# 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 comes from my former student Mario Acosto. His topic, from Pre-Algebra: multiplying fractions.

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.)

Word problems can be a good way to make your students start to think about topics. I feel like giving students worksheets isn’t a good way for them to learn new material because it’s just boring and makes the students not be excited about the topic. Some word problems can be very interactive such as the example that I have right here. (https://www.pbslearningmedia.org/resource/mket-math-ee-vidgoliathbeetle/beetle/#.W4nobvZFxu0).

This video shows a great way for students to first visualize on how to multiply fractions and it also gives the students something new to know about beetles. I really love this video because of how detail the images are and gives the students an example problem at the end of the video. For the example that is given at the end of the video, I will make the students pair up and let them try to solve the example for at least 10 mins and then go over it together as a class. If the student has a hard time seeing fractions then this is a good way to see them.

How have different cultures throughout time used this topic in their society?

Knowing that different cultures used multiplying fractions in their own way is so satisfying because each one is so different to where it makes you think that a new math concept of multiplying fractions could come up within the next century. This article gives amazing examples of how each culture used multiplying fractions in their own way but end up having the same mindset.    http://www.math.wichita.edu/history/topics/num-sys.html

To engage my students into learning about this I would make a chart of the different cultures there are in the above link and make the students choose one culture as a group and let them learn about it. After they have learned more about the culture then I will tell them to come up with a way to teach what they have learned to the class. This should take about half of the class time but not all. It’s a good way for students to learn something new while still being engaged.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is the best way to teach students in this generation because many of the students are high tech with their devices. So, having a website that teaches your students about multiplying fractions but in an engaging way. (  http://math.rice.edu/~lanius/fractions/frac5.html )

# Engaging students: Square roots

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 Julie Thompson. Her topic, from Algebra: square roots.

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

When I think of square roots my mind immediately takes me to the very popular movie ‘The Wizard of Oz’. In a scene near the end of the movie, the scarecrow incorrectly states the Pythagorean Theorem. He states it so fast that some people may not have time to process what he is saying is incorrect. The theorem he states is as follows: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” There are a couple things wrong with this statement. First of all, the Pythagorean Theorem is based on right triangles, not isosceles. Secondly, we take the square of two specific sides and set it equal to the square of the third side, not the square root of ‘any two sides’ equal to the square root of the remaining side.

As an engage, I think it would be very interesting to first show the clip of the movie to capture my students’ attention, and then have a discussion about why the theorem is wrong and what the correct theorem actually is!

Also, I found an awesome worksheet from Mathbits that is all about this scene from the movie and goes through a couple examples that shows why his theorem can’t work, and also allows students to prove why it is false!!

Click to access OzMath.pdf

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

A very engaging website that was actually introduced to me in college: KAHOOT! I first played Kahoot in my TNTX 1200 class here at UNT. It was very exciting and fun for me, as a college student, to play, so I know middle and high school students will love it as well. Kahoot is an online quiz game where students use their own technology to join in to the game with a game pin provided by the teacher. Students get to give themselves a game nickname which makes it fun to be able to see their name pop up on the scoreboard. Then a variety of questions on the topic are asked, one at a time, with a time limit for the students to answer in (usually about 20-30 seconds). This is a quick game that can be used as an engage at the beginning of class to get students thinking and excited about the topic for the day. In this case…square roots! I found a great Kahoot created by ‘remangum’ that focuses on finding square roots of numbers (it throws in a couple cube roots). Once you get passed about 7 questions, they throw some variables into the mix. One of the question asks to find d:

Sqrt (d*d)=9, where * is multiplication. In this case, d=9 because sqrt(81)=9. I like this because it allows the students to think a little harder and problem solve.

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

Many students who enter middle school/ early high school wonder why they have to learn all these pointless concepts such as square roots and the order of operations. They might even think to themselves, “When will I ever need to know this in the future when I have a job?” According to homeschoolmath.net, “The answer is that you need algebra in any occupation that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade, commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You also need algebra to take your SAT test or GED.” This is very important to let students know, but they may not believe you or care. For instance, they may say that’s true for math and science professions, but they are planning to major in something totally different and they won’t need math. Math actually can be useful in other fields, but for the sake of this question, I will stick to math and science.

In their future classes, such as Algebra II, they will be using things such as the quadratic formula. This will involve plugging in and simplifying things under a radical, as well as dealing with square roots in whole equations rather than just on their own. Also, understanding the nature of square roots will help them in future courses such as PreCalculus when they must study all the characteristics of the square root function. As an engaging aspect to all of this, I may mention that, “Studying algebra also has a benefit of developing logical thinking and problem solving skills. Algebra can increase your intelligence! (Actually, studying any math topic — even elementary math — can do that, if it is presented and taught in such a manner as to develop a person’s thinking.)”

References

A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card). (2012, January 15). Retrieved September 09, 2016, from https://reflectionsinthewhy.wordpress.com/2012/01/15/a-visual-approach-to-simplifying-radicals-a-get-out-of-jail-free-card/

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Buncombe, A. (16, April 4). Square Root Day: There are only nine days this century like this. Retrieved September 09, 2016, from http://www.independent.co.uk/news/world/americas/square-root-day-there-are-only-nine-days-this-century-like-this-a6967991.html

Fowler, D., & Robson, E. (n.d.). Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Historical Mathematica, 366-378. Retrieved September 9, 2016, from https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf.

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/

# Engaging students: Solving word problems of the form “a is p% of b”

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 Christian Oropeza. His topic, from Algebra: solving word problems of the form “a is p% of b.”

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.)

Students would be able to answer word problems that involve real world applications. For example, a student could be asked: “Sam went to Academy to buy clothes, sports equipment, and fishing gear. At the register the total of Sam’s transaction before tax is \$141.32. Given that the sales tax is 8.25%, what would Sam’s total be after tax?” These type of word problems would be relatable to students, which would show them the importance of this topic in life. Students always ask the question, “how is this used in everyday life?”, and with these type of word problems students may be able to generalize the concept more easily. When students cannot relate to a topic in math they become easily discouraged, give up, and stop paying attention in class, but with problems like these the students would be able to incorporate the topic into their own lives. Some other problems that students could be asked could involve any type of scenario where there is a percentage to be found between two numbers (Reference 1 & 4).

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

This topic can be used in different scenarios for math and science, but Chemistry is an excellent example. In chemistry, there is a topic that covers calculating percent composition. The basic idea of this topic is to calculate the percentage of each element’s mass in regard to a molecule’s total molecular mass. An example would be, “Calculate the mass percent composition of each element in a potassium ferricyanide, K3Fe(CN)6 molecule.” (Reference 2). These types of problems would help students understand how much a certain element or compound is in a particular molecule. Another example of how this topic can be used, is in math when a student has to convert between fractions, decimals, and percentages in a word problem. An example could be, “Mia has a basket full of fruit. In this basket she has 1/5 apples, 2/3 oranges, and 2/15 bananas. What percent of each fruit does she have in relation to the basket?” Students would be able to work on their converting skills to enhance their understanding of multiple representations of the same number (Reference 3).

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 4) is an excellent piece of technology to introduce or review this topic to the students because the website goes through visual representations of how a percentage of a whole looks like. Also, the website has a section where a student can input a number and a slider that allows the student to move it around to see what number would represent a certain percentage of the number inputted. Another example of effective technology is the website Khan Academy (Reference 1) because it has real world problems that are relatable. The website also gives hints and step-by-step solutions for each question in case a student is stuck and does not know what to do next. The use of multiple websites is good for students to have a variety to choose from in case one is easier to understand than another.

References:

# Engaging students: Box and whisker plots

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 submission comes from my former student Chris Brown. His topic: how to engage students when teaching box and whisker plots.

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

My all-time favorite TV show as a child was Pokémon. This show is still a staple amongst the young and even adult generation of today. The activity that I have created, was designed to take place after a formal lesson over how to create Box and Whisker plots. For this activity, students will be given a labeled bar graph of the Pokémon Type Distribution for generations 1 through 6 of Pokémon, which I have listed an online data source below. The students will be tasked with identifying the top 7 Pokémon types and creating a Box and Whiskers plots for each of those types. They will then go through and analyze the consistency of the creation of Pokémon for that specific type and then compare contrast this same box plot to any other box plot of their choice. The students will then make predications for the number of Pokémon for each of the top 7 Pokémon types, for generation 7 and base their reasoning in the box plots they created. Then the student will finally research the type distributions for the 7th generation of Pokémon, and discuss how the actual number compares to their prediction.

This is the online source for the type distributions for generations 1 – 6:

https://plot.ly/~powersurge360/6.embed

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

From my experience, Box Plots are first taught in the early middle school years, in 6th or 7th grade. When constructing box plots by hand, in its essence, box plots require knowledge of how to order sets of numbers from least to greatest; an understanding and ability to find the maximum, minimum, median, and mean of a data set; and lastly, critical thinking and analytic skills developed from general course content. Box plots allow students to combine each of these skills to effectively analyze data sets with ease and compare different data sets with precision and accuracy. If any or all of these skills are not quite up to par, students will have an opportunity to develop them through box plots as they spend time creating them. For all students no matter their level, they will still gain better insight on how to properly analyze data and grow as analytical thinkers as they take the represented data and turn it into meaningful interpretations.

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

In a classroom, I personally believe that Desmos is a wonderful online tool that can aid students in the understanding of how box and whisker plots function, and also a great place to check their work. Desmos, which is linked below, gives students the ability to list as many data points as they need to, and concurrently creates a box plot as they do so. In this way, students are able to see how singular data points can skew the data in significant and insignificant amounts. What I also love about Desmos is that, the list of data points does not have to be in any kind of order, so students do not have to worry about that tedious step! Desmos also lists the 5-point summary in two different places, on the box plot itself, and also on a drop-down menu, which is super convenient. Lastly, I love how Desmos also displays the mean of the data set as well, students can calculate the skew of the data, and definitively determine how it is skewed. This is a super visual, and interactive tool that will allow the student to manipulate box plots so seamlessly they will not be focused on the tediousness of the setup and solely on the concept.

The link to the Desmos setup is here: https://www.desmos.com/calculator/h9icuu58wn

# 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 Cameron Story. His topic, from Algebra: ratios and rates of change.

What interesting word problems using this topic can your students do now?

Since the most relatable example of a ratio is speed (meters per second, miles per hour, etc.), it’s easy to see how a teacher can make an interesting or engaging word problem out of this. First, however, let us take a look at an infamous word problem involving ratios/rates of change that is not inherently interesting on its own.

“Train A, traveling 70 miles per hour (mph), leaves Westford heading toward Eastford, 260 miles away. At the same time Train B, traveling 60 mph, leaves Eastford heading toward Westford. When do the two trains meet? How far from each city do they meet?” (“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.)

This is a distance-over-time that most students or past students are familiar with, but why is this problem still being used? There are a few issues I have with this example. Firstly, I cannot think of very many students who could honestly get excited about trains, especially now in the modern era of vehicular travel. I am willing to bet that most of your high school math students have never even been on a train; and if they have, it was most likely an underwhelming experience. This example also lacks creativity. Giving the trains actual names or having them traveling between real world places would have been a step in the right direction.

So how can we change this example to become engaging to students? Firstly, let’s replace the trains with modern cars, and crank up the speed. Every student is familiar with cars, and fast-moving cars (in my opinion) is much more exciting. One could easily imagine using modern rockets as the vehicle as well, and replacing the towns with interplanetary destinations. Next, instead of naming the cars Car A and Car B, we can use actual modern electric cars such as the Model 3 from Tesla Motors. Take a look of the following word problem I came up with instead (you may notice the stakes of the situation described is objectively more engaging then a problem about train travel):

“Tesla is hoping to feature one of its new cars in a commercial, in which a car attempts to race underneath a falling refrigerator in dramatic fashion. In the commercial, the car must travel at top speed, traveling over 25 meters of track from start to finish. As soon as the car passes the starting line, the fridge is dropped from 10 meters up in the air above the finish line, at a rate of 20 meters per second. The top speeds (in meters per second) of the Tesla Model 3 and the Tesla Roadster are shown below. Which car should Tesla pick to safely beat the falling fridge?”

The reason a creative approach works better is that it increases the student engagement; students do not want to do word problems, so it is our job as teachers to make them interesting.

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

Creating an activity around rates of change allows for a lot of creativity. For example, one could take a physical approach, in which students record how fast they can run (only requires a stop watch and a set distance) and using that to plot their data on a distance vs. time graph.

It is important to remember that ratios can represent far more than just speed. Some relatable examples include rate of hair growth, number of hours studied per week, or even how many gallons of water drank in a day. For my Tesla commercial word problem, I used a website (desmos.com) to flesh out this one problem into an engaging classroom activity. Having your classroom activities on interactive platforms that evoke teamwork and cooperation in your students is key to student engagement.

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

Desmos Classroom Activities (at teacher.desmos.com) is an incredibly useful tool that teachers can use to quickly create any activity for their students. These activities can even be done on smart phones, which removes some of the hassle of getting computers in the classroom. When creating an activity, teachers also have access to a wide range of tools including (but not limited to) animation, student inputs, information slides (for presentation), and even interactive functions that allow students to modify given equations.

The main benefit of using Desmos for classroom activities is that the teacher has full and complete access to viewing student progress. Instead of walking around the room trying to hunt down students who need help, the teacher can view which students are stuck on which problems. The teacher can then approach the issue fully prepared, and know exactly which students are having problems before their hands even hit the air.

I created a Desmos activity available for use in a lesson about ratios or rates of change (link: https://teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0) which uses the same Tesla commercial word problem I gave before. Using this website, I was able to build this world problem into a somewhat-realistic and animated simulation, asking critical questions in order to build upon the underlying mathematical concepts. Feel free to adapt my lesson (Desmos has a copy/edit feature for activities) for any vehicle, scenario, or speed.

References:

“Desmos Classroom Activities.” Desmos Classroom Activities, 2010, teacher.desmos.com/.

Story, Cameron. “Ratios and Rates of Change Activity.” Desmos Classroom Activities, 30 Aug. 2018, teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0.

“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.