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Wednesday Wisdom
Teaching Tip #10
Wednesday, October 9, 2024
Integrate interdisciplinary connections to demonstrate how math is interconnected with other subjects and the real world. Math is what makes the world go round. It is EVERYWHERE! But you wouldn’t always know it by the way we teach the subject. Students need to see how math is connected with so many areas of life, including their other classes, their potential careers, and their day-to-day life. So in this week’s blog, I am going to give you practical examples from a few different resources, and show you how you can extend each problem if you want to get students thinking more deeply about the concept and its application.
Online Resources
If there is a particular topic that you would like to connect math to, try doing an online search for “high school math problems related to …” Depending on the topic, you may get some great options at the top of the list, or you may need to do a little digging. I did a search for math and business and one of the first websites that appeared was Dartef.com. If you click on the “32 Examples of Real-World Math Problems” you get a list of problems that come from biology, business, healthcare, and more. Here are a couple of problems I particularly liked.
Tangent (Trigonometry) – An Architecture & Construction Example
What is the optimal size of a roof overhang? You can read through the explanation on the webpage, which is very clear and provides the method and the answer. At the end of the problem, there is a link to a worksheet. This is what you hand out to students. They will use the tangent of a right triangle to determine how long the overhang should be to provide shade to the window during the summer.
Extension: Once students are able to arrive at the answer for the given problem, you could ask them questions like:Would the length of the optimal overhang be different in Iowa (or some northern US state) compared to Florida (the state used in the worksheet)? If you leave this open, students will have to investigate and make an assumption for the angle of the summer sun is in the designated state.
What if you have a two story house and you would like the overhang to shade the first story windows? Then what would the optimal length be? Do you think that is a reasonable overhang on a house? Again, if you leave this open, students will have to investigate and make an assumption for how high the roof line is on a two-story house.
Linear Function – An Entertainment Example
This problem has a short YouTube video (1:29) included to show how a lighting engineer uses math in their work. There is also an interactive simulation that students can explore. This one does not have a worksheet, all of the information is on the web page.
Extension: Once students understand how the equations control the lights, you could ask them questions like:Using the first equation where the light increased 2% each second, write a piecewise function for the light to fade in, stay at full brightness for 5 seconds, and then fade out at the same rate as the fade in.
In the image, you can see a set of three lights that fade in and fade out in a sequence, one after the other. Magine you are a lighting specialist and you have been asked to program the set of three lights so that each light fades in 20% more each second and then immediately fades out at the same rate. Each light should begin fading in 2 seconds after the previous light begins. Write the piecewise function for each light in the sequence. Graph all three piecewise functions on the same coordinate plane in three different colors.
Another great online resource is Next Generation Personal Finance (ngpf.org). They have developed an entire free curriculum for personal finance, including a Financial Algebra course. I have actually taught the course as an elective at a previous school. But even if you cannot teach the whole course, there are some good activities that can be used to make math applicable to real-life. Here is just one example:
Card Sort: Graphing Systems of Equations – Saving and Systems of Equations
You can find the activity by clicking on the link. There are four sets of four cards that students must match. One set has the verbal scenario, one set shows the graph, one set contains the system of equations, and the final set has the solutions.
Extension: Once the students have matched the cards correctly, you could ask them additional questions like:In the problem with Maryam and Malak, who will have $10 in their piggy bank first and how much sooner? Who will have $20 in their piggy bank first and how much sooner? Explain how to get these answers graphically and algebraically.
In the problem with Elijah and James, imagine that three weeks into saving, Elijah’s parents start to give him an extra $1 each week to put in his piggy bank. Now when will Elijah and James have the same amount of money? How much will they each have?
Textbooks
Textbooks can be a great resource for problems that connect math to other topics. Most textbooks these days have “word problems” in a majority of their lessons that try to tie the math to another area. You can use these problems as is, or you may want to use them as a starting place. Take away some of the given information and you can turn the problem into mathematical modeling, where students are responsible to make reasonable assumptions (see Blog #1). Or you can use the problem in the textbook, but ask additional questions that get students to think more deeply. The following examples are problems from textbooks on OpenStax.org.
From Algebra 2 & Trigonometry (Jay Abramson), Lesson 5.8 Modeling Using Variation, Problem #53 – Music Example
The rate of vibration (Hz) of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?
Extension:If we assume that the above conditions are for a piano, what is the length of the string needed to play middle C (about 262 Hz) on the keyboard?
On an 88-key keyboard , the highest note is played on a sting about 5 cm long. How many vibrations per second does this note have?
On a guitar, the strings are similar lengths, implying length of string is not the only factor when playing notes. What are some other factors to consider?
Reworded from Algebra 2 & Trigonometry (Jay Abramson), Lesson 6.7 Exponential and Logarithmic Models, Problems #28 & 29 – Medicine Example
You are a doctor at a clinic when your patients, Mr. and Mrs. Jones come to see you. Mrs. Jones has the flu and feels terrible. You prescribe 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug? Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram.
Extension:You also suggest that Mrs. Jones take acetaminophen (Tylenol) to reduce her fever and achiness. Since you know Mrs. Jones is normally a healthy adult, it is reasonable to assume that the acetaminophen will have a normal half-life of about 2 hours in her system. The label gives the adult dosage as 2 caplets every 6 hours. Each caplet is 500 mg. How much acetaminophen will still be in Mrs. Jones’ body just before taking her next dose in 6 hours?
Mr. Jones has come in because he strained his back when he was exercising. After examining him, you determine that there is no serious damage and suggest that he ice his back 3 – 4 times a day for 20 minutes each time. You also want to recommend that he takes acetaminophen for the discomfort. However, you know that Mr. Jones has some liver disfunction, so you expect the half-life to be 3 hours for his system. If you want Mr. Jones to have the same amount of acetaminophen in his system as Mrs. Jones in the previous problem before he takes his next dose, how long should he wait between doses?
Contemporary Mathematics (Donna Kirk), Lesson 5.11 Linear Programming, Problem #19 – Business Example
Randy’s RV Storage stores two types of Recreational Vehicles (RVs), The Xtra RV (𝑥) takes up 400 square feet of space, while the Yosemite RV (𝑦) takes up 600 square feet of space. Randy has 55,000 square feet of storage space. By local law, he is only allowed to have a maximum of 100 RVs on his property at any one time. He charges $60 a month to store an Xtra RV, and $80 a month to store a Yosemite RV. How many of each should he store in order to maximize his profit?
Extension:Explain a scenario where Randy may not be able to store this many RVs.
There you have it. Six examples of how you can connect math to other areas, and how to extend the problems for deeper thinking if you so desire. I hope these problems give you some ideas, not only about how math can be applied in a myriad of scenarios, but also how you can take problems that are relatively “simple” and extend them to make them more thought-provoking. Hope you are having a great week! As always, contact me if you have any questions or comments. See you next Wednesday!!
(Graphs provided were created by me on Desmos.com)
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