# Engaging students: Defining the words acute, right, and obtuse

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 Johnny Aviles. His topic: how to engage geometry students when defining the words acute, right, and obtuse.

A1.) How could you as a teacher create an activity or project that involves your topic?

To have the students get engaged with the topic of Defining the terms acute, right, and obtuse, I will begin with having the classroom set up into groups of 4-5. Within their group they will create 10 examples of where each acute, right, and obtuse angles or triangles can be found in the classroom or in the real world in general. For example, the letter Y, end of a sharpened pencil, and the angle under a ladder can be used. They will be given about 10-15 minutes depending on how fast they can all finish. This is a great activity as the students can work together to try to come up with these examples and can familiarize themselves with amount of ways these terms are used in life. I will tell them before I begin the activity that the group that comes up with the most examples will be given extra credit in the next exam or quiz. This will give them extra incentive to stay on task as I am well aware that some groups may finish earlier than the rest and may take that extra time to cause disruptions.

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

In previous courses, students have learned had some exposure to these types of angles. Most students have been familiar with the use of right triangles and have learned methods like the Pythagorean theorem. When we extend the terms acute, right, and obtuse in geometry, it begins to be more intensified. These angles then extend in terms of triangle that will then have many uses. Students will then be expected to not only find missing side lengths but also angles. Students will then be exposed to methods later like, law of sines and cosines, special right triangles, triangle inequality theorem and triangle congruency in. This topic essentially is the stepping stone for a large part of what is soon to be learned. Other courses will use a variety of other was to incorporate the terms acute, right, and obtuse. Geometry, precalculus and trigonometry will essentially have a great deal of uses for these terms for starters and can then also be extended in many higher-level math courses in universities.

E1.) How can technology be used to effectively engage students with this topic?

An effective way to teach this topic using technology and the terms acute, right, and obtuse would be games. There is a magnitude of game that involve angles and be beneficial in the understanding of these angles. I have found this one game called Alien Angles. In this game, you are given the angle of where the friendly alien at and you have to launch your rocket to rescue them. the purpose of the game is for students to be familiar with angles and how to find them. after you launch the rocket, you are given a protractor that shows the angles and I believe this is beneficial for students as they can also be more familiar with the application of protractors. I can post this on the promethean board and have students identify what the angle I need to rescue the aliens. I can then call for volunteers to go on the board and try to find the correct angle to launch the rocket.

https://www.mathplayground.com/alienangles.html

# Engaging students: Defining the words acute, right, and obtuse

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 Katelyn Kutch. Her topic: how to engage geometry students when defining the words acute, right, and obtuse.

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

As a teacher I think that a fun activity that is not too difficult but will need the students to be up and around the room is kind of like a mix and match game. I will give a bunch a students, a multiple of three, different angles. And then I will give the rest of the students cards with acute, obtuse, and right triangle listed on them. The students with the angles will then have to get in groups of three to form one of the three triangles. Once the students are in groups of three, they will then find another student with the type of triangle and pair with them. They will then present and explain to rest of the class why they paired up the way that they did. I think that it would be a good way for the students to be up and around and decide for themselves what angles for what triangles and then to show their knowledge by explaining it to the class.

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

The topic of defining acute, right, and obtuse triangles extend what my students should already know about the different types, acute, right, and obtuse, angles. The students should already know the different types of angles and their properties. We can use their previous knowledge to build towards defining the different types of triangles. I will explain to the students that defining the triangles is like defining the angles. If they can tell me what angles are in the triangle and then tell me the properties of the triangles then they can reason with it and discover which triangle it is by looking at the angles.

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

I found an article that I like that was written about a soccer club, FC Harlem. FC Harlem was getting a new soccer field as part of an initiative known as Operation Community Cup, which revitalizes soccer fields in Columbus and Los Angeles. This particular field, when it was opened, had different triangles and angles spray painted on the field in order to show the kids how soccer players use them in games. Time Warner Cable was the big corporation in on this project.

References:

http://www.twcableuntangled.com/2010/10/great-day-for-soccer-in-harlem/

# Engaging students: Defining the words acute, right, and obtuse

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 Lisa Sun. Her topic: how to engage geometry students when defining the words acute, right, and obtuse.

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

I believe a scavenger hunt will be a great activity for the students to help concrete their knowledge of acute, right, and obtuse angles. It will be a take home activity rather than an activity that they’ll complete in school. I’ve created this scavenger hunt to take place outside of the classroom so students will understand that what we learn in math class takes place in our everyday lives outside of the walls of school.

This scavenger hunt activity requires students to observe their surroundings everywhere they go. I want them to find 10 acute angles, 10 right angles, and 10 obtuse angles. Along with that, they must take a picture or sketch accordingly to which angle the image has. (For example, picture/sketch of a corner of book shelf – right angle). To spark some motivation and interest, I will announce to the students that if they are able to find 15 of each angle instead of 10, I will add 2 points to their next exam grade.

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

Archimedes and Euclid are the mathematicians who have discovered and developed the idea of the types of angles that we have today. As a student, when my teachers related the topic with the brilliant minds who made such discoveries, I felt that the topics that I was learning were more relatable and I had gained a deeper understanding of the topic. I hope to do the same for my students with this topic. Here are the following interesting facts about Archimedes and Euclid to keep the students enlightened for geometry.

Interesting facts about Archimedes:

• 1 of 3 most influential and important mathematician who ever lived (other two are Isaac Newton and Carl Gauss)
• Rumors that he was considered to be of royalty because he was so respected by the King during his time
• Invented the odometer

Interesting facts about Euclid:

• “Father of Geometry”
• His book “Elements” is one of the most powerful works in history of mathematics
• His name means “Good Glory” in Greek

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

Above is a link that I would present, on replay, as students are walking into my classroom to set the tone of the classroom for the day. Once they are all seated, I will tell them to get out their interactive journal and write at least 5 facts that are new to them as I play the video for them once more. By doing so, we’re keeping the students engaged as they are reinforcing what they just heard in writing. Once students are done with this task, I will select students randomly to state one fact that they had just learned from the video. Guide the students to know and remember the “take home message” which are the following:

• Definition of Angle: The amount of turn between two rays that have a common end point, the vertex
• Angles are measured in degrees
• Angles are seen everywhere
• Acute angles: 0 – 89 degrees
• Right angles: 90 degrees
• Obtuse angles: 91-180

References:

https://www.mathsisfun.com/definitions/angle.html

http://www.yurtopic.com/society/people/archimedes-facts.html

http://www.10-facts-about.com/Euclid/id/382

# Engaging students: Defining the words acute, right, and obtuse

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 Jesse Faltys. Her topic: how to engage geometry students when defining the words acute, right, and obtuse.

E. TECHNOLOGY: How can technology be used to effectively engage students with this topic?

ACUTE, OBTUSE, and RIGHT Angles Song

This is a great video for the end of the lesson when first introducing acute, right, and obtuse angles.  A little corny but it’s always helpful to link new knowledge to a song.  Music brings back memories or in this situation recognition.  By using creative things, you are helping the students reinforce new ideas.  Just hearing words will not help us retain the information, but adding the words to a song help reinforce the reminder for the information.  We can remember anything if we just put our minds to it.  The kids in the video are singing lyrics about right, obtuse and acute angles to the song Old McDonald Had a Farm.  The video helps the students to summarize their understanding of the three new terms and a way to retain it for future use.

http://www.watchknowlearn.org/Video.aspx?VideoID=2446

D. HISTORY: How have different cultures throughout time used this topic in their society?

In Egypt as far back as 1500BC, measurements were taken of the Sun’s shadow against graduations marked on stone tables. These measurements are just different angles used to show time with some degree of accuracy.  Gromas were used for the purpose of construction in ancient Egypt.  Gromas were right-angle devices that the ancient Egyptians used when they began construction project by surveying an area. They could sketch out long lines at right angles.  The Romans will actually use the same tool to sketch out their roads.  1,713 years ago they were using right angles.  This might be important.

http://www.fig.net/pub/cairo/papers/wshs_01/wshs01_02_wallis.pdf

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

Angry-Birds: “Use the unique powers of the Angry Birds to destroy the greedy pigs’ fortresses!“ Angry-Birds is an app that is played by a large percentage of children on a daily basis.  Birds are positioned on a slingshot and launched at pigs that are resting on different structures.  We create a zero plane from the bird sitting in the slingshot, releasing the bird, and mark the maximum height reached. We now have an angle. The bird has created an angle with its path.  Can we classify the majority of these angles as acute, right or obtuse?

Bubble Shooter:  A Puzzle game that will help you stay busy for a while!

The point of the game is to remove all the spheres by matching like colors.  The “cannon” at the bottom of the page is your tool to directing the sphere were you want it to go.  You can directly shot the sphere or you can bounce off the edge of the wall.  Here is the trick, what kind of angle do you need to deliver your sphere.  One of the helpful hints from the website, “you can use the left and right border to bounce new balls in more advanced angles.” These advanced angles can be denoted as acute, right or obtuse.

http://www.shooter-bubble.com/

# Engaging students: Solving exponential 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 comes from my former student Jesse Faltys. Her topic: solving exponential equations.

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

Once your students have learned how to solve exponential equations, they can solve many different kinds of applied problems like population growth, bacterial decay, and even investment earning interest rate.    (Examples Found: http://www.education.com/study-help/article/pre-calculus-help-log-expo-applications/)

Examples

1. How long will it take for $1000 to grow to$1500 if it earns 8% annual interest, compounded monthly?

$A = P \left( 1 + \displaystyle \frac{r}{n} \right)^{nt}$

• $A (t) = 1500$, $P = 1000$, $r = 0.08$, and $n = 12$.
• We do not know $t$.
• We will solve this equation for $t$ and will round up to the nearest month.
• In five years and one month, the investment will grow to about \$1500.

2. A school district estimates that its student population will grow about 5% per year for the next 15 years.  How long will it take the student population to grow from the current 8000 students to 12,000?

• We will solve for t in the equation $12,000 = 8000 e^{0.05t}$.

$12,000 = 8000 e^{0.05t}$

$1.5 = e^{0.05t}$

$0.05t = \ln 1.5$

$t = \displaystyle \frac{\ln 1.5}{0.05} \approx 8.1$

• The population is expected to reach 12,000 in about 8 years.

3. At 2:00 a culture contained 3000 bacteria.  They are growing at the rate of 150% per hour.  When will there be 5400 bacteria in the culture?

• A growth rate of 150% per hour means that $r = 1.5$ and that $t$ is measured in hours.

$5400 = 3000 e^{1.5t}$

$1.8 = e^{1.5t}$

$1.5t = \ln 1.8$

$t = \displaystyle \frac{\ln 1.8}{1.5} \approx 0.39$

• At about 2:24 ($0.39 \times 60 = 23.4$ minutes) there will be 5400 bacteria.

A note from me: this last example is used in doctor’s offices all over the country. If a patient complains of a sore throat, a swab is applied to the back of the throat to extract a few bacteria. Bacteria are of course very small and cannot be seen. The bacteria are then swabbed to a petri dish and then placed into an incubator, where the bacteria grow overnight. The next morning, there are so many bacteria on the petri dish that they can be plainly seen. Furthermore, the shapes and clusters that are formed are used to determine what type of bacteria are present.

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

The students need to have a good understanding of the properties of exponents and logarithms to be able to solve exponential equations.  By using properties of exponents, they should know that if both sides of the equations are powers of the same base then one could solve for x.  As we all know, not all exponential equations can be expressed in terms of a common base.  For these equations, properties of logarithms are used to derive a solution.  The students should have a good understanding of the relationship between logarithms and exponents.  Logs are the inverses of exponentials.  This understanding will allow the student to be able to solve real applications by converting back and forth between the exponent and log form.  That is why it is extremely important that a great review lesson is provided before jumping into solving exponential equations. The students will be in trouble if they have not successfully completed a lesson on these properties.

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

1. Khan Academy provides a video titled “Word Problem Solving – Exponential Growth and Decay” which shows an example of a radioactive substance decay rate. The instructor on the video goes through how to organize the information from the world problem, evaluate in a table, and then solve an exponential equation. For our listening learners, this reiterates to the student the steps in how to solve exponential equations.

2. Math warehouse is an amazing website that allows the students to interact by providing probing questions to make sure they are on the right train of thought.

For example, the question is $9^x = 27^2$ and the student must solve for $x$.  The first “hint” the website provides is “look at the bases.  Rewrite them as a common base” and then the website shows them the work.  The student will continue hitting the “next” button until all steps are complete. This is allowing the visual learners to see how to write out each step to successfully complete the problem.

# Engaging students: Solving one-step and two-step 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 first student submission comes from my former student Jesse Faltys (who, by the way, was the instigator for me starting this blog in the first place). Her topic: how to engage students when teaching one-step and two-step inequalities.

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

1. Index Card Game: Make two sets of cards. The first should consist of different inequalities. The second should consist of the matching graph. Put your students in pairs and distribute both sets of cards.  The students will then practice solving their inequalities and determine which graph illustrates which inequality.
2. Inequality Friends: Distribute index cards with simple inequalities to a handful of your students (four or five different inequalities) and to the rest of the students pass of cards that only contain numbers. Have your students rotate around the room and determine if their numbers and inequalities are compatible or not. If they know that their number belongs with that inequality then the students should become “members” and form a group. Once all the students have formed their groups, they should present to the class how they solved their inequality and why all their numbers are “members” of that group.

Both applications allow for a quick assessment by the teacher.  Having the students initially work in pairs to explore the inequality and its matching graph allows for discover on their own.  While ending the class with a group activity allows the teacher to make individual assessments on each student.

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

In a previous course, students learned to solve one- and two-step linear equations.  The process for solving one-step equality is similar to the process of solving a one-step inequality.  Properties of Inequalities are used to isolate the variable on one side of the inequality.  These properties are listed below.  The students should have knowledge of these from the previous course; therefore not overwhelmed with new rules.

Properties of Inequality

1. When you add or subtract the same number from each side of an inequality, the inequality remains true. (Same as previous knowledge with solving one-step equations)

2. When you multiply or divide each side of an inequality by a positive number, the inequality remains true. (Same as previous knowledge with solving one-step equations)

3. When you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. (THIS IS DIFFERENT)

There is one obvious difference when working with inequalities and multiply/dividing by a negative number there is a change in the inequality symbol.  By pointing out to the student, that they are using what they already know with just one adjustment to the rules could help ease their mind on a new subject matter.

C. CultureHow has this topic appeared in pop culture?

Amusement Parks – If you have ever been to an amusement park, you are familiar with the height requirements on many of the rides.  The provide chart below shows the rides at Disney that require 35 inches or taller to be able to ride. What rides will you ride?

(Height of Student $\ge$  Height restriction)

 Blizzard Beach Summit Plummet 48″ Magic Kingdom Barnstormer at Goofy’s Wiseacres Farm 35″ Animal Kingdom Primeval Whirl 48″ Blizzard Beach Downhill Double Dipper 48″ DisneyQuest Mighty Ducks Pinball Slam 48″ Typhoon Lagoon Bay Slide 52″ Animal Kingdom Kali River Rapids 38″ DisneyQuest Buzz Lightyear’s AstroBlaster 51″ DisneyQuest Cyberspace Mountain 51″ Epcot Test Track 40″ Epcot Soarin’ 40″ Hollywood Studios Star Tours: The Adventures Continue 40″ Magic Kingdom Space Mountain 44″ Magic Kingdom Stitch’s Great Escape 40″ Typhoon Lagoon Humunga Kowabunga 48″ Animal Kingdom Expedition Everest 44″ Blizzard Beach Cross Country Creek 48″ Epcot Mission Space 44″ Hollywood Studios The Twilight Zone Tower of Terror 40″ Hollywood Studios Rock ‘n’ Roller Coaster Starring Aerosmith 48″ Magic Kingdom Splash Mountain 40″ Magic Kingdom Big Thunder Mountain Railroad 40″ Animal Kingdom Dinosaur 40″ Epcot Wonders of Life / Body Wars 40″ Blizzard Beach Summit Plummet 48″ Magic Kingdom Barnstormer at Goofy’s Wiseacres Farm 35″ Animal Kingdom Primeval Whirl 48″ Blizzard Beach Downhill Double Dipper 48″ DisneyQuest Mighty Ducks Pinball Slam 48″ Typhoon Lagoon Bay Slide 52″

Sports – Zdeno Chara is the tallest person who has ever played in the NHL. He is 206 cm tall and is allowed to use a stick that is longer than the NHL’s maximum allowable length. The official rulebook of the NHL state limits for the equipment players can use.  One of these rules states that no hockey stick can exceed160 cm.  (Hockey stick $\le$ 160 cm) The world’s largest hockey stick and puck are in Duncan, British Columbia. The stick is over 62 m in length and weighs almost 28,000 kg.  Is your equipment legal?

Weather – Every time the news is on our culture references inequalities by the range in the temperature throughout the day.  For example, the most extreme change in temperature in Canada took place in January 1962 in Pincher Creek, Alberta. A warm, dry wind, known as a chinook, raised the temperature from -19 °C to 22 °C in one hour. Represent the temperature during this hour using a double inequality. (-19 < the temperature < 22) What Inequality is today from the weather in 1962?