Engaging students: Introducing variables and expressions

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 Caitlin Kirk. Her topic, from Pre-Algebra: introducing variables and expressions.

To keep track of some of the coldest things in the universe, scientist use the Kelvin temperature scale that begins at 0 Kelvin, or Absolute Zero. Nothing can ever be colder than Absolute Zero because at this temperature, all motion stops. The table below shows some typical temperatures of different systems in the universe.

Table of Cold Places

Temp.(K)	Location
183	Vostok, Antarctica
160	Phobos- a moon of Mars
128	Europa in the summer
120	Moon at night
88	Miranda surface temp.
81	Enceladus in the summer
70	Mercury at night
55	Pluto in the summertime
50	Dwarf Planet Quaoar
33	Pluto in the wintertime
1	Boomerang Nebula
0	ABSOLUTE ZERO

You are probably already familiar with the Celsius (C) and Fahrenheit (F) temperature scales. The two formulas below show how to switch from degrees-C to degrees-F.

$C = \frac{5}{9} (F-32)$

$F = \frac{9}{5} C + 32$

Because the Kelvin scale is related to the Celsius scale, we can also convert from Celsius to Kelvin (K) using the equation:

$K = 273 + C$

Problems

Use these three equations to convert between the three temperature scales:

Problem 1: 212 F converted to K

Problem 2: 0 K converted to F

Problem 3: 100 C converted to K

Problem 4: Two scientists measure the daytime temperature of the moon using two different instruments. The first instrument gives a reading of +107 C while the second instrument gives +221 F.

a. What are the equivalent temperatures on the Kelvin scale?

b. What is the average daytime temperature on the Kelvin scale?

Problem 5: Humans can survive without protective clothing in temperatures ranging from 0 F to 130 F. In what, if any, locations from the table above can humans survive?

Solutions

Problem 1: First convert to C: C = 5/9 (212-32) = +100 C. Then convert from C to K: K = 273 + 100 = 373 Kelvin.

Problem 2: First convert to Celsius: 0 = 273 + C so C = -273. Then convert from C to F: F = 9/5 (-273) + 32 = -459 Fahrenheit.

Problem 3: K = 273 – 100 = 173 Kelvin.

Problem 4:

a. 107 C becomes K = 273 + 107 = 380 Kelvin. 221 F becomes C = 5/9(221-32) = 105 C, and so K = 273 + 105 = 378 Kelvin.

b. (380 + 378)/2 = 379 Kelvin

Problem 5:

First convert 0 F and 130 F to Celsius so that the conversion to Kelvin is quicker. 0 F becomes C = 5/9(0-32) = -18 C (rounded to the nearest degree) and 130 F becomes C = 5/9 (130-32) = 54 C (rounded to the nearest degree).

Next, convert -18 C and 54 C to Kelvin. -18 C becomes K = 273-18 = 255 and 54 C becomes k = 273 + 54 = 327 K.

None of the locations on the table have temperatures between 255 K and 327 K, therefore humans could not survive in any of these space locations.

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

This topic is one of the first experiences students have with algebra. Since algebra is the point from which students dive into more advanced mathematics, this topic will be used in many different areas of future mathematics. After mastering the use of one variable, with the basic operations of addition, subtraction, multiplication, and division, students will be introduced to the use of more than one variable. They may be asked to calculate the area of a solid whose perimeter is given and whose side lengths are unknown variables. Or in a more advanced setting, they may be asked to calculate how much money will be in a bank account after five years of interest compounded continuously. In fact, the use of variables is present and important in every mathematics class from Algebra I through Calculus and beyond. There very well may never be a day in a mathematics students’ life where they will not see a variable after variables have been introduced.

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

In basic arithmetic, probably in elementary or early middle school math classes, students learn how to do calculations with numbers using the four basic operations of addition, subtraction, multiplication and division. They also learn simple applications of these basic operations by calculating the area and perimeter of a rectangle, for example. Introducing variables and expressions is a continuation of those same ideas except that one or more of the numbers is now an unknown variable. Students can rely on the arithmetic skills they already possess when learning this introduction to algebra with variables and expressions.

Students are familiar with calculating the area and perimeter of figures like the one on the left before they are introduced to variables. Later, they may see the same figure with the addition of a variable, as shown on the right. The addition of the variable will come with new instructions as well.

The difficulty of problems using variables is determined by the information given in the problems. For instance, the problem on the right can be a one step equation if an area and perimeter are given so that students only need to solve for w. The difficulty can be increased by giving only a perimeter so that students must solve for w and then for the area.

Engaging students: Computing the determinant of a matrix

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 Caitlin Kirk. Her topic: computing the determinant of a matrix.

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

Students learn early in their mathematical careers how to calculate the area of simple polygons such as triangles and parallelograms. They learn by memorizing formulas and plugging given values into the formulas. Matrices, and more specifically the determinant of a matrix, can be used to do the same thing.

For example, consider a triangle with vertices $(1,2)$ , $(3, -4)$ , and $(-2,3)$ . The traditional method for finding the area of this circle would be to use the distance formula to find the length of each side and the height before plugging and chugging with the formula $A = \frac{1}{2} bh$ . Matrices can be used to compute the same area in fewer steps using the fact that the area of a triangle the absolute value of one-half times the determinant of a matrix containing the vertices of the triangle as shown below.

First, put the vertices of the triangle into a matrix using the x-values as the first column and the corresponding y-values as the second column. Then fill the third column with 1’s as shown:

Next, compute the determinant of the matrix and multiply it by ½ (because the traditional area formula for a triangle calls for multiplying by ½ to account for the fact that a triangle is half of a rectangle, it is necessary to keep the ½ here also) as shown:

Obviously, the area of a triangle cannot be negative. Therefore it is necessary to take the absolute value of the final answer. In this case $|-8| = 8$ , making the area positive eight instead of negative eight.

The same idea can be applied to extend students knowledge of the area of other polygons such as a parallelogram, rectangle, or square. Determinants of matrices are a great extension of the basic mathematical concept of area that students will have learned in previous courses.

D. History: What are the contributions of various cultures to this topic?

The history of matrices can be traced to four different cultures. First, Babylonians as early as 300 BC began attempting to solve simultaneous linear equations like the following:

There are two fields whose total area is eighteen hundred square yards. One produces grain at the rate of two-thirds of a bushel per square yard while the other produces grain at the rate of on-half a bushel per square yard. If the total yield is eleven hundred bushels, what is the size of each field?

While the Babylonians at this time did not actually set up matrices or calculate any determinants, they laid the framework for later cultures to do so by creating systems of linear equations.

The Chinese, between 200 BC and 100 BC, worked with similar systems and began to solve them using columns of numbers that resemble matrices. One such problem that they worked with is given below:

There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?

Unlike the Babylonians, the Chinese answered this question using their version of matrices, called a counting board. The counting board functions the same way as modern matrices but is turned on its side. Modern matrices write a single equation in a row and the next equation in the next row and so forth. Chinese counting boards write the equations in columns. The counting board below corresponds to the question above:

1 2 3

2 3 2

3 1 1

26 34 39

They then used what we know as Gaussian elimination and back substitution to solve the system by performing operations on the columns until all but the bottom row contains only zeros and ones. Gaussian elimination with back substitution did not become a well known method until the early 19^th century, however.

Next, in 1683, the Japanese and Europeans simultaneously saw the discovery and use of a determinant, though the Japanese published it first. Seki, in Japan, wrote Method of Solving the Dissimulated Problems which contains tables written in the same manner as the Chinese counting board. Without having a word to correspond to his calculations, Seki calculated the determinant and introduced a general method for calculating it based on examples. Using his methods, Seki was able to find the determinants of 2×2, 3×3, 4×4, and 5×5 matrices.

In the same year in Europe, Leibniz wrote that the system of equations below:

$10+11x+12y=0$

$20+21x+22y=0$

$30+31x+32y=0$

has a solution because

$(10 \times 21 \times 32)+(11 \times 22 \times 30)+(12 \times 20 \times 31)=(10 \times 22 \times 31)+(11 \times 20 \times 32)+(12 \times 21 \times 30)$ .

This is the exact condition under which the matrix representing the system has a determinant of zero. Leibniz was the first to apply the determinant to finding a solution to a linear system. Later, other European mathematicians such as Cramer, Bezout, Vandermond, and Maclaurin, refined the use of determinants and published rules for how and when to use them.

Source: http://www-history.mcs.st-and.ac.uk/HistTopics/Matrices_and_determinants.html

B. Curriculum: How can this topic be used in you students’ future courses in mathematics or science?

Calculating the determinant is used in many lessons in future mathematics courses, mainly in algebra II and pre-calculus. The determinant is the basis for Cramer’s rule that allows a student to solve a system of linear equations. This leads to other methods of solving linear systems using matrices such as Gaussian elimination and back substitution. It can also be used in determining the invertibility of matrices. A matrix whose determinant is zero does not have an inverse. Invertibility of matrices determines what other properties of matrix theory a given matrix will follow. If students were to continue pursuing math after high school, understanding determinants is essential to linear algebra.

Collaborative Mathematics: Challenge 05

My colleague Jason Ermer is back from summer hiatus and has posted his fifth challenge video, shown below.

Video responses can be posted to his website, http://www.collaborativemathematics.org. In the words of his website, this is a unique forum for connecting a worldwide community of mathematical problem-solvers, and I think these unorthodox but simply stated problems are a fun way for engaging students with the mathematical curriculum.

Engaging students: Distinguishing between inductive and deductive reasoning

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 Caitlin Kirk. Her topic, from Geometry (and proof writing): distinguishing between inductive and deductive reasoning.

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

Inductive and deductive reasoning are often used on TV, radio, or in print in the form of advertising.

Deductive Reasoning

Man: What’s better, faster or slower?

All kids: Faster!

Man: And what’s fast?

Boy: My mom’s car and a cheetah.

Girl: A space ship.

Man: And what’s slow?

Boy: My grandma’s slow.

Man: Would you like her better if she was fast?

Boy: I bet she would like it if she was fast.

Man: Hmm, maybe give her some turbo boosters?

Boy: Or tape a cheetah to her back.

Man: Tape a cheetah to her back, it seems like you’ve thought about this before.

Narrator: It’s not complicated, faster is better. And iPhone 5 downloads fastest on AT&T 4G.

Deductive reasoning, which applies a general rule to specific examples, can be seen in advertisements like the AT&T commercial above. The kids establish in their conversation that faster things are better. The narrator says that iPhone 5 downloads fastest on AT&T 4G. Thus the viewer is left with the conclusion that AT&T 4G is better. This commercial’s deduction can be summed up as follows:

Faster things are better.

AT&T 4G is faster.

AT&T 4G is better. (conclusion)

Inductive Reasoning

Hotch: Sprees usually end in suicide. If he’s got nothing to live for, why wouldn’t he end it?

Reid: Because he’s not finished yet.

Reid: He’s obviously got displaced anger and took it out on his first victim.

Hotch: The stock boy represented someone. We need to know who. What about the other victims.

Reid: Defensive.

Hotch: Was he military?

Garcia: Negative.

Hotch: He’s lashing out. There’s got to be a reason. Rossi and Prentiss, dig through his house. Reid and JJ, get to the station. Morgan and I will take the crime scene. This guy’s got anger, endless targets and a gun. And from the looks of it, he just got started.

Inductive reasoning, which uses specific examples to make a general rule, can be seen frequently in episodes of TV shows or movies that involve crime scene investigation. The show Criminal Minds features a special unit of the FBI that profiles criminals. They do this by interviewing criminals who have already been caught and then inducing general rules about all criminals in order to catch the one they are looking for. Conversations among the profilers, like the one above, lead to inductive reasoning that can be summed up as follows:

He has nothing to live for.

He doesn’t want to commit suicide.

He wasn’t in the military.

He has displaced anger.

He has endless targets.

He has a gun.

He is a dangerous man who will hurt more people. (conclusion)

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

When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature’s God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among these are Life, Liberty, and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed. That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.

-The Declaration of Independence

July, 4, 1776

The Declaration of Independence was drafted as a deductive argument as to why the United States can and should be a country independent of Great Britain. Thomas Jefferson drafted the declaration with a series of premises leading to four different conclusions.

George III is a tyrant
The colonies have a right to be free and independent states
All political connections between Britain and the colonies should be dissolved
The “united states” have the right to do all things that free nations do

These four conclusions then serve as premises for the final conclusion that the United States is now an independent country. The declaration is a great example of deductive reasoning because it takes specific examples, such as the 27 grievances against the monarch, and makes logical conclusions, such as “George III is a tyrant,” from the examples. Its deduction can be plainly seen.

The Declaration of Independence is a great example of high culture to use in the classroom because every student who is educated in the United States will have some knowledge of this document. Therefore learning to analyze it “mathematically” in terms of deductive versus inductive reasoning, is a great engagement tool.

E. Technology: How can technology be used to effectively engage students with this topic?

Crime Scene Games & Deductive Reasoning: https://sites.google.com/a/wcsga.net/mock-trial/crime-scene-games-deductive-reasoning

This website contains links to several crime scene investigation games. Several of the games require students to collect clues, compare evidence, and then determine who is responsible for committing a given crime. These games are great for having students use their deductive skills. A couple of the other games require students to review given qualities of a criminal and inductively decide who the criminal in a scenario is based on these broad statements.

This website could be used to engage students easily. Having students play a game, especially one like these where they cannot pick out the mathematical skill they are using, is a great way to get students to abandon their potential distaste for a topic and be involved. After the students have completed a game and solved their crime, the teacher can smoothly transition into a geometrical lesson on inductive and deductive reasoning. The teacher will have activated the students’ knowledge of reasoning through a fun game. They will then be in a better position to learn a new, mathematical application of the reasoning they just used.

Engaging students: Finding the area of a square or rectangle

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 Alyssa Dalling. Her topic, from Geometry: finding the area of a square or rectangle.

D. How have different cultures throughout time used this topic in their society?

For three thousand years, the Great Pyramid of Giza was the world’s tallest man-made structure. It is also the oldest structure of the Seven Wonders of the Ancient World. It was built by cutting huge stones into rectangles then placing each stone into place to create the base. It is believed by many that the pharaoh Khufu had his vizier Hemon create the design for the great Pyramids. What is amazing about the design of the Pyramid of Giza is that each of the four sides of the base has an average error of only 58 millimeters in length. Meaning the base is almost a perfect square!
It would be fun to start the engage with introducing the Pyramid of Giza and explaining the facts above. Then students would be given the dimensions of other pyramids where they would have to find the area of the base to see whether they created a square or rectangular pyramid. This would get them excited about this topic because students would be exploring math that has actually been used in real life.

The Mesoamericans also built pyramids with square and rectangular bases. The picture above is in a city known as Chechen Itza which is located in the Mexican state of Yucatan. It is called El Castillo, and also known as the Temple of Kukulkan. Unlike the Egyptian pyramids though, the Mayan pyramids were usually meant as steps to get to a temple on top. The pyramids consisted of several square bases stacked onto each other with steps up each side. El Castillo consists of nine square terraces each about 8.4 feet tall. The main base of the pyramid is approximately 55.3 meters (181 feet).
What would be fun to do is have students find the area of each level and compare it to all the levels on the pyramid. I feel students would have fun seeing just how big this type of structure is and understanding the planning it took to create the different levels in this pyramid.

Sources: http://en.wikipedia.org/wiki/Great_Pyramid_of_Giza and http://en.wikipedia.org/wiki/Pyramids#Nigeria

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

Finding the area of squares and rectangles will be used a lot in Algebra and Algebra II. One example in Algebra is when students start solving for unknown variables. A student would be asked to find the area of a square when they have two unknown sides.
The following is an example engage problem students would use the finding the area of a square or rectangle to solve.

Principal Smith has decided the school needs a new practice basketball court. The current practice court is a square with an area of 144 square feet. She wants the new court to be a rectangle twice as long as it is wide. Find the length of all the sides of both the old court and the new court and find the area of the new court.

$x^2 = 144$

So $x = 12$

Then $x(2x) = 2x^2 = 2(12)^2 = 288$

The square court has sides of 12.

The rectangular court has sides of 12×24 and an area of 288 square feet.

Engaging students: Equation of a circle

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 Alyssa Dalling. Her topic, from Precalculus: the equation of a circle.

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

A fun way to engage students and also introduce the standard form of an equation of a circle is the following:

Start by separating the class into groups of 2 or 3
Pass each group a specific amount of flashcards. (Each group will have the same flashcards)
Each flashcard has a picture of a graphed circle and the equation of that circle in standard form
The students will work together to figure out how the pictures of the circle relate to the equation

This will help students understand how different aspects of a circle relate to its standard form equation. The following is an example of a flashcard that could be passed out.

Source: http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php

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

Circles have been used through history in many different works of art. One such type is called a tessellation. The word Tessellate means to cover a plane with a pattern in such a way as to leave no region uncovered. So, a tessellation is created when a shape or shapes are repeated over and over again. The pictures above show just a few examples of how circles are used in different types of art. A good way to engage students would be to show them a few examples of tessellations using circles.

Source: http://mathforum.org/sum95/suzanne/whattess.html

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

Khan Academy has a really fun resource for using equations to graph circles. At the beginning of class, the teacher could allow students to play around with this program. It allows students to see an equation of a circle in standard form then they would graph the circle. It gives hints as well as the answer when students are ready. The good thing about this is that even if a student goes straight to the answer, they are still trying to identify the connection between the equation of the circle and the answer Khan Academy shows.

http://www.khanacademy.org/math/trigonometry/conics_precalc/circles-tutorial-precalc/e/graphing_circles

Advertising for slide rules, from 1940

I’m about to begin a series of posts concerning how previous generations did complex mathematical calculations without the aid of scientific calculators.

Courtesy of Slide Rule Universe, here’s an advertisement for slide rules from 1940. This is a favorite engagement activity of mine when teaching precalculus (as an application of logarithms) as well as my capstone class for future high school math teachers. I have shown this to hundreds of college students over the years (usually reading out loud the advertising through page 5 and then skimming through the remaining pictures), and this always gets a great laugh. Enjoy.

test

Height

I’m about to do a series of posts concerning square roots and logarithms. So I thought that this picture would be an appropriate introduction to the topic. (One of these days, I’ll spring for the wall poster version of this picture to hang in my office.)

Source: http://www.xkcd.com/482/

Engaging students: Solving logarithmic equations

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 Caitlin Kirk. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

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

Logarithms are a topic that appears at multiple levels of high school math. In Algebra II, students are first introduced to logarithms when they are asked to identify graphs of parent functions including f (x) = log_ax. Later in the same class, they learn to formulate equations and inequalities based on logarithmic functions by exploring the relationship between logarithms and their inverses. From there, they can develop a definition of a logarithm.

Solving logarithmic equations extends what students learned about logarithms in Algebra II. Once a proper definition of logarithms has been established, along with a graphical foundation of logs, students learn to solve logarithmic equations. Properties of logarithms are used to expand, condense, and solve logarithms without a calculator in Pre Calculus. Practical applications of the logarithmic equation also follow from previous skills. Students learn to calculate the pH of a solution, decibel voltage gain, intensity of earthquakes measure on the Richter scale, depreciation, and the apparent loudness of sound using logarithms.

C. Culture: How has this topic appeared in the news?

One application of logarithmic equations is calculating the intensity of earthquakes measured on the Richter scale using the following equation:

$R = \log(A/P)$

where $A$ is the amplitude of the tremor measured in micrometers and $P$ is the period of the tremor (time of one oscillation of the earth’s surface) measured in seconds.

Reports of earthquake activity appear in the news often and are always accompanied by a measurement from the Richter scale. One such report can be found here: http://www.bbc.co.uk/news/world-asia-20638696. As the story says, a 7.3 magnitude earthquake struck off the coast of Japan in December of 2012, and created a small tsunami. There were six aftershocks of this quake whose Richter scale measurements are also given. The article also explains how Japan has been able to enact an early warning system that predicts the intensity of an earthquake before it causes damage. All of the calculations given in this story, and almost all others involving earthquakes, involves the use of the Richter scale logarithmic equation.

D. History: What are the contributions of various cultures to this topic?

The development of logarithms saw contributions from several different countries beginning with the Babylonians (2000-1600 BC) who developed the first known mathematical tables. They also introduced square multiplication in which they simply but accurately multiplied two numbers using only addition and subtraction. Michael Stifel, of Germany, was the first mathematician to use an exponent in 1544. He developed an early version of the logarithmic table containing integers and powers of 2. Perhaps the most important contribution to logarithms came from John Napier in Scotland in 1619. He, like the Babylonians, was working with on breaking multiplication, division, and root extraction down to only addition and subtraction. Therefore, he created the “logarithm” $L$ of a number $N$ defined as follows:

$N = 10^7 (1-10^{-7})^L$

for which he wrote $\hbox{NapLog}(N) = L$ .

Napier’s definition of the logarithm led to the following logarithmic identities that are still taught today:

$\hbox{NapLog}(\sqrt{N_1N_2}) = \frac{1}{2} (\hbox{NapLog} N_1 + \hbox{NapLog} N_2)$

$\hbox{NapLog}(10^{-7} N_1 N_2) = \hbox{NapLog} N_1 + \hbox{NapLog} N_2$

$\hbox{NapLog} \left( 10^{-7} \displaystyle \frac{N_1}{N_2} \right)= \hbox{NapLog} N_1 - \hbox{NapLog} N_2$

Henry Briggs, in England, published his work on logarithms in 1624, which included logarithms of 30,000 natural numbers to the 14th decimal place worked by hand! Shortly after, back in Germany, Johannes Kepler used a logarithmic scale on a Cartesian plane to create a linear graph the elliptical shape of the cosmos. In 1632, in Italy, Bonaventura Cavalieri published extensive tables of logarithms including the logs of trig functions (excluding cosine). Finally, Leonhard Euler made one of the most commonly known contributions to logarithms by making the number $e = 2.71828\dots$ the base of the natural logarithm (which was also developed by Napier). While it is untrue, as is commonly believed, that Euler invented the number $2.71828\dots$ , he did give it the name $e$ . He was interested in the number because he wanted to calculate the amount that would result from continually compounded interested on a sum of money and the number $2.71828\dots$ kept appearing as a constant in his equation. Therefore he tied $e$ to the natural logarithm that was not as widely used because it did not have a base.