Engaging students: Square roots

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 Julie Thompson. Her topic, from Algebra: square roots.

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

When I think of square roots my mind immediately takes me to the very popular movie ‘The Wizard of Oz’. In a scene near the end of the movie, the scarecrow incorrectly states the Pythagorean Theorem. He states it so fast that some people may not have time to process what he is saying is incorrect. The theorem he states is as follows: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” There are a couple things wrong with this statement. First of all, the Pythagorean Theorem is based on right triangles, not isosceles. Secondly, we take the square of two specific sides and set it equal to the square of the third side, not the square root of ‘any two sides’ equal to the square root of the remaining side.

As an engage, I think it would be very interesting to first show the clip of the movie to capture my students’ attention, and then have a discussion about why the theorem is wrong and what the correct theorem actually is!

Also, I found an awesome worksheet from Mathbits that is all about this scene from the movie and goes through a couple examples that shows why his theorem can’t work, and also allows students to prove why it is false!!

https://mathbits.com/MathBits/MathMovies/OzMath.pdf

 

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

A very engaging website that was actually introduced to me in college: KAHOOT! I first played Kahoot in my TNTX 1200 class here at UNT. It was very exciting and fun for me, as a college student, to play, so I know middle and high school students will love it as well. Kahoot is an online quiz game where students use their own technology to join in to the game with a game pin provided by the teacher. Students get to give themselves a game nickname which makes it fun to be able to see their name pop up on the scoreboard. Then a variety of questions on the topic are asked, one at a time, with a time limit for the students to answer in (usually about 20-30 seconds). This is a quick game that can be used as an engage at the beginning of class to get students thinking and excited about the topic for the day. In this case…square roots! I found a great Kahoot created by ‘remangum’ that focuses on finding square roots of numbers (it throws in a couple cube roots). Once you get passed about 7 questions, they throw some variables into the mix. One of the question asks to find d:

Sqrt (d*d)=9, where * is multiplication. In this case, d=9 because sqrt(81)=9. I like this because it allows the students to think a little harder and problem solve.

Here is the link: https://play.kahoot.it/#/?quizId=8ed37283-e3fc-4389-a8ed-ff7200993731

 

 

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

Many students who enter middle school/ early high school wonder why they have to learn all these pointless concepts such as square roots and the order of operations. They might even think to themselves, “When will I ever need to know this in the future when I have a job?” According to homeschoolmath.net, “The answer is that you need algebra in any occupation that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade, commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You also need algebra to take your SAT test or GED.” This is very important to let students know, but they may not believe you or care. For instance, they may say that’s true for math and science professions, but they are planning to major in something totally different and they won’t need math. Math actually can be useful in other fields, but for the sake of this question, I will stick to math and science.

In their future classes, such as Algebra II, they will be using things such as the quadratic formula. This will involve plugging in and simplifying things under a radical, as well as dealing with square roots in whole equations rather than just on their own. Also, understanding the nature of square roots will help them in future courses such as PreCalculus when they must study all the characteristics of the square root function. As an engaging aspect to all of this, I may mention that, “Studying algebra also has a benefit of developing logical thinking and problem solving skills. Algebra can increase your intelligence! (Actually, studying any math topic — even elementary math — can do that, if it is presented and taught in such a manner as to develop a person’s thinking.)”

Quotations from:    https://www.homeschoolmath.net/teaching/why_need_square_roots.php

 

 

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/

 

Engaging students: Solving word problems of the form “a is p% of b”

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 Christian Oropeza. His topic, from Algebra: solving word problems of the form “a is p% of b.”

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Students would be able to answer word problems that involve real world applications. For example, a student could be asked: “Sam went to Academy to buy clothes, sports equipment, and fishing gear. At the register the total of Sam’s transaction before tax is $141.32. Given that the sales tax is 8.25%, what would Sam’s total be after tax?” These type of word problems would be relatable to students, which would show them the importance of this topic in life. Students always ask the question, “how is this used in everyday life?”, and with these type of word problems students may be able to generalize the concept more easily. When students cannot relate to a topic in math they become easily discouraged, give up, and stop paying attention in class, but with problems like these the students would be able to incorporate the topic into their own lives. Some other problems that students could be asked could involve any type of scenario where there is a percentage to be found between two numbers (Reference 1 & 4).

 

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

This topic can be used in different scenarios for math and science, but Chemistry is an excellent example. In chemistry, there is a topic that covers calculating percent composition. The basic idea of this topic is to calculate the percentage of each element’s mass in regard to a molecule’s total molecular mass. An example would be, “Calculate the mass percent composition of each element in a potassium ferricyanide, K3Fe(CN)6 molecule.” (Reference 2). These types of problems would help students understand how much a certain element or compound is in a particular molecule. Another example of how this topic can be used, is in math when a student has to convert between fractions, decimals, and percentages in a word problem. An example could be, “Mia has a basket full of fruit. In this basket she has 1/5 apples, 2/3 oranges, and 2/15 bananas. What percent of each fruit does she have in relation to the basket?” Students would be able to work on their converting skills to enhance their understanding of multiple representations of the same number (Reference 3).

 

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

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 4) is an excellent piece of technology to introduce or review this topic to the students because the website goes through visual representations of how a percentage of a whole looks like. Also, the website has a section where a student can input a number and a slider that allows the student to move it around to see what number would represent a certain percentage of the number inputted. Another example of effective technology is the website Khan Academy (Reference 1) because it has real world problems that are relatable. The website also gives hints and step-by-step solutions for each question in case a student is stuck and does not know what to do next. The use of multiple websites is good for students to have a variety to choose from in case one is easier to understand than another.

 

 

 

References:

  1. https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-percent-word-problems/e/percentage_word_problems_1
  2. https://www.thoughtco.com/how-to-calculate-mass-percent-609502
  3. http://www.aaamath.com/pct.htm#topic7
  4. https://www.mathsisfun.com/percentage.html

 

 

Engaging students: Ratios and rates of change

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 Cameron Story. His topic, from Algebra: ratios and rates of change.

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

Since the most relatable example of a ratio is speed (meters per second, miles per hour, etc.), it’s easy to see how a teacher can make an interesting or engaging word problem out of this. First, however, let us take a look at an infamous word problem involving ratios/rates of change that is not inherently interesting on its own.

“Train A, traveling 70 miles per hour (mph), leaves Westford heading toward Eastford, 260 miles away. At the same time Train B, traveling 60 mph, leaves Eastford heading toward Westford. When do the two trains meet? How far from each city do they meet?” (“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.)

 

This is a distance-over-time that most students or past students are familiar with, but why is this problem still being used? There are a few issues I have with this example. Firstly, I cannot think of very many students who could honestly get excited about trains, especially now in the modern era of vehicular travel. I am willing to bet that most of your high school math students have never even been on a train; and if they have, it was most likely an underwhelming experience. This example also lacks creativity. Giving the trains actual names or having them traveling between real world places would have been a step in the right direction.

So how can we change this example to become engaging to students? Firstly, let’s replace the trains with modern cars, and crank up the speed. Every student is familiar with cars, and fast-moving cars (in my opinion) is much more exciting. One could easily imagine using modern rockets as the vehicle as well, and replacing the towns with interplanetary destinations. Next, instead of naming the cars Car A and Car B, we can use actual modern electric cars such as the Model 3 from Tesla Motors. Take a look of the following word problem I came up with instead (you may notice the stakes of the situation described is objectively more engaging then a problem about train travel):

“Tesla is hoping to feature one of its new cars in a commercial, in which a car attempts to race underneath a falling refrigerator in dramatic fashion. In the commercial, the car must travel at top speed, traveling over 25 meters of track from start to finish. As soon as the car passes the starting line, the fridge is dropped from 10 meters up in the air above the finish line, at a rate of 20 meters per second. The top speeds (in meters per second) of the Tesla Model 3 and the Tesla Roadster are shown below. Which car should Tesla pick to safely beat the falling fridge?”

The reason a creative approach works better is that it increases the student engagement; students do not want to do word problems, so it is our job as teachers to make them interesting.

 

 

 

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

Creating an activity around rates of change allows for a lot of creativity. For example, one could take a physical approach, in which students record how fast they can run (only requires a stop watch and a set distance) and using that to plot their data on a distance vs. time graph.

It is important to remember that ratios can represent far more than just speed. Some relatable examples include rate of hair growth, number of hours studied per week, or even how many gallons of water drank in a day. For my Tesla commercial word problem, I used a website (desmos.com) to flesh out this one problem into an engaging classroom activity. Having your classroom activities on interactive platforms that evoke teamwork and cooperation in your students is key to student engagement.

 

 

 

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

Desmos Classroom Activities (at teacher.desmos.com) is an incredibly useful tool that teachers can use to quickly create any activity for their students. These activities can even be done on smart phones, which removes some of the hassle of getting computers in the classroom. When creating an activity, teachers also have access to a wide range of tools including (but not limited to) animation, student inputs, information slides (for presentation), and even interactive functions that allow students to modify given equations.

The main benefit of using Desmos for classroom activities is that the teacher has full and complete access to viewing student progress. Instead of walking around the room trying to hunt down students who need help, the teacher can view which students are stuck on which problems. The teacher can then approach the issue fully prepared, and know exactly which students are having problems before their hands even hit the air.

I created a Desmos activity available for use in a lesson about ratios or rates of change (link: https://teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0) which uses the same Tesla commercial word problem I gave before. Using this website, I was able to build this world problem into a somewhat-realistic and animated simulation, asking critical questions in order to build upon the underlying mathematical concepts. Feel free to adapt my lesson (Desmos has a copy/edit feature for activities) for any vehicle, scenario, or speed.

 

References:

“Desmos Classroom Activities.” Desmos Classroom Activities, 2010, teacher.desmos.com/.

 

Story, Cameron. “Ratios and Rates of Change Activity.” Desmos Classroom Activities, 30 Aug. 2018, teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0.

 

“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.

 

 

 

 

Powers Great and Small

I enjoyed this reflective piece from Math with Bad Drawings about determining whether a^b or b^a is larger. The final answer, involving the number e, was a complete surprise to me.

Short story: e is the unique number so that e^x > x^e for all positive x.

Powers Great and Small

Engaging students: Equations of two variables

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 Trent Pope. His topic, from Algebra: equations of two variables.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

I found a website that has many word problems where students can solve for two variables. An example of one of these problems is “If a student were to buy a certain number of $5 scarfs and $2 hats for a total amount of $100, how many scarfs and hats did they buy?”. This example would give students a real world application of how we use two variable equations. It would show students that there are multi variable problems when we go to the store to shop for things, like food or clothing. An instance for food would be when a concession stand sells small and large drinks at a sporting event and want to know how many drinks they have sold at the end of the night. After using a two variable linear equation and knowing the price of the cups, total amount earned, and total cups sold, students would be able to solve for the number of small cups as well as large cups sold.
https://sites.google.com/site/harlandclub/Home/math/algebra/word2var

 

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B2. How does this topic extend what your students should have learned in previous courses?

This topic extends on the students’ ability to graph and solve a linear equation, which should have been taught in their previous classes. The only difference is that the variable, y, that you solved for in Pre-Algebra is now on the same side as the other variable. For instance, the equation y =(-1/4) x + 4 is the same as x + 4y = 16. We see that we solve for the same variables, but they are both on the same side. This is because you are solving the same linear equation. A linear equation can be written in multiple forms, as long as the forms have matching solutions. This is something that students could prove to you by graphing and solving the equations. They would solve the equations to see that they have the same variables. This makes students more aware that they need to be able to compute for other variables besides x if the question asks for it.

 

 

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E1. 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.

The most effective way to engage a student about this topic is by using a graphing calculator. This is to help students make the visual connection with the topic and check to see if they have graphed the equations the correct way. Students learn more effectively through visual demonstration. Because students are the ones to solve for the equation and plug it into the calculator to check their work, they are going to be able to make that connection, and we will be able to verify that they understand the material. As teachers, we need to incorporate more technology into the ways of learning because we are surrounded by it daily. Using graphing calculators would be a great way to show and check the work of a two variable equation. This gives students a chance to see what mistakes they have made and what lose ends need to be tied up.

References

Solving Word Problems using a system with 2 variables. n.d. <https://sites.google.com/site/harlandclub/Home/math/algebra/word2var&gt;.

Engaging students: The quadratic formula

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 Megan Termini. Her topic, from Algebra: the quadratic formula.

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The Quadratic Formula came about when the Egyptians, Chinese, and Babylonian engineers came across a problem. The engineers knew how to calculate the area of squares, and eventually knew how to calculate the area of other shapes like rectangles and T-shapes. The problem was that customers would provide them an area for them to design a floor plan. They were unable to calculate the length of the sides of certain shapes, and therefore were not able to design these floor plans. So, the Egyptians, instead of learning operations and formulas, they created a table with area for all possible sides and shapes of squares and rectangles. Then the Babylonians came in and found a better way to solve the area problem, known as “completing the square”. The Babylonians had the base 60 system while the Chinese used an abacus for them to double check their results. The Pythagoras’, Euclid, Brahmagupta, and Al-Khwarizmi came later and all contributed to what we know as the Quadratic Formula now. (Reference A)

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A2. How could you as a teacher create an activity or project that involves your topic?

A great activity that involves the Quadratic Formula is having the students work in groups and come up with a way to remember the formula. It could be a song, a rhyme, a story, anything! I have found a few examples of students and teachers who have created some cool and fun ways of remembering the Quadratic Formula. One that is commonly known is the Quadratic Formula sung to the tune of “Pop Goes the Weasel” (Reference B). It is a very catchy song and it would be able to help students in remembering the formula, not just for this class but also in other classes as they further their education. Now, having the students create their own way of remembering it will benefit them even more because it is coming from them. An example is from a high school class in Georgia. They created a parody of Adele’s “Rolling in the Deep” to help remember the Quadratic Formula (Reference C). It’s fun, it gets everyone involved, it engaging, and it helps student remember the Quadratic Formula.

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E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is a great way of engaging students in today’s world. Many students now have cell phones or the school provides laptops to be used during class. Coolmath.com is a great website for students to use to learn about the quadratic formula and great way to practice using it. They show you why the formula works and why it is important to know it because not all quadratic equations are easy to factor. There are a few examples on there and then they give the students a chance to practice some random problems and check to see if they got the right answer. This website would be good for student in and out of the classroom (Reference D). Khan Academy is another great way for students to learn how to use the quadratic formula. They have many videos on how to use the formula, proof of the formula, and different examples and practices of applying the quadratic formula (Reference E). Students today love when they get to use their phones in class or computers, so technology is a great way to engage students in learning and applying the quadratic formula.

 

References:

A. Ltd, N. P. (n.d.). H2g2 The Hitchhiker’s Guide to the Galaxy: Earth Edition. Retrieved September 14, 2017, from https://h2g2.com/approved_entry/A2982567
B. H. (2011, April 04). Retrieved September 14, 2017, from https://www.youtube.com/watch?feature=youtu.be&v=mcIX_4w-nR0&app=desktop
C. E. (2013, January 13). Retrieved September 14, 2017, from https://www.youtube.com/watch/?v=1oSc-TpQqQI
D. The Quadratic Formula. (n.d.). Retrieved September 14, 2017, from http://www.coolmath.com/algebra/09-solving-quadratics/05-solving-quadratic-equations-formula-01
E. Worked example: quadratic formula (negative coefficients). (n.d.). Retrieved September 14, 2017, from https://www.khanacademy.org/math/algebra/quadratics/solving-quadratics-using-the-quadratic-formula/v/applying-the-quadratic-formula

 

 

 

 

Engaging students: Graphs of linear equations

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 Saundra Francis. Her topic, from Algebra: graphs of linear equations.

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B1. How can this topic be used in your students’ future courses in mathematics and science?

Learning how to graph linear equations is the basis for many topics that students will learn later in Algebra and future mathematics and science courses. Students will now be able to solve word problems using graphs to model the situation describe in the problem. Being able to graph linear equations will help students graph non-linear equations since they will be able to apply the steps they learn on how to graph to different types of equations, Students will also be able to graph inequalities to find solutions for an equation since graphing equations is the first step in graphing inequalities. Another application of graphing linear equations is when students need to make graphs when completing science labs, many times students need to graph their data collected and find an equation that represents the data.

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C3. How has this topic appeared in the news?

Graphs of linear equations are displayed in the markets sections on The New York Times. Segments of different linear equations can be put together match the graphs that display the rise and fall of different markets and stocks. Time is displayed on the x-axis while the y-axis list the price of the stock. The slope of the line is the percent change in the price of the stock and can be positive or negative depending if the price rose or fell. The y-intercept would be the price that the stock or market was at before the percent change. This will engage students because it is an example of how graphs of linear equations is displayed in the real world and they get a chance to see how they can use this concept in the future. This could also be made into an activity where students discover the linear equations that are combined to make a certain market or stock graph.

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D1. What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

René Descartes was born in 1596 and was a French scientist, philosopher, and mathematician. He is thought to be the father of modern philosophy. Descartes started his education at age nine and by the time he was twenty-two he had earned a degree in law. Then Descartes tried to understand the natural world using mathematics and logic, which is when he discovered how to visually represent algebraic equations. Descartes was the first to use a coordinate system to display algebraic equations. In 1637 Descartes published La Géométrie, which was where he first showed how to graph equations. He linked geometry and algebra in order to represent equations visually. While thinking about the nature of knowledge and existence Descartes stated, “I think; therefore I am”, which is one of his most famous thoughts. Students will gain interest in graphing equations when they are told about Descartes since he was an interesting person and he discovered things not only in the field of mathematics but philosophy too.

References
https://www.biography.com/people/ren-descartes-37613
http://www.classzone.com/books/algebra_1/page_build.cfm?content=links_app4_ch4&ch=4
https://markets.on.nytimes.com/research/markets/overview/overview.asp

 

 

Engaging students: Multiplying binomials

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 Sarah McCall. Her topic, from Algebra: multiplying binomials.

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B2. How does this topic extend what your students should have learned in previous courses?

My hope is that this topic may be easier to understand if student’s can first recall an easier concept that they have already mastered, and then build upon that foundation to learn new skills. For example, at this point students should have already learned the distributive property. To introduce this new concept, I would begin by writing 4(x-5)=4 on the board and asking students what the very first step would be to solve for x. They should know to start by distributing the four to both x and -5, to get 4x-20=4. After completing a few similar examples as a class and/or in groups, then the idea of multiplying binomials would be introduced. This way, students are less intimidated when presented with new material, and they will have a good understanding of how to distribute to each term.

 

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Teaching students some of the history behind what they are learning can be a great engaging tool. In this case it is helpful to know where the foil method first originated. I would incorporate this by discussing how it first was used in 1929; in William Bentz’ Algebra for Today. In Algebra for Today, Bentz was the first person to mention the “first terms, outer terms, inner terms, last terms” rule. Students should be knowledgeable about the history behind the math they are using, so that they realize the importance of this method. I also believe that it will be cool for students to see how a method developed is still relevant 88 years later. This technique was created in order to provide a memory aid, or “mnemonic device” to help students learn how to multiply binomials. The fact that it is still being used even today proves what an influential concept it was at its time, and throughout the years.

 

green lineE1. 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 am a huge fan of incorporating technology in the classroom, and YouTube is especially great because most students already use YouTube outside of school. The following clip (stopped at 1:48) provides a clear, concise explanation and demonstration of the FOIL method for multiplying binomials. It explains how factoring and foiling are related, and shows students which order to distribute in (first, outer, inner, last). The acronym FOIL is easy for students to remember, and gives them something that they can write down each time they complete a problem to help them distribute properly. Additionally, the clip is just under two minutes, which is the perfect time to ensure that students don’t zone out or lose interest before the end of the video. I would choose to follow up this video by completing a few examples as a class, emphasizing the four steps of foiling as mentioned in the video and how to use them.

References

http://pballew.blogspot.com/2011/02/origin-of-foil-for-binomial.html

Engaging students: Graphing inequalities

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 Lyndi Mays. Her topic, from Algebra: graphing inequalities.

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A1. Once students get to the point where they’re graphing inequalities, they should have a pretty good understanding of how to graph regular functions. I’ve noticed that where students have issues graphing inequalities is knowing which side of the graph should be shaded. Students get confused thinking that the graph should be shaded depending on the direction of the line instead of checking specific points. One activity that I would like to try in the classroom is giving them a worksheet where they graph inequalities on a small graph and when all the little graphs are graphed and shaded it creates a mosaic picture. I feel like there needs to be some sort of pattern or picture so that the students are sure that they’re doing the questions correctly. Another reason I like this activity is because it reaches the intelligence of artistic students. It’s not often that a math lesson can reach artistic intelligences.

 

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C1. One thing the students might find interesting about linear inequalities is that they appeared in the popular TV series, Numbers. In this particular episode, there is a blackout from attacks on an electrical substation. In order to figure out where the attack was located they mapped out where the blackouts were happening. Once they filled in all the different places that were blacking out, they realized it was one big section. Then they drew lines as if the map was on the coordinate plane. From there they are able to target the location where the attack happened.
Students also might be interested in knowing that this is also the way that policeman use to locate a cell phone. They mark the three closest cell towers that the cell phone pinged off of and are then able to draw a section and use linear functions to find the cell phone.

 

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E1. https://us.sofatutor.com/mathematics/videos/graphing-linear-inequalities

This video shows students how to solve for a variable and graph with inequalities. I liked the way it was set up because it was a word problem set up like a story and then solved. I know that students can become intimidated by having to learn new material and then having to apply it to a word problem. But this video kind of walks them through it which I believe could be helpful. Another thing was that the thing we were solving for was very realistic and might help students see why they would need to know how to graph linear equations in the future. The video also showed what x represented (cookies) and what y represented (lemonade). This lets the students know that x and y actually mean something instead of just being an arbitrary variable. I also liked that the video checked for specific points for the shading portion since many students forget that that’s a possibility and end up guessing where to shade.

 

References:
Sayfan, Sayfan. Graphing Linear Inequalities. https://us.sofatutor.com/mathematics/videos/graphing-linear-inequalities.

 

 

 

Engaging students: Slope-intercept form of a line

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

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

This student submission comes from my former student Jessica Williams. Her topic, from Algebra I: the point-slope intercept form of a line.

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A.2 How could you as a teacher create an activity or project that involves your topic?

In order to teach a lesson regarding slope intercept form of a line, I believe it is crucial to use visual learning to really open the student’s minds to the concept. Prior to this lesson, students should know how to find the slope of a line. I would provide each student with a piece of graph paper and small square deli sheet paper. I would have them fold their deli sheet paper into half corner to corner/triangle way). I would ask each student to put the triangle anywhere on the graph so that it passes through the x and the y-axis. Then I will ask the students to trace the side of the triangle and to find two points that are on that line. For the next step, each student will find the slope of the line they created. Once the students have discovered their slope, I will ask each of them to continue their line further using the slope they found. I will ask a few students to show theirs as an example (picking the one who went through the origin and one who did not). I will scaffold the students into asking what the difference would look like in a formula if you go through the origin or if you go through (0,4) or (0,-3) and so on. Eventually the students will come to the conclusion how the place where their line crosses the y-axis is their y intercept. Lastly, each student will be able to write their equation of the line they specifically created. I will then introduce the y=mx+b formula to them and show how the discovery they found is that exact formula. This is a great way to allow the students to work hands on with the material and have their own individual accountability for the concept. They will have the pride of knowing that they learned the slope intercept formula of a line on their own.

 

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E.1 How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Graphing calculators are a very important aspect of teaching slope-intercept form of a line. It allows the students to visually see where the y-intercept is and what the slope is. Also, another good program to use is desmos. It allows the students to see the graph on the big screen and you can put multiple graphs on the screen at one time to see the affects that the different slopes and y intercept have on the graph. This leads students into learning about transformations of linear functions. Also, the teacher can provide the students with a graph, with no points labeled, and ask them to find the equation of the line on the screen. This could lead into a fun group activity/relay race of who can write the formula of the graph in the quickest time. Also, khan academy has a graphing program where the students are asked to create the graph for a specific equation. This allows the students to practice their graphing abilities and truly master the concept at home. To engage the students, you could also use Kahoot to practice vocabulary. For Kahoot quizzes, you can set the time for any amount up to 2 minutes, so you could throw a few formula questions in their as well. It is an engaging way to have each student actively involved and practicing his or her vocabulary.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Learning slope intercept form is very important for the success of their future courses and real world problems. Linear equations are found all over the world in different jobs, art, etc. By mastering this concept, it is easier for students to visualize what the graph of a specific equation will look like, without actually having to graph it. The students will understand that the b in y=mx+b is the y-intercept and they will know how steep the graph will be depending on the value of m. Mastering this concept will better prepare them to lead into quadratic equations and eventually cubic. Slope intercept form is the beginning of what is to come in the graphing world. Once you grasp the concept of how to identify what the graph will look like, it is easier to introduce the students to a graph with a higher degree. It will be easier to explain how y=mx+b is for linear graphs because it is increases or decreases at a constant rate. You could start by asking,
1.What about if we raise the degree of the graph to x^2?
2.What will happen to the graph?
3.Why do you think this will happen, can you explain?
4.What does squaring the x value mean?
It really just prepares the students for real world applications as well. When they are presented a problem in real life, for example, the student is throwing a bday party and has $100 dollars to go to the skating rink. If they have to spend $20 on pizza and each friend costs $10 to take, how many friends can you take? Linear equations are used every day, and it truly helps each one of the students.

References:
https://www.khanacademy.org/math/algebra/two-var-linear-equations/graphing-slope-intercept-equations/e/graph-from-slope-intercept-equation