# Finding the Regression Line without Calculus

Last month, my latest professional article, Deriving the Regression Line with Algebra, was published in the April 2017 issue of Mathematics Teacher (Vol. 110, Issue 8, pages 594-598). Although linear regression is commonly taught in high school algebra, the usual derivation of the regression line requires multidimensional calculus. Accordingly, algebra students are typically taught the keystrokes for finding the line of best fit on a graphing calculator with little conceptual understanding of how the line can be found.

In my article, I present an alternative way that talented Algebra II students (or, in principle, Algebra I students) can derive the line of best fit for themselves using only techniques that they already know (in particular, without calculus).

For copyright reasons, I’m not allowed to provide the full text of my article here, though subscribers to Mathematics Teacher should be able to read the article by clicking the above link. (I imagine that my article can also be obtained via inter-library loan from a local library.) That said, I am allowed to share a macro-enabled Microsoft Excel spreadsheet that I wrote that allows students to experimentally discover the line of best fit:

http://www.math.unt.edu/~johnq/ExploringTheLineofBestFit.xlsm

I created this spreadsheet so that students can explore (which is, after all, the first E of the 5-E model) the properties of the line of best fit. In this spreadsheet, students can enter a data set with up to 10 points and then experiment with different slopes and $y$-intercepts. As they experiment, the spreadsheet keeps track of the current sum of the squares of the residuals as well as the best guess attempted so far. After some experimentation, the spreadsheet can also provide the correct answer so that students can see how close they got to the right answer.

# Engaging students: Solving two-step algebra problems

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 Bonney. Her topic, from Pre-Algebra: solving two-step algebra problems.

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

A great activity to use in the classroom with students for this topic would have to be algebra tiles. The tiles are a good manipulative that can be used to improve the students’ understanding and offer contact to representative manipulation for students that are more kinesthetic learners. The algebra tiles can be used to help justify and explain the process of solving two-step equations. They were developed on the basis of two ideas: (1) we can isolate variables by using “zero pairs” and (2) equations don’t change when equal amounts of tiles are used on both sides of the equation. Algebra tiles come in different colors and sizes, which can be used to represent different parts of an equation that can help students solve two-step algebra problems.  I think this would be a fun and interactive activity to help students learn and understand how to go about solving these types of problems.

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

Once a student gets to a certain grade level, they constantly start building upon what they learn. This material can be carried into high school and even college level courses.  Before a student learns two-step equations, they must master one-step equations, and even before that they need to understand basic arithmetical operations. Once mastery has been achieved, students will move onto solving larger polynomials, which can later be used in future algebra, geometry, and calculus courses. Another interesting use for two-step algebra problems is for future science and even computer science courses. In science, let’s say physics or chemistry, the students can use the two-step method for solving how fast a ball fell from a rooftop or for solving how fast a chemical evaporated at a certain temperature. Now in computer science students can learn how to develop algebraic functions in a computerized setting.

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

Rene’ Descartes, born in March of 1596, was a French mathematician, philosopher, and scientist. He is widely known for the statement, “I think, therefore I am,” deriving it from the foundation of intuition that, when he thinks, he exists. After obtaining a degree in law, his father wanted him to join Parliament, but sadly he was only 20 and the minimum age to join was 27. In turn, he moved to the Netherlands where he was influenced to study science and mathematics. During this time he formulated a common method of logical reasoning, centered on mathematics, which can be related to all sciences. This method is discussed in Discourse on Method, and is comprised of four rules: “(1) accept nothing as true that is not self-evident, (2) divide problems into their simplest parts, (3) solve problems by proceeding from simple to complex, and (4) recheck the reasoning.” We use these rules everyday when directly apply them to mathematical procedures.

References:

“Rene Descartes”. Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia

Britannica Inc., 2016. Web. 07 Sep. 2016 <https://www.britannica.com/biography/Rene-

Descartes>.

# My Favorite One-Liners: Part 77

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

At the end of every semester, instructors are often asked “What do I need on the final to make a ___ in the course?”, where the desired course grade is given. (I’ve never done a survey, but A appears to be the most desired course grade, followed by C, D, and B.) Here’s the do-it-yourself algorithm that I tell my students, in which the final counts for 20% of the course average.

Let $F$ be the grade on the final exam (as I write a big F on the chalkboard). [groans] After all, final starts with $F$, and it’s important to assign variable names that make sense.

Also, let $D$ be the up-to-date course average prior to the final. [more groans]

This gives us the course average. Just to be nice, let’s call that $A$. [sighs of relief]

So $A = 0.2F + 0.8D$.

More seriously, here’s a practical tip for students to determine what they need on the final to get a certain grade (hat tip to my friend Jeff Cagle for this idea). It’s based on the following principle:

If the average of $x_1, x_2, \dots x_n$ is $\overline{x}$, then the average of $x_1 + c, x_2 + c, \dots, x_n + c$ is $\overline{x} + c$. In other words, if you add a constant to a list of values, then the average changes by that constant.

As an application of this idea, let’s try to guess the average of $78, 82, 88, 90$. A reasonable guess would be something like $85$. So subtract $85$ from all four values, obtaining $-7, -3, 3, 5$. The average of these four differences is $(-7-3+3+5)/4 = -0.5$. Therefore, the average of the original four numbers is $85 + (-0.5) = 84.5$.

So here’s a typical student question: “If my average right now is an $88$, and the final is worth $20\%$ of my grade, then what do I need to get on the final to get a $90$?” Answer: The change in the average needs to be $+2$, so the student needs to get a grade $+2/0.2 = +10$ points higher than his/her current average. So the grade on the final needs to be $88 + 10 = 98$.

Seen another way, we’re solving the algebra problem

$88(0.8) + x(0.2) = 90$

Let me solve this in an unorthodox way:

$88(0.8) + x(0.2) = 88 + 2$

$88(0.8) + x(0.2) = 88(0.8+0.2) + 2$

$88(0.8) + x(0.2) = 88(0.8) + 88(0.2) + 2$

$x(0.2) = 88(0.2) + 2$

$x = \displaystyle \frac{88(0.2)}{0.2} + \frac{2}{0.2}$

$x = 88 + \displaystyle \frac{2}{0.2}$

This last line matches the solution found in the previous paragraph, $x = 88 + 10 = 98$.

# My Favorite One-Liners: Part 21

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Sometimes, just every once in a blue moon, something in mathematics doesn’t appear right to students at first glance. For example, take the common notation

$(a,b)$

What does this symbol mean? Sadly, it depends on the context.

Sometimes, it means a point in the Cartesian plane whose first coordinate is $a$ and whose second coordinate is $b$.

Other times, it could mean the set $\{x : a < x < b\}$, or the interval between $a$ and $b$ that does not contain the endpoints.

You’d think that, by now, mathematicians would’ve figure out a way to not denote these two completely different things with the same symbol. Indeed, I’ve seen textbooks that use $]a,b[$ to denote the open interval between $a$ and $b$ to avoid this duplication; however, this notation hasn’t been widely adopted by the mathematical community.

So here’s my quip when something like this comes up. Sometimes, a young child will come crying to her parents to complain about the injustices in the world, and the child may be right. But all the parent can say is, “Sorry, sweetheart, but sometimes life isn’t fair.” And I’ll act this out, talking to an imaginary child as I look down to the floor.

To complete the quip, I’ll then turn to my class and conclude, “Sorry, sometimes life isn’t fair.” It doesn’t make much sense, but we’re stuck with it.

# My Favorite One-Liners: Part 8

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

1. Arithmetic/Algebra: $a \cdot (b + c) = a \cdot b + a \cdot c$. This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
2. Algebra: If $a,b > 0$, then $\sqrt{ab} = \sqrt{a} \sqrt{b}$.
3. Algebra: If $a,b > 0$ and $x$ is any real number, then $(ab)^x = a^x b^x$.
4. Precalculus: $\displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i$.
5. Precalculus: $\displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i$.
6. Calculus: If $f$ is continuous at an interior point $c$, then $\displaystyle \lim_{x \to c} f(x) = f(c)$.
7. Calculus: If $f$ and $g$ are differentiable, then $(f+g)' = f' + g'$.
8. Calculus: If $f$ is differentiable and $c$ is a constant, then $(cf)' = cf'$.
9. Calculus: If $f$ and $g$ are integrable, then $\int (f+g) = \int f + \int g$.
10. Calculus: If $f$ is integrable and $c$ is a constant, then $\int cf = c \int f$.
11. Calculus: If $f: \mathbb{R}^2 \to \mathbb{R}$ is integrable, $\iint f(x,y) dx dy = \iint f(x,y) dy dx$.
12. Calculus: For most differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ that arise in practice, $\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}$.
13. Probability: If $X$ and $Y$ are random variables, then $E(X+Y) = E(X) + E(Y)$.
14. Probability: If $X$ is a random variable and $c$ is a constant, then $E(cX) = c E(X)$.
15. Probability: If $X$ and $Y$ are independent random variables, then $E(XY) = E(X) E(Y)$.
16. Probability: If $X$ and $Y$ are independent random variables, then $\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$.
17. Set theory: If $A$, $B$, and $C$ are sets, then $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
18. Set theory: If $A$, $B$, and $C$ are sets, then $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

1. Algebra: $(a+b)^x \ne a^x + b^x$ if $x \ne 1$. Important special cases are $x = 2$, $x = 1/2$, and $x = -1$.
2. Algebra/Precalculus: $\log_b(x+y) = \log_b x + \log_b y$. I call this the third classic blunder.
3. Precalculus: $(f \circ g)(x) \ne (g \circ f)(x)$.
4. Precalculus: $\sin(x+y) \ne \sin x + \sin y$, $\cos(x+y) \ne \cos x + \cos y$, etc.
5. Precalculus: $\displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right)$.
6. Calculus: $(fg)' \ne f' \cdot g'$.
7. Calculus $\left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}$
8. Calculus: $\int fg \ne \left( \int f \right) \left( \int g \right)$.
9. Probability: If $X$ and $Y$ are dependent random variables, then $E(XY) \ne E(X) E(Y)$.
10. Probability: If $X$ and $Y$ are dependent random variables, then $\hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y)$.

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

I wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

# 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 Matthew Garza. His topic, from Algebra: defining a function of one variable.

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

Being able to define a function of one variable is necessary for creating a model that describes the most basic phenomenon in math and science. In math, understanding these parent functions is crucial to understanding more complicated functions and, by considering some variables as temporarily fixed, multivariable equations and systems of equations can be easier to understand. In science, we often observe functions of a single variable.  In fact, even if there are multiple variables coming into play, a good lab will likely control all but one variable, so that we can understand the relationship with respect to that single variable – a function.

Consider in science, for example, the ideal gas law: PV = nRT, where P is pressure, V is volume, n is the quantity in moles of a gas, R is the gas constant, and T is temperature.  This law, taught in high school chemistry, is not taught from scratch.  The proportional, single-variable functions that make up the equation are observed individually before the ideal gas law is introduced. Students will probably be taught Boyle’s, Charles’, Gay-Lussac’s, and Avogadro’s laws first. Boyle’s law states pressure and volume are inversely proportional (for a fixed temperature and quantity of gas).  This law can be demonstrated in one lab by clamping a pipette with some water and air inside, thus fixing all but two variables.  Pressure is applied to the pipette and the volume of air is measured using the length of the air column in the pipette.  Students must then evaluate volume V as a function of the single variable pressure P.  It should be noted that the length of the air column is measured, while the diameter of the pipette is fixed, thus volume must be calculated as a function of the single variable length.  Understanding the single variable, proportional and inversely proportional relationships is crucial to understanding the ideal gas law itself.

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.

Generally speaking, Khan Academy has great videos to help understand math concepts.  Although it’s a little dry, this “Introduction to Functions” video is clear, concise, and touches on several ideas that I was having trouble tying in to every example.  This introductory video begins with the basic concept of a function as a mapping from one value to another single value.  The first examples it uses are a piece-wise function and a less computational function that returns the next highest number beginning with the same letter.  At first I didn’t like that these functions were discontinuous, but this actually gives something else to discuss.  The video links back prior knowledge, explaining that the dependent variable y that students are familiar with is actually a function of x, and represents the two in a table.  The last couple minutes of the video address the fundamental property that a function must produce unique outputs for each x, or it is a relationship.

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

One idea might be to examine any function in which time is the independent variable.  Basic concepts of motion in physics can supplement an activity – Have groups evaluate position and speed with respect to time of, say, a marble or hot wheels car rolling down a ramp.  Using a stop watch and marking distance on an inclined plane, students could time how long it took to reach certain points and create a graph over time of displacement.  This method might result in some students graphing time as a function of displacement, which could lead to an interesting discussion on independence and dependence, and why it might be useful to view change as a function of time.

Technology could supplement such a lesson as to avoid confusion over whether distance is a function of time or vice versa.  Using motion sensor devices to collect data, such as the CBR2, students can use less time collecting and plotting data and more time examining it.  Different trials resulting in different graphs can lead to discussion on how to model such motion as a function of time – letting an object sit still would result in a constant graph, something rolling down an incline will give a parabolic graph (until the object gets too close to a terminal velocity).

To add variety, students can examine what a graph looks like if they move toward and away from the CBR2 or try to reproduce given position graphs.  This may result in the same position at different times, but since an object can be in only one position at a given time, the utility of using position as a function of time can be represented. Sporadic motion, including changes in speed and direction (like moving back and forth and standing still) also allow discussion of piecewise functions, and that functions don’t necessarily have to have a “rule” as long as only one output is assigned per value in the domain.

# Engaging students: Graphs of 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 Anna Park. Her topic, from Algebra: graphs of linear equations.

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

• Have the students enter the room with all of the desks and chairs to the wall, to create a clear floor. On the floor, put 2 long pieces of duct tape that represent the x and y-axis. Have the students get into groups of 3 or 4 and on the board put up a linear equation. One of the students will stand on the Y-axis and will represent the point of the Y-Intercept. The rest of the students have to represent the slope of the line. The students will be able to see if they are graphing the equation right based on how they form the line. This way the students will be able to participate with each other and get immediate feedback. Have the remaining groups of students, those not participating in the current equation, graph the line on a piece of paper that the other group is representing for them. By the end of the engage, students will have a full paper of linear equation examples. The teacher can make it harder by telling the students to make adjustments like changing the y intercept but keeping the slope the same. Or have two groups race at once to see who can physically graph the equation the fastest. Because there is only one “graph” on the floor, have each group go separately and time each group.
• Have the students put their desks into rows of even numbers. Each group should have between 4 and 5 students. On the wall or white board the teacher has an empty, laminated graph. The teacher will have one group go at a time. The teacher will give the group a linear equation and the student’s have to finish graphing the equation as fast as possible. Each group is given one marker, once the equation is given the first student runs up to the graph and will graph ONLY ONE point. The first student runs back to the second student and hands the marker off to them. That student runs up to the board and marks another point for that graph. The graph is completed once all points are on the graph, the x and y intercepts being the most important. If there are two laminated graphs on the board two groups can go at one time to compete against the other. Similar to the first engage, students will have multiple empty graphs on a sheet of paper that they need to fill out during the whole engage. This activity also gives the students immediate feedback.

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

Sir William Rowan Hamilton was an Irish mathematician who lived to be 60 years old. Hamilton invented linear equations in 1843. At age 13 he could already speak 13 languages and at the age of 22 he was a professor at the University of Dublin. He also invented quaternions, which are equations that help extend complex numbers. A complex number of the form w + xi + yj + zk, where wxyz are real numbers and ijk are imaginary units that satisfy certain conditions. Hamilton was an Irish physicist, mathematician and astronomer. Hamilton has a paper written over fluctuating functions and solving equations of the 5th degree. He is celebrated in Ireland for being their leading scientist, and through the years he has been celebrated even more because of Ireland’s appreciation of their scientific heritage.

Culture: How has this topic appeared in pop culture?

An online video game called “Rescue the Zogs” is a fun game for anyone to play. In order for the player to rescue the zogs, they have to identify the linear equation that the zogs are on. This video game is found on mathplayground.com.

References

https://www.teachingchannel.org/videos/graphing-linear-equations-lesson

https://en.wikipedia.org/wiki/William_Rowan_Hamilton

http://www.mathplayground.com/SaveTheZogs/SaveTheZogs.html

# 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 Madison duPont. Her topic, from Algebra: equations of two variables.

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

Problem: It’s tax free weekend (clothes are tax free) and you want to spend exactly $15 (so you can get$5 back from a $20 bill) on only shirts and shorts. Shirts are on sale for$4 and shorts are on sale for $3. 1. Write an equation to model this situation. 2. Determine how many shorts and shirts you should buy to spend exactly$15.

This problem does a good job of introducing a relatable and realistic situation that can be written as an equation with 2 unknowns. The mathematical portion of solving this is also approachable using conceptual strategies such as drawings, counting in groups, or more calculative tactics like trial and error with multiplication and addition, or even more advanced concepts like knowledge of division algorithm. The use of traditional variables is not even necessary to write an equation as the students can use pictures or words next to the coefficients to represent the unknowns. Because there are multiple levels of approaching the problem both in creating an equation and in finding the unknowns, this is a good exercise to have them explore the topic and gain conceptual understanding.

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

Activity: Have students sit in groups (2-4). Have 10 di-cut images of apples and 10 di-cut images of bananas (or oranges, etc.) in the center of the group to serve as manipulatives. On each of the apple di-cuts write $.10 in the center and on each of the banana (or other fruit) write$.20. Tell the students they need to find a way to spend exactly \$1.00 (using at least one of each fruit).

This activity allows students to explore the concept of considering two unknowns in the same situation in a tactile and conceptual way before encountering the mysterious algebraic equation. Students sharing answers can demonstrate that there are different possibilities and therefore the number of fruits is truly variable and can be written as an equation.

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

An equation of two variables will be the stepping stone to linear equations and functions. When the equation is solved for “y” in terms of “x” you will get a linear function. Having a decent conceptual understanding of two-variable equations and being familiar with manipulating the equations will help students begin to understand notions of inputs and outputs and to see that having one variable will allow you to find the other. All of those topics will lead to the graphing of functions and taking algebraic work to a visual type of mathematics. Equations of three variables will also be a future topic related to this one as well as solving systems of equations for both two variable and three variable equations. Knowing how much will be built off of this topic makes equations of two variables much more appealing for teachers to teach the topic well and for students to learn conceptual and mathematical components of this topic well.

# Engaging students: Multiplying binomials

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 Lucy Grimmett. Her topic, from Algebra I: multiplying binomials like $(a+b)(c+d)$.

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

There are tons of activities that could be created with this topic. The first thing that came to mind was giving each student a notecard when they walked in the room. Each notecard would have a binomial on it. Students would be asked to find a partner in the classroom and multiply their binomials together. They would be able to assist one another, discuss possible misconceptions, and ask questions that they might not want to ask in front of an entire class. This could be a quick 5-minute warm up at the beginning of class, or could turn into a longer activity depending on how many partners you want each student to have. This wouldn’t involve much work on the teacher’s part; all you would have to do is create 30 differing binomials. If you feel the need to create a cheat sheet with answers to every possible pair you can, but that would involve more work then necessary.

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

In previous courses and chapters in algebra, students are set up with knowledge of combining like terms. The most common idea of combining like terms is adding or subtracting, for example 2-1=1 or 2+1=3. Students don’t realize that in the elementary school they are combining like terms. This is a key tool used when multiplying binomials. As future math teachers, we know that when we see 2x + 3x we can quickly combine these numbers to get 5x. This simplifies an equation. Students will struggle with this at first because they will not be used to having a variable, such as x, mixed into the equation, literally. This will be a similar issue when discussing multiplying binomials. Students will have to get used to seeing  (4x+1)(3x-8) and turning it into the longer version $12x^2+3x-32x-8$ and then finding the like terms to simplify again, creating the shorter version $12x^2-29x-8$. This is an extension of like-terms.

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

Algebra tiles are a great tool for students and teachers to use. Even better is an online algebra tile map. This allows a teacher to show students how to use algebra times from a main point, such as a projector, rather then walking around the room and individually showing them. Teachers can have students work individually with their iPad’s (if they have them) or use actual algebra tiles. This would be a great engagement piece for a day when students are recapping distributing or “FOIL” as many teachers like to call it. This can also be a great discovery lesson when students are learning how to multiply binomials. This all depends on if students have used algebra tiles before, and how comfortable the teacher is with implementing a lesson like this in the classroom. Another idea is pairing students and giving them binomials to multiply, which they will present to the class in a short presentation using their online algebra tile tool.

Here’s the link for the online algebra tiles:

http://technology.cpm.org/general/tiles/

# Engaging students: Parabolas

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 Lisa Sun. Her topic, from Algebra II: parabolas.

How has this topic appeared in high culture?

Parabola is a special curve, shaped like an arch. Any point on a parabola is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). Today, I will be presenting the parabolas’ unique shape to the class. Parabolas are everywhere in our society today. Students just don’t know it yet because no one has informed them. Parabolic structures can be seen in buildings, mosaic art, bridges, and many more. One that I’m going to share with the class is going to be roller coasters. Similar to this image below:

This specific roller coaster is The Behemoth. It is a steel coaster located in Canada’s Wonderland in Vaughan, Ontario, Canada. I will first present this photo to the class and ask the following:

• What do you notice that’s repeating in this roller coaster?
• Do you think you’ve seen this similar structure anywhere else? Where?

–Present definition of Parabola–

• Does this roller coaster have any parabolic structure? Where?

With these guiding questions, I want the students to be familiar with how a parabola looks like and that we can see them in our real world other than school.

How has this topic appeared in the news?

https://www.sciencedaily.com/releases/2016/07/160713143146.htm

This link above is a recent article from Science News on how an engineer from the University of Warwick discovered how to build bridges and buildings to enhance the safety and long durability without the need for repair or restructuring by the use of inverted parabolas. Using inverted parabolas and a design process called “form finding”, engineers will be able to take away the main points of weakness in structures. I believe this is a remarkable discovery that must be shared with students. Math is truly used in our everyday life and can definitely benefit the society today by how fast our technology is advancing.

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

Prezi favors visual learning and works similar to a graphic organizer or a mind map. It helps students to explore a canvas of small ideas then turning it into a bigger picture or vice versa. Prezi is a great tool to maintain an interactive classroom and creates stunning visual impact on students keeping them engaged in the lecture.

http://www.bbcactive.com/BBCActiveIdeasandResources/UsingPreziInEducation.aspx

Above is a link of a Prezi presentation of parabolas in roller coasters. This is a great example as to what I would create for my students to provide them the information of a parabola.

Behemoth

https://www.mathsisfun.com/definitions/parabola.html

https://www.sciencedaily.com/releases/2016/07/160713143146.htm

https://prezi.com/pwkzfddbu4bu/parabolas-in-roller-coasters/

http://www.bbcactive.com/BBCActiveIdeasandResources/UsingPreziInEducation.aspx