Another poorly written word problem (Part 10)

The current women’s world record for the long jump is 7.52 meters, or 24 feet, 8 inches.

Source: https://www.facebook.com/FloTrack/photos/a.432324889444/10156740897914445/?type=3&theater

My Favorite One-Liners: Part 114

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.

I’ll use today’s one-liner whena step that’s usually necessary in a calculation isn’t needed for a particular example. For example, consider the following problem from probability:

Let X be uniformly distributed on \{-1,0,1\}. Find \hbox{Cov}(X,X^2).

The first step is to write \hbox{Cov}(X,X^2) = E(X \cdot X^2) - E(X) E(X^2) = E(X^3) - E(X) E(X^2). Then we start computing the expectations. To begin,

E(X) = (-1) \cdot \displaystyle \frac{1}{3} + 0 \cdot \displaystyle \frac{1}{3} + 1 \cdot \displaystyle \frac{1}{3} = 0.

Ordinarily, the next step would be computing E(X^2). However, this computation is unnecessary since E(X^2) will be multiplied by E(X), which we just showed was equal to 0. While I might calculate E(X^2) if I thought my class needed the extra practice with computing expectations, the answer will not ultimately affect the final answer. Hence my one-liner:

To paraphrase the great philosopher The Rock, it doesn’t matter what E(X^2) is.

P.S. This example illustrates that the covariance of two dependent random variables (X and X^2) can be zero. If two random variables are independent, then the covariance must be zero. But the reverse implication is false.

My Favorite One-Liners: Part 113

I tried a new wisecrack when teaching my students about Euler’s formula. It worked gloriously.

Source: https://www.facebook.com/MathematicalMemesLogarithmicallyScaled/photos/a.1605246506167805.1073741827.1605219649503824/2062654510427000/?type=3&theater

My Favorite One-Liners: Part 112

This was also the story of my childhood.

Source: https://www.facebook.com/MathematicalMemesLogarithmicallyScaled/photos/a.1605246506167805.1073741827.1605219649503824/2116636955028755/?type=3&theater

My Favorite One-Liners: Part 111

I tried a new wise-crack in class recently, and it was a rousing success. My math majors had trouble recalling basic facts about tests for convergent and divergent series, and so I projected onto the front screen the Official Repository of all Knowledge (www.google.com) and searched for “divergent series” to “help” them recall their prior knowledge.

Worked like a charm.

https://www.google.com/search?q=divergent+series

My Favorite One-Liners: Part 110

I overheard the following terrific one-liner recently. A teacher was about to begin a lecture on exponential growth. His opening question to engage his students: “What does your bank account have to do with bacteria… other than they both might be really tiny?”

My Favorite One-Liners: Part 109

I tried a new joke in class recently; it worked gloriously.

I wrote on the board a mathematical conjecture that has yet to be proven or disproven. To emphasize that nobody knows the answer yet despite centuries of effort, I told the class, “If you figure this out, call me and call me collect,” writing my office phone number on the board.

To complete the joke, I said, “Yeah, this is crazy. So here’s my number…”

I thoroughly enjoyed my students’ coruscating groans before I could complete the punch line.

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 again comes from my former student Sarah McCall. Her topic, from Algebra: multiplying binomials.

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B2. How does this topic extend what your students should have learned in previous courses?

My hope is that this topic may be easier to understand if student’s can first recall an easier concept that they have already mastered, and then build upon that foundation to learn new skills. For example, at this point students should have already learned the distributive property. To introduce this new concept, I would begin by writing 4(x-5)=4 on the board and asking students what the very first step would be to solve for x. They should know to start by distributing the four to both x and -5, to get 4x-20=4. After completing a few similar examples as a class and/or in groups, then the idea of multiplying binomials would be introduced. This way, students are less intimidated when presented with new material, and they will have a good understanding of how to distribute to each term.

 

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Teaching students some of the history behind what they are learning can be a great engaging tool. In this case it is helpful to know where the foil method first originated. I would incorporate this by discussing how it first was used in 1929; in William Bentz’ Algebra for Today. In Algebra for Today, Bentz was the first person to mention the “first terms, outer terms, inner terms, last terms” rule. Students should be knowledgeable about the history behind the math they are using, so that they realize the importance of this method. I also believe that it will be cool for students to see how a method developed is still relevant 88 years later. This technique was created in order to provide a memory aid, or “mnemonic device” to help students learn how to multiply binomials. The fact that it is still being used even today proves what an influential concept it was at its time, and throughout the years.

 

green lineE1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I am a huge fan of incorporating technology in the classroom, and YouTube is especially great because most students already use YouTube outside of school. The following clip (stopped at 1:48) provides a clear, concise explanation and demonstration of the FOIL method for multiplying binomials. It explains how factoring and foiling are related, and shows students which order to distribute in (first, outer, inner, last). The acronym FOIL is easy for students to remember, and gives them something that they can write down each time they complete a problem to help them distribute properly. Additionally, the clip is just under two minutes, which is the perfect time to ensure that students don’t zone out or lose interest before the end of the video. I would choose to follow up this video by completing a few examples as a class, emphasizing the four steps of foiling as mentioned in the video and how to use them.

References

http://pballew.blogspot.com/2011/02/origin-of-foil-for-binomial.html

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 again comes from my former student Kelly Bui. Her topic, from Algebra: multiplying polynomials.

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A2. How could you as a teacher create an activity or project that involves your topic?
The main idea of the activity will be finding an expression to represent the area of the border given the dimensions of the outer rectangle and the inner rectangle. The students will need to know how to multiply binomials and add or subtract polynomials. Therefore, this activity would be towards the end of the unit. Students will be asked to roam around the classroom or the hallway in search of items that already have dimensions labeled. For example, in the hallway there may be a bulletin board with the dimensions (2x² – 7) for the length and (3x – 4) for the width. Inside of the bulletin board, there will be the dimensions of a smaller rectangle. The question will be asked: What expression will represent the area I want to cover if I want to cover the only the border with paper?
Students may work in partners or groups to put minds together to solve this problem. Every object labeled with dimensions will be in the shape of a rectangle and the math involved will require students to multiply binomials and subtract polynomials.

 

green lineB2. How does this topic extend what your students should have learned in previous courses?
Before students learn to add and subtract polynomials, they learn how to combine like terms such as 3x and 5x. When we add and subtract polynomials, it is very similar to combining like terms in algebraic expressions. Students will need be familiar with the concept of combing like terms before they add or subtract polynomials. To introduce the topic of combining polynomials, it can be set up horizontally.

Such as: (3x² – 5x + 6) – (6x² – 4x + 9)

By setting it up this way, students can determine which terms can be combined and which terms need to be left alone. Additionally, students will build on the concept of combining like terms as it applies to this process as well. Setting it up horizontally will also increase the chance of preventing the mistake of forgetting to distribute the negative sign throughout the second polynomial. Once students are comfortable doing it this way, the addition and subtraction can be set up as a vertical problem where students must now take the step to align the like terms together in order to add or subtract. By taking the step to set up the polynomials horizontally before vertically, it will give the students a deeper understanding of what concept is actually behind adding and subtracting polynomials.

 

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E1. How can technology be used to effectively engage students with this topic?

The website http://www.quia.com provides a multitude of activities relating to different subjects. The game I chose to correlate adding and subtracting polynomials is identical to the actual game Battleship. The game can be played by anyone with access to the internet and Adobe. This game is interactive because you won’t have to perform math on every single shot fired at the enemy. If the student does hit one of the vessels, in order to actually “hit” the enemy’s ship, the student must successfully add or subtract two polynomials. If a student hits a vessel but is unable to solve the polynomial correctly, the game will highlight the hit area so that the student can try again. This game can either engage the students to see who can sink all of the enemy’s ships first, or it can be assigned as a homework assignment that requires showing work and screenshotting the end result of the game. Lastly, you can choose the level of difficulty of the game. For example, on the hard level, you must determine the missing addend or minuend to the expression, or add or subtract polynomials of different degrees.
The website also offers an option to create your own activities, so if Battleship isn’t panning out as desired, it is possible to create your own game for your students.
Game: https://www.quia.com/ba/28820.html

References:

Area of the Border: https://www.sophia.org/concepts/adding-and-subtracting-polynomials-in-the-real-world

Combining Like Terms: https://courses.lumenlearning.com/boundless-algebra/chapter/introduction-to-polynomials/

Battleship: https://www.quia.com/ba/28820.html

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

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Rachel Delflache. Her topic, from Algebra: using the point-slope equation of a line.

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A2: How could you as a teacher create an activity that involves the topic?

An adaptation of the stained-glass window project could be used to practice the point-slope formula (picture beside). Start by giving the students a piece of graph paper that is shaped like a traditional stained-glass window and then let they students create a window of their choosing using straight lines only. Once they are done creating their window, ask them to solve for and label the equations of the lines used in their design. While this project involves the point slope formula in a rather obvious way, giving the students the freedom to create a stained-glass window that they like helps to engage the students more than a normal worksheet. Also, by having them solve for the equations of the lines they created it is very probable that the numbers they must use for the equation will not be “pretty numbers” which would add an addition level of difficulty to the assignment.

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B2: How does this topic extend what your students should have learned in previous courses?

The point-slope formula extends from the students’ knowledge of the slope formula

m = (y2-y1)/(x2-x1)
(x2-x1)m = y2-y1
y-y1 = m(x-x1).

This means that the students could solve for the point-slope formula given the proper information and prompts. By allowing students to solve for the point-slope formula given the previous knowledge of the formula for slope, it gives the students a deeper understanding of how and why the point-slope formula works the way it does. Allowing the students to solve for the point-slope formula also increases the retention rate among the students.

 

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C1&3: How has this topic appeared in pop culture and the news?

Graphs are everywhere in the news, like the first graph below. While they are often time line charts, each section of the line has its own equation that could be solved for given the information found on the graph. One of the simplest way to solve for each section of the line graph would be to use point slope formula. The benefit of using point slope formula to solve for the equations of these graphs is that there is very minimal information needed—assuming that two coordinates can be located on the graph, the linear equation can be solved for. Another place where graphs appear is in pop culture. It is becoming more common to find graphs like the second one below. These graphs are often time linear equation for which the formula could be solved for using the point slope formula. These kinds of graphs could be used to create an activity where the students use the point slope formula to solve to the equations shown in either the real world or comical graph.

 

 

References:

Stained glass window-
http://digitallesson.com/stained-glass-window-graphing-project/

iPhone sales-
https://www.usatoday.com/story/tech/news/2017/06/28/iphones-smartphone-revolution-4-graphs/103216746/

Halloween graph-
https://www.buzzfeed.com/agh/halloween-charts-and-graphs?utm_term=.hpXrNWPm9#.qpvwGmxp0