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 18

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. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert  x = 0.\overline{432} = 0.432432432\dots into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number x.

1000x = 432.432432\dots

x = 0.432432\dots

Notice that the decimal parts of both x and 1000x are the same. Subtracting, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time.

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you will believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

TI1637

See also my fuller post on this topic as well as the index for the entire series.

 

My Favorite One-Liners: Part 14

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. This quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of \sec x, the first step is far from obvious:

\displaystyle \int \sec x \, dx = \displaystyle \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \, dx

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

x = 0.\overline{432} = 0.432432432\dots

into a decimal, the first step is to multiply x by 1000:

1000x = 432.432432\dots

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and poof! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting x from 1000x, the decimal parts cancel, leaving

999x = 432

or

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}.

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

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.

green lineI 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.

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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.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

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.

Source:  https://www.khanacademy.org/math/algebra/algebra-functions/intro-to-functions/v/what-is-a-function

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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: Multiplying 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 Daniel Herfeldt. His topic, from Algebra: multiplying polynomials.

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

Activities for multiplying polynomials are endless. An activity that I would do with my students is a game called polynomial dice. To do this, you would first is to get several blank dice and write random polynomials on each side of the dice. Then in class, divide the students into groups of no more than three. Each group will get a pair of dice. Have the students roll the dice and they should have two different polynomials. Once they have rolled, have them multiply the polynomials together. This is best done with groups so that the students can share their work with their partners to see if they both got the same answer. If they did not get the same answer, they can go back through each other’s steps to see where they went wrong. If you want to make the game a bit harder, you can add more dice to make them multiply three polynomials, or maybe even more. This is a great game because it can be used for multiplying polynomials, as well as dividing, adding and subtracting. It could be a great review game before a major test to have students remember how to do each individual property. For example, have the students roll the dice, then with the two polynomials they get, they first add the polynomials, followed by the difference, then the product, and finally the quotient.

 

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

Multiplying polynomials is used all over mathematics. It is first introduced in Algebra I and Algebra II. Multiplying polynomials can be very difficult for students and make them not want to do the work. This is due to there being so much work for one problem. Since there is so much work, there is a lot of room for mistakes. This topic is used is Algebra I, Algebra II, Algebra III, Pre-Calculus, Calculus and just about every higher math course. If a student is looking to go into an architecture or engineering field, they will have to apply their knowledge of polynomials. Due to this, the topic is one of the most important topics that students need to understand. Knowing how to multiply polynomials also makes it easier to divide polynomials. If a student is struggling with dividing polynomials, you can go back to showing them how to multiply them. Once a student sees the pattern of multiplying polynomials, they are more likely to get the hang of dividing them.

 

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

I believe this video would be a great engage for the students when you, as a teacher, are teaching the students how to multiply polynomials for the first time. This video helps students remember what exactly is a polynomial. Although there is only three types of polynomials in the video (monomial, binomial, and trinomial), it uses the three main types that students will be using in a high school level. Another great thing in the video is that it shows how to tell the degree of the polynomial. Although it seems easy to just say the power of x is the same as the degree, students still might forget how to do it. For example, a student might think that a digit by itself and with no variable has a degree of one, but is really a degree of zero. The final point that is key to this video is that it shows students how to line up the terms. Some students might put 6+x^2+3x, and although that is still correct, it will be better written as x^2+3x+6.

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 Brittany Tripp. Her topic, from Algebra: using the point-slope equation of a line.

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

The point-slope equation of a line can be used in a variety of different ways in mathematics classes that some students may encounter later on. It is used in Calculus when dealing with polynomials. For instance, “key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions.” It is also seen when dealing with Linear Approximations. “A differentiable function is one for which there is a tangent line at each point on the graph. In an intuitive sense, the tangent to a curve at a point is the line that looks most like the curve at the point of tangency. Assuming that f is differentiable at a, the tangent line to the graph y = f(x) at the point (a,f(a)) is given by the equation.

y – f(a) = f ‘(a)(x – a)

This equation arises from the point-slope formula for the line passing through (a,f(a)) with slope f ‘(a).” In Pre-Calculus with discussing horizontal and vertical shifts you can easily relate back to point-slope equation of a line. You can relate point-slope equation of a line to the definition of derivative where the equation is rewritten with limits to describe the slope as the derivative. This is just a few ways that point-slope pops up in later mathematics courses. It is important to be able to form the point-slope equations of a line, as well as slope-intercept form, and being about to understand it well enough to build off of it when leading into harder concepts.

 

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

Point-slope equation of a line is used in movies in a huge way that most people probably never even realize. Point-slope equation of a line is used in pinhole cameras. A pinhole camera “is a simple optical imaging device in the shape of a closed box or chamber. In one of its sides is a small hole which, via the rectilinear propagation of light, creates an image of the outside space on the opposite side of the box.” In other words, let’s say we had an object, there is light constantly bouncing off the object. In the case of a pinhole camera, there is a small hole in the nearest wall/barrier which only allows light to pass through the hole. The light that makes it through the hole then hits the far wall, or image plane, creating a projection of the original image. The way point-slope equation of a line is used is first by adding a coordinate plane that has the origin centered at the pinhole. We can imagine that our scene is off to the right of the origin and the image plane is off to the left of the origin. We can choose some point in our scene to be a coordinate point in our coordinate plane. Some of the light bouncing off of that point in our scene will pass through the pinhole and land somewhere on our image plane. One of the ways we can find where it lands in our image plane is by using slope-intercept equation of a line. There is a really cool video on the khan academy website that talks all about the mathematics behind pinhole cameras. There is actually an entire curriculum called Pixar in a Box that goes through a variety of different topics and subject matter that is involved in the making of Pixar movies.

 

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

There are a ton of games online that involve point-slope equation of a line. One website that I found that has a variety of games on it is called Websites for Math. I went and tried out some of these games myself and found them to be fun and entertaining, but somewhat challenging at the same time. The website has links to different games that pertain to slope and equation of a line. You can choose games specifically by what form of an equation of a line you want to practice, among other things. The first game I tried was Algebra Vs. Cockroaches. It pops up with a coordinate plane with a cockroach on it and you have to type in the equation of the line in order to kill the cockroach, but if you take too long the cockroaches start to multiple. I liked this game because it started with just having you identify the y-intercept before leading into harder equations. However, this game focused more on slope-intercept equation of a line than point-slope equation. There were games specifically designed for point-slope equation of a line. One of those games being point-slope jeopardy. If you choose a questions for 300 points you are given a coordinate point and a slope and asked to write the point-slope equation that fits for the given data. If you choose a question for 600 points you are given two coordinate points and asked to write the point-slope equation of the line that fits the given data. Therefore, you must first use the coordinate points to calculate the slope and then plug that into your equation. What I also like about this game is that you can either play by yourself or with a friend. The things I enjoy most about this website is that it has games that don’t only pertain to slope-intercept equation of a line. There are games that focus on slope specifically, graphing equations, slope-intercept form, etc. That way if you are having issues with any of the topics that may have been discussed previously, to point-slope equations of a line, you can find a game that might help refresh your memory.

 

References:

http://matheducators.stackexchange.com/questions/9907/should-i-be-teaching-point-slope-formula-to-high-school-algebra-students

http://calculuswithjulia.github.io/precalc/polynomial.html

https://www.khanacademy.org/partner-content/pixar/virtual-cameras/depth-of-field/v/optics6-final

http://www.pinhole.cz/en/pinholecameras/whatis.html

https://www2.gcs.k12.in.us/jpeters/slope.htm

http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

 

 

 

15-square puzzle

From the category “This Is Completely Useless”: here’s what a 15-square puzzle looks like when you arrange the tiles in order of how many factors they have.

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Engaging students: Determining the largest fraction

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 Perla Perez. Her topic, from Pre-Algebra: determining which of two fractions is largest if the denominators are unequal.

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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.)

 

Students are introduced to fractions in elementary school, but at a certain point this topic can become tedious. Trying to introduce new concepts to a topic they’ve seen and practiced for a while can be a challenge. A good idea can be to give them a problem at the start of class that they can answer after the day’s lesson is done. Students are given a word problem such as:

“James was arguing with John that he could eat more pizza than him, while John without a doubt believed the opposite. It got to the point where everyone in class had established their own opinions on it. So Nancy came up with a solution and ordered two large pizzas to see who could eat the most. Well, when the pizzas arrived they noticed that one pizza was cut into 10 equal pieces and the other into 16 equal pieces. After they devoured all that they could, John had eaten 7/10 and James had eaten 13/16. Now, who at the most pizza?”

After giving the students to time to think about the problem without any more information, get a show of hands to see who they think ate the most. Write up the number of students who voted for James and John somewhere visible. Then, at the end of the lesson, give and explain the answer.

 

 

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How has this topic appeared in the news?

 

The euro currently cost .890646 or 445323/500000 of a dollar. The British Pound .753423 or 753423/1000000 of a dollar. Now which currency is cheaper? If the fraction were only given to a student, some might be able to say the British pound because the 7 is greater that the 8 while others might say euro because of the 5 in denominator, and some that have no idea. There’s actually a formula to find out which is: AMOUNTto=(AMOUNTfrom X RATE from)/RATEto Although most times currency exchange is shown in decimal form, it gives a broader sense of how a simple concept relates to big-world topics. It is important for students to be able to determine if 3/7 is greater or less than 4/5, so that one day they can apply it to their daily lives. The exchange rate is just one example of different fractions being used in today’s society; in this case how the use of decimals and fractions translate to foreign relations. By relating the outside world to a classroom, educators can show students that there is more to numbers than just a grade in a class. These real world concepts can help students better understand the application of the material.

References:

http://www.mathinary.com/currency_conversion.jsp

http://www.x-rates.com/table/?from=USD&amount=1

 

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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.)

 

To students it may seem as if fractions have always been there. Some may have not thought much of its origin. A brief interesting part of history can be shared to spark some light in the matter. Well although there were contributions from the Babylonians, Arabs, and Ancient Rome, it was the Egyptians in 1800 BC seem to be the ones already using them. But interesting enough it isn’t like how it is seen today. Rather than seeing a fraction be an integer over another they used hieroglyphics and base ten.

For example, “The Egyptians wrote all their fractions using what we call unit 1 as its numerator (top number). They put a mouth picture (which meant part) above a number to make it into a unit fraction.”

It would be represented like,

Because of this method it was difficult to compute so they had to use numerous tables. Although our methods have changed one thing still remains the same; the way we use manipulatives in showing how fractions with different denominators compare. For instance, we have circle pictures that visually show fractions with different denominators can ease student into understanding them better.

Babylonians, though found a simpler way of representing fractious with symbols. All in all, it is interesting how visual description can be helpful still in today’s society.

 

References:

https://nrich.maths.org/2515

Engaging students: Adding and subtracting a mixture of positive and negative integers

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 Marissa Arevalo. Her topic, from Algebra: adding and subtracting a mixture of positive and negative integers.

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

An activity written by Kim Claryon The National Council of Teachers of Mathematics involves students understanding what it means when adding a negative integer, subtracting a positive integer, etc. This activity is called Zip, Zilch, Zero. Students are set in a group of 3 to 4 and dealt seven cards each, where the rest of the cards are left as a draw pile with a single card in the discard pile. Red cards are negative values and black cards are positive where Ace is equal to one, Jack is equal to eleven, Queen is equal to twelve, and King is equal to thirteen. Each student must draw a card from the top of the draw/discard pile. The point of the game is to add cards together to make a “Zip” or equal zero. the object of the game is that when a player plays the last card in their hand, all of the hands are scored by subtracting the absolute value of the sum of the cards in the hands from the absolute value of the cards played in a “Zip”. The winner has the highest score. Do note that the rules may be very tricky to understand as first and should be read aloud in class to help the games to go smoothly. Rules can be found on the website given below.

 

 

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

 

In a lot of idealizations of math by students, they do not associate STEM subjects with that of art. However, as a student who likes to paint and draw, I know that the arts involve a lot of mathematical logic in its creation, so one way to get the students involved, is to show that math is in everything. Therefore, I found a website with a video that discusses positive and negative space in a picture. In the video there is a black and white image of a tree on a flat landscape without anything in the background. The white space of the photo is referred to as the negative space and the black is the positive space as it is the subject and area of interest. In the video, the narrator describes that as the image is made smaller and larger that the value of the negative and positive space increases or decreases.

This can be a great engage as far as to asking the students to observe what happens when you make the subject area smaller or larger and whether or not that means if the negative space has decreased or increased. This could lead to a discussion as to how this relates to numbers and how the values of an integer change based on adding or subtracting from it.

 

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

 

A video I found on a blog “Embrace the Drawing Board” when I was looking through Pinterest had a very entertaining video that demonstrates what happens when adding or subtracting positive and negative integers. On the video the positive integers were green army men and the negative integers were red army men that were fighting in a “War of the Integers”. For example, in each battle, an equal amount of red and green army men will die on both sides when combining, or adding, the red and green men onto the battle field.

 

This is a great beginning to a sort of game between the students in which two students can play with one as the negative army and the other as the positive army. They can take turns to roll a pair of die where that number is the number of army they are brining to battle. Both students take turns deciding whose value goes first in the equation and then constructs the equations on a sheet of paper to figure out which side won the battle. Then, after about five to ten minutes of addition, the operation switches to subtraction, and the students continue to switch in whose number goes where in the equations.

 

_____   +  ______ = ________

 

_____   —  ______ = ________

 

 

Afterwards the teacher allow a student lead discussion by asking them what happened when subtracting a negative, adding a negative, etc. Then students can create their own theories and develop their own theories as to why they happened before the teacher can address any misconceptions.

 

References:

 

http://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/6-8/ZZZ-AS-RulesandRecord.pdf

http://thevirtualinstructor.com/positive-and-negative-space.html

http://mrpiccmath.weebly.com/blog/category/lesson%20ideas