Engaging students: Adding and subtracting decimals

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

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

This student submission comes from my former student Elizabeth (Markham) Atkins. Her topic, from Pre-Algebra: adding and subtracting decimals.

Applications

Adding and subtracting decimals is a fun subject to learn about. Decimals are everywhere in the world! Sports use decimals when timing people. Let’s try this problem: “Billy Joe ran a lap in 61.7 seconds the first time and 59.3 seconds the second time. How long did both laps take Billy Joe?” We use decimals to measure rainfall. “On Monday it rained a total of 1.27 inches, measured in a rain gauge. By Tuesday .23 inches had evaporated. Tuesday night’s big storm gave us another 3.58 inches. How much rain was in the rain gauge after Tuesday’s big storm?” We also use decimals with money! “Let’s say you found a lost cat. You return it to its owner for a reward of $50.00.Then you receive your allowance of $50.00. You then get your pay check from work which states you earned $108.75 for a week after taxes were taken out. It’s been a good week! You decide to spend a little money. You put $10.03 of gas in your car. You then by three items: Shoes ($51.99), jeans ($71.27) and gun ($0.97). How much do you have left?”

Technology

Technology is an awesome tool that we have to use to engage your students. On YouTube there is a song called the decimal song about how to add, subtract, multiply, and divide decimals. There is also a website where you can buy mathematical songs like his YouTube hit the Rappin’ Mathematician Decimals. He has a catchy way to grab student’s attention and they still learn. Technology can be used to enhance a lesson, an anchor video for example. Many website provide games. Mathgamesfun.net is a good example. Calculators are not a good enhancement tool because students can simply have the calculator do all the work for them. Calculators are a good technology to use to check a student’s work! Math.harvard.edu provides examples of math in movies. This way a student can see how math is used in the world. Learnalberta.ca/content/mesg.html/math6web/index.html?page=lessons&lesson=m6lessonshell01.swf is a website devoted to fractions. Another good technology for the teacher’s advantage is kaganonline.com. It is a website of different tools to use when teaching mathematics!

Curriculum

Decimals, along with fractions, numbers, and other basics, are a key foundational mathematical stepping stone to schooling and in life. Students will use math every day of their lives. In their science classes students will use decimals in measurement, weights, and time. Also when the student learns about scientific notation, they will use decimals. Students will use decimals to answer half-life questions. Decimals are used in economy. All of economy deals with money. Money deals with decimals. When learning about the stock market they use decimals. When looking at the mileage on their car, they use decimals. Students will have to learn decimals to help with percentages, sales, interest, sales tax, loans, and any sort of measurements in everyday life. Percentages are just decimals with a fancy symbol. If the students want to save money they need to know how to add and subtract decimals. Decimals are all around us we just have to teach the students how to see and use them!

Ohm

Source: http://www.xkcd.com/643/

Engaging students: Solving quadratic 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 comes from my former student Elizabeth (Markham) Atkins. Her topic, from Algebra II: solving quadratic equations.

D. History: Who were some of the people who contributed to the discovery of this topic?

Factoring quadratic polynomials is a useful trick in mathematics. Mathematics started long ago. http://www.ucs.louisiana.edu/~sxw8045/history.htm stated that the Babylonians “had a general procedure equivalent to solving quadratic equations”. They taught only through examples and did not explain the process or steps to the students. http://www.mytutoronline.com/history-of-quadratic-equation states that the Babylonians solved the quadratic equations on clay tablets. Baudhayana, an Indian mathematician, began by using the equation $ax^2+bx=c$ . He provided ways to solve the equations. Both the Babylonians and Chinese were the first to use completing the square method which states you take the equation $ax^2+bx+c$ . You take $b$ and divide it by two. After you divide by two you square that number and add it to $ax^2+bx$ and subtract it from $c$ . Even doing it this way the Babylonians and Chinese only found positive roots. Brahmadupta, another Indian mathematician, was the first to find negative solutions. Finally after all these mathematicians found ways of solving quadratic equations Shridhara, an Indian mathematician, wrote a general rule for solving a quadratic equation.

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

USA today (http://www.usatoday.com/news/education/2007-03-04-teacher-parabola-side_N.htm) had a news article that talks about students who used quadratic equations to cook marshmallows. A teacher had students in teams choose a quadratic equation. The teams then used the quadratic equation choosen to build a device to “harness solar heat and cook marshmallows”. http://www.kveo.com/news/quadratic-equations-no-problem talks about a 6 year old who learned to solve quadratic equations. Borland Educational News (http://benewsviews.blogspot.com/2007/03/memorize-quadratic-formula-in-seconds_3620.html) talks about someone who came up with a song for the quadratic formula, which is a way to solve a quadratic equation. They sing the following words to the tune of Pop Goes the Weasel: “X is equal to negative B plus or minus the square root of B squared minus 4AC All over 2A.” It may be an elementary way to solve the equation, but it sure does work. Mathematics is all around us. It is in our everyday lives. We use it without even knowing it sometimes!

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

Lesson Corner (http://www.lessoncorner.com/Math/Algebra/Quadratic_Equations) is an excellent resource for finding lesson plans and activities for quadratic equations. One lesson (http://distance-ed.math.tamu.edu/peic/lesson_plans/factoring_quadratics.pdf) talking about engaging the students with a game called “Guess the Numbers”. The students are given two columns, a sum column and a product column. They are then to guess the two numbers that will add to get the sum and multiply to get the product. This is an excellent game because it gets the students going and it is like a puzzle to solve. Learn (http://www.learnnc.org/lp/pages/2981) has a lesson plan for a review of quadratic equations. The students are engaged by playing “Chutes and Ladders”. The teacher transformed it. The procedures are as follows:

Draw a card.
Roll the dice.
If you roll a 1 or a 6, then solve your quadratic equation by completing the square.
If you roll a 2 or 5, then solve your quadratic equation by using the quadratic formula.
If you roll a 3, then solve your quadratic equation by graphing.
If you roll a 4, then solve your quadratic equation by factoring if possible. If not, then solve it another way.
If you solve your equation correctly, then you may move on the board the number of spaces that corresponds to your roll of the die.
If you answer the question incorrectly, then the person to your left has the opportunity to answer your question and move your roll of the die.
The first person to reach the end of the board first wins the game!
Good luck!!

I think this is an excellent idea because it brings back a little of the students’ childhood!

Complex knowledge

I took a statistics course at MIT. I would go study and do problems, and have high confidence that I understood the material. Then I’d go to the lecture, and be more confused than I was when I entered the classroom. Thus, I discovered that some teachers were capable of conveying negative knowledge, so that after listening to them, I knew less than I did before.

It was also clear that knowledge varies considerably in quantity among people, and this convinced me that real knowledge varies over a very wide range.

Then I encountered people who either did not know what they were talking about, or were clearly convinced of things that were wrong, and so I learned that there was imaginary knowledge.

Once I understood that there was both real and imaginary knowledge, I concluded that knowledge is truly complex.

– Hillel J. Chiel, Case Western Reserve University

Source: American Mathematical Monthly, Vol. 120, No. 10, p. 923 (December 2013)

Engaging students: Truth tables

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 Elizabeth (Markham) Atkins. Her topic, from Geometry: truth tables.

D. History: Who were some of the people who contributed to the development of this topic?

In “Peirce’s Truth-Functional Analysis and the Origin of Truth Tables” it is said that Charles Peirce was the first to start studying truth tables or rather developing the idea. He created the truth table in 1893. Peirce stated “the purpose of reasoning is to establish the truth or falsity of our beliefs, and the relationship between truth and falsity”. Nineteen years later, two mathematicians developed the truth table as we know it today. Ludwig Wittgenstein and Bertrand Russell both knew of truth tables but formalized them into the form we know today. In “The Genesis of the Truth-Table Device” it is said that George Berry stated “Peirce developed the technique, but not the device”. Wittgenstein developed the terminology that we today associate with truth tables. All in all it is the work of many people that finally developed the truth tables that we know today.

APPLICATIONS: What interesting word problems using this topic can your students do now?

Truth tables state that if P is true and Q is true then both P and Q are true. If either P or Q or both are false then P and Q are false. So I could have the students construct many truth tables to demonstrate their knowledge of the subject or I could come up with some interesting word problems. Word problems such as “True or false: If Billy Joe graduated and Shawn graduated then both Billy Joe and Shawn graduated.” There are not many word problems you could create that would deal with truth tables. You can have the students begin to think logically. You could give them a statement to complete such as, “Good apples are red. Granny Smith apples are green. Thus ____” This enables the teacher to get the students in the logical process of thinking in order for them to correctly understand truth tables.

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

By teaching my students truth tables and how to use them correctly it prepares them for future classes and for everyday life. In high schools now the students are learning twenty first century skills. To learn truth tables it will help with the twenty first century skills. When you learn truth tables you learn to think logically. The students need to learn logical thinking for science and economics. In Science, they need to learn logical thinking for when they do experiments. It will allow them to process, “well if I do this then this might happen.” In economics students need logical thinking so that when they learn to invest money they can weigh their options. In everyday life students make decisions that they need to think about. Teenagers in the modern day are moving so fast that they often do and say things without thinking. If they learn to think logically then they might be able to think, “If I say or do this then this might happen.”

Irving H. Anelli’s

PEIRCE’S TRUTH-FUNCTIONAL ANALYSIS AND THE ORIGIN OF TRUTH TABLES http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

THE GENESIS OF THE TRUTH-TABLE DEVICE http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal

Engaging students: Solving exponential 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 comes from my former student Elizabeth (Markham) Atkins. Her topic, from Precalculus: solving exponential equations.

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

Exponential equations can be different topics. You can use exponential equations for bacterial growth or decay, population growth or decay, or even a child eating their Halloween candy. Another example would be minimum wage. A good word problem would be at one point minimum wage was $1.50 an hour. Use A= $1.6 e^{rt}$ to figure out when minimum wage will reach $10.25 an hour. Another good word problem would be Billy Joe gets a dollar on his first day of work. Every day he works his salary for that day doubles. How much money does he have at the end of 30 days? A good money example would also be banking. “Use the equation $A=Pe^{rt}$ . Shawn put $100 in a savings account, which has a rate of 5% per year. How long will it take for his savings to grow to $1000? There are many ways to show exponential growth and decay.

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

Exponential equations can be used in science and life for many years from now. Students will see exponential equations when they begin to study bacteria. They will have to find the decay of growth. Students will also have to see population growth and decay throughout history. They may be asked to find out what the population will be in twenty years. When students take economics, or do their own banking, they will need to calculate interest and principal. Students will also need to do the stock market which uses exponential equations. If students go into field where they are concerned with the population of species that may be becoming extinct then the student would predict when the species would become distinct by using an exponential formula. They could also calculate how long until a certain species may take over the world, such as tree frogs or rabbits. Exponential equations are everywhere in the world and in other subjects, besides mathematics.

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

Exponential equations are used with technology everyday and every which way. Khan Academy has a few examples of exponential growth and exponential decay. Youtube has many great examples of exponential equations. Crewcalc’s exponential rap is an excellent example. They are very creative high school who found a way to express a mathematical concept through music.

Zombie Growth shows another interesting way to portray the mathematical concept of exponential equations. They use the phenomenon of zombies to demonstrate how exponential equations work.

Math project on Youtube showed another way to demonstrate how exponential equations work. They posed a problem and then stated the steps to solve the problem. Students need to use graphing calculators to check whether or not they have the right graph based on information given. They also need calculators to calculate equations and check their equations.

Football and the hypotenuse

An amusing video from Training Camp 2013 for illustrating that each side of a triangle (including right triangles) is shorter than the sum of the other two sides. The speaker is Jason Garrett, head coach of the Dallas Cowboys.

Factors

I thought my daughter would have been a little older than 7 before she asked me a math question that I couldn’t immediately answer. I was wrong. Here was her question, asked innocently over breakfast one morning:

$72$ has $12$ factors, and $12$ is also a factor of $72$ . How many numbers are there that are like that?

It took me about 15 minutes before I could definitely give her an answer.

Rather than spoiling the fun for my readers, I’ll just leave this one unanswered and let you think about it.

Fun lecture on geometric series (Part 5): The Fibonacci sequence

Every once in a while, I’ll give a “fun lecture” to my students. The rules of a “fun lecture” are that I talk about some advanced applications of classroom topics, but I won’t hold them responsible for these ideas on homework and on exams. In other words, they can just enjoy the lecture without being responsible for its content.

In this series of posts, I’m describing a fun lecture on generating functions that I’ve given to my Precalculus students. In the previous post, we looked at the famed Fibonacci sequence

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$

We also looked at that (slightly less famous) Quintanilla sequence

$1, 1, 3, 5, 11, 21, 43, 85, \dots$

which is defined so that each term is the sum of the previous term and twice the term that’s two back in the sequence. Using the concept of a generating function, we found that the $n$ th term of the Quintanilla sequence is

$Q_n = \displaystyle \frac{2^{n+1} + (-1)^n}{3}$

To close out the fun lecture, I’ll then verify that this formula works by using mathematical induction. As seen below, it’s a lot less work to verify the formula with mathematical induction than to derive it from the generating function.

$n=0$ : $Q_0 = \displaystyle \frac{2+1}{3} = 1$ .

$n = 1$ : $Q_1 = \displaystyle \frac{2^2-1}{3} = 1$ .

$n-1$ and $n$ : Assume the formula works for $Q_{n-1}$ and $Q_n$ .

$n+1$ :

$Q_{n+1} = Q_n + 2 Q_{n-1}$

$Q_{n+1} = \displaystyle \frac{2^{n+1} + (-1)^n}{3} + 2 \cdot \frac{2^n + (-1)^{n-1}}{3}$

$Q_{n+1} = \displaystyle \frac{2^{n+1} + (-1)^n + 2 \cdot 2^n + 2 \cdot (-1)^{n-1}}{3}$

$Q_{n+1} = \displaystyle \frac{2^{n+1} + 2^{n+1} + (-1)^n - 2 \cdot (-1)^n}{3}$

$Q_{n+1} = \displaystyle \frac{2 \cdot 2^{n+1} - (-1)^n}{3}$

$Q_{n+1} = \displaystyle \frac{2^{n+2} + (-1)^{n+1}}{3}$

That’s the formula if $n$ is replaced by $n+1$ , and so we’re done.

Let me note parenthetically that the above simplification is not all intuitive when encountered by students for the first time — even really bright students who know the laws of exponents cold and who know full well that $x + x = 2x$ and $x - 2x = -x$ . That said, I’ve found that simplifications like $2^{n+1} + 2^{n+1} = 2 \cdot 2^{n+1} = 2^{n+2}$ are usually a little intimidating to most students at first blush, though they can quickly get the hang of it.

By this point, I’m usually near the end of my 50-minute fun lecture. Since students are not responsible for replicating the contents of the fun lecture, I’ve found that most students are completely comfortable with this pace of presentation.

Then I ask my students which way they’d prefer: generating functions or mathematical induction? They usually respond induction. However, they also are able to realize that the thing that makes mathematical induction is also the challenge: they have to guess the correct formula and then use induction to verify that the formula actually works. On the other hand, with generating functions, there’s no need to guess the correct answer… you just follow the steps and see what comes out the other side.

Finally, to close the fun lecture, I tell them that the above steps can be used to find a closed-form expression for the Fibonacci sequence. (I devised the Quintanilla sequence for pedagogical purposes: since the denominator of its generating function easily factors, the subsequent steps aren’t too messy.) I won’t go through all the steps here, so I’ll leave it as a challenge for the reader to start with the generating function

$f(x) = \displaystyle \frac{1}{1-x-x^2}$ ,

factor the denominator by finding the two real roots of $1 - x - x^2 = 0$ , and then mimicking the above steps. If you want to cheat, just use the following Google search to find the answer: http://www.google.com/#q=fibonacci+%22generating+function%22

I conclude this post with some pedagogical reflections. I taught this fun lecture to about 10 different Precalculus classes, and it was a big hit each time. I think that my students were thoroughly engaged with the topic and liked seeing an unorthodox application of the various topics in Precalculus that they were learning (sequences, series, partial fractions, factoring polynomials over $\mathbb{R}$ , mathematical induction). So even though they would likely receive a fuller treatment of generating functions in a future course like Discrete Mathematics, I liked giving them this little hint of what was lying out there for them in the future.

I covered the content of this series of five posts in a 50-minute lecture. I’d usually finish the proof by induction as time expired and then would challenge them to think about how to similarly find the formula for the Fibonacci sequence. The rules of a “fun lecture” were important to pull this off — I made it clear that students would not have to do this for homework, so the pressure was off them to understand the fine details during the lecture. Instead, the idea was for them to appreciate the big picture of how topics in Precalculus can be used in future courses.

Fun lecture on geometric series (Part 4): The Fibonacci sequence

In this series of posts, I’m describing a fun lecture on generating functions that I’ve given to my Precalculus students. In the previous post, we looked at the famed Fibonacci sequence

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$

We also looked at that (slightly less famous) Quintanilla sequence

$1, 1, 3, 5, 11, 21, 43, 85, \dots$

which is defined so that each term is the sum of the previous term and twice the term that’s two back in the sequence. We also used the Bag of Tricks to find that the generating function is

$Q(x) = \displaystyle \frac{1}{1-x-2x^2}$

To get a closed-form definition of the Quintanilla sequence, let’s find the partial-fraction decomposition of $Q(x)$ . Notice that the denominator factors easily, so that

$Q(x) = \displaystyle \frac{1}{(1+x)(1-2x)}$

To find the partial fraction decomposition, we need to find the constants $A$ and $B$ so that

$\displaystyle \frac{A}{1+x} + \frac{B}{1-2x} = \displaystyle \frac{1}{(1+x)(1-2x)}$ ,

$A(1-2x) + B(1+x) = 1$

Perhaps the easiest way of finding $A$ and $B$ is by substituting conveniently easy values of $x$ .

If $x = \displaystyle \frac{1}{2}$ , then we obtain $\displaystyle \frac{3}{2} B = 1$ , or $B = \displaystyle \frac{2}{3}$ .
If $x = -1$ , then we obtain $3A =1$ , or $A = \displaystyle \frac{1}{3}$ .

Therefore,

$Q(x) = \displaystyle \frac{1}{3} \cdot \frac{1}{1+x} + \frac{2}{3} \cdot \frac{1}{1-2x}$

Finally, let’s write the rational functions on the right-hand side as infinite series. Using the formula for an infinite geometric series, we find

$Q(x) = \displaystyle \frac{1}{3} \left(1 - x + x^2 - x^3 + x^4 - x^5 \dots \right) + \frac{2}{3} \left( 1 + 2x + 4x^2 + 8x^3 + 16x^4 + 32 x^5 \dots \right)$

Notice that this matches the terms of the Quintanilla sequence! For example, the coefficient of the $x^5$ term is

$\displaystyle -\frac{1}{3} + \frac{2}{3}(32) = \displaystyle \frac{63}{3} = 31$ ,

which is a term of the Quintanilla sequence.

In general, the coefficient of the $x^n$ term is

$\displaystyle \frac{(-1)^n}{3} + \frac{2 \cdot 2^n}{3} = \displaystyle \frac{2^{n+1} + (-1)^n}{3}$

This is the long-awaited closed-form expression for the Quintanilla sequence. For example, we quickly see that the 12th term is $\displaystyle \frac{2^{13} + 1}{3} = 2731$ , which was obtained without knowing the 10th and 11th terms.