Jordan Massoud: Blog #4: Teaching linear functions

Hello class, my name is Professor Massoud and today we will learn about linear functions. To begin with, don’t get intimidated by the terms linear or function, I assure you each one of you has real world experience with both of these concepts.

First, in order to grasp the concept of a “linear function,” an explanation of a function is in order. For example, take an automated baseball-pitching machine. This simple, yet illustrative example is a function. How you might ask? Well, the machine pitches a ball for every ball that is fed into it. The machine in this example is a function. From this example we can label a few terms. The input of a function is the value you insert into a function. The answer or value that pops out, in the example it would be the ball flying you at 90 mph, is the corresponding output. Just look at the beginning of both words to keep these two concepts straight—input is what you put in, and output is what a function spits out. The formal definition of a function is a relationship in which one output is paired with exactly one input. It is also important to know that a function can be represented in four different ways: a graph, table, formula, or words.

Why learn about input/output or even functions for that matter? Well for mathematics of course, but also it is important to be able to explain the world around you in a systematic and consistent way. These concepts are also the building blocks for many mathematics courses you all will have to take.

Now that we have a basic understanding of the second part of the phrase “linear function” let’s examine the term linear. To do so, let’s think of running on a treadmill. When you get on the treadmill you speed up to your desired speed and let’s say you run for 25 minutes at that pace. In this scenario we have two variables: time spent running and miles ran. To show this, let’s look at a table.

          Time Spent Running (x)                                               Miles Ran (y)

5 minutes	.5 miles
10 minutes	1.0 miles
15 minutes	1.5 miles
20 minutes	2.0 miles
25 minutes	2.5 miles

Here, our input is time spent running and our output is the number of miles ran. As you can see, at 5 minutes John Doe has ran .5 miles, and at 10 minutes he has ran 1.0 miles, and so on.

Now I’d like to introduce another concept—rate of change, or more commonly known as slope. Rate of change simply measures the changes in the y values relative to the x-values. The rate of change or slope is a crucial tool in mathematics. It is used extensively in finding the formula to be able to graph a line. Just from looking at a graph, one can tell if the slope is positive, negative, less than one or greater than one. The slope is the single most defining feature of any graph.

 In the table above, we have a set of x values and a set of y-values. The formula for the rate of change is:

Change in y-values/Change in x values

                        Or

(Y2-Y1)/(X2-X1)

In the second equation, Y2/Y1 and X2/X1 represent an ordered pair of x,y values. So, in our example we could have the ordered pairs: (10,1) and (25, 2.5). Let’s practice calculating the rate of change between these two points.

(2.5-1)/(25-10)= 1.5/15=   .1 miles/minute

So, you might be thinking this is great, but what does this have to do with linear functions? Well, a linear function is a function where the rate of change at every interval is the same. An interval is two sets of points. In other words, for our example, this function can only be linear if we continue to get .1 miles/minute when we compare different sets of x and y values. Let’s see:

(1.5-1)/(15-10)= .5/5= .1 miles/minute

(2.0-.5)/(20-5)= 1.5/15= .1 miles/minute

As you can see, we continually get .1 miles/minute for the rate of change. This tells us that the example we started class with is indeed a linear function.

Now, try this problem and determine whether this function is linear or not and explain why/why not.

**Remember: functions can be represented as words, graphs, tables, or formulas

Let’s say that you and your parents go out for an ice cream cone… aw so cute. From start to finish, the trip takes you 20 minutes. The ice cream parlor is a few miles away so you decide to drive. Below is a table representing your trip.

         Time in minutes (x)                                   Speed of car in mph (y)

5 minutes	25 mph
10 minutes	42 mph
15 minutes	21 mph
20 minutes	0 mph

As you should have discovered this function is not linear because the rate of change at every interval is changing. This is because the car is not traveling at the same speed the entire length of the trip, due to stop signs, speed limits, and other factors. As a result, this table still models a function where for every output there is one corresponding input, that function is just not linear.

Blog 1 Alex Freedgood

Blog Post 1

Alex Freedgood

70 inches

19 years old

10-11 am

Butterfinger

Pringles

Dillon Francis

Sports, hockey, trap shooting

High school Precalculus

What is your comfort level with solving math word problems, math application problems, and conceptual math problems?

C

What is your comfort level with math computations and calculations/using formulas and following steps and procedures?

C

Equations VC

Factoring  VC

Adding/subtracting/multiplying/dividing fractions/rational expressions UC

Simplifying expressions C

Simplifying rational expressions (fractions and fractional polynomials) UC

Operations on roots and fractional exponent expressions UC

Operations involving whole number exponent expressions C

Finding common denominators VC

Inequalities VC

Functions VC

Absolute value C

Linear functions C

Systems of equations C

Domain and range UC

Inverse functions UC

Transformations (shifting, scaling, stretching, etc) UC

Quadratic functions C

Completing the square VC

Exponential functions C

Logarithmic functions C

Graphing (trigonometric, quadratic, linear, exponential, and logarithmic functions)

Trigonometric concepts C

Have you taken American university’s math 151 (finite math) class? If yes, was your grade a “b” or better? No

Have you taken American university’s math 160 (applied precalculus) class?

No

Which social media formats are you familiar with?

Texting, facebook, Instagram

Which social media formats do you use most frequently?

Texting and Facebook

What is your major?

Business

Why are you taking this math class?

So that I can take Calc

Will you take other math classes in the future at American University or will this be your only math class? If yes, which classes do you anticipate taking?

Yes, stats and calc

Seth Brennock blog

Something From Nothing written by Phoebe Gilman is a child’s book about a blanket that gets old and changed into something new. This is done many times until there is no more material to work with.  As the boy named Joseph grows and the blanket wears out, the old tailor recycles it, in succession fashioning a jacket, a vest, a tie and, finally, a cloth-covered button. Finally Joseph goes back to the tailor and says that he lost the button, at this point the tailor can’t make it into anything.  The tailor who is joseph grandfather tells him, “Joseph, I can’t make something out of nothing.” I chose this book because it was my favorite book growing up and it was read to me when I was little.

This Jewish folktale is a cute way of explaining that you as things go away you cant get them back. This childs story uses area to show that as the area of something decreases there is less material to utilize. As josephs blanket turned into a jacket, then a vest, the things his grandfather could make became smaller and smaller items. The area of these items diminished in as the size of the items and things his grandfather could make got smaller.

I think that a childs book is a good way to teach these basic math ideas. Although it is not a complex theory or idea, it is important to know that as things get smaller the area decreases. Just learning basic concept while maybe not even realizing the child is learning is an essential part of children stories.

Laura Romero's Blog Post # 3

Rock, Brock, and the Savings Shock, by Sheila Bair, is a book about two twin brothers who earn money for ten weeks in the summer from doing errands for their grandfather. Rock was a tidy boy who liked to buy things, and Brock was messy and lazy, but he liked to save. The first week, their grandfather gave them one dollar, and promised them that every Saturday he would match the amount of money they had saved. Rock spent his dollar every week, while Brock saved the money and every week he would double his savings. The brothers learned the importance of savings in those ten weeks of summer.
The book illustrates the concept of exponential growth in a way that kids can clearly understand. Both brothers started with one dollar each, the second week, the brother who saved his previous dollar earned another one, so having 2 dollars, the following week he gained two dollars, making his total savings $4. The 3rd week the amount doubled to $8, and on the 4th week it went up to $16, and so on it kept doubling until the 10th week, when he reached $512. This book represents the growth pattern 2^n for n greater than or equal to zero. The  book creatively explains the concept of doubling and exponential growth by teaching kids a valuable lesson about managing and saving money.
I believe that literature is a great way to explain mathematical concepts to kids because it simplifies complicated concepts such as exponential growth, and presents it in a way that kids will understand and be able to apply to their everyday lives. It also makes learning math easier at school.

Neda Ghias Blog 3

1. The children's book, A Fair Bear Share, is written by Stuart J. Murphy. This picture book is about Mama Bear wanting to make her special Blue Ribbon Blueberry Pie, but it can only be done if her four cubs go and gather enough nuts, berries, and seeds. The four cubs go and then add up all they have collected by arranging it in groups of tens and ones. After, they realize they don’t have enough and it’s because one of the cubs spent the day skipping, running, and doing cartwheels instead of helping out. It is up to her to to go collect her share of ingredients so Mama Bear can make her special Blue Ribbon Blueberry Pie. 

2. The mathematical concept that was presented in the book was input/ output. This is because the amount of ingredients the cubs gather depends on the output, of Mama Bear producing the pie and how much she can produce. In other words the input  would be the ingredients that the cubs gathered and the output would be the amount of pie that can be produced by Mama Bear. If the bears don’t gather all the ingredients needed then they won’t be able to have the pie they wanted. This all depends on the one cub to do their share to have all that was needed. 

3. The book, A Fair Bear Share is an effective way to teach math because it’s an easier way to lay out the steps in a way that children can relate to at that age or are interested in. In this book specifically it raised the question of whether the cubs had enough ingredients for Mama Bear to make the pie. By regrouping their berries, nuts, and seeds by tens and ones helped show them that one of the cubs did not do their part. The use of pictures and easy language makes it easy for children to follow along but also learn a lesson at the end. 

Alexander, Who Used To Be Rich Last Sunday

Carlos Torres 

Blog Post 3

1. Alexander, Who Used To Be Rich Last Sunday is a short story that tells of a young boy named Alexander, who thought he was rich after his grandparents gave him some money after coming over to visit. His family encourages him to save his money. They tell him that if he does save his money then one day he will be able to buy himself a radio, but the temptation to buy other things overwhelms him. After making various unnecessary purchases, he runs out of money, and his family tells him that he would need to go out on a cage every time he went out due to his poor money management skills. After a while he tries to regain some of the money he lost but is unsuccessful, he tries to sell some of his teeth and then he tries to look for money on phone booths. He also tries selling things, but these are unsuccessful as well. Alexander finishes broke, with only a handful of unnecessary purchases to his name and a couple of bus tokens. 

2. I think that this story in particular expresses the mathematical concept of Profit/Revenue. Alexander forgot about the basic principles of being profitable. In order for him to have stayed "rich" he should have focused on gaining more revenue and cutting cost as much as possible. If Alexander would have focused on this, then maybe he would have even turned a profit after trying to sell other things, such as things he no longer found useful. 

3. I think that this story exemplifies how one should always be careful on how to manage his/her money.  It helps us visualise in a clear and concise way the concept of profit/revenue and helps us avoid making the same mistakes Alexander made. 

Blog Post #3 Who sunk the boat?

"Who sunk the Boat? by Pamela Allen, Paffin 1996

Summary:
Once upon a time there were 5 friends, a cow, a donkey, a pig, a sheep and a mouse. One day the five friends decided to go on a boat trip. They entered the boat from the biggest to the smallest causing it every time to go lower and lower into the water. The question is...Who sunk the boat?
Concepts:
The first and most significant mathematical idea in this story is the probability. To be more specific, it talks about the chances that each animal had in sinking the boat. Each animal had a 1/5 probability to have sunk the boat. But the exciting part of this story is that not all the animals weighted the same. This leads to some other mathematical models such as addition, subtraction and division of number so we can find the weight of the animals and which one would most likely be responsible for the sinking. The heaviest animals, such as the cow and the donkey, would have more chances in sinking the boat than the mouse. Also, the story could be an example of input and output. As an animal was entering the boat (input), what was the outcome(output)? Did it sink or not? The last mathematical model that this story could be an example of would be a function with the quantity of the animals on the x-axis and the weight in pounds or kilos on the y-axis. The graph in this case would increase and concave up until the point where the boat sunk.

Literature can be very helpful in learning and understanding mathematical models and mathematics in general. Some people prefer, or they find easier, to learn complex methods through a simple story or images that they can identify with. Also, with the use of real life scenarios and examples, math becomes way more easy to understand, and helpful for everyday life as well.

Blog 3 "Once Upon A Dime "

Marek Niesiobedzki

Once Upon A Dime

By Nancy Kelly and Alan Doyle  

Part 1 – Beginning  One day a young farmer who’s name is Worth notices a tiny tree in a place on his land where nothing has ever grown before.

Middle-  Worth has some local friend farmers. He also realizes that with the use of organic fertilizer money grows on the tree. His friends including Lewis and Cluck provide chicken dropping fertilizer, Mooly Pitcher provides cow patties and   Mr. Oinkhower provides Worth with Pig droppings.

End- Worth notices that each fertilizer provides and makes the tree grow different amounts of money using each fertilizer however the farmer realizes that not everything is about money, and when dimes grew on the tree the sound it made was beautiful and decided to use only fertilizer that produces dimes , so that the whole neighborhood can enjoy the beautiful sound they make in the winds, while his young helper just wants more money.

Part 2- The mathematical concept is input and output. Based on what fertilizer is used on the money tree, the different coins grow.

At first, after using just one of the natural fertilizers chicken poop”, Nickels grow,  Cow patties  produces pennies and Pig droppings lead to quarters. However, when all of them are combined nickels grow, which are not the highest and most profitable coin to grow but provide a beautiful sound in the air which the farmer enjoys and the rest of the community. This book perfectly describes how a certain input leads to a certain output, demonstrating the concept of functions.

Part 3- The reason why I think that this literature is a spectacular way of teaching math to youngsters is because it demonstrates input and output in a very fun and creative way. Input and output concepts will be used all through our lives, with investments, mortgages, interest rates etc,