Engaging students: Midpoint 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 Chi Lin. Her topic, from Geometry: deriving the midpoint formula.

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.)

To let students engage in the topic, as teachers, we want to create some good examples for students to let them interested in doing it. We need to know what students are interested in or students realize they can use this knowledge in the real life. For example, if students like eating pizza, then I will create some examples about pizza or some delicious food and using pizza representation to raise their attention. In this topic, since we are going to talk about the midpoint formula, one of the real-world problems that I can come up with is using Google Maps. I will show a big Google map of the US in the class, and I will ask students question that “Miss Lin is planning a road trip from Dallas to Arizona on Thanksgiving. However, she wants to split the driving into two days. Now Miss Lin needs your help to figure out what is the middle city (midpoint) between Texas (Dallas) to Arizona.” After students talk with their groupmates, I will invite students to come to the map and circle the city that their think is the middle city between Texas (Dallas) to Arizona and explain their thoughts as well.

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

Khan Academy shows that what students show know before we learn how to derive the midpoint formula. It gives some details which help the teacher to prepare the lesson. First, students should know points in the coordinate plane. Students should require describing every point on the plane with an ordered pair in the form correctly. Second, students have learned how to use addition, subtraction, and square with negative numbers. Students need to know the distance and slope between points on the coordinate plane, how to represent points on the left or below the original point. Third, students have learned the distance and displacement between points to calculate the slope. Students need to understand what absolute value is as well. The last thing I think students should have learned in the previous class is the slope and square root.

Reference:

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-distance-and-midpoints/a/getting-ready-for-analytic-geometry

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.

Khan Academy is a good resource for students to study themselves when they want to study this topic. It tells students what they need to know and why this topic is important before they get to learn. Students might think about deriving the midpoint formula is just figuring out some points in the coordinate plane. However, Khan Academy shows that knowing the midpoint formula is not only for figuring out the points in the coordinate plane but also related to the distance formula. Also, Khan Academy provides online tutoring videos to help students understand the materials. If students don’t understand or forget the materials, they can always go back to check the videos. Khan Academy also provides practices for students to do after each topic, it helps students do the self-checking. I recommend this website because, since the covid, we realize that online learning is also one of the ways for students to learn. However, sometimes it is hard for teachers to check students’ understanding through the screen, and we couldn’t make sure that every student is on the same page with us. Khan Academy does provide detailed explanations on their website, so I will suggest students check this website with this topic if my class is online.

Reference:

https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-distance-and-midpoints/a/midpoint-formula

Engaging students: Defining intersection

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

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

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.

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 angles, alternate 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