# Engaging students: Word problems involving inequalities

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 Jillian Greene. Her topic, from Algebra: word problems involving inequalities.

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

The students, in pairs, are stranded on a deserted island. There’s another island nearby that has various items that they need to survive, but that island is overrun by snakes and is virtually uninhabitable. The have one canoe to get to the island and back, but it was damaged and will only last for two roundtrip voyages to the other island. Luckily, the students possess a certain clairvoyance that tells them the weight that the canoe can hold, as well as the weight of each supply. The numbers will vary for each group, but the canoe will hold something like up to 37 lbs (after the weight of the person on the canoe) for the first trip, and 25 lbs for the second trip. There will be weight for individual fire-building supplies, food, water, an old radio, weapons, etc. and will then be left to the students to find the different combinations they can transfer. They then have to choose which items, how many of each item, and what combination they think would benefit them the most. To add a fun element, the teacher might even have a correct answer as to which materials will save them. This activity would be a fun way for student to take numbers given to them and organize them in a way that they’re excited about.

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

If this is an algebra 1 class, this concept will be new to them. If this is algebra 2, then they should have seen this in geometry already. However, this is a fun way to look at how inequalities help us with very base level geometry. Assuming this is algebra 1, the students will discover the triangle inequality theorem and then be informed that is a big concept that they will discuss next year in geometry. They can do the activity where they’re given uncooked spaghetti noodles and break a piece into three pieces and see if it makes a triangle. They can measure the pieces and see when a triangle does work and when it doesn’t, describe their findings using words, and try to formulate the necessary inequality from that (the third side must be less than or equal to the sum of the other two sides.) If the students are learning this in Algebra II, then they can see how the description connects to the equation, and it will be interesting for them to build off of prior knowledge.

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

This activity is not as much deciphering the inequality from a word problem as it is understanding what inequalities mean in a graphical sense. However, it is indeed a situation involving inequalities, and a TV clip. I do not have the clip available to me right now (legally) but it’s in an episode of Numb3rs called “Blackout” where an attacker is causing blackouts throughout the city and then committing the crimes during the blackouts. The investigators found a code for where the attacks take place and they’re given two inequalities that they need to graph to find region in which it might take place.

https://mathstrategies.wordpress.com/numb3rs-activities/

This will not only allow for a solid practice on how inequalities look on a graph, but for the (kind of) practical application of using things like this. The teacher can ask a few fun questions, like “why do you think the attacker is choosing this region?” or “how would it affect the graph if all of the area between Ramirez St and Gateway Plz was closed due to construction?” This will make the “less than” and “greater than” signs actually hold some amount of meaning.

# Happy Pythagoras Day!

I’d like to wish everyone a Happy Pythagoras Day! Today is 12/20/16 (or 20/12/16 in other parts of the world), and $20^2 = 12^2 + 16^2$.

Bonus points if you can figure out (without Googling) when the next three Pythagoras Days will be.

# Engaging students: Polynomials and non-linear 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 comes from my former student Jessica Martinez. Her topic, from Algebra II: polynomials and non-linear functions.

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

There is a Ted Talk video showing the math behind professional basketball player Michael Jordan’s hang time. The video connects a popular sport and player with mathematics by using quadratic equations to explain how MJ stays in the air as long as he does. You can see that the video is aimed at a younger audience since it’s done with cartoon animation, and it’s fairly easy to follow along as it explains the math. The video explains how they derived the formula for MJ’s jump shot by using his initial velocity and the force of gravity along with the variable of time. It also provides a great visual representation of how jumping into the air resembles a parabola of a quadratic function when they place MJ jumping against a graph. The video shows how applicable quadratics are by explaining that the roots of the parabola of MJ’s jump shot are the spots where he jumps and where he lands again. We could also calculate the maximum height of MJ’s jump by finding the vertex of the parabola and I could modify the equation as a problem for my students to solve. For example, we could look up the world record for highest jump and I could ask my students to calculate what the initial velocity would be for that person to get the highest jump using MJ’s hang time.

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

As a student, the first couple of times I looked at the graphs of polynomials I always thought, “Huh, those kind of look like rollercoasters”. I did some researching and I found a project where students are asked to use polynomials to analyze and design rollercoasters. As a teacher, I could introduce this project with a short video or advertisement of a popular theme park (like Six Flags) to get their interest and show some of the cool rollercoasters in action. Then I would have the students answer word problems about rollercoasters and their polynomial functions to find the local max/min of the coasters, where the function is increasing/decreasing (riding down or up on the rollercoaster), and what type of function best models certain parts of the coaster (quadratic, cubic or quartic). After my students have worked some polynomial function problems, I would have them pair up or work in groups to design their own rollercoaster using polynomials. I would also like to collaborate with a physics teacher as well; by using physics, my students could test the equations of their coasters with velocity, force and acceleration and see if they are realistic or not (and they could also see how this topic extends to other courses).

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

Whenever a new infectious disease begins to spread rapidly (like Ebola or the Zika virus), there is coverage of the spread all over the news, making this topic highly relevant for my students. The spread of infectious disease can be modeled through non-linear functions such as exponential functions. I could create multiple word problems about the Ebola outbreak in Africa; for example, I could have my students pretend that scientists have developed a vaccine for the Ebola virus but now the problem is distributing the vaccine to all of the infected people. I would have my students pretend they were a disease control team trying to race against the spread of this disease in order to vaccinate the people before it was too late. By using actual date on reported cases in a specific country in Africa (like Liberia or Sierra Leone), my students could find the exponential function that best represents their data. They could then use that function to estimate the time it would take for all of the population in their country to be infected and compare that to the rate and time it would take to distribute all of the vaccines to the people (making estimates based on research of the country and how it has handled disease spread in the past). Since actual data won’t always match precisely with a mathematical function, I would have my students discuss what other variables and factors could affect their calculations as well.

References

Dawdy, T. (n.d.). Roller Coaster Polynomials. Retrieved September 23, 2016, from http://betterlesson.com/lesson/435674/roller-coaster-polynomials

Honner, P. (2014, November 05). Exponential Outbreaks: The Mathematics of Epidemics. Retrieved September 23, 2016, from http://learning.blogs.nytimes.com/2014/11/05/exponential-outbreaks-the-mathematics-of-epidemics/?_r=0

TEDEd. (2015, June 04). The math behind Michael Jordan’s legendary hang time – Andy Peterson and Zack Patterson. Retrieved September 23, 2016, from https://www.youtube.com/watch?v=sDbmcPnzwy4

# Engaging students: Completing the square

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 Deborah Duddy. Her topic, from Algebra: completing the square.

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

Applying what is learned in the class is very vital in fact it is a process TEKS that teachers need to use to maximize student’s understanding. “When are we going to use this in real life?” and “Why do we need to know this?” are questions that students ask on a daily basis. Connecting material to the real world helps engage students and develops critical thinking. Describing a path of a ball, how far an item can be tossed in the air and how to maximize profits for a company are just some examples of how quadratics can be used in the real world.

One important event happens during high school; students receive their driver’s license. In their written driver’s test, students must know the distance needed to stop a car at certain speed limits. Using an example like the one below will be interesting for the students and help connect lesson material and real life.

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

To begin class and get students involved with their learning, the class will participate in an activity. Each pair of students will have two different cards such as (x+2)^2 and x^2+4x+4, and any variations of these problems. They can only look at the (x+2)^2 card. Students will work out the problem on paper. Students will be asked to remember how to find the area of a square and then set up a square with the dimensions matching the first card. From there, the pairs would use algebra tiles (after knowing what each tile stands for) and attempt to “complete the square”. This activity will be used as an engage and a beginning explore for the students. This activity will help students see completing a square geometrically.

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

Completing the square is another way of solving/factoring the equation. The process of completing the square is to turn a basic quadratic   equation of ax^2 + bx + c = 0 into a(x-h)^2 + k = 0 where (h,k) is  the vertex of the parabola. Therefore this process is very beneficial because it helps students graph the quadratic equation given. In order to find h and k, students should be able to factor, square a term, find the square root and manipulate the equation.

In solving the equation by completing the square is to subtract the constant off the left side and onto the right side. Then students take the coefficient off the x-term divide it then square it. Students then add this number to both sides of the equations. By simplifying the right side of the equation, students give the perfect square. Then solve the equation left by taking the square root of both sides and determining x.

References:

# Engaging students: Multiplying polynomials

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 Daniel Herfeldt. His topic, from Algebra: multiplying polynomials.

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

Activities for multiplying polynomials are endless. An activity that I would do with my students is a game called polynomial dice. To do this, you would first is to get several blank dice and write random polynomials on each side of the dice. Then in class, divide the students into groups of no more than three. Each group will get a pair of dice. Have the students roll the dice and they should have two different polynomials. Once they have rolled, have them multiply the polynomials together. This is best done with groups so that the students can share their work with their partners to see if they both got the same answer. If they did not get the same answer, they can go back through each other’s steps to see where they went wrong. If you want to make the game a bit harder, you can add more dice to make them multiply three polynomials, or maybe even more. This is a great game because it can be used for multiplying polynomials, as well as dividing, adding and subtracting. It could be a great review game before a major test to have students remember how to do each individual property. For example, have the students roll the dice, then with the two polynomials they get, they first add the polynomials, followed by the difference, then the product, and finally the quotient.

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

Multiplying polynomials is used all over mathematics. It is first introduced in Algebra I and Algebra II. Multiplying polynomials can be very difficult for students and make them not want to do the work. This is due to there being so much work for one problem. Since there is so much work, there is a lot of room for mistakes. This topic is used is Algebra I, Algebra II, Algebra III, Pre-Calculus, Calculus and just about every higher math course. If a student is looking to go into an architecture or engineering field, they will have to apply their knowledge of polynomials. Due to this, the topic is one of the most important topics that students need to understand. Knowing how to multiply polynomials also makes it easier to divide polynomials. If a student is struggling with dividing polynomials, you can go back to showing them how to multiply them. Once a student sees the pattern of multiplying polynomials, they are more likely to get the hang of dividing them.

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 believe this video would be a great engage for the students when you, as a teacher, are teaching the students how to multiply polynomials for the first time. This video helps students remember what exactly is a polynomial. Although there is only three types of polynomials in the video (monomial, binomial, and trinomial), it uses the three main types that students will be using in a high school level. Another great thing in the video is that it shows how to tell the degree of the polynomial. Although it seems easy to just say the power of x is the same as the degree, students still might forget how to do it. For example, a student might think that a digit by itself and with no variable has a degree of one, but is really a degree of zero. The final point that is key to this video is that it shows students how to line up the terms. Some students might put 6+x^2+3x, and although that is still correct, it will be better written as x^2+3x+6.

# Engaging students: Solving systems of linear inequalities

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 Heidee Nicoll. Her topic, from Algebra: solving linear systems of inequalities.

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

I found a fun activity on a high school math teacher’s blog that makes solving systems of linear inequalities rather exciting.

The students are given a map of the U.S. with a grid and axes over the top, and their goal is to find where the treasure is hidden.  At the bottom of the page there are six possible places the treasure has been buried, marked by points on the map.  The students identify the six coordinate points, and then use the given system of inequalities to find the buried treasure.  This teacher’s worksheet has six equations, and once the students have graphed all of them, the solution contains only one of the six possible burial points.  I think this activity would be very engaging and interesting for the students.  Using the map of the U.S. is a good idea, since it gives them a bit of geography as well, but you could also create a map of a fictional island or continent, and use that as well.  To make it even more interesting, you could have each student create their own map and system of equations, and then trade with a partner to solve.

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

If students have a firm understanding of inequalities as well as linear systems of equations, then they have all the pieces they need to understand linear systems of inequalities quite easily and effectively.  They know how to write an inequality, how to graph it on the coordinate plane, and how to shade in the correct region.  They also know the different processes whereby they can solve linear systems of equations, whether by graphing or by algebra.  The main difference they would need to see is that when solving a linear system of equations, their solution is a point, whereas with a linear system of inequalities, it is a region with many, possibly infinitely many, points that fit the parameters of the system.  It would be very easy to remind them of what they have learned before, possibly do a little review if need be, and then make the connection to systems of inequalities and show them that it is not something completely different, but is simply an extension of what they have learned before.

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

Graphing calculators are sufficiently effective when working with linear systems of equations, but when working with inequalities, they are rather limited in what they can help students visualize.  They can only do ≥, not just >, and have the same problem with <.  It is also difficult to see the regions if you have multiple inequalities because the screen has no color.  This link is an online graphing calculator that has several options for inequalities: https://www.desmos.com/calculator.  You can choose any inequality, <, >, ≤, or ≥, type in several equations or inequalities, and the regions show up on the graph in different colors, making it easier to find the solution region.  Another feature of the graphing calculator is that the equations or inequalities do not have to be in the form of y=.  You can type in something like 3x+2y<7 or solve for y and then type it in.  I would use this graphing calculator to help students visualize the systems of inequalities, and see the solution.  When working with more than two inequalities, I would add just one region at a time to the graph, which you can do in this graphing calculator by clicking the equation on or off, so the students could keep track of what was going on.

References

Live.Love.Laugh.Teach.  Blog by Mrs. Graves.  https://livelovelaughteach.wordpress.com/category/linear-inequalities/

Graphing calculator https://www.desmos.com/calculator

# Engaging students: Using the point-slope equation of a line

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 Brittany Tripp. Her topic, from Algebra: using the point-slope equation of a line.

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

The point-slope equation of a line can be used in a variety of different ways in mathematics classes that some students may encounter later on. It is used in Calculus when dealing with polynomials. For instance, “key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions.” It is also seen when dealing with Linear Approximations. “A differentiable function is one for which there is a tangent line at each point on the graph. In an intuitive sense, the tangent to a curve at a point is the line that looks most like the curve at the point of tangency. Assuming that f is differentiable at a, the tangent line to the graph y = f(x) at the point (a,f(a)) is given by the equation.

y – f(a) = f ‘(a)(x – a)

This equation arises from the point-slope formula for the line passing through (a,f(a)) with slope f ‘(a).” In Pre-Calculus with discussing horizontal and vertical shifts you can easily relate back to point-slope equation of a line. You can relate point-slope equation of a line to the definition of derivative where the equation is rewritten with limits to describe the slope as the derivative. This is just a few ways that point-slope pops up in later mathematics courses. It is important to be able to form the point-slope equations of a line, as well as slope-intercept form, and being about to understand it well enough to build off of it when leading into harder concepts.

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

Point-slope equation of a line is used in movies in a huge way that most people probably never even realize. Point-slope equation of a line is used in pinhole cameras. A pinhole camera “is a simple optical imaging device in the shape of a closed box or chamber. In one of its sides is a small hole which, via the rectilinear propagation of light, creates an image of the outside space on the opposite side of the box.” In other words, let’s say we had an object, there is light constantly bouncing off the object. In the case of a pinhole camera, there is a small hole in the nearest wall/barrier which only allows light to pass through the hole. The light that makes it through the hole then hits the far wall, or image plane, creating a projection of the original image. The way point-slope equation of a line is used is first by adding a coordinate plane that has the origin centered at the pinhole. We can imagine that our scene is off to the right of the origin and the image plane is off to the left of the origin. We can choose some point in our scene to be a coordinate point in our coordinate plane. Some of the light bouncing off of that point in our scene will pass through the pinhole and land somewhere on our image plane. One of the ways we can find where it lands in our image plane is by using slope-intercept equation of a line. There is a really cool video on the khan academy website that talks all about the mathematics behind pinhole cameras. There is actually an entire curriculum called Pixar in a Box that goes through a variety of different topics and subject matter that is involved in the making of Pixar movies.

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

References:

http://matheducators.stackexchange.com/questions/9907/should-i-be-teaching-point-slope-formula-to-high-school-algebra-students

http://calculuswithjulia.github.io/precalc/polynomial.html

http://www.pinhole.cz/en/pinholecameras/whatis.html

https://www2.gcs.k12.in.us/jpeters/slope.htm

http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

# Engaging students: Finding the slope of a line

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 Brianna Horwedel. Her topic, from Algebra: finding the slope of a line.

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

Algebra vs. the Cockroaches is a great way to get students engaged in learning about slopes. The object of the game is to kill the cockroaches by figuring out what the equation of the line that they are walking on is. It progresses from simple lines such as y=5 to more complicated equations such as y=(-2/3)x+7. It allows the students to quickly recognize y-intercepts and slopes. Once finished, you can print out a “report” that tells you how many the student got correct and how many tries it took them to complete a level. This game could even be used as a formative assessment for the teacher.

http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

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

Last year, I was placed in an eighth grade classroom that was learning about slope. One of the things that really stuck out to me was that the teacher gave a ski illustration to get the students talking about slope. The illustration starts off with the teacher going skiing. She talks about how when she is going up the ski lift she is really excited and having a “positive” experience which correlates to the slope being positive. Once she gets off of the ski lift, she isn’t going up or down, but in a straight line. She talks about how she doesn’t really feel either excited or nervous because she is on flat ground. This corresponds to lines that have a slope of 0. She then proceeds to talk about how when she starts actually going down the ski slope, she hates it! This relates to the negative slope of a line. She also mentions how she went over the side of a cliff and fell straight down. She was so scared she couldn’t even think or “define” her thoughts. This is tied to slopes that are undefined. I thought that this illustration was a great way of explaining the concept of slope from a real world example. After sharing the illustration, the students could work on problems involving calculating the slope of ski hills.

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

Understanding how to find the slope of a line is crucial for mathematics courses beyond Algebra I and Algebra II. Particularly, knowing how to find the slope of a line is essential for finding tangent lines of curves. This comes in handy for Calculus when you have to use limits to determine the slope. If a student does not have a strong grasp of what slope means and what its relationship is with the graph and the equation in Algebra I, then they will have a difficult time understanding slopes of lines that are not straight.

# The Math Behind Super Mario

Forbes magazine had a great piece that can be shared with students about how mathematics is used when programming video games: http://www.forbes.com/sites/quora/2016/10/21/this-is-the-math-behind-super-mario/#57f7cf532a5c