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.

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.

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.

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: Solving systems of linear inequalities

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 Heidee Nicoll. Her topic, from Algebra: solving linear systems of inequalities.

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

I found a fun activity on a high school math teacher’s blog that makes solving systems of linear inequalities rather exciting.

Link: (https://livelovelaughteach.files.wordpress.com/2013/09/treasure-hunt1.pdf)

The students are given a map of the U.S. with a grid and axes over the top, and their goal is to find where the treasure is hidden. At the bottom of the page there are six possible places the treasure has been buried, marked by points on the map. The students identify the six coordinate points, and then use the given system of inequalities to find the buried treasure. This teacher’s worksheet has six equations, and once the students have graphed all of them, the solution contains only one of the six possible burial points. I think this activity would be very engaging and interesting for the students. Using the map of the U.S. is a good idea, since it gives them a bit of geography as well, but you could also create a map of a fictional island or continent, and use that as well. To make it even more interesting, you could have each student create their own map and system of equations, and then trade with a partner to solve.

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

If students have a firm understanding of inequalities as well as linear systems of equations, then they have all the pieces they need to understand linear systems of inequalities quite easily and effectively. They know how to write an inequality, how to graph it on the coordinate plane, and how to shade in the correct region. They also know the different processes whereby they can solve linear systems of equations, whether by graphing or by algebra. The main difference they would need to see is that when solving a linear system of equations, their solution is a point, whereas with a linear system of inequalities, it is a region with many, possibly infinitely many, points that fit the parameters of the system. It would be very easy to remind them of what they have learned before, possibly do a little review if need be, and then make the connection to systems of inequalities and show them that it is not something completely different, but is simply an extension of what they have learned before.

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

Graphing calculators are sufficiently effective when working with linear systems of equations, but when working with inequalities, they are rather limited in what they can help students visualize. They can only do ≥, not just >, and have the same problem with <. It is also difficult to see the regions if you have multiple inequalities because the screen has no color. This link is an online graphing calculator that has several options for inequalities: https://www.desmos.com/calculator. You can choose any inequality, <, >, ≤, or ≥, type in several equations or inequalities, and the regions show up on the graph in different colors, making it easier to find the solution region. Another feature of the graphing calculator is that the equations or inequalities do not have to be in the form of y=. You can type in something like 3x+2y<7 or solve for y and then type it in. I would use this graphing calculator to help students visualize the systems of inequalities, and see the solution. When working with more than two inequalities, I would add just one region at a time to the graph, which you can do in this graphing calculator by clicking the equation on or off, so the students could keep track of what was going on.

References

Live.Love.Laugh.Teach. Blog by Mrs. Graves. https://livelovelaughteach.wordpress.com/category/linear-inequalities/

Graphing calculator https://www.desmos.com/calculator

Engaging students: Using the point-slope equation of a line

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.

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.

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.

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

Engaging students: Finding the slope of a line

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 Brianna Horwedel. Her topic, from Algebra: finding the slope of a line.

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

Algebra vs. the Cockroaches is a great way to get students engaged in learning about slopes. The object of the game is to kill the cockroaches by figuring out what the equation of the line that they are walking on is. It progresses from simple lines such as y=5 to more complicated equations such as y=(-2/3)x+7. It allows the students to quickly recognize y-intercepts and slopes. Once finished, you can print out a “report” that tells you how many the student got correct and how many tries it took them to complete a level. This game could even be used as a formative assessment for the teacher.

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

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

Last year, I was placed in an eighth grade classroom that was learning about slope. One of the things that really stuck out to me was that the teacher gave a ski illustration to get the students talking about slope. The illustration starts off with the teacher going skiing. She talks about how when she is going up the ski lift she is really excited and having a “positive” experience which correlates to the slope being positive. Once she gets off of the ski lift, she isn’t going up or down, but in a straight line. She talks about how she doesn’t really feel either excited or nervous because she is on flat ground. This corresponds to lines that have a slope of 0. She then proceeds to talk about how when she starts actually going down the ski slope, she hates it! This relates to the negative slope of a line. She also mentions how she went over the side of a cliff and fell straight down. She was so scared she couldn’t even think or “define” her thoughts. This is tied to slopes that are undefined. I thought that this illustration was a great way of explaining the concept of slope from a real world example. After sharing the illustration, the students could work on problems involving calculating the slope of ski hills.

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

Understanding how to find the slope of a line is crucial for mathematics courses beyond Algebra I and Algebra II. Particularly, knowing how to find the slope of a line is essential for finding tangent lines of curves. This comes in handy for Calculus when you have to use limits to determine the slope. If a student does not have a strong grasp of what slope means and what its relationship is with the graph and the equation in Algebra I, then they will have a difficult time understanding slopes of lines that are not straight.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 9: Mentally computing $n$ given $n^5$ if $10 \le n \le 99$ .

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a recent picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

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

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.

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.

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.

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:

Click to access ZZZ-AS-RulesandRecord.pdf

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

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

Engaging students: Finding points on the coordinate plane

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 Lucy Grimmett. Her topic, from Pre-Algebra: finding points on the coordinate plane.

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

As a hands-on learner, I love activities that require me to be up and moving. I have always heard about the human coordinate plane. The teacher creates a life size coordinate plane on the floor of their classroom. The teacher would label the x and y axis and place contact paper on the floor marking coordinate points. Students would walk into class and write their name on a point. Later they would return to their point and the activity begins. The teacher or another student would stand at the origin and ask how she would get to “insert name” point. Students would discover the coordinates of their spot or point. They would then return to their seats, after the activity, and complete a journal entry in their notebook. The teacher would then have notes and discussion with the class. During note taking time I would tell student about the elevator idea. The idea is that you must walk to the elevator before you can go up or down. This is a great reminder for students to remember what order the plot the numbers in, and what direction.

See this link for more detailed instructions: http://www.cpalms.org/Public/PreviewResourceLesson/Preview/49870

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

Plotting points and finding points on a coordinate plane is a very necessary topic for many courses. If you cannot find points adequately on a coordinate plane then students will not be able to graph equations of lines through plugging in. They will not be able to find slope of a line. They will not be able to graph vectors in both mathematics and physics. There are endless needs for plotting points on a graph. When you move into polar coordinates in mathematics, it is necessary to know how to plot points as well. Polar coordinates then lead into trigonometry, which leads into graphing equations of trig functions, which leads into calculus, which also requires graphing equations and using the graph to visualize where the tangent line would be on the coordinate plane. Being familiar with coordinate points and how to find them and plot them is going to be a lesson student’s take with them forever. Even if they are an art major, they will still used coordinates.

How has this topic appeared in the news?

The idea of plotting points is used in graphs, charts, diagrams and many other visual aids in the news. The news is constantly using these visual aids to make data look more dramatic. Another idea is during the weather segment, meteorologist have to mark the temperatures on the city. Now it is very computerized, but back in the day, they had to use latitude and longitude to find the city, which is very similar to plotting points on a coordinate plane. Another idea is stock news. This news is typically on websites rather then broadcasted segments, however, stocks are a great idea of graphing points. You merely plot the day or time and the rate at which the stock was increasing or decreasing. The points are then connected with lines to show how it the stock goes up and down. This is a good idea when students start to learn about slope of lines as well.

Engaging students: Order of operations

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 Lisa Sun. Her topic, from Pre-Algebra: order of operations.

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

Given that my students have knowledge on the topic of Order of Operations, I will provide them a project where they must apply their knowledge and present it in front of their peers. Students will each receive a number from me and they must create a mathematical problem, an equation, using all of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction).

Students will then present individually in front of their peers at the board. The presenter’s role is to be the teacher. To have the ability to clearly vocalize his/her thought process to achieve their given number with the use of PEMDAS. As each student presents, the audience will be following along on a sheet of paper where they must also solve the equation that the presenter created. This paper must be turned in along with their own project to document that they were paying attention. The audience’s role is to be the grader. To make sure that the presenter’s use of PEMDAS was correct to achieve the number that was given to them. If the presenter’s use of PEMDAS is incorrect, I will select an audience member to explain. The presenter will then have to come present their project again to me before or after school so that I can make sure there is no misconception regarding the Order of Operations.

To help motivate the students’ to be precise with their project, I would state that if all students were able to display their use of PEMDAS correctly, everyone would receive 5 extra points on the upcoming test. I believe that this project would be great for students to strengthen their knowledge on Order of Operations. As they are taking up on their roles as the grader, they are physically and mentally reinforcing their knowledge by solving problems after problems. As the teacher, they are verbally reinforcing their knowledge.

How has this topic appeared in pop culture?

Figure 1: Pokémon Center Lego

Pokémon Go is the craze among society today and I believe it would be fitting to engage the class with both Pokémon and Legos. I would present this to the class, preferably one that is physically available to the class and ask the following questions:

When building this Lego figure, do you think procedures need to be followed sequentially?
What happens if they are not? (Display to the class what the Lego figure would look like).

I would discuss why doing things in order is important tying it with Orders of Operations. Display a problem with Orders of Operations but solve it by not following PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) and state that the solution comes out to be incorrect. Similar to how the Lego that was built in the wrong order didn’t match up with the picture on the box.

I believe YouTube can be a great learning tool in the classroom when it comes to engaging students. People of all demographics post helpful tools on this site that are so easily relatable to students today. Below is a video of a PEMDAS rap song.

I will be playing this PEMDAS rap song as students are walking into class to quickly engage the students. Once class has officially started, I would play this video again as students are reading the lyrics and following along the two examples the video provided. This video is to aid the students to remember the Orders of Operations by the use of PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction). To engage the students even more, I would have the students sing along the chorus. “Parentheses first, exponents next, multiplication and division in the same step. Addition and subtraction, if you got the nerve, from left to right, first come first serve”. Hopefully, this song will be catchy enough for the students to have it be stuck in their head for a while.

References:

http://www.purplemath.com/modules/orderops.htm

http://www.pbslearningmedia.org/resource/mgbh.math.oa.ooo/order-of-operations/

Engaging students: Fractions, percents, and 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 again comes from my former student Kim Hong. Her topic, from Pre-Algebra: fractions, percents, and decimals.

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

I think making the students create a foldable, a short and quick project, would be a good and concrete activity for teaching fractions, decimals, and percents. Each flap is a topic. There is a definition and example. On the back of the foldable the students could create a table going between fractions, decimals and percents with many “harder” values.

The foldable is portable and quick, and can be a helpful and quick resource.

The students can also draw pictures inside the flaps. E.g A pizza and its slices to show fractions.

http://smithcurriculumconsulting.com/m4m_foldable/

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

This topic can be used in a students’ future course when they come across proportions and rates. They could see proportions when it appears in physics such a changes in time and speed. They could see rates of change when it appears in calculus involving derivatives. These values are factions that can be changed to decimals and percents because everything is a part of a whole.

Also, fractions, which are numbers over a whole, are the same as the term rational quantities. Rational quantities are numbers that can be written as a ratio that is a fraction. There is a subset of the Reals that are called the Rationals. In advanced logic and math courses, students will be able to work with this subset of the Reals.

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.

I found this really awesome website the students could play around with for the first minutes of class to get their juices flowing. Basically the objective of the game is to group the equal values in circles. There is a check answer option as well.

It starts off very simple with very easy mental math and then with each level, the difficulty increases.

http://www.mathplayground.com/Decention/Decention.html

Engaging students: Square roots

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 Jessica Martinez. Her topic, from Algebra: square roots.

How has this topic appeared in the news?

There is a (sort of) holiday for square root days; sort of because square root days only come 9 time every century and this year we celebrated 4/4/16. Since it’s not as frequent as Pi Day, it’s a lesser known “holiday”, but even then, it still pops up in the news. I found this online article for a UK news site that described other square root-related fun facts in history. It also included a post from Good Morning America with the hashtag #squarerootday, which gave me this idea: I would like to encourage my students to participate in all of the fun square root-related activities that celebrate this day (if there was one that school year). The founder of square root day has suggestions that include but are not limited to: square dancing, drinking root beer out of square glasses, or even taking a drive on route 66. In the days leading up to this fantastic math-related day, I would consider giving my kids an extra credit point for posting a picture of themselves doing something square root related on the class twitter with the tag #squarerootday (or a post on some other class social media). If there wasn’t a square root day during that academic year, I still think it would be fun to tell my students about this holiday.

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

My students should have already learned about perfect squares and their multiplication tables up to 12 or 13, at least. For a simple refresher, I could have my students color/highlight perfect squares on multiplication tables. Then taking the square root of something is the inverse of creating perfect squares, unless what’s under the square root sign isn’t a perfect square. Then what’s under the radical is something that they need to divide into its prime factors so that they can simplify. My students should have also at least learned about prime numbers, if not prime factoring. A way to solve square roots would be pairing up the prime factors under the square root so that you can “take it out” from under the radical; for my students, I could have them think of the square root sign as a jail cell, and the only way that the numbers could “get out” of the cell is if they had a “prime partner” to escape with (i.e. a pair of 2s, a pair of 3s etc.).

green line

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

So one of the oldest records of square roots in history would be The Old Babylonian tablet YBC 7289, which dates back anywhere from 2000-1600 BC. It depicts a square with two diagonals drawn and on the diagonals are numbers; when they are calculated, you get a very close approximation of the square root of 2 for the diagonal. Their value for the square root of two was about 1.41421297; I could have my students quickly calculate the square root of two (about 1.41421356) and mention to my students that this is pretty impressive for a civilization without modern day technology. The fact that they used clay tablets for math calculations shows how little they had to work with. Yet Babylon was also one of the most famous ancient cities in Mesopotamia; it’s mentioned multiple times in the bible and they were pretty advanced in mathematics for their area, despite the lack of resources we have today. They used a sexagesimal number system, which is base 60; they could solve algebra problems and work with what we now call Pythagorean triples; they could also solve equations with cubes.

References

A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card). (2012, January 15). Retrieved September 09, 2016, from https://reflectionsinthewhy.wordpress.com/2012/01/15/a-visual-approach-to-simplifying-radicals-a-get-out-of-jail-free-card/

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Buncombe, A. (16, April 4). Square Root Day: There are only nine days this century like this. Retrieved September 09, 2016, from http://www.independent.co.uk/news/world/americas/square-root-day-there-are-only-nine-days-this-century-like-this-a6967991.html

Fowler, D., & Robson, E. (n.d.). Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Historical Mathematica, 366-378. Retrieved September 9, 2016, from https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf.

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/