Engaging students: Defining intersection

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 Ethan Gomez. His topic, from Geometry: defining intersection.

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

In geometry, students gain a better conceptual understanding of what an intersection is in mathematics. Particularly, by the end of geometry, students should be able to understand that different figures in mathematics can intersect, and depending on the nature of those figures, could intersect at more than one place. In Algebra II, students begin learning about rational polynomials. Often, the graphical representation of rational polynomials contains either vertical, horizontal, or slant asymptotes (these are the common asymptotes in Algebra II). Students could make a connection between what an asymptote is and the definition of intersection. Namely, an asymptote is some sort of “invisible line” that a function cannot intersect. Thus, by understanding what intersections are in geometry, they are able to better understand the idea of a lack of intersection. This characteristic of asymptotes should then be intuitive by students, so all they would need to learn is that the functions approach the asymptotes but never cross it, i.e., intersect it. This is the new knowledge they can add to their prior knowledge.

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

Students will most likely have taken Algebra I before Geometry. Thus, students should have discussed solving systems of linear equations. Visually, they should understand that the solution to the system of equations should be a single point in the cartesian plane, particularly a point of intersection. So, students are aware that figures in mathematics can intersect. In geometry, we introduce more figures instead of just dealing with lines. Thus, these figures can intersect, and depending on the figures, they may intersect in more than one point. Up to this point, students have not seen figures in mathematics that could intersect in more than one point, thus extending their idea of what intersections may look like.

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

I wasn’t able to find anything online; if I had time, I’d create a Desmos activity that reflects the ideas I’m about to propose (since I know Desmos has a lot of cool features). On Desmos, your can use sliders to adjust different variables. Thus, I would write two slope-intercept linear equations with slider-variables for the slopes and the y-intercepts. Additionally, I would write an equation for a circle with sliders-variables for the radius and center coordinate. Student would then be able to manipulate the location of the two lines and the circle, and they will be able to see the different kinds of intersections — intersections that they may not have seen in Algebra I. For example, a line can either intersect a circle at two points, one point, or no points; students would be able to visually see what each of those cases looks like. Additionally, students could make the lines perpendicular and make the circle tangent to both lines just to get them thinking about different theorems of circles and lines.

Engaging students: Finding the area of a right triangle

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 Ashlyn Farley. Her topic, from Geometry: finding the area of a right triangle.

Music is a large part of entertainment in today’s society, thus bringing in music to the classroom can help students relate to the material more. There have been studies that shows music activates both the left and right brain, which can maximize learning and improve memory. Along with the fact that it’s easier to memorize lyrics to a song than a fact, music-based learning can be engaging and impactful. It’s the same reason why musicians put a hook in their songs; brains look for patterns to better understand and process information. For the area of a triangle there are two examples, one is a rap by PBS, “Area of a Triangle Musically Interpreted,” the other is a pop parody, “Half It Baby.” By having multiple types of songs, students who have a variety of musical interest can each make a personally connection, and having a parody makes memorizing the lyrics even easier since the students will already have a reference of the melody in their brains. These songs, and other types, can be found on YouTube.

Origami is heavily based in geometry, so many lessons, such as finding area, can be created. One activity that could be engaging for the students, and have the students find the area of a triangle themselves, is with origami. The idea is that the students will create their own origami figures, after taking the area of the paper they are working with. After folding the shape, the students are to find the area of each shape, which should add up to be to total of the paper. Therefore, this project, applies the ideas of finding the area of a triangle, and finding the area of composite figures. Since origami is mainly quadrilaterals and triangles, the students are using what they know and see to figure out what the triangles’ areas equal. Because the students get to choose the origami figures, the material becomes personalized by their choices. However, this can be a difficult task if not scaffolded correctly, thus the teacher should take precautions. Done correctly, this project can be done as PBL if desired, not just group work.

Finding the area of a triangle, as well as many other shapes, is very important in architecture. However, architecture, and its designers, have very different understandings of the triangle’s meaning. A basis for all architecture, is the fact that triangles are common because the design and symmetry aid in distributing weight. Some examples of famous long-standing triangles in architecture are the Egyptian pyramids, The east Building in the National Gallery of Art in Washington, the Hearst Tower in Manhattan, the Louve in France, and the Flatiron Building in New York City. Some of these designs are using triangles as support, while others are used for decoration. However, according to Feng Shui, the triangle should be avoided, both in terms of architecture and interior design. The triangle is associated to fire energy which is chaotic energy. When triangles are used, they should point upward, implying the upward movement of energy. As seen, there are many times that the area of triangle is needed in architecture.

Resources:

Engaging students: Writing if-then statements in conditional form

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 Bri Del Pozzo. Her topic, from Geometry: writing if-then statements in conditional form.

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

There are numerous examples of conditional statements in pop culture including movies, tv shows, and video games. I think that a fun activity to introduce students to conditional statements is to have students play a matching card game where they match the “if” strand of a famous quote to the “then” strand. For example, students would match the phrase: “If you’re happy and you know it” to “then clap your hands!” This would allow the opportunity for students to discover if-then statements in a fun and interactive way! A couple more examples that I would consider including would be from Justin Bieber’s “Boyfriend”: “If I was your boyfriend, (then) I’d never let you go.” I would also include a line from the famous children’s book, “If You Give a Mouse a Cookie.” I want to include relatable and fun examples that also help students get a clear idea of what a conditional statement is. After the matching activity, I would have students pair up and determine the definition of a conditional statement and what their general structure looks like. Including pop culture references is a fantastic way to keep the lesson fun while engaging students in the lesson material.

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

As an introduction to writing inverses, converses, and contrapositives, I could help students create graphic-organizer. Conditional statements can start to get confusing when introducing inverses, converses, and contrapositives, so a graphic organizer would be a fantastic way for students to differentiate the vocabulary and the structures of each type of statement. I would encourage students to include examples (possibly from the card sort activity), drawings, and the mathematical representation of each type of statement. The graphic organizer can also serve as a guide for students as they work through practice problems and start to develop their skills in writing conditional statements in a geometric context. As students progress through the content, I would allow students the time to go back to their organizer and include geometric examples and pictures. The organization of concepts serves as an excellent scaffold for more difficult concepts and serves as a fun way for students to practice their statement writing.

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.

This Desmos Activity can be an effective resource for students to gain some practice with conditional statements. What I like most about this website is that the questions come in different formats and ask students to utilize different skills. It is beneficial to students’ development in the subject matter that some questions ask them to write conditional statements and their converse, inverse, or contrapositive, and other questions that ask students to underline keywords. This activity would fit into this lesson topic after students have learned conditional statements, inverses, converses, and contrapositives. The interactive Desmos Activity would go well with the foldable and students can complete both lesson components simultaneously. Additionally, the interactive Desmos Activity includes examples of the different types of statements with symbols included. The combination of visuals and words is very beneficial to students who may have trouble understanding the difference between the different types of statements. Finally, the card sort activity can encourage students to work in pairs and complete an activity similar to their entry activity.

(Here is the link to the Desmos Activity https://teacher.desmos.com/activitybuilder/custom/5b909548262be93b79d1e056)

Engaging students: Defining the terms corresponding angles, alternate interior angles, and alternate exterior angles

A quick programming note: I am transitioning to another administrative role at my university, and I expect that I’ll have much less time to post original content to this blog in the future. For this reason, I’ll only be posting on Fridays for the foreseeable future.

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 Sydney Araujo. Her topic, from Geometry: defining the terms corresponding anglesalternate interior angles, and alternate exterior angles

.

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

Geogebra is a great source for this topic. It’s an interactive program where students can make their own geometric shapes. Geogebra even has ready-made animations and programs that correspond with different geometry concepts. I found several ready-made explorations and animations that explain and visually show corresponding, alternate interior, and alternate exterior angles. Some of them come with questions for students to answer which would be a great activity for students to do. They have the ability with the program to adjust angles, shapes, and see how much of a difference a small change makes. It’s great for students for them to make their own discoveries and they have the ability to with this program and the different activities available. Instead of students simply being told about these angles and doing a simple worksheet, they can explore on their own which is more organic and engaging for them.

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

Using the program Geogebra that I describe, there’s several different activities already prepared on the website that can used to define corresponding, alternate interior, and alternate exterior angles. Because of the technology resources available, I could either do a jigsaw activity or a stations activity. Using a jigsaw activity I could have students form groups of 3 and each student would be in charge of learning one of the three angles. They would each complete a Geogebra activity that corresponds to their topic they are responsible for. Then after they have mastered their topic they will come back to their original groups and teach the other group members what they have learned. They could also do a stations activity where they rotate around during the class time doing a Geogebra activity for corresponding, alternate interior, and alternate exterior angles.

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

Euclid is known as the father of geometry and wrote The Elements. He was a Greek mathematician who lived from 325 BC to 265 BC. The Elements is divided into 13 books is widely famous and used among mathematicians, even in current times. It is quite amazing the discoveries Euclid made and proved during that time. In total, The Elements contains 465 theorems and proofs in which Euclid only used a compass and a straight edge. He reworked the math concepts of his predecessors, like Plato and Hippocrates, into a whole which would later become known as Euclidean geometry. Which still holds today, 2,300 years later. We actually see his proof of alternate angles in Book 1 of The Elements, it is proposition 29. It is actually the first proposition in The Elements that depends on the parallel postulate.

References

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 Emma White. Her topic, from Algebra: multiplying binomials.

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

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

Around 600-700 AD, the Hindu mathematicians had taken the Babylonia methods of approaching equations a step further when it came to introducing unknowns, sometimes more than one unknown in a single problem. It wasn’t until the Medieval times did the Islamic mathematicians discuss the variable x and how important it was. This is when the binomials theorems evolved. Furthermore, the Islamic mathematicians were able to use many operations on polynomials and soon binomials, such as multiplication, division, finding roots, and more! One thing I find highly fascinating is the Islamic mathematicians advanced the study of algebra, which “flourished during the golden age”. Evermore so, private collections were found in a lost Islamic library, which was destroyed in the 13th Century. These private collections “altered the course of mathematics.” An example of a concept that was furthered studied was the Fibonacci sequence (which is, in my opinion, one of the most fascinating things in math history and how it relates to the world and finding mathematics around us, but that is for another time…). All I can say is the Babylonians, the Hindu and Islamic mathematicians were a driven and mathematically inclined people and it blows my mind how far these people brought the world of mathematics.

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

When it comes to finding ways to use technology for multiplying binomials, I truly believe visuals are essential. I’m a little biased since I was introduced to a way of multiply binomials just last semester in one of my teaching classes and it BLEW MY MIND. I wish I knew how to do this earlier in high school!  Essentially, this online source allows the student to use algebra tiles without having them physically in front of them. Therefore, they can use this source if they have technology capable of doing so (such as a phone, computer, tablet, etc.). This source is visual and easy for students to understand and manipulate. The student starts by placing the corresponding tiles for one binomial across the top like a table (would be 4 x-tiles and 2 1-tiles). Along the left side, the other binomial is represented (long ways/up-and-down). You then multiply corresponding values and where they meet in the open area (example: where an x-tile and another x-tile meet, it would become since x times x is ). Algebra tiles can also be used for upcoming topics the students would learn, such as completing the square. For a student who may have trouble grasping the idea of multiplying binomials and struggling to understand the concept of abstracts, using algebra tiles will hopefully help with the misunderstandings and confusion. All I’m saying is if this concept of online algebra tiles assisted a college student and made the topic MUCH easier to visualize and explain, I’m sure most high school students will find the use of technology in their math class interesting. Who knows, some students may come to love math more because of it!

Reference(s):

“Multiplying Binomials by the FOIL Method” by Professor Dave Explains:

“History of Polynomials”: https://polynomialshistory.weebly.com/history.html

“How modern mathematics emerged from a lost Islamic libray”: https://www.bbc.com/future/article/20201204-lost-islamic-library-maths

Algebra Tiles: https://technology.cpm.org/general/tiles/

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 Morgan Mayfield. His topic, from Algebra: graphs of linear equations.

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.)

Given a rather vague statement such as ”Graphs of Linear Equations”, I was unsure if it meant only the technique of analyzing graphs or being able to have the ability to build a graph of a linear equation. In A1, I attempt to rely on analysis. Here are the problems I encountered on Space Math @ NASA:

• Problem 1 – Calculate the Rate corresponding to the speed of the galaxies in the Hubble Diagram. (Called the Hubble Constant, it is a measure of how fast the universe is expanding).
• Problem 2 – Calculate the rate of sunspot number change between the indicated years.

Space Math has these problems listed as “Finding the slope of a linear graph”, the two key phrases being “Finding the slope” and “linear graph”. The students must be able to do both. Students are given three sets of graphical data to analyze (shown below). I am not an expert in any of these fields, but I suspect these graphs were made using real data scientists collected. The Space Math team gave students two points on the data to aid in calculations. What makes these graphs interesting is the fact that they are messy, but real compared to a graph of a linear equation in a classroom. These graphs can be analyzed further than the problems Space Math offered. Students could see how that data can be collected and put into a scatterplot, like in the case of graph 2, and have an approximately linear correlation. Sadly, most things don’t follow a neat model of what we see in our math class, yet we can still derive meaning from real-world phenomena because of what we learn in math class. Scientists use their understanding of graphs of linear equations and linear models to analyze data and come to conclusions about our real-world environment. In graph 2, a scientist would clearly see that there is a linear proportional relationship between the speed and distance from the Hubble space telescope of a galaxy, or more meaningfully understood as a rate, 76 km/sec/mpc. Now, if a scientist encountered a new galaxy, they could determine an approximate speed or distance of the galaxy given the other variable.

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

Students will formalize learning about graphing linear function in Algebra I. Graphs of linear equations are important in solving linear inequalities in two variables, solving systems of linear inequalities, solving systems of linear equations, and solving systems of equations involving linear and nonlinear equations which are all topics in Algebra I and II. Solving systems can be done algebraically, but graphing systems give students a more concrete way in finding a solution and is an excellent way of conveying information to others. If a student ever found themself in a business class, they may be asked to make “business decisions” on a product to buy. If I were the student explaining my decision to my teacher and potential “investors”, I would be making a graph of linear systems to help explain my “business decisions”. Generally, a business class would also introduce “Supply and Demand” graphs, where the solution is called the “equilibrium”. Many graphs in an intro class depict supply and demand as a system of linear equations.

In the high school sciences, a student will come across many linear equations. Students in a regular physics course and an AP physics course will come across simplified distance vs. time graphs to represent velocity, velocity vs. time graphs to represent acceleration, and force vs. distance graphs to represent work and energy (khan academy link included below). Note, just because many of the examples used in a physics class are graphs of linear equations, real life rarely works out like this.

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

We are shown data daily that our climate is changing fast through infographics on social media, posters set up by environmentalists, and news broadcasting. Climate change is one of the most important issues that society faces today and is on the collective consciousness of my generation as we have already seen the early consequences of climate change. Climate change, like most real-world data collecting does not always follow a good linear fit or any other specific fit with 100% accurately. However, a way scientists and media want to convey a message to us is to overlay a “trend line” or a “line of best fit” over the graphed data. Looking at the examples below, we can clearly understand that average global temperatures have been on the rise since 1880 despite fluctuations year-to-year and comparisons to the expected average global temperature. The same graph also gives insight on how the same data can also be cherrypicked to fit another person’s agenda. From 1998 – 2012, the rate of change, represented by a line, is lower than both 1970 – 1984 and 1984 – 1998. In fact, the rate is dramatically lower, thus climate change is no more! Not so, this period of slowing down doesn’t immediately refute the notion of climate change but could be construed as so. Actually, in the NOAA article linked below and its corresponding graph actually finds that we were using dated techniques that led to underestimates and concluded that the IPCC was wrong in it’s original analysis of 1998-2012 and that the trend was actually getting worse, indicated by the trend line on the second graph, as the global temperature departed from the long-term average.

Look at how much information could be construed by a few linear functions represented on a graph and some given rate of changes.

References:

(or Problem 226 from https://spacemath.gsfc.nasa.gov/algebra1.html)

https://d1yqpar94jqbqm.cloudfront.net/documents/Gateway5A1VAChart.pdf (or grade 5 – Algebra II Vertical Alignment https://www.texasgateway.org/resource/vertical-alignment-charts-revised-mathematics-teks)

https://bim.easyaccessmaterials.com/index.php?location_user=cchs

https://www.ncdc.noaa.gov/news/recent-global-surface-warming-hiatus

https://www.climate.gov/news-features/climate-qa/did-global-warming-stop-1998

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 Fidel Gonzales. His topic, from Algebra I: the point-slope intercept form 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?

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 advancing right in front of us. Using it in the classroom can be a tool that allows students to have a more hands on experience in the classroom. When I was in middle school, the only tool that we had to learn slope intercept form of a line was using a ti-inspire calculator. However, schools are receiving more funding and can provide students with tablets or computers to assist in their academic career. Gizmos is a website that contains many user-friendly programs that a student can use to learn a concept, or an educator can present to reinforce a skill. For the topic of slope intercept form of a line, the gizmo has two sliding parts that allows the user to change the values of the equation. One for the slope and one for the y- intercept. The student can adjust the values of both and observe the changes that occur to the line. This experience is more user friendly since it only allows the person to change those two aspects compared to having to input the equation each time into the graphing calculator. The reason that students would be more likely to be engaged is because they are already used to technology and there is still a need to incorporate technology into the classroom. So, students would prefer using a computer compared to the traditional paper and pencil. Imagine them having to graph by hand each graph to compare differences!

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

Slope intercept form is a way that data can be displayed. The data is usually continuously decreasing or continuously increasing. There is a magnitude of activities that can be used to help students gather a better understanding of the topic. As an educator, I would create a scavenger hunt that displayed either a word problem or a graph. Both will ask for the student to represent the information as slope intercept form. For each problem, there will be 4 answer choices that the student could choose for their answer. On their worksheet, there will be fill in the blanks that will be filled up from the letter that is in front of the correct answer. As the student progresses to the next problem, they will be filling out the letter blanks in a random order. So, if the person does the activity correctly, they should end up with the correct word phrase. The word phrase will be a math pun to add to the magic. This activity will allow students to switch from graph and word problems to slope intercept form.

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

As educators, we want to ensure that our students have the proper foundation to continue advancing their mathematic skills. Slope intercept form is an algebra base lesson. The skills that students used to reach this topic is addition. At a young age, students learn to count numbers in repeated increments. An example of this is when a student keeps adding 5 until they reach a certain number. Displaying this as slope intercept would be a line with no y intercept and a slope of 5. We have even used y intercepts in context to adding in past classes. An example of this would be a person wanting to sell 200 dollars’ worth of tickets that are worth 5 dollars each and they already started with 57 dollars. If they were to solve the problem using slope intercept form, they would put 200 as the y value and 57 as the y intercept of the problem. The slope would be 5. In the past, they would add 5 to 57 until they reach their goal. Slope intercept form is a way for students to display data with a constant increasing or decreasing value. It is more convenient for students to use slope intercept form compared to how they displayed the pattern in the past. They use it now since they learned why it works before they reach algebra.

References:

Engaging students: Finding the slope 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 again comes from my former student Bri Del Pozzo. Her topic, from Algebra: finding the slope of a line. How could you as a teacher create an activity or project that involves your topic? As a teacher I would likely introduce a very popular and well-received project to my students, the project where students draw an angular image on a graph and then calculate the slope of 20 lines from their image. I love this project because it allows students to connect mathematics to art and encourages them to express themselves creatively. As a precursor to the project, I would introduce students to the types of slopes and their characteristics using a tool that I learned in my Algebra Class, Mr. Slope Guy (pictured below).
In the image the positive slope is indicated by the left eyebrow and above the plus-sign eye, the negative slope is the right eyebrow by the negative sign eye, the nose represents the undefined slope and is denoted by a vertical line and a “u” for undefined, and the mouth represents the zero-slope shown by a horizontal line and two zeros. The students could use this resource while completing their projects to serve as a reminder of the types of slopes. The main focus of the project, however, would focus on the process of how to find slope given two points on a line. (This project is based on an example from: https://kidcourses.com/slope/ #5) How does this topic extend what your students should have learned in previous courses?             In the grade 7 TEKS for mathematics, students are expected to “represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form .” This creates a foundation for finding the slope of a line by introducing students to multiple representations of what slope looks like. When discussing how to find the slope of a line, I think that the tabular representation is a great tool for students to visualize the meaning behind slope. In seventh grade math, students were able to conceptualize slope without using the formula. When finding slope in early algebra, I would encourage students to look at graphs from a new lens, noticing features such as the sign of the line, the steepness of the line, the difference in x’s and y’s at different points on the line, and the slope itself. When looking at a table, I would ask students to calculate the difference in x’s and y’s as they go down the rows of the table and have them compare those numbers to those that they saw in the graph. How has this topic appeared in the news?             As many of us know, over the past 18 months or so, the number of Covid-19 Cases in the United States has been on the rise. For a long time, the total number of cases in the United States was growing exponentially and very quickly. As more research has been done by the Centers for Disease Control and Prevention, we have learned that there is a way to flatten the curve and reduce the number of daily cases. This initiative to flatten the curve has resulted in the growth of cases to resemble liner growth rather than exponential growth. As mathematicians, we can calculate the slope of the line that represents the (linear) growth of Covid-19 cases per day. We can make comparisons between growth rates in different states and use that data to make predictions about effectiveness of Covid-19 prevention procedures.

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 Taylor Bigelow. Her topic, from Algebra: equations of two variables.

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

This topic is perfect for word problems, you can make a lot of interesting word problems using 2 variables. Here are some examples of word problems.

● Sam is mowing lawns for money over the summer. They charge \$10 an hour. They have a family discount of 20% per hour. If they mow non-family members laws for 10 hours this week and mowed family members laws for 3 hours, how much money did they make this week?
○ 10N+8F=?
○ N=10 and F=3
○ 10(10)+8(3)=124

● John is buying blue and yellow gummy bears at the store. He has \$20 to spend on candy. Blue gummy bears come in bags of 20 for \$1 each, and Yellow gummy bears come in bags of 50 for \$3 each. He knows we want exactly 100 Blue gummy bears. How many yellow gummy bears can he buy?
○ B=Blue gummy Bears Y=Yellow gummy Bears
○ 20=B+Y
○ B= 100/20= \$5 for 100 gummy bears
○ 20= 5+Y so Y=\$15
○ With \$15 he can buy 5 bags of yellow gummy bears. 5*50=250. So he can buy
250 yellow gummy bears

● Alex is building a fence for her backyard. She is building it in a rectangular shape, and she wants the length of the fence to be twice as long as the width of the fence. If the area of her backyard is 200 feet, how long is the width, and how long is the length?
○ L=length W=width
○ L*W=200
○ L=2W
○ 2W*W=200
○ 2W^2=200
○ W^2=100
○ W=10
○ So L=2(10)=20

These are just 3 examples I came up with on the spot. You can create a lot more, and
with a variety of difficulties.

How does this topic extend what your students should have learned in previous courses?
This topic builds on knowledge from elementary school and extends into almost all future math. It starts with kids understanding multiplication and addition, then to them being introduced to solving equations in middle school, and then is heavily used in high school math classes, and any math class that requires basic algebra skills in the future. I looked through some of the teks to find references to two-variable equations and found it only referenced in algebra 1 and 2. I also went back through 6th, 7th, and 8th grade and found where they were using one-variable equations since that is the prior knowledge that they are building onto with two-variable equations.

○ (9) Expressions, equations, and relationships. The student applies mathematical
process standards to use equations and inequalities to represent situations. The
student is expected to:
■ (A) write one-variable, one-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, one-step equations and
inequalities on number lines; and
■ (C) write corresponding real-world problems given one-variable,
one-step equations or inequalities.
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use equations and inequalities to solve
problems. The student is expected to:
■ (A) model and solve one-variable, one-step equations and inequalities
that represent problems, including geometric concepts; and
■ (B) determine if the given value(s) make(s) one-variable, one-step
equations or inequalities true.
○ (10) Expressions, equations, and relationships. The student applies
mathematical process standards to use one-variable equations and inequalities
to represent situations. The student is expected to:
■ (A) write one-variable, two-step equations and inequalities to represent
constraints or conditions within problems;
■ (B) represent solutions for one-variable, two-step equations and
inequalities on number lines; and
■ (C) write a corresponding real-world problem given a one-variable,
two-step equation or inequality.
○ (11) Expressions, equations, and relationships. The student applies
mathematical process standards to solve one-variable equations and inequalities.
The student is expected to:
■ (A) model and solve one-variable, two-step equations and inequalities;
■ (B) determine if the given value(s) make(s) one-variable, two-step
equations and inequalities true
○ Expressions, equations, and relationships. The student applies mathematical
process standards to use one-variable equations or inequalities in problem
situations. The student is expected to:
■ (A) write one-variable equations or inequalities with variables on both
sides that represent problems using rational number coefficients and
constants;
■ (B) write a corresponding real-world problem when given a
one-variable equation or inequality with variables on both sides of the
equal sign using rational number coefficients and constants;
■ (C) model and solve one-variable equations with variables on both
sides of the equal sign that represent mathematical and real-world
problems using rational number coefficients and constants
● Algebra 1
○ (2) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using properties of linear functions to
write and represent in multiple ways, with and without technology, linear
equations, inequalities, and systems of equations. The student is expected to:
■ (B) write linear equations in two variables in various forms, including y
= mx + b, Ax + By = C, and y – y1 = m (x – x1 ), given one point and the
slope and given two points;
■ (C) write linear equations in two variables given a table of values, a
graph, and a verbal description;
■ (H) write linear inequalities in two variables given a table of values, a
graph, and a verbal description
○ (3) Linear functions, equations, and inequalities. The student applies the
mathematical process standards when using graphs of linear functions, key
features, and related transformations to represent in multiple ways and solve,
with and without technology, equations, inequalities, and systems of equations.
The student is expected to:
■ (D) graph the solution set of linear inequalities in two variables on the
coordinate plane;
■ (F) graph systems of two linear equations in two variables on the
coordinate plane and determine the solutions if they exist;
■ (G) estimate graphically the solutions to systems of two linear
equations with two variables in real-world problems; and
■ (H) graph the solution set of systems of two linear inequalities in two
variables on the coordinate plane.
○ (5) Linear functions, equations, and inequalities. The student applies the
mathematical process standards to solve, with and without technology, linear
equations and evaluate the reasonableness of their solutions. The student is
expected to:
■ (C) solve systems of two linear equations with two variables for
mathematical and real-world problems.
● Algebra 2
○ (3) Systems of equations and inequalities. The student applies mathematical
processes to formulate systems of equations and inequalities, use a variety of
methods to solve, and analyze reasonableness of solutions. The student is
expected to:
■ (C) solve, algebraically, systems of two equations in two variables
consisting of a linear equation and a quadratic equation;
■ (D) determine the reasonableness of solutions to systems of a linear
equation and a quadratic equation in two variables;
■ (E) formulate systems of at least two linear inequalities in two variables;
■ (F) solve systems of two or more linear inequalities in two variables; and
■ (G) determine possible solutions in the solution set of systems of two or
more linear inequalities in two variables.

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

Algebra is a really old concept, dating back almost 4 thousand years ago. (So kids have been doing the same thing in classes for millennia.) The Babylonians were the first to use algebra in the 1900s. The Egyptians also used algebra around the same time, but they focused on linear algebra, while the Babylonians did quadratic and cubic equations. The ancient Greeks used geometric algebra around 300 BC. They solved algebra equations using geometry, and their methods are very different from the ones we use today. A thousand years later, around 800 AD, Muhammad ibn Musa al-Khwarizmi became the father of modern algebra. The middle east used Arabic numerals (the numbers 0-9 which we still use today). The word algorithm is even derived from his name. Algebra started thousands of years ago to solve problems and has been developed over time into what it is today.

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 Sydney Araujo. Her topic, from Algebra: the quadratic formula.

D4. What are the contributions of various cultures to this topic?

The quadratic formula can be traced all the way back to the Ancient Egyptians. The ancient Egyptians knew how to calculate the area of different shapes but did not know how to calculate the length of the sides of a shape. Moving forward, it is speculated that the Babylonians developed the completing the square method to solve problems involving areas. The Babylonians used a more similar number system to the one we use today. Instead, they used hexagesimal which made addition and multiplication easier. We can also see a similar method used by the Chinese around the same time. Pythagoras and Euclid were some of the first to attempt to find a more general formula to solve quadratic equations, both using a geometric approach. They’re ideas differ slightly, Pythagoras observed that the value of a square root is not always an integer but he refused to allow for proportions that were not rational. Whereas Euclid proposed that irrational square roots are also possible. At the time, the ancient Greeks did not use the same number system that we use, so it was impossible to calculate square roots by hand. It wasn’t until the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic formula. This is because Indian mathematics used the decimal system as well as zero which had a massive advantage over the Egyptians and Greeks. Brahmagupta was the one that recognized that there are two roots in the solution to the quadratic equation and described the quadratic formula.

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

One of my fondest high school memories is from my junior year physics class. It was the famous Punkin’ Chunkin’ project. Students were put in groups and asked to build a trebuchet or catapult that could launch a pumpkin across a field. The only requirement was for the device to work, the distance was just fun extra credit. For this project we had to predict the pumpkins trajectory using different variables like the pumpkin’s weight, force, momentum, etc. However, by the time we were juniors, we had either taken Algebra 2 or were currently in it. So, our physics and algebra teacher were working together so that by the time this project came around we were working on quadratic equations in algebra. As the shape of the trajectory of a pumpkin was a parabola. Because of this experience, I can create an activity or even a similar project with the physics teacher. This way students see the different applications of quadratic equations and have a tangible real world math experience.

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

As the quadratic formula is taught in Algebra 1, students have only seen linear equations prior to that point. Students recognize that when they are solving these equations, they are looking for one solution, no solution, or infinitely many solutions. The one solution being a singular ordered pair and then they are done. What students then must extend on when they reach quadratic equations, and the quadratic formula is that they’re now looking for two separate solutions. So, at this point they know how to solve for x and understand inverses which is important when it comes to quadratic equations. During the solving process of a quadratic equation, students may have to take the square root of both sides of the equation which will give you a plus or minus sign in front of the square root. Which makes the connection on why there are two solutions to a quadratic equation and the quadratic formula, because a parabola has two roots.

Works Cited:

Brahambhatt, Rupendra. “Quadratic Formula: What, Why, and How It Changed Mathematics.” Interesting Engineering, Interesting Engineering, 16 July 2021, interestingengineering.com/quadratic-formula-what-why-and-how-it-changed-mathematics.