# Engaging students: Factoring quadratic 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 comes from my former student Chris Brown. His topic, from Algebra: factoring quadratic polynomials.

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

The ability to factor quadratic polynomials is at the essence of many two-dimensional kinematic word problems that students will encounter in the future physics courses. One specific word problem that students can now solve, is, “In a tied game between the Golden State Warriors and the Houston Rockets, Steph Curry has the ball for his team. If Steph Curry is 20ft away from the basketball hoop and throws the basketball up in the air at a velocity of 3 m/s, will he be able to make the shot if 3 seconds is left on the clock and win the game for his team? Consider this to be an isolated system.” This special type of problem gives them initial distance, final distance, initial velocity, and acceleration. He student then needs to solve for time, which turns this into a quadratic scenario that requires factoring. I feel like this problem situation is super relevant to the high school age group as it seems to be popular amongst that age group, and with this problem they can extend it to any real-world scenario that searches for time when given distance and velocity.

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

When factoring quadratic equations, one of the universal methods of factoring is called factoring by grouping. Let’s identify a quadratic equation to be ax2 + bx + c = 0. When factoring by grouping, the students must first multiply ‘a’ and ‘c,’ and then find factors of the product which sum to ‘b’. Let’s call these specific factors ‘n’ and ‘m’. Thus far, this brings in students abilities to create factor trees from 3rd grade mathematics. The next step requires students to replace ‘b’ with the factors ‘n’ and ‘m,’ such that we now have ax2 + nx + mx + c = 0. Now the students have to group the ‘ax2’ term and ‘c’ with either the ‘nx’ and ‘mx’ terms in such a way that when the greatest common divisor is pulled away, what’s left is identical for each group. The ability to identify the greatest common divisor between two terms stems from what they learned in 5th grade mathematics. Then, the last step would be to factor out the common term. This entire process, which was not completed here, has used two very fundamental skills from elementary mathematics.

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

I believe Symbolab is an amazing website, that the students can use to aid them in the understanding of the process of factoring quadratic polynomials. I chose this website, because it focuses on the process of factoring and uses common language to explain their steps which the students should be aware of. Lastly, I love this website because it gives students the option to hide the steps and just see the answer. With this, the students can type in random quadratics and work towards the solution, and if they get stuck, they can see all the steps. All in all, it is an amazing way to practice the skill of factoring quadratic equations for as long as they please!

Here is the link to Symbolab: https://www.symbolab.com/solver/factor-calculator/factor%20x%5E%7B2%7D-4x%2B3%3D0

# Engaging students: Solving linear systems of equations by either substitution or graphing

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 Cameron Story. His topic, from Algebra II: solving linear systems of equations by either substitution or graphing.

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

In algebra I, students are most likely to focus on a system of two equations with two unknown variables. Teachers incorporating two differently priced objects into a word problem works great as a real-world financial problem. However, these tend to be self-similar and are arguably uninspired. More importantly, students working to discover how to solve these systems are more challenged and engaged than those who are just handed the rulebook on systems of equations.

Suppose you place your students in the place of a farmer in ancient history. They have 25 different plots of land in their field, and each plot can either have a corn plant OR a wheat plant. However, suppose the farmer requires 4 times as many corn plants than wheat plants. Task your students to find out how many corn plants and how many wheat plants are in the 25-plot field, using any method they chose.

What is interesting is that there are multiple ways to solve this problem. Students could fill a 5×5 grid with labels C and W for corn and wheat. Then, making sure that they add 4 C’s for every W, they can simply count the squares in the grid to find the answer. Just from the information given to them, they could conclude that  and that . Students could then use substitution to arrive at the answer.

While many other methods arrive at the same solution, graphing these two equations on a W vs C graph reveals the answer to the student visually. After solving each equation for C in terms of W, the intersections of the two lines is the solution. Note that when showing this solution to your students, it is an opportune time to introduce what a system of equations with no solutions (parallel lines) or infinite solutions (two of the same line) look like.

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

Students are introduced to linear equations with the usual . In this equation, we have the one dependent variable y, whose value depends on the one independent variable x. When you first introduce a system of equations with two unknown variables, whose solution is some coordinate (x, y), the learning curve could be steep the students lack the conceptual understanding to connect linear equations to systems of linear equations.

You can then reveal to your students, or have them discover on their own, that you can take a system of two linear equations, write them in such a way that they represent two separate lines in point-slope form, and then find their intersection. If they intersect, then this is your (x, y) solution. Students should know that there is no coincidence here; just manipulation of something new into something more familiar.

How can technology (graphing calculator websites or phone apps) be used to effectively engage students with this topic?

Say a student is solving a word problem that results in the following system of linear equations:

$x-y=-1$

$x-4y=-2$

First the student is required to graph this system on an x vs y graph. On a typical handheld graphing calculator, this system cannot simply be punched into the calculator as is. The student might not know this yet, but their calculator could graph it after converting to point-slope form. However, the Geogebra (https://www.geogebra.org/graphing) website and mobile-app can take the equations as shown above as inputs directly without conversion. What I like most about having the students obtain the graph first is that it takes the system and transforms it into a 2-D graph of two intersecting lines. Students should know that each of these lines can be written as  . At this point with some further guidance, the relationship between the system of equations and the lines they represent in 2 dimensions should become apparent to the students through their own independent discovery.

References:

“Free Math Apps – Used by over 100 Million Students & Teachers Worldwide.” GeoGebra, http://www.geogebra.org/.

# 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 Austin Carter. His topic, from Algebra: solving linear systems of inequalities.

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

System of equations can be solved in several ways. Changing from linear systems to systems of inequalities only means there is a range of viable answers, but the processes for solving them remain the same; graphing, elimination, substitution, or matrices. Learning how to deal with inequalities will also give us access to more interesting real world problems, because we don’t always need an exact value; sometimes we need at least this much or no more than a certain amount. For example:

• In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?
• Virenas Scout troop is trying to raise at least \$650 this spring. How many boxes of cookies must they sell at \$4.50 per box in order to reach their goal?
• The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?

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

Systems of inequalities are most easily understood with visual aid. Different colors for each equation, dotted line vs. solid line, and shading are all major components of inequalities and being able to see how each shaded region overlaps is invaluable to understanding the answer. In my experience, the easiest tool to visualize all these components is the desmos online calculator. Desmos is very user friendly and will accept equations in any form. Also, it assigns different colors to each equation entered, allows students to zoom in and out to see detail on any scale, and allows students to “click and drag” and equation line to see the (x,y) components at that location. Desmos could be used to have students create their own graphs and explain the limiting factors of their picture.

Application/Technology

Sensors are how our electronics interact with the real world. Just think about a car, and how many things are being measured and monitored constantly. Every one of those sensors is responsible for measuring something specific and making sure that measurement stays within an acceptable range. What happens if your car gets too hot? What happens if you don’t buckle your seatbelt? As autonomous vehicles come online, what happens if that vehicle gets too close to another object? All of these things are measured by sensors, and those measurements are constantly being run through software to make sure those measurements stay within an acceptable range. But how does the software determine what an acceptable range is? The software uses system of inequalities to make sure every single measurement stays within an acceptable range, and if it doesn’t it alerts the driver. The world as we know it would come crumbling down without the sensors we rely on daily, but the information those sensors collect would be useless if we didn’t have systems of inequalities to make the data meaningful.

References:

Solve Real-World Problems Using Inequalities. (2015, July 7). Retrieved September 14, 2018, from https://students.ga.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/1798/12938/Algebra_ReasoningwithEquationsandInequalities7.html

# 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 Victor Acevedo. His topic, from Pre-Algebra: probability and odds.

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

There is an online interactive game in which students practice their knowledge on probability. The game is called “Beat the Odds” and it is on PBS’s learning media website. There are two game modes: training and competition. In training mode, students must answer questions about finding the probability of various events. (rolling a die, picking from a deck of cards, etc.) For each correct answer, students earn digital money and the questions scale in difficulty. After the students feel that they have earned enough money, they can switch over to competition mode. Competition mode allows students to bet money against other bot players to see who can answer questions the most accurately. Students are asked various questions and whoever is the closest to the correct answer wins the money in the “pot.”  Students can keep playing either until they lose all their money or until they decide to get out while they are ahead.

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

Probability is an integral part to sports analysis. In baseball, batting averages are used to determine a player’s batting ability by dividing the number of successful hits by the number of at bats. This statistic can be used to determine the probability that a player may hit a ball during their next at bat. For example, a player that has a .400 would have roughly a 40% chance of hitting the ball during their next at bat. By using a player’s batting average and other stats, teams can decide how to set up their line up for going up to bat. Typically, the players with the highest batting averages take up the first 5 spots in the lineup. The first three players need to be able to make it on to a base, while the fourth player needs to be a heavy hitter than can possibly have everyone score runs. Coaches consider every players’ batting averages, as well as other stats, to help them determine their best lineup and chances of winning.

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

Quantum theory is a branch of physics that focuses on studying the different properties of atoms and particles. The most famous application of probability in quantum theory is the concept of the wave-particle duality of light. A thought experiment with Schrodinger’s cat helps to illustrate this idea in terms that most can comprehend. A cat is trapped in a box with a poison gas that is randomly released. As an observer, you cannot tell whether that is dead or alive unless you open the box. Schrodinger theorized that until the box is open, the cat is neither dead nor alive but rather in between. The concept of wave-particle duality states that light and other quantum sized particles can behave as either waves or particles depending on the observer. Theoretical physicists have concluded that this idea of fluctuating realities is an underlying truth of all probabilities. Because of this, physicists believe that either we must accept this as truth and hold true the possibility of multiple universes, or that there may be something wrong with the theory as it currently stands.

References

Fell, A. (2013, February 5). Does probability come from quantum physics? Retrieved from https://www.ucdavis.edu/news/does-probability-come-quantum-physics/

Freudenrich, C., Ph.D. (2000, July 10). How Light Works. Retrieved from https://science.howstuffworks.com/light6.htm

# Engaging students: Line Graphs

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 Tinashe Meki.  His topic, from Pre-Algebra: line graphs.

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

An engaging activity to introduce line graphs is to compare the height of boys v. girls in the classroom. I would pick 6 girls and 6 boys from the class and line each group up separately from shortest to tallest in front of the same board. Then, mark their heights by placing a point above their heads. Connect the points of the height of the boys in one color and the height of the girls in another. After the activity, I would ask students probing question to analyze and compare the data in the graph: Which group had the shortest height? Which group had the tallest height? Which group’s height increased the most? and Which two points has the greatest increase in height?  Then, create a x and y axis to provoke discussion on the naming of the axes.

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

Line graphs are the foundation for many other subjects within mathematics. For example, graphing linear equations in Algebra builds concepts of connecting two or more separate points to form a line. The ability to visualize relationship between points further enhances students’ understanding of linear equations. Understanding how to interpret a line graph based on data prepares students to be able to interpret linear equations. Topics such as slope would be easier to introduce to students who understand the concept of the “change in values”. Students would be able differentiate between increasing or decreasing slope. Although a line graph’s main purpose is to compare data, subtle lessons help students understand algebraic equations also.  Students could apply this line graph to slope by plotting different points on a coordinate plane. The students can randomly connect two points and compare the relationship of the lines they have created. They could differentiate how different lines are increasing and decreasing based on their direction. They could also compare the different rates of change between the lines.

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.

I open the weather app on my phone every morning after getting ready for class. Prior to this assignment, I never noticed how the predicted weather was displayed on my phone. The app uses a line graph to show the different temperature levels during the day and week. Weather apps and websites show students how line graphs can be used for scientific purposes. An engaging activity could be to observe how line graphs are utilized to predict change in different parts of the United States. To make things more interesting, students can be assigned different cities in the U.S to search on the weather website. Once the students have analyzed the graph, they can take turns sharing interesting trends about their cities temperature line graph.

# Engaging students: Circle Graphs

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 Phuong Trinh.  Her topic, from Pre-Algebra: circle graphs.

How has this topic appeared in pop culture?

Circle graphs, or pie charts, are regularly used to visualize data and information. As technology advances, pie charts do not appear only in statistic or scientific documents anymore. They have started to show up more regularly on social media as a mean for the younger generation to express themselves. One can easily type “funny circle graphs” into Google and get back plenty of results on various.

While the students might not be familiar with the formal documents, they can easily put themselves into the situation described in Figure 1. The students can discuss what the colors from the picture represent, as well as the meanings of their proportion. From there, the students can make connection to the data and information from more formal subjects such as statistic or science.  On other hands, showing them a funny example not only will get a chuckle out of them, it can also pique their interest in the topic.

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

Circle graphs can be used in many projects and activity.  An activity that can get the students to engage in the topic is having the students create circle graphs about themselves, more specifically, how they spent their time on an average day. The students will be given a circle graph that is divided into 24 equal sectors. Each sector represents one hour. The students will use different colors to record their activities for one day (24 hours), and provide a key to show which activity is presented by each color. The proportions of each activity will be different, depends on how much time they spent for each activity. Once the graphs are completed, the students will share and explain their circle graphs with their shoulder partner. With this activity, the students will learn how to create and interpret a circle graph while sharing who they are.

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

In this day and age, most students are familiar with technology. It is a great way to engage the students into the lesson.  The National Center for Education Statistics (NCES) provides a good website for getting students to understand the relationship between data and circle graphs (Reference A). The layout of the website is fairly simple and easy to understand with 2 tabs on the left side and 5 on the right. The left tabs include “Help” tab, which provides explanation for each element that appear on the right side of the page, and “Example” tab, which provides examples of how different types of graphs look like. The tabs on the right include “Design”, “Data”, “Labels”, “Preview”, and “Print/Save”. With the pie chart design, the site allows us to adjust the data amount, or “slices”, as well as input data as needed. On other notes, under the “Labels” tab, we can choose the type of value that will be shown (For example, value or % of total). As they explore the site, the students can compare their data with the graphs in order to make connection to how the arc length of each slice is proportional to the data it represents.

References: