of calculus and in a way that students would easily understand. If a student is able to understand the idea behind the rate of change, then they can understand a derivative. Likewise, the teacher can introduce certain applications of percentages outside of mathematics in order to tie in other topics.

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

As mentioned previously, one method to engage students is through real world applications. Both rate of change and percentages can be found in compound interest. There is a link to a video on YouTube which illustrates how powerful compound interest really can be. The use of graphics and other visuals within the video would allow for student to grasp how large the rate of change is, even after starting with small numbers.

Another useful tool that could be used in the classroom is an online calculator to observe the rate of change. If students have the ability to access the internet, then they could access the URL listed below. The website allows for students to put in different dollar amounts to observe the rate of change in regards to investment. While there is certainly a time to teach students how to calculate this without the website, this could be something that the students use to gain insight into how quickly compound interest can occur. It also gives students the opportunity to observe how different values change the final total and therefore make observations about how compound interest works. The link is: https://www.calculatestuff.com/financial/compound-interest-calculator.

References:

https://www.youtube.com/watch?v=immQX0RKFY0

https://www.calculatestuff.com/financial/compound-interest-calculator

Engaging students: Solving two-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 comes from my former student Jessica Williams. Her topic, from Pre-Algebra: solving two-step algebra problems.

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

Learning two-step algebra problems can difficult for students at first glance of the equation; therefore I believe using a hands on manipulative to demonstrate is a big help, especially for your visual/kinesthetic learners. I’ve recently helped demonstrate this lesson I’ve seen online to my sister, who is in 7th grade. It worked marvelously with her; therefore I would definitely do it in my actual classroom. To teach this lesson, I would bring in cups and colored chips for each student to use to demonstrate the equation given. For starters, present the students with an equation to solve. (2x+3=9) Next, present the students of the guide lines/rules of the cups and chips. Let them know that if the variable is a positive number, to place the cup facing upwards. Similarly, if the variable is a negative number, tell them to place the cup facing down. Let the students know that the coefficient of the variable is what lets you know how many cups to use. Next, you would guide the students with questioning but asking them to display what 2x is using their cups. They should each have two cups facing upwards. Next, they will place 3 chips next to their cups to represent the +3 and have an equal sign with 9 chips on the other side. This would lead into asking the students what they could do to get rid of the 3 chips on one side, which results in having to get rid of 3 on the side with 9 as well. This will lead the students to 2x=6, and you can ask the students if 2 cups equals 6 chips, then how many does only one cup equal. They should get to x=3, with enough scaffold questioning. Then the teacher could provide multiple more examples to do on their own with the objects in front of them. This allows for the students to visual see why solving the two-step equations work the way they do. It shows students how you have to “do to one side what you do to another.”

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

Prior to learning about two-step equations the students will have worked with one step equations. They are used to seeing simple equations, such as x-2=6 or 2x=10, where they only have to complete one step. Before teaching the students two-step equations, the teacher should allow a couple practice problems to access the student’s prior knowledge. The two-step algebra problems are only a slight extension to what they have practiced. It also extends on basic addition, subtraction, multiplication, and division. They know how to do all of these things; however adding a variable to the mix is quite an extension in the perspective of the students. They have to be taught the meaning and definition of a variable and how it has a specific value that needs to be solved for in order for the equation to be correct. Learning this topic will also help prepare the students for more difficult math such as solving quadratic equations, word problems, etc.

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

Technology can always be made fun for the kids, however it can be very good for extra practice as well. For example, http://www.math-play.com/Two-Step-Equations-Game.html is an amazing way to practice and have fun at the same time. The game requires player one and player two to go against each other. Each player has to answer a two-step equation correctly in order to shoot his or her ball at the basketball hoop. The player at the end with the most points wins the game! This game is extremely engaging for the students because it involves competition. What does every student love to do? WIN! It boosts their confidence. If the student would rather work alone, that’s fine as well. It still benefits every student why keeping their mind in the game and focusing on answering correctly. The students can also answer question on Khan academy or watch videos for refreshment before the next class. There are so many ways technology can be beneficial. In previous lessons, I have used Kahoot and plickerz. Both require 100% engagement from each student and they both require individual accountability.

References:

http://www.math-play.com/Two-Step-Equations-Game.html

Annenberg Learner: https://www.learner.org/workshops/algebra/workshop1/lessonplan2.html

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.

Now Add or Subtract, Now Add or Subtract.”

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