Engaging students: Defining a function of one variable

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 Algebra: defining a function of one variable.

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

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life.  For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is $1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

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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.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus $0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

 

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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 are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

 

 

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

 

 

 

 

 

Engaging students: Slope-intercept form 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 comes from my former student Michael Garcia. His topic, from Algebra I: the point-slope intercept form of a line.

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

Writing linear function in slope-intercept form is an important topic in Algebra I. The slope-intercept form of a line consists of an independent and dependent variable, a slope, and a y-intercept. In previous courses, the students should have learned slope. They may not have learned specifically about the word “slope,” but they should have been introduced to the topic of rate of change. The students also should have been introduced to the topic of graphing, specifically graphing a point on a Cartesian coordinate plane. Finally, the students should have learned about independent and dependent variables.

The slope-intercept form of a line extends the concept of rate of change, graphing, and independent/dependent variables by “putting it all together.” Students now have their first glimpse into the world of graphing equations. They can now visually see the representation of the rate of change (or slope) between the different points of a line. The students can see how the slope is a constant rate that goes through our points of data.

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How has this topic appeared in the news?

On September 12, 2018, Apple held its annual news conference. They announced plenty of new gear and updates to their IOS, but everyone tuned in to hear about the new iPhone. The world freaked out when the new iPhone XS, iPhone XS Max, and iPhone XR were announced. So, how does this announcement relate to the slope-intercept form of a line? If we wanted to purchase a new iPhone and have a cell service plan with it, we can write a linear equation.

According to the Apple website, you can purchase the iPhone XS for $999, while you can purchase the iPhone XS Max for as little as $1,099. However, those price points are for the 64 GB model. If we are going to purchase an iPhone, we are going to buy the biggest and flashiest model. Since the iPhone XR is not currently taking pre-orders, we are going to purchase an iPhone XS Max with 512 GB of storage for $1,449. Since most people cannot afford to spend $1,449 on a single item, we are going to have monthly payments of $68.66.

According to an Apple sales representative I spoke on the phone with, there would not be a down payment on this iPhone model.* Also, according to my mom’s phone bill, it would cost $40 a month for one cell phone line that has unlimited talking, but 0.05 cents per text (my mom doesn’t text) . Our linear equation would be y= 0.05x +68.66 + 40, which is the same as y= 0.05x + 108.66.

This is a great way of viewing linear functions in slope-intercept form because it makes the problem more personable to the student.

 

*given that the customer signs up for the Apple iPhone Upgrade program

 

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

With new technology coming out every day, there are plenty of resources available for teachers. One tool that can be used to engage students with the slope-intercept form of a line is an application called GeoGebra. GeoGebra is “dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package.” There are online resources already created (very similar to Desmos), but I would mainly use their external app. I would use GeoGebra because of the many possibilities that are provided within the app.

The beauty of GeoGebra is student can utilize this app in their studying time as well. When you plot two points, the application automatically writes the equation associated with the line. This is a great way for the student to check their work when graphing/writing linear equation in slope-intercept form.

References:

https://www.apple.com/shop/buy-iphone/iphone-xs/6.5-inch-display-512gb-space-gray-sprint#01,11,31,42

https://www.geogebra.org/about

 

 

 

 

 

Engaging students: Finding x- and y-intercepts

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 Lissette Molina. Her topic, from Algebra: finding x- and y-intercepts.

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

Using this topic, students can now use word problems that involve two variables in our everyday lives. One problem that many scientists often use is population growth. In population growth, we can usually see a trend of a line and determine the slope. We initially begin with a certain population in a certain year, this is considered the y-intercept, since we start at the initial year that we consider to be at x=0. Using the slope of the line when we are speaking in terms of population decay, we may then set our y=0 to find when a population would be equal to zero. We can also consider other examples such as the depreciation of a car, or when a business’s grows out of debt and begins to profit. Word problems include, but are not limited to, problems that involve a trend and wanting to find where that trend will lead to at a certain point, x, when we are given an initially amount or reverse this operation.

 

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

This topic crosses multiple courses in mathematics. In general, knowing the x and y-intercepts of equations help students start outlining what the graph of the function might look like. This gives part of the visual representation needed to complete part of the graph. These intercepts usually also give a prediction of what the shape of the graph may look like. A fun assignment would be giving a student two points on the graph and along with the intercepts of that equation that the points belong to. Along with this, these intercepts give us the solutions of the equations. When there are not x or y-intercepts, we would now know that the solutions do not exist or at least are imaginary. Overall, x and y-intercepts help us get a better understanding of what the graphs of almost all equations must look like. This is essentially especially when we are graphing by hand.

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

 

Graphing calculators is one fun essential way of finding intercepts as well as learning functions on a calculator. When a student graphs a function on a graphing calculator, for example, the sine function, we can ask the student where they believe the graph would intercept with the x-axis. We would then ask them to find the intercepts using the calculator by pressing [2nd][trace][4] function and proceed to find the approximated x-intercepts. The student would then find that the intercepts occur at every npi/2. Essentially, using this function is an interesting way of estimating the intercepts along the graph in an interactive way. Other online graphing calculators may do this as well and give students a better understanding of where the intercepts occur.

Engaging students: Equations of two variables

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: equations of two variables.

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What interesting things can you say about the people who contributed to the discovery/and or development of this topic?

4000 years ago the Babylonians knew how to solve systems of two linear equations with two variables. The most basic notion in all of math, most of us would say, is the equation- you manipulate both sides until you obtain what you were solving for. It sounds very simple to us these days, but it actually took a LONG time to develop as a concrete concept. Not until the 16th century was the concept of an equation taken as its own mathematical entity!

Early text from Egypt, specifically in the Rhind Papyrus, 1650 BC, shows that Egyptians were able to solve linear equations of one variable. Then as late as 300 BC evidence shows that the Egyptians also know how to solve equations with TWO unknowns!

In their society, equations of two variables could be used for very useful things such as finding the length and width of their field given the area and perimeter. There were no symbols at the time, so all calculation was done mentally and verbally. Linear equations with two unknowns have been discovered and used for a very long time!

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

A popular question among students is, “How does this relate to the real world?” The unit on equations of two variables is one of the best ways to show students how equations really do help you solve real life problems. We solve them in our heads all the time without even realizing it! For example, if we are at a store shopping for clothes, and shirts cost $5 and pants cost $10, but we only have a $50 budget, we mentally set up 5x+10y=50 in our minds and try to find how many of each item we could buy that satisfies our equation. When introduced in a math class, it may not look so interesting. That is why as a teacher I would want to have a project where students can have fun setting up equations of two variables modeling a real life situation (maybe something they could even save and use for the future)! The idea is having students plan their ideal vacation! There are many variables that are in play when planning a trip. The students must consider gas, lodging, food, transportation, activity cost, etc. The idea is to have students do research and actually plan a trip they would want to take, while considering their budget and making decisions based on what their calculations allow them to do. An example of an equation they would write involves a rental car and gas. Let’s say that a rental car costs $50 per day and gas costs 2.33/gallon. Then the total cost, y, for their transportation after arriving at their destination PER DAY will be modeled by y=50+2.33x, where x is the number of gallons of gas they use in a day. Hopefully by the end of the project they will be able to make the connection between the topic and the real world, and even have a trip planned that they can take one day!

 

 

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As I was reading the news regarding hurricanes, I ran across this article that is comparing the European model with the American model in regards to which model is tracking the hurricane more accurately. “The European model collects data continuously over several hours at various observation points, measured in 4D. It does this before making a prediction for the next 10 days. This is updated two times per day.” Contrastingly, the American model only collects data 4 times per day in 3D. The part that especially caught my attention and relates to my topic is, “Additionally, ensembles are used to test different variables in forecasting equations. Since the European model uses more than twice the number of ensembles than the GFS, it is able to plug-in more numbers, thus generating more outcomes for potential hurricane paths.” The actual process may be a lot more complicated than what students are introduced to in Algebra I or II, but when reading this brief news article, students are exposed to content that they have learned in class and may be able to make a real-world connection. ‘Testing different variables in equations’ is exactly what students are doing when writing and solving equations with two variables. They are trying to come up with an equation to model a situation and find possible solutions. This is relatable to what the European model is doing with the hurricane- testing different variables in the equation and coming up with possible paths for Florence. Although it is more complicated, it is still the same concept in action!

 

References:

https://www.britannica.com/science/algebra#ref761896

https://www.coast2coast.me/carl/2015/01/28/the-road-trip-project-made-to-support-a-linear-equations-unit/

https://www.13newsnow.com/article/news/tracking-florence-euro-vs-american-model-what-is-the-difference/291-593842154

Engaging students: Adding and subtracting 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 Christian Oropeza. His topic, from Algebra: multiplying polynomials.

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

The activity would consist of each student being given a bowl with 20 pieces of candy, which has multiple colors (e.g., Skittles or M&M’s) and a worksheet, which by the end will show students how to add and subtract polynomials(Reference 1). The objective for each student is to group all of the pieces of candy by the same color. Once this has been completed, the students will write down on the worksheet for “Part 1”, how many pieces of candy are in each group. Next, the students would be given 10 more pieces of random colored candy. Then, the students will regroup the new pieces of candy and write down the new number of candies in each group for “Part 2”. For “Part 3”, students will eat(or put away) 10 of their candies randomly. Finally, the students will write down the new number of candies in each group. Then the students would be asked, “What did each one of you do to put the candies in groups?”, “what operation was used for Part 2 of the worksheet”, and “what operation was used for Part 3 of the worksheet”. The students’ responses should be somewhere along the lines of “group the candies by the same color”, “addition”, and “subtraction”. Then the students would be told to relabel each group of colored candies into a different variable. For example, green=x, red=x2, yellow=k, blue=y, etc. Knowing the previous information, the students will next repeat the Part 1, 2, and 3, but using the assigned variables instead of the colors. The purpose of this activity is to show students that each variable in a polynomial must be grouped by like terms when performing addition or subtraction.

 

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

This topic relates to previous math classes by activating students’ prior knowledge on the concept of adding and subtracting integers. This means knowing the rules of addition and the rules of subtraction. For example, students should know that a 3+2=5=3+2, but 3-2=1\ne2-3 (i.e., commutative property). Students should also know that the when subtracting a negative integer, the signs cancel out and all that is left is the addition of a positive integer (e.g., -(-2)=2). Students should also be familiar with grouping anything into specific groups. For example, if students were given colored tiles, then the students should be able to group the tiles into different colored groups. The distributive property is a topic the students should have covered before, which helps out when trying to simplify an expression involving parenthesis (e.g., 2(3+a)=6+2a. The idea of closure for integer properties and operations is the key to adding and subtracting polynomials, so students must have understood this concept prior in order to use the operation of addition and subtraction on like terms.

 

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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 2) is an excellent piece of technology to introduce this topic to the students because the website breaks down the idea of adding and subtracting polynomials piece by piece in easy manner that will help students see patterns and activate prior knowledge. With the inclusion of examples and non-examples students will learn where to minimize their potential errors. Some of the examples are animated with colors to help the more visual students understand and recognize the pattern for each problem. Another example of effective technology is the website Khan Academy (Reference 3,4,5). Khan Academy has great videos that thoroughly explains this topic. Reference 3 defines the word “polynomial” in math language by breaking the word into two words, which will help students remember and recognize this topic more easily. Also, Reference 2 goes over the vocabulary associated with adding and subtracting polynomials (e.g., coefficients, monomial, binomial, trinomial, and degree). Reference 4 goes over an example of adding a polynomial by going through step by step procedures. Reference 5 does the same thing as Reference 4, but over an example of subtracting polynomials.

References:

  1. http://www.cpalms.org/Public/PreviewResourceLesson/Preview/47832
  2. https://www.mathsisfun.com/algebra/polynomials-adding-subtracting.html
  3. https://www.khanacademy.org/math/algebra/introduction-to-polynomial-expressions/introduction-to-polynomials/v/polynomials-intro
  4. https://www.khanacademy.org/math/algebra/introduction-to-polynomial-expressions/adding-and-subtracting-polynomials/v/adding-and-subtracting-polynomials-1
  5. https://www.khanacademy.org/math/algebra/introduction-to-polynomial-expressions/adding-and-subtracting-polynomials/v/subtracting-polynomials

 

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.

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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.

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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.

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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.

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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.

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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.

 

 

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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.

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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?

 

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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.

 

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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.

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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.

 

 

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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.

 

 

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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

Beat the Odds. (n.d.). Retrieved from https://kera.pbslearningmedia.org/resource/mgbh.math.sp.beatodds/beat-the-odds/#.W4ndKuhKhPY

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.

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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.

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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.

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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.