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

 

 

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 again comes from my former student Deetria Bowser. Her topic, from Algebra: finding x- and y-intercepts. Unlike most student submissions, Maranda’s idea answers three different questions at once.

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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? 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.

One example of an engaging form of technology that involves finding x- and y-intercepts of lines is mangahigh.com. Under the algebra section, there is a tab for finding x and y intercepts which once clicked provides an option to start a game (“Algebra.”). In this game, the student is expected to look at lines and quickly decipher what is known about the x and y intercepts of the line in question. Before the game begins, the student is able to choose the difficulty of the game as well as the number of questions. After the game is completed students are able to review their answers. Implementing this website into the classroom will help students gain quickness in identifying x and y intercepts. Additionally, this game is also a quick and fun way to evaluate students understanding of x and y intercepts, without forcing them to take a quiz.

green lineD1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

The topic of x and y intercepts falls under a much broader topic called analytical geometry.The article “Analytic geometry” defines analytical geometry as “[a] mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry” (D’Souza). One of the people who discovered this topic was René Descartes. René Descartes was actually a french modern philosopher who also made discoveries in the realms of science as well as mathematics. Descartes “dismissed apparent knowledge derived from authority,” meaning that he made his discoveries based on what he thought rather than taking ideas from scientists, philosophers and mathematicians (Watson). He discovered analytical mathematics (along with Fermat) in the 1630s (D’Souza). He also “he stressed the need to consider general algebraic curves—graphs of polynomial equations in x and y of all degrees” (D’Souza). Mentioning Descartes in class, and explaining his accomplishments in Mathematics as well as modern philosophy and science, will encourage students to realize that they can succeed in more than one subject . Also, Descartes can be used as an influence in the building of ideas in the classroom, since he did not just accept ideas already created.

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C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The topic of x and y intercepts appeared on a “pop culture blog” called the comeback.com. In an article posted in November 2016, a former UCLA and current Cleveland Indians baseball player named Trevor Bauer helped one of his fans with her math homework (Blazer). This article describes a girl asking Bauer for help determining the slope of a line and the y – intercepts via Twitter. Her specific question involves the equation 2y=x (Blazer). He then explains that “for every 1 unit on the x axis go 2 units on the y axis. y intercept is where it crosses the y axis. Make y 0 and figure x” (Blazer). Since Bauer is a professional baseball player, he already has a great influence over people. Showing students this article about Bauer will show students that even people who play baseball for a living still have the knowledge of Algebra.

 

References
“Algebra.” Mangahigh.com – Algebra,
http://www.mangahigh.com/en-us/math_games/algebra/straight_line_graphs/find_the_x_and_y_intercepts_of_lines. Accessed 15 Sept. 2017.

Blazer, Sam, et al. “Trevor Bauer helped a fan do their math homework on Twitter.” The
Comeback, 13 Nov. 2016,
thecomeback.com/mlb/trevor-bauer-twitter-math-homework.html. Accessed 15 Sept.
2017.

D’Souza, Harry Joseph, and Robert Alan Bix. “Analytic geometry.” Encyclopædia Britannica,
Encyclopædia Britannica, inc., 6 June 2016,
http://www.britannica.com/topic/analytic-geometry. Accessed 15 Sept. 2017.

Watson, Richard A. “René Descartes.” Encyclopædia Britannica, Encyclopædia Britannica, inc.,
27 Jan. 2017, http://www.britannica.com/biography/Rene-Descartes. Accessed 15 Sept. 2017.

Engaging students: Factoring 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 Brittnee Lein. Her topic, from Algebra: factoring polynomials.

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

There are many great websites that can help to provide students a conceptual framework for factoring polynomials in lieu of simple lecture. This website lets students explore polynomial equations with online algebra tiles.

https://illuminations.nctm.org/activity.aspx?id=3482

Algebra tiles are effective in teaching factoring because they provide a visual representation of abstract concepts and allow students to understand that the symbol “=” in an equation really means equivalence (i.e. what you do to one side of the equation, you must do to the other side). I also think algebra tiles are very beneficial in teaching students about zero pairs. There are other websites –such as wolfram alpha– that are especially great supplements to go alongside topics such as factoring polynomials because students can see the graphical meaning of the roots of a quadratic equation. When combined, these websites can aid students in gaining a both conceptual and procedural understanding of the topic.

 

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

There is an activity called “Factor Draft” where students set up a ‘playing field’ of cards. In this field, there are factor cards such as (x+2), (x-12), etc. sum (5x), (12x), etc., and product cards (1), (42), and so on. The goal of the game is to draw a winning hand of two factor cards and a corresponding sum and product card. Each card is color coded to their type. Each turn a player draws one card from the field of face up cards. The player must pay mind to not only his/her own cards but also those of their opponent’s –as the first person to get two factor cards and their corresponding sum and product card wins. This activity is beneficial in furthering student understanding between the relationships between each term in a quadratic polynomial. For example (x+4)(x-3) = x^2 + 1x - 12 and the corresponding factor cards would be (x+4) and (x-3) the sum card would be (1x) and the product card would be (-12). This activity allows students to intuitively get a sense of the process of factoring and gives them practice multiplying out polynomials.

 

 

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2. How can this topic be used in your students’ future courses in mathematics or science?
• Factoring polynomials is used in many important future science and mathematics concepts. When a quadratic equation cannot be factored simply, teachers must introduce the quadratic formula. This slides into the introduction of complex roots of an equation and complex numbers. When factoring polynomials of higher degree than 2, synthetic division (another topic in high school mathematics) is useful in finding the roots of the equation. If a student is able to understand the meaning of the roots of an equation, that will aid in solving many interesting physics and mathematics problems. Factoring is used quite often to find the domain of a rational equation such as f(x) = (x+2)/ (x^2+ 4x+3). A student must also have a strong basis in factoring polynomials to learn concepts such as completing the square.

References

• National Library of Virtual Manipulatives, nlvm.usu.edu/en/nav/vlibrary.html.

• Cleveland, James. “The Factor Draft.” The Roots of the Equation, 23 May 2014, rootsoftheequation.wordpress.com/2014/05/22/the-factor-draft/.

Cheat Sheet

The news clip below shows why, when I allow my students to use a 3×5 card on an exam, I specify — “that’s in inches. It must be handwritten. And no magnifying glasses.”

Student Outwits Prof By Bringing 3×5 FOOT Cheat Sheet To Exam

What Do You Do With 11-to-13-Year-Olds?

I greatly enjoyed this very thoughtful post about the unique joys and challenges of teaching middle school mathematics: https://mathwithbaddrawings.com/2017/07/05/what-do-you-do-with-11-to-13-year-olds/

Engaging students: Solving one-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 Deetria Bowser. Her topic, from Algebra: solving one-step algebra problems.

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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.)

To create a successful word problem that would both interest, and engage students, the teacher must “know his class.” Knowing one’s class involves knowing the many different students your students have. For example, if one knows that there are a lot of baseball players in the classroom, then creating word problems that involve baseball would be engaging for these students.

Additionally, to benefit all students you could do problems that involve finances. Including more “finance problems” will help students realize the importance of math, and how they can apply it in everyday life. An example of such problem would be “Damon’s earnings for four weeks from a part time are shown in the table. Assume his earnings vary directly with the number of hours worked. Damon has been offered a job that will pay him $7.35 per hour worked. Which job is better pay (Tucker, A.)? Including word problems that students can relate to now or in the future can help students stay engaged while learning, and answer the question that is most commonly asked by students: “When will I ever use this in real life?”

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

As a teacher, creating engaging activities and/or projects can prove to be quite difficult for word problems that are one- or two-step algebra problems, due to the fact that most students completely shut down once a word problem is presented to them. To combat this I have found that making it into a cooperative game can help soothe the anxiety caused by word problems. One game that is great to play with one or two step algebra problems is called rally coach. In this game, students are paired off. Student A is expected to work on solving the problem, while Student B is expected to watch, listen, check, and praise just as a coach would. Once the students think they have the correct answer, they will raise their hand so that the teacher may check it. If they get the answer correct, then the teacher will give them another problem (this time Student A and Student B switch roles). If the answer is incorrect, they must continue working on the problem. The end goal of the game is to answer as many questions as possible before time runs out. By playing this game students are able to help each other solve one or two step word problems.

 

 

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

In future courses many problems will involve one or two step algebra problems. For instance, in science courses like chemistry and physics, one will need to know how to solve for different variables of equations. For example, if one is in a chemistry course and is given a word problem (i.e If a 3.1g ring is heated using 10.0 calories, its temperature rises 17.9°C. Calculate the specific heat capacity of the ring) that provides heat energy (Q) mass of a substance (m) and change in temperature (deltaT), but is asked to solve for the specific heat, students will need to know how to solve for the specific heat either by isolating the variable in the beginning (Cp=Q/mdeltaT) or plugging in the givens and isolating the variable (Daniell, B).

References

Daniell, B. (n.d.). Energy Slides 3 [Powerpoint that contains Specific Heat problem].

Tucker, A. (2016). Direct Variation. Retrieved September 01, 2017, from

http://www.showme.com/sh/?h=PQvPbm4