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 again comes from my former student Kelley Nguyen. Her topic, from Algebra: slope-intercept form of a line.

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

 

The slope-intercept form of a line is a linear function. Linear functions are dealt with in many ways in everyday life, some of which you probably don’t even notice.

 

One example where the slope-intercept form of a line appears in high culture is through music and arts. Suppose a band wants to book an auditorium for their upcoming concert. As most bands do, they meet with the manager of the location, book a date, and determine a payment. Let’s say it costs $1,500 to rent the building for 2 hours. In addition to this fee, the band earns 20% of each $30 ticket sold. Write an equation that determines whether the band made profit or lost money due to the number of tickets sold – the equation would be y = 0.2(30)x – 1500, where y is the amount gained or lost and x is the number of tickets sold that night. This can also help the band determine their goal on how many tickets to sell. If they want to make a profit of $2,000, they would have to sell x-many tickets to accomplish that.

 

In reality, most arts performances make a profit from their shows or concerts. Not only do mathematicians and scientists use slope-intercept of a line, but with this example, it shows up in many types of arts and real-world situations. Not only does the form work for calculating cost or profit, it can relate to the number of seats in a theatre, such as x rows of 30 seats and a VIP section of 20 seats. The equation to find how many seats are available in the theatre is y = 30x + 20, where x is the number of rows.

 

 

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

 

A great way to engage students when learning about slope-intercept form of a line is to use Geometer’s Sketchpad. After opening a graph with an x- and y-axis, use the tools to create a line. From there, you can drag the line up or down and notice that the slope increases as you move upward and decreases as you move downward. Students can also find the equation of the line by selecting the line, clicking “Measure” in the menu bar, and selecting “Equation” in the drop-down list. This gives the students an accurate equation of the line they selected in slope-intercept form. Geometer’s Sketchpad allows students to experiment and explore directions of lines, determine whether or not it has an increasing slope, and help create a visual image for positive and negative slopes.

 

Also, with this program, students can play a matching game with slope-intercept equations and lines. You will instruct the student to create five random lines that move in any direction. Next, they will select all of the lines, go to “Measure” in the menu bar, and click “Equation.” From there, it’ll give them the equation of each line. Then, the student will go back and select the lines once again, go to “Edit” on the menu bar, hover over “Action Buttons,” and select “Hide/Show.” Once a box comes up, they will click the “Label” tab and type Scramble Lines in the text line. Next, the lines will scramble and stop when clicked on. Once the lines are done scrambling, the student could then match the equations with their lines. This activity gives the students the chance to look at equations and determine whether the slope is increasing and decreasing and where the line hits the y-axis.

 

 

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

 

With this topic, I could definitely do a project that consists of slope-intercept equations, their graphs, and word problems that involve computations. For example, growing up, some students had to earn money by doing chores around the house. Parents give allowance on daily duties that their children did.

 

The project will give the daily amount of allowance that each student earned. With that, say the student needed to reach a certain amount of money before purchasing the iPad Air. In part one of the project, the student will create an equation that reflects their daily allowing of $5 and the amount of money they have at the moment. In part two, the student will construct a graph that shows the rate of their earnings, supposing that they don’t skip a day of chores. In part three, the students will answer a series of questions, such as,

  • What will you earn after a week?
  • What is your total amount of money after that week?
  • When will you have enough money to buy that iPad Air at $540 after tax?

 

This would be a short project, but it’s definitely something that the students can do outside of class as a fun activity. It can also help them reach their goals of owning something they want and making a financial plan on how to accomplish that.

 

References

 

 

 

 

Engaging students: Approximating data by a straight 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 Delaina Bazaldua. Her topic, from Algebra: approximating data to a straight line.

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

One of my favorite shows to watch is How I Met Your Mother. I specifically chose this topic for this class because of how it relates to an episode of the show. A piece of the episode that I’m referring to is shown in the YouTube video:

Barney, one of the main characters, describes the graph as the Crazy/Hot Scale. According to him, a girl cannot be crazier than hot which means she has to be above the diagonal straight line. This relates to the topic because one can approximate data by the straight line that Barney gives the viewer. I think the students will be able to relate to this and also find it humorous. Because this video has both of these characteristics, they will be able to remember the concept for upcoming homework and tests which is ultimately the most important part of math: understanding it and being able to recall it.

 

 

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How has this topic appeared in the news?

Most lines are drawn for the purpose of seeing if there is a relationship between the x and y axis and trying to figure out if you can approximate data from the straight line that is drawn. Graphs like this are found all over the news, and they often relate to natural disasters. For example, this linear regression, http://d32ogoqmya1dw8.cloudfront.net/images/quantskills/methods/quantlit/bestfit_line.v2.jpg, describes floods. In http://serc.carleton.edu/mathyouneed/graphing/bestfit.html, where the picture is found, describes more activities that can be used to create a linear regression which can be converted into a straight line. These examples of straight lines can be used to find more data that isn’t necessarily shown from the points that are plotted. The examples the website gave are: flood frequency curves, earthquake forecasting, meteorite impact prediction, earthquake frequency vs. magnitude, and climate change. This is beneficial for math because it allows students to realize that math isn’t abstract like it is often perceived to be, but rather, it is used for something very important and something that occurs several times a year such as natural disasters and weather.

 

 

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

One of the purposes for teachers to teach is for students to learn what they should for the following year so they can be successful in the particular topic. When it comes to approximating data based on a straight line, the knowledge a student learns in algebra will carry them through statistics, physics, and other higher math and science classes. Linear regression is shown in statistics as one can see in this statistics website: http://onlinestatbook.com/2/regression/intro.html while physics is represented in the physics website: http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_DataAnalysisMethods.xml. A lot can be predicted from these straight lines which is why these graphs aren’t foreign to upper level math and science classes. As I stated before, a lot can be predicted from the graph where data points aren’t necessarily on the trend the data is setting which allows students to expect what would occur at a particular x or y value. A background in this area can help students through the rest of school and perhaps even the rest of their life in some cases.

 

References:

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

http://serc.carleton.edu/mathyouneed/graphing/bestfit.html

http://onlinestatbook.com/2/regression/intro.html

http://dev.physicslab.org/Document.aspx?doctype=3&filename=IntroductoryMathematics_DataAnalysisMethods.xml

 

Engaging students: Solving one-step linear equations

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 Jessica Trevizo. Her topic, from Pre-Algebra: solving one-step linear equations.

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A2. How could you as a teacher create an activity or project that involves your topic?

            Many students have played “Around the World” at one point in their elementary childhood, or have at least heard of the game. Around the World is an activity that is commonly used by elementary school teachers when they are teaching multiplication. Students are supposed to sit in the form of a circle. One person is chosen to attempt to go around the world. He/she will stand behind a student and will compete against the student that is sitting down. Once both students are ready the teacher holds up a multiplication card. The student who responds with the correct answer first gets the chance to move on to the next person. If the student who is standing up loses then he/she gets to sit down while the other student who obtained the correct answer advances. Every person has to attempt the problem on a sheet of paper, but they are not allowed to call out the answer. The student who “goes around the world” first is the winner. If a student is not able to complete the entire circle then the student who advanced the farthest is the winner. The same idea will be used after the students have learned how to solve one step linear equations.  After having a deep conceptual understanding of the topic it is very important for the students to keep practicing problems.  Around the World allows the students to keep practicing in an entertaining way. The students should be able to solve the equations within 30 seconds since it only requires one step to solve. The ability to use calculators with this activity will vary depending on the level of difficulty of the problems as well as the teacher.

 

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B1. How can this topic be used in your students’ future courses in mathematics or science?

            Being able to solve one step linear equations is an important skill that every student should acquire. After the students learn how to solve one step linear equations they are expected to be able to solve multi-step equations, solve absolute value equations, solve inequalities, finding the side lengths of a shape given a certain area in geometry, etc. If the students are not able to master solving one step linear equations then they will have a very difficult time in other math courses.

In geometry the Pythagorean Theorem requires the skill to solve one step equations. Students are expected to solve for the missing variable in order to find the missing side length of a right triangle. In Algebra II the students are required to manipulate equations in order to solve systems of linear equations through substitution. Also this basic skill is necessary when finding the inverse of a function. This topic is also used in physics. For example, if the student is asked to find the acceleration of an object given only the force and the mass, then it involves using Newton’s second law which states Force=mass*acceleration.

 

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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This website is an amazing tool that allows the students to visualize how to solve linear equations using algebra tiles. If the teacher decides to teach this lesson using algebra tiles in the classroom, then this website will allow the students to continue to practice at home. Also, the website automatically lets the student know if he/she responded correctly. Obtaining quick results allows the student to know whether or not they truly understand how to solve the equations as opposed to having a worksheet with 50 problems for homework and not knowing if the same mistake was repeated.  Also, by using the online algebra tiles the students are able to understand the zero pair concept and see how it is being applied. This website can also be used for other algebra topics such as factoring, the distributive property, and substitution.

http://illuminations.nctm.org/Activity.aspx?id=3482