Engaging students: Order of operations

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 Theresa (Tress) Kringen. Her topic, from Pre-Algebra: order of operations.

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

Order of operations is commonly used in most mathematics problem that involve more than one operation or when parenthesis are involved. It would be easy to show the students what the answer to a given problem, say 5+20/5, would be when using the proper order of operations, then solve the problem by solving left to right as you would read a book. It is clear, to a math major, that the answer is 9. For someone who does not know the order of operations, they most likely would come up with the answer of 5. The difference in the correct answer and the incorrect answer is only 4, but the problem is only working with numbers less than or equal to twenty. It would then be beneficial to point out that when dealing with more complex problems, that this answer may become even larger. If the class was working on given problems, I would give them a few word problems to solve. Once they solved them on their own, I would show them that the difference between the correct way to answer the given problem and the incorrect way to answer the problem to help them connect the concept to why it is important to compute answers in the way.

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

This topic extends what students should have previously learned by allowing them to use their skills of multiplication, division, exponents, addition, and subtraction to solve more complex problems. When learning how to solve problems more complicated than what they have been given in the past, they use this topic to guide them through to the next step. They must already be familiar with all of the operations by themselves prior to using the order of operations to solve a problem. Once they are accustomed to using the order of operations, the will be given more challenging problems and their math skills will build upon itself. It is clear that if a student is unable to solve a simple problem, such as an exponent problem or a more complicated division problem, they will not be able to use the order of operations for problems that contain what they have not learned.

How did people’s conception of this topic change over time?

It is believed that the idea of using multiplication before addition became a concept adopted around the 1600s and was not disagreed about. The other operations took their place in the order over time, beginning in the 1600s. It seems that although it was not documented well, most mathematicians agreed upon the same order. It wasn’t until books stated being published that it was important to document the order of operations. The notation may have been different depending on who was writing on the subject, but the concept was the same. It seems that although it was not documented well, most mathematicians agreed upon the same order. Once books were being published, the order, PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction), was put into print. Now, teachers use the phrase Please Excuse My Dear Aunt Sally as a way for students to remember the acronym and are able to put it to use.

http://jeff560.tripod.com/operation.html

http://mathforum.org/library/drmath/view/52582.html

Art of pi

The picture below is just one of many creative artistic representations of $\pi$ and $e$ found at http://mkweb.bcgsc.ca/pi/

In the words of the author, “Numerology is bogus, but art based on numbers have a beautiful random quality.”

Engaging students: Powers and exponents

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 Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: powers and exponents.

A) Applications: What interesting word problems using this topic can your students do now?

I chose the problem below from http://www.purplemath.com because I think that solving a problem that deals with disease would be interesting to my students. People have to deal with sickness and disease everyday and I think that solving a real world problem would entice the students into wanting to learn more.

A biologist is researching a newly-discovered species of bacteria. At time t = 0 hours, he puts one hundred bacteria into what he has determined to be a favorable growth medium. Six hours later, he measures 450 bacteria. Assuming exponential growth, what is the growth constant “k” for the bacteria? (Round k to two decimal places.)

For this exercise, the units on time t will be hours, because the growth is being measured in terms of hours. The beginning amount P is the amount at time t = 0, so, for this problem, P = 100. The ending amount is A = 450 at t = 6. The only variable I don’t have a value for is the growth constant k, which also happens to be what I’m looking for. So I’ll plug in all the known values, and then solve for the growth constant:

$A = Pe^{kt}$

$450 = 100 e^{6k}$

$4.5 = e^{6k}$

$\ln(4.5) = 6k$

$k = \displaystyle \frac{\ln(4.5)}{6} = 0.250679566129\dots$

The growth constant is 0.25/hour.

I think this kind of problem would be beneficial to students because it would help them understand how bacteria grows and how easily they can get catch something and get sick.

C) Culture: How has this topic appeared in pop culture?

Exponents and powers are everywhere around us without the students knowledge. Many movies and video games have ideas related to powers and exponents. Take, for example, the movie Contagion that was released in September 2011. This movie is about “the threat posed by a deadly disease and an international team of doctors contracted by the CDC to deal with the outbreak” (http://www.imdb.com/title/tt1598778). In this movie, there is a scene where the doctors are using mathematical equations with exponents to find out how fast the disease spreads and how much time they have left to save the majority of the population. There are many movies like this that involve powers and exponents, Contagion is just one example. There are also popular video games that deal with the spread of disease. For example, in the video game Call Of Duty: World At War the player is a soldier in WWII and his mission is to kill zombies, and zombie populations grow exponentially. Now, my brother plays this game and I know for a fact that he doesn’t think about the mathematics behind it, but I think talking about pop culture while teaching would really bring some excitement to the classroom and get the students thinking.

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

Exponents and powers have been among humans since the time of the Babylonians in Egypt. “Babylonians already knew the solution to quadratic equations and equations of the second degree with two unknowns and could also handle equations to the third and fourth degree” (Mathematics History). The Egyptians also had a good idea about powers and exponents around 3400 BC. They used their “hieroglyphic numeral system” which was based on the scale of 10. When using their system, the Egyptians expressed any number using their symbols, with each symbol being “repeated the required number of times” (Mathematics History). However, the first actual recorded use of powers and exponents was in a book called “Artihmetica Integra” written by English author and Mathematician Michael Stifel in 1544 (History of Exponents). In the 14^th century Nicole Oresme used “numbers to indicate powering”(Jeff Miller Pages). Also, James Hume used Roman Numerals as exponents in the book L’Algebre de Viete d’vne Methode Novelle in 1636. Exponents were used in modern notation be Rene Descartes in 1637. Also, negative integers as exponents were “first used in modern notation” by Issac Newton in 1676 (Jeff Miller Pages).

Works Cited

Ayers, Chuck. “The History of Exponents | eHow.com.” eHow | How to Videos, Articles & More – Discover the expert in you. | eHow.com. N.p., n.d. Web. 25 Jan. 2012. http://www.ehow.com/about_5134780_history-exponents.html.

“Contagion (2011) – IMDb.” The Internet Movie Database (IMDb). N.p., n.d. Web. 25 Jan. 2012. http://www.imdb.com/title/tt1598778/.

“Exponential Word Problems.” Purplemath. N.p., n.d. Web. 25 Jan. 2012. http://www.purplemath.com/modules/expoprob2.htm.

“Mathematics History.” ThinkQuest : Library. N.p., n.d. Web. 25 Jan. 2012. http://library.thinkquest.org/22584/.

juxtaposition.. “Earliest Uses of Symbols of Operation.” Jeff Miller Pages. N.p., n.d. Web. 25 Jan. 2012. http://jeff560.tripod.com/operation.html.

World’s Most Accurate Pie Chart

Source: Houghton College Facebook page, https://www.facebook.com/photo.php?fbid=10151682563311916&set=a.127025151915.116224.22485036915&type=1&theater

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 Kayla (Koenig) Lambert. Her topic, from Geometry: finding the area of a square or rectangle.

B) Curriculum: How can this topic be used in student’s future courses in math or science?

Finding the area of a square or rectangle can be applied in many other subjects throughout a student’s school career. This topic is learned around 4^th or 5^th grade, and around this time students will just be using the formulas to find the areas. In middle school, they might be finding the areas by way of more difficult problems, like word problems. The real fun for this subject, in my opinion, doesn’t start until high school. In high school you can use the area of squares and rectangles to find the solutions to many problems. In high school geometry, the Pythagorean Theorem is taught. The area of squares is related to this depending on how the teacher presents this to the student. The Pythagorean Theorem states that “in any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs of the right triangle” (Square-geometry).

In college, possibly high school calculus, students will learn to approximate the total area under a curve (or integral) using the Riemann Sum. To approximate the integral, you find the area of each rectangle, and all of the rectangles areas added together give you the approximated integral. The area of rectangles is also used in Statistics. When creating a histogram, you multiply the height (density) and width of the bars (rectangles). Then adding the areas (relative frequencies) of all of the bars should be equal to one. Students will also need to use the area of squares and rectangles on college placement exams and standardized testing.

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

In my opinion, anything and everything is a form of art, so the area of squares and rectangles can appear in an infinite amount of high culture. M.C. Escher has used squares and rectangles to create tessellations and “portrayed mathematical relationships among shapes, figures and space” (MC Escher). The area of a rectangle was used to Polykeitos the Elder who was a Greek sculptor. He used the area of a rectangle to create the perfect ratio for the human body. Painters also needed to figure out how to depict 3D scenes onto 2D canvas during the Renaissance (Mathematics and Art).

However, one of the more well-known applications of mathematics in art is the Golden Rectangle, which just so happens to involve the area of squares and rectangles. The Golden Rectangle is the area of the original rectangle to the area of the square, which is also the Golden Ratio. In other words, the Golden Rectangle is a rectangle wherein the ratio of its length to its width is the Golden Ratio (Golden Rectangle). Many ancient art and architecture have incorporated the Golden Rectangle into designs. The Golden Rectangle was used in the floor plans and design of the exterior of The Parthenon, which was a Greek temple dedicated to goddess Athena in 5^th century BC (Mathematics and Art). Leonardo DaVinci also used the Golden Rectangle in his work. When painting the Mona Lisa, he used this to “draw attention to the face of the woman in the portrait” (Mathematics and Art). DaVinci also used the Golden Rectangle in the Last Supper using it to create a “perfect harmonic balance between placement of characters in the background” and also used it to arrange the characters around the table (Mathematics and Art).

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

Finding the area of squares and rectangles didn’t just come out of the blue; we can thank geometry and ancient mathematics for the development of this topic. One person in particular who contributed to the development of this topic was Euclid, or Euclid of Alexandria, who was a Greek mathematician and known as the “Father of Geometry” (Euclid). He was said to revolutionize geometry and his book The Elements is considered the most influential textbook of all time (History of Mathematics). The collection of his books, all thirteen of them, contain all traditional school geometry (Solomon).

However, Euler wasn’t the only one to contribute to this topic. Pythagoras and his students discovered most of what high school students learn in geometry today (History of Mathematics). In the classical period, Aryabhata wrote a treatise including the computation of areas. From the kingdom of Cao Wei, Liu Hui edited and commented on The Nine Chapters of Mathematics Art in 179 AD (History of Mathematics). There are so many people who contributed to this topic, and people are still contributing and developing to the area of squares and rectangles today!

Works Cited

“Euclid – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/Euclid.

“Golden Rectangle.” Logicville : Puzzles and Brainteasers. 20 Feb. 2012. http://www.logicville.com/sel26.htm.

“M. C. Escher – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/M._C._Escher.

“Mathematics and art – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://www.en.wikipedia.org/wiki/Mathematics_and_art.

Solomon, Robert. The Little Book Of Mathematical Principals, Theories and Things. New York: Metro Books, 2008.

“Square (geometry) – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/Square_(geometry).

“History of mathematics – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/History_of_mathematics.

Engaging students: Solving proportions

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 Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: solving proportions.

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

Solving proportions, or the idea of a proportion being solved, appears in the news more often than not. One specific example that can be used is the effect of the economy on real estate companies. Say we are given 25% of 16 real estate companies that have closed their businesses due to poor economy. We can use proportions to determine the number of real estate companies that closed. We know that the percent is 25 and that the whole is 16. Therefore $25/100 = x/16$ which gives us 4 real estate companies that closed (Review of Proportions). Proportions can also be used to determine how many miles we can drive on a certain amount of gas, and gas prices are constantly on the news. Also, this will be relevant to high school students who drive and need to find how much money they need to buy gas for the week, etc.

We can also use proportions to find the unit price of an item at a grocery store, or if an item costs a certain amount, you can find out how many of those items you can buy with a fixed amount of money you have. Buying items and saving money are also all over the news. If you find the unit price you can compare items therefore saving money by buying the item that you get the most out of your money. Another way solving proportions can appear on the news is by the stock market. You can use proportions to find out how much the stock market will rise in a given amount of days given the current amount of points it has raised in a certain amount of days. Making a proportion problem for students to solve is relatively easy and can be related to anything that is on the news. We can use this to our advantage to get the students to be a little more interested in proportions (and mathematics) so they can see different ways it is related to real life.

D. History: How was this topic adopted by the mathematical community?

The idea of proportions was adopted and used by many in the mathematical community. Proportions were used by Greek writers, including one named Nicomachus, who include proportions and ratios in arithmetic (Math Forum). Proportions were also adopted by Exodus who used them in geometry and by Theon of Smyrna who used proportions in music (Math Forum). In 2000 B.C., the Babylonians adopted proportions to represent place value notation (Pythagoras – Geometrical Algebra). Using proportions was accepted by mathematicians and was used to solve so many different equations used for so many different ideas, and is still used today. Early proportions were adopted by the Egyptians and were used to calculate fractions and measurement of farmland (Mathematics History). Later, proportions were adopted by so many more in the mathematical community like in Greece, China, India, and Babylonia in order to learn geometry. Greeks, like Plato, adopted proportions in order to study them with the Egyptians. I think that proportions were well liked by mathematicians and were adopted by many because you can use proportions to solve so many things.

D. History: How did people’s conception of proportions change over time?

From the beginning, people have used proportions. Early humans used proportions to see if one tribe was twice as large as another or if one leather strap is only half as long as another (Math Forum). It is obvious that the idea solving proportions hasn’t really changed that much, but what we can use proportions to solve has changed. In 2000 B.C. Babylonians used proportions to evolve place value notation by allowing arbitrarily large numbers and fractions to be represented (An Overview of Egyptian Mathematics). Around 1600 B.C. in Egypt, proportions were used to calculate the fraction and superficial measure of farmland (Mathematics History). Egyptians then used proportions to find volumes of cylinders and areas of triangles.

Vitruvius thought of proportions in terms of unit fractions for their architecture calculations (Proportion (architecture)). Also, scribes used “unit fractions” for their calculations in Egypt and Mesopotamia. Egyptians based proportions on parts of their body and their symmetrical relation to each other; like fingers, palms, hands, etc. Multiples of body proportions would be found in the arrangement of fields and buildings people lived in (Proportion (architecture)) and from here, proportions evolved. In 600 B.C., the idea of using proportions evolved and was then used for geometry (Mathematics History). Proportions are still used in geometry, like in architecture and land, like it was 3000 years ago. When you think about it, proportions have evolved, but the use of proportions has evolved even greater. There are so many topics we can now solve using proportions!

Works Cited

“Math Forum – Ask Dr. Math.” The Math Forum @ Drexel University. 7 Mar. 2012. <http://www.mathforum.org/library/drmath/view/64539.html>.

“Mathematics History.” ThinkQuest : Library. 7 Mar. 2012. <http://library.thinkquest.org/22584/>.

“Proportion (architecture).” Wikipedia, the free encyclopedia. 7 Mar. 2012. <http://en.wikipedia.org/wiki/Proportion_%28architecture%29>.

“Review of Proportions.” Self Instructional Mathematics Tutorials. 7 Mar. 2012. <http://www.cstl.syr.edu/fipse/decunit/ratios/revprop.htm>.

“An Overview of Egyptian Mathematics.” 7 Mar. 2012. < http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_mathematics.html >

Engaging students: Adding and subtracting decimals

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!

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!

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.