# Large number formats

A great explanation of the comic can be found at https://www.explainxkcd.com/wiki/index.php/2319:_Large_Number_Formats.

# Engaging students: Using the point-slope equation 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 Johnny Aviles. His topic, from Algebra: using the point-slope equation of a line.

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

On the 1987 NBA Dunk Contest, Michael Jordan won by dunking all the way from the free throw line. (I will play them a clip). Now can anyone tell me how high the hoop is from the ground? And how far is the free throw line from the base of the hoop? So, in total he went 10 feet in the air while jumping 15 feet! This is incredibly difficult and was why he won the contest. Now lets just compute that slope. With rise/ run we get that the slope was 2/3. Another example I can use is the time I took to get to school. I live 30 miles away and it took me 40 minutes to get to school. would anyone be able to find the average speed? (45 MPH) Then I will make it more complex and say I went 60 miles an hour for the first 20 minutes, how fast was I going the last 20 minutes?(30 MPH) Then I will have a round robin activity where I will give 5 min for my students to discuss amongst their groups where they can create a scenario where they can use point-slope equation of a line.

C3 How has this topic appeared in the news?

We all have many factors that interest us and the news’ job is to keep us updated. For many people, the stock market is a very serious subject of interest. Everything is shown in charts and done on points and percentages for simplicity reasons. This uses the concept of point-slope equation of a line to create this data. The news also covers may other topics like the rise of current temperature from given years to see if factors like global warming may have played a role to create the next leading story. The data from previous years can create point-slope equation that can predict the rain and snow fall amount for a given city or town. The weather initially can use point-slope equation of a line to predict all factors all data collected over decades. There is a copious amount of data that the news has to be used in all aspects of the news, one that has been shown is the rise of mass shootings. This is a very controversial matter as many people seek reform of the second amendment. Overall, point-slope equation of a line is widely used in many platforms of our news programs.

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

Architecture has been the biggest contribution that point-slope equation of a line and has to be applied. Various cultures have their own specific style of how they have their cities, towns and neighborhoods but all will apply the basics of point-slope equation of a line. For example, when creating a building, they use materials with large mass and need to be supported. If the slope of a beam is even slightly off, it can generally cause the building to collapse under its own weight causing the lives of many. Every aspect of the building needs to be measured in a precise way to create a solid structure. Styles then range from all cultures and can have tilted and rounded with elaborate beams to add more diversity. Overall, all cultures have their own specific style of houses that all require the same point-slope equation of a lines that contributes them to remain standing.

# How to picture an exponent

While I’m easily amused by math humor, I rarely actually laugh out loud after reading a comic strip. That said, I laughed heartily after reading this one.

Source: https://xkcd.com/2283/

# Engaging students: Finding points on the coordinate plane

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 Tiger Hersh. His topic, from Pre-Algebra: finding points on the coordinate plane.

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

To find a point on a 2-D coordinate plane we would need to have an x-axis and y-axis. Many things in the real world could act as a coordinate plane and that could also be used to create an activity or project. One of those things could be where the students could use a Nerf gun and fire it at a wall with a coordinate plane. This activity would not only be engaging for students but also help them understand how to plot the points on a coordinate plane, but also show students how to find the point on the coordinate plane.

Students will group up and take turns firing darts at a wall that would have a coordinate plane on it. Each group will have different color darts to indicate where each group has plotted their point. Each student in each group will fire two darts at the coordinate plane; After each student has finished plotting their points they will approximate the point and record it down on their worksheet.

Curr1 : How can this topic be used in your students’ future courses in mathematics or science?

Plotting points on a 2-D coordinate plane is used in almost every future course in mathematics. You can observe the usage of 2-D coordinate planes in Geometry, Algebra 1, Algebra 2, Pre-Cal, and so on.
In Geometry you can plot the points of a triangle on the coordinate plane to then find the distance between them with the distance formula or you could find the midpoint between each point using the midpoint formula. These are only some examples that plot points on the 2-D coordinate plane.

In Algebra 1/2 you can see that you can find the slope between two points using the slope equation. You can also use this concept to plot points for equations that involve the slope-intercept form, polynomials, the unit circle, shapes, etc. The points that are plotted could also show what is happening over a period of time and also give us an idea what the equation is trying to tell us.

In Pre-cal you plot points on a coordinate plane in the equation $x^2+y^2=1$ to form the unit circle and also plot points when you have to rotate or transform a shape or equation.

Cul1 : How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The game Starcraft 2 is a real-time strategy (RTS) game where you have to build an economy to fuel an army and beat the opponent by destroying their infrastructure, economy, or army. Interestingly when you build your building you notice that you are building on a 2-D coordinate plane.

The game itself is in its own 2-D coordinate plane where you have to plan where to move at certain points and also place your buildings at certain points to either block off a ramp or create a concave for your units so that they are able to deal more damage towards the opponent. There are also times in the game where you have to keep in mind about key parts in the map where your opponent is, where your next bases are, where proxies are, and where to set up counter attacks on your opponent.

# Visualizing One Million vs. One Billion

From the YouTube description: “There are lots of ways to compare a million to a billion, but most of them use volume. And I think that’s a mistake, because volume just isn’t something the human brain is great at. So instead, here’s the difference between a million and a billion, in a more one-dimensional way: distance.

The video is more than an hour long, which is the point. In the last minute of the video, he mentions what a trillion would be in the same scenario.

# Engaging students: The field axioms

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 Andrew Sansom. His topic, from Pre-Algebra: the field axioms of arithmetic (the distributive law, the commutativity and associativity of addition and multiplication, etc.).

Algebra, from one perspective, is the use of numbers’ and operations’ properties to manipulate expressions. Some of these properties, called the field axioms, are crucial to being able to easily solve equations. These properties include associativity, commutativity, distributivity, identity, and inverse. To better appreciate how these properties are so helpful in algebra, it is useful to explore some examples of operations that do not obey these laws.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Example 1: The Average (Mean) is Not Associative

Part 1
A math teacher Mrs. Taylor instructs a class of three students: Alice, Bob, and Charlie. The class took an exam last week, but Charlie was sick and missed the test, so he took it today. Mrs. Taylor promised the class that if the class average on the exam was high enough, she would give them all candy. If Alice scored a 96 and Bob scored an 83, what was the class average (the average of those two students) after the first day of the exam?

mean(A,B)= $\frac{(A+B}{2}$=

Part 2
After Charlie took the exam (he scored an 89), Mrs. Taylor wanted to know if she had to calculate the average from scratch (i.e. add all three scores and divide by three), or if she could just average the previous mean and Charlie’s score (i.e. add your answer from part 1 and Charlie’s score and divide by 2), since she already had done some arithmetic and didn’t want to waste time. Would she find the same answer if she tried both methods? If not, which one is correct? Why?
mean(mean(A,B),C)= $\frac{ \frac{A+B}{2} +C}{2}$ =

mean(A,B)= $\frac{A+B+C}{3}$=

Part 3
After her discovery in Part 2, Mrs. Taylor is curious if she first found the mean of Bob and Charlie’s grades, then averaged it with Alice’s grade, if it would be the same as an answer above. Is it? Why or why not?

mean(A,mean(B,C))=$\frac{A+ \frac{B+C}{2} }{2}$=

Part 4
What does it mean for an operation to be associative? How does this activity show that the average (mean) is not associative? Why does this mean you have to be extra careful when solving problems with averages?

Example 2: Subtraction is Not Commutative

Part 1
Mrs. Taylor likes to visit Alaska during the summer. When she arrived in Anchorage, it was 10F, but a snowstorm caused the temperature to drop by 21F. Write an equation with subtraction to find the new temperature the next day.

The next summer, when Mrs. Taylor arrives in Anchorage, it is 21F but the temperature drops 10F. Write an equation with subtraction to find the new temperature the next day.

Part 2
What does it mean for an operation to be commutative? Based on what you found in Part 1, is subtraction commutative? Why or why not? Why does that mean you need to be extra careful when solving problems with subtraction?

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

Prior to pre-algebra, students should be proficient in arithmetic. In that study, they should have been exposed to fact families, which are simple examples of the inverse elements of addition and multiplication. The field axioms generalize these ideas to other objects. Students also should have realized that subtraction and division do not commute, though they likely never used that name. They also likely realized that addition by 0 or multiplication by 1 do not affect the value of the other element. By learning the names of these different properties, students build upon their prior experience to be able to label and acknowledge when these properties appear in other contexts.

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

Although high school students will spend most of their time working in fields, instead of other algebraic structures such as non-Abelian groups or noncommutative rings, an appreciation and awareness of the field axioms while studying pre-algebra will prepare them for solving equations involving exponents (for example, intuitively questioning whether 2^x=x^2, which are trivially different, but not obvious to the novice). Furthermore, most Algebra II classes do briefly study Matrix Algebra, which is noncommutative (i.e. matrix multiplication does not commute), which causes many interesting conundrums for the uninitiated student while trying to solve problems. This appreciation of the field axioms prepares them for later study in Linear Algebra and Abstract Algebra. Outside of their math classes, vector fields form a critical part of physics, even at the high school level. Although most high school students do not realize it, they have to use the field axioms all the time to solve physics problems.

References:
Use of the mean as a simple example of a non-associative operation courtesy of StackExchange user “Accumulation” on the thread “Non-Associative Operations” (https://math.stackexchange.com/a/2892589)

# What’s bigger: 1/3 pound burgers or 1/4 pound burgers?

I recently enjoyed reading about an unanticipated failed marketing campaign of the 1980s. Here’s the money quote:

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “¼,” larger than the “3” in “⅓,” led them astray.

# A Father Transformed Data of his Son’s First Year of Sleep into a Knitted Blanket

This is one of the more creative graphs that I’ve ever seen. From the article:

Seung Lee tracked the first year of his baby’s sleep schedule with the BabyConnect app, which lets you export data to CSV. Choosing to work with six minute intervals, Lee then converted the CSVs into JSON (using Google Apps Script and Python) which created a reliable pattern for knitting. The frenetic lines at the top of the blanket indicate the baby’s unpredictable sleep schedule right after birth. We can see how the child grew into a more reliable schedule as the lines reach more columnar patterns.

# Fun with Proportions and Atoms

I came across this fun video on proportions, imagining how large some objects would be if atomic (and subatomic) length scales were magnified to the size of a tennis ball.

# Engaging students: Defining a function of one variable

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 Phuong Trinh. Her topic, from Algebra: defining a function of one variable.

How have different cultures throughout time used this topic in their society?

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life.  For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is$1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

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.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus$0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

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

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.