N Ways to Apply Algebra With The New York Times

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Teaching ideas based on New York Times content.

This summer, the political scientist Andrew Hacker’s New York Times Op-Ed essay “Is Algebra Necessary?” set off a national debate on mathematics education. Here is how the essay begins:


The debate inspired us to suggest several algebraic opportunities right here in the pages of The Times. After considering them, and reading Mr. Hacker’s essay in full, we invite you to tell us what you think. Is algebra necessary?

1. Mathematically Modeling Mortgages

Use algebra to evaluate the housing market. Find some homes for sale in your area (for example in Atlanta), research current interest rates for mortgages and apply this formula to determine the required monthly mortgage payment.

Experiment with different home prices, different interest rates and different mortgage lengths to explore the impact of each variable on the resulting monthly mortgage payment. Use the Real Estate section’s built-in mortgage calculator to check your work.

For a given house, explore questions like “How much would you end up paying over the entire term of the mortgage, and how does this compare to the value of the house?” and “How big is the difference in monthly payments between a 15-year mortgage and a 30-year mortgage?”

Do you think it is more cost effective to rent an apartment or buy a house? Read this article and play around with this Times Interactive to explore the answer.

2. Evaluating Colleges

Check out how algebra is used to rank colleges. Read this article about the history of the Academic Index and then check out this informational graphic detailing the formulas that are used to create college rankings.

Calculate and compare the Academic Index of colleges like SUNY Binghamton and the University of Southern California by finding the appropriate data here. Are your results consistent with those of the U.S. News & World Report rankings, or those of The Princeton Review?

Use other available data — like tuition, acceptance rates and faculty information — to create your own algebraic formula for evaluating colleges. Can you create a ranking system that puts SUNY Binghamton ahead of U.S.C.? Can you create a different ranking that puts U.S.C. ahead of SUNY Binghamton? Can you create a ranking that puts your favorite college at the top of the list?

3. Calculating Car Costs

Use data from The Times’s Automobiles section to create a model for how quickly cars lose their value.

Use the Used Car Search to collect price information on specific makes and models of cars. Select your automobile, get your list of prices (for example, there are nearly 20,000 Ford Explorers for sale nationwide), and create a scatter plot with “year” on the horizontal access and “price” on the vertical access.

Use the graph and your data to explore questions like “How much value does a used car lose per year?” and “When does an automobile depreciate fastest?” Use technology like a graphing calculator or a computer spreadsheet to generate a possible function that models this data. Use the model to project the value of a new car two, three or 10 years from now. Set up equations to solve questions like “After how long will the car be worth 50 percent of its original value?”

Ratchet up the reality and complexity by compensating for the differences in new car prices and inflation. Go deeper by exploring questions like “Do SUVs lose value faster than sedans?” and “Which cars are the best long-term investments?”

4. Algebra of the Election

To win the presidency, a candidate must secure more than half of the 538 electoral votes. Take a look at the current breakdown of electoral votes by state, and find combinations of states that add up to 270 (or more) electoral votes. Each solution to this equation (or inequality) represents a potential path to the presidency.

Compare your possibilities to those demonstrated in this Electoral Map informational graphic. Take a look at some possible situations and interact with the mathematics by moving states around and exploring other solutions to this equation. If you were running for president, where would you focus your campaign resources?

5. Do the MetroCard Math

Is it better to buy a card that gives you unlimited access to public transit, or should you just pay ride by ride? Check out this article on the mathematics of New York City MetroCards. Currently, a seven-day unlimited MetroCard costs $29 and a 30-day unlimited MetroCard costs $104. If a single ride costs $2.25, under what circumstances would you be better off buying the weekly unlimited? The monthly unlimited? Set up simple linear equations and inequalities to answer these questions. Who should buy the 30-day unlimited?

Transfer over to some harder problems by factoring in the 7 percent bonus you get when you prepay for a pay-per-ride card. How does this discount affect the above calculations? And for even more real-world complexity, consider the effect that pretax commuter benefits can have on the equation. Check your numbers against the Metropolitan Transportation Authority.’s own comparisons here, and research the commuter cards in your area to find your best local options.

6. Olympic Algebra

Use mathematics to explore the wealth of Olympic data The Times has to offer. Take a look at the world records set this year in London, and check out the history of world records for events in track, field, swimming and more at the bottom of the page. Use simple formulas like Distance = Rate * Time to compare and contrast the average speeds of athletes over time, across events and even by gender. How much faster is the fastest sprinter now than 50 years ago? How much faster is the fastest sprinter than the fastest long-distance runner? How do swimmers, runners and cyclists compare in speed?

Pick an event, like the 100-meter freestyle swim, and take a look at the history of the world-record times. Create a scatter plot of the data with the horizontal axis representing the year and the vertical axis representing the world-record time. Experiment with some simple linear equations to find a line that fits the data, and use this equation to project the possible world-record times in 2016, or to hypothesize when the fastest time might drop below 43 seconds. If appropriate, use computing technologies to generate regression equations to compare to your own work.

Be sure to check out how beautifully mathematical results can be visualized and presented by watching this amazing video showing the Jamaican sprinter Usain Bolt running against every other Olympic medalist in history.

7. Redo Those Recipes

Search this Times Topics page for some tasty recipes with your favorite ingredients like chicken with lemon and rosemary or midnight pasta. Suppose you had twice as many people coming for dinner: what would the new recipe look like? Turn your recipe into an algebra problem by converting all amounts into some standard measure (like grams) and creating expressions that tell you exactly how much you need for any number of people. How many grams of chicken will you need if N guests show up for dinner?

Use a calorie counter to determine the number of calories per serving of your dishes. Using this United States Department of Agriculture guide, plan out meals for breakfast, lunch and dinner that keep you in the healthy range for caloric intake. Or list the 10 foods you most like to eat and find out how many calories are in each of them. Set up inequalities to figure out how many you should, or shouldn’t, consume to keep within your healthy range.

8. Solve for Stocks

Choose some of your favorite companies and use Times resources to collect historical stock price information: for example, check out Ford Motor Company, Apple or Amazon.com. Look at the prices over some fixed period of time, say the past two years, and determine the growth of the value of the stock over that period. If you had invested $1,000 at the beginning of that time period, how much would you have at the end?

Using the formula for compound interest, A = P(1+(r/n))nt, run the numbers on some hypothetical investments of the same principle investment over the same time period, and compare the results to the performance of the stock. Did the stock do better than a 5 percent annual rate of return? How about 8 percent? Set A equal to the final value of the stock, and solve for the rate, r, to figure out the exact rate of growth of your chosen stock.

Find the rates of return for several stocks, and put together a hypothetical investment portfolio for the school year. Watch your stocks through the year and compare their performance to your calculations. Does past performance guarantee future returns?

9. Population Growth, Football, the Economy and More

Use existing Learning Network resources to apply algebraic tools to rank sports teams, chart population growth analyze economic measures or even succeed in fantasy football.

10. Is Algebra Necessary?

Read Mr. Hacker’s original essay and decide for yourself. Do you use internal equations to help plan out your schedule? Or inequalities when you’re spending your money? Or graphs when you are plotting your performance? Or maybe you use algebra to figure out if that frequent shopper program is worth sticking to.

Find the algebra that’s around you and write a response to Professor Hacker. Maybe algebra isn’t necessary, but it sure can be useful.

This lesson addresses the following Common Core domains, standards and practices under Algebra:

Create equations that describe numbers or relationships.
Solve equations or inequalities in one variable.
Represent and solve equations graphically.
Interpret the structure of expressions.


Seeing Structure in Expressions
1. Interpret expressions that represent a quantity in terms of its context.
4. Derive the formula for the sum of a finite geometric series and use the formula to solve problems. For example, calculate mortgage payments.

Creating Equations



Reasoning With Equations and Inequalities

3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

C.C.S.S. Practices

Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

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Gas mileage. We give gas mileage in mpg, and use algebra to figure out our gas mileage when we buy gas (those of use with old cars) and to figure out how much we have to buy to make it when driving cross country and stopping somewhere with brutal gas prices.

These examples are pretty awful. No one would ever calculate mortgage payments using the actual formula. No one.

That level of algebra is so tedious that it amounts to little more than an accounting problem. Plugging numbers into an algebraic formula requires absolutely no understanding. It is the mathematical equivalent of “painting by numbers.”

I do not believe algebra should be mandatory in our high schools. Many kids my age do not even know their multiplication tables and other simple math. Why should we teach the students algebra when 90% of jobs do not even require it in that field. I feel that there is better classes to teach in that time, that will educate us in a superior way that will help us in the near future.
I feel we subject our students through this ordeal to make things in life more difficult. But lets face it colleges would rather see a high school kid take algebra class than multiplication studies.

@Hari, no one? Not only do I use the formula (and a much more complex version of it), I have been paid $250/hr to do so.

I think algebra should definitely be taught. I personally love algebra, and I’ve found algebra really useful to solve real-life problems I would have been stumped on using plain arithmetic. Algebra makes my life so much easier. I may not ever use it in my job, but there is such a thing as life outside of your job, and algebra is really useful there. Calculus on the other hand…

By all means, remove the requirements for algebra from schools. It will lead to my kids having a vast lead in life over everyone else who can’t understand the basics of adult life that algebra helps to define. Just teach them the phrase, “would you like fries with that” instead. It will serve them well.

It surprises me that in a year of election where dealing with the deficit is of utmost importance, I ask: how can you access the truth behind economic claims?

Requesting no algebra education is the same as declaring you don’t mind having a ignorant society. Because that is what you will get when citizens don’t cast their vote based on truth.

You have missed the point entirely. We do not teach mathematics simply because of its application to daily life. We teach mathematics because mathematics teaches us how to think.
Algebra at its core is symbolic manipulation. Algebra is most students first introduction to abstract thinking, to true, formalized abstraction. You will never mistake a person who has learned the important concept of a variable, from one who has not. You will never mistake a person who is capable of understanding abstractions, for one who is not . Algebra is more than simply solving for x, it is an entirely different, and extremely important way of thinking. This is why it is so difficult, this is why it is so important, .
Algebra is not important simply because it is applicable here or there, in a simple straightforward manner. Algebra is critically important, because algebra truly teaches us the entire concept of abstraction. Algebra teaches us one of the most important paradigms of human thought ever developed, and every single person incapable of grasping this is a detriment to himself and society.

I love math – got my degree in it – but the argument that math “teaches us to think” and therefore must be part of the curriculum is baloney, used by math-lovers who haven’t been in a real h.s. classroom in ages.

Algebra is the most common reason that intelligent educated people hate math. It clicks with a few kids but turns off far, far more of them, and those never go back.

That’s partly because algebra seems pointless; the above exercises are excellent ways to attach it to reality. This might help kids for whom it doesn’t click to stick with algebra long enough to enjoy it.

Excellent column. I hope teachers around the country are printing it out.

Algebra is absolutely needed to understand the sciences and technologies that our civilization is based, and to create public support for solutions to problems that these technologies are creating. To see this we need look no further than the American debate on climate change and how evolution is questioned.

To prove to themselves that science works, students only need to be able to understand the experiments they typically do in high school chemistry and physics classes. This quantitative understanding is only possible with a firm understanding of algebra and geometry.

It is a shame that children start with algebraic looking equations
such as 3 + 5 = 11, and 3 x 5 = 15 and then instead of going to equations like Y = 3X + 5 and Y = X^2.,

we burden them for years memorizing industrial age calculation routines for adding long columns of numbers and multiplying and dividing large numbers..

When students finally reach algebra, they are well past their prime language learning period. Children who have been deprived of language during this period never get it quite right, it’s how the biology works.

Algebra, geometry and calculus of high school are so much simpler than the spoken language that it’s ridiculous to think students cannot learn them without the pain they’ve been going through.

The problem cannot be with students since all humans are born to learn languages. The difficulty in learning mathematics is caused more by when it is taught than how .

Today’s system of math education works well for those children who are gifted or inherited activated math genes or are introduced to math concepts much earlier than normal.

If our goal is to reach a prosperous, sustainable information age civilization algebra is necessary and no child should be left behind.

Math is poorly taught.. especialy Algebra.

However, withouth basic Algebra you cannot really move forward with sciences in general. So not exposing the general population to sciences such as Physics, Chemistry, etc.. is a big mistake.

Furthermore, many fields require math. It may be that there are people that get by without it.. but at the same time one could argue that many get by without applying what they learned about English, History or Geography.

The original article is absurd. And if applied it would set the US back way behind many third world countries.

@Dave Brooks:
I am a recent college graduate with a degree in mathematics. I have seen the inside of a high school classroom rather recently.

Algebra certainly does train the way you think. Try explaining a variable to someone who has never taken algebra or had a literary education. It is almost impossible. People who have not had the training to think in abstract ways, often find themselves simply incapable of it. A metaphor and a simile serve the same function, which is why we also teach literature.

Algebra is a core, and fundamental way of instructing people how to decompose a problem and how one solution here is the same as another there. It teaches us abstraction. It is absolutely teaching you to think.

How many students are weeded out of every level of school by the demanding world of mathematics? “Lots” is the statistical term applicable to this question. We need an alternative way to teach these dropouts whatever it is math is supposed to teach. Apart from the obvious career opportunities derived from becoming a certified math wizard what differentiates the wizard from the dullard? Since math departments have failed on a colossal level to get their point across to lots of students, maybe we should try something different? How about a few billion dollars to figure out what can be done to save our math souls? Another enormous thousand page textbook is hardly the answer, we have plenty of these as it is and they only work for a small fraction.

Clearly, it makes no sense teaching algebra. That would show many people how broke they really are and won’t get into more debt (buy a new car, pay minimum credit card balances).

Obviously, math is not useful, we’ve seen the US is in terrible debt and nothing has collapsed yet. Which clearly supports the idea that numbers mean nothing.

Yay for math!

Despite Mr. Hacker’s and others’ “hack jobs” on math, kids do need to learn their times tables, they need to work with positive and negative numbers, they need algebra, they need practical examples, they need the theory …

If a customer tries to be helpful and realizes “Oh! I have 17 cents!” after you’ve started to ring up “fries with that,” you’ll wish you’d learned your math so you don’t have to stand behind the counter embarrassed because you can’t add it up in your head.

Try building a house. What angle do you set your saw at to cut a rafter for a 5:12 roof pitch, if you don’t know any trig? How long would you cut that rafter for a 12-foot horizontal run? How many 4’x8′ sheets of plywood do you need to sheathe that roof? How many 100-sq.ft. units of roofing do you need to order?

I don’t care what kind of a job it is. Learn a little math and do it right. Then learn a little more math and figure out how far your paycheck goes so you can reach your financial goals.

The anti-math activists tell us computer programmers don’t need to know math, either. I don’t know where they come up with these ideas. Is it really the intent to dumb down the entire next generation and pull the wool over their eyes?

54 years old. Previously a teacher of English, and now an IT manager for a small company, by choice. (I walked away from a manager’s job at Cisco 11 years ago to get back to doing it all for smaller companies.)

Successful at my trade? I suppose I could argue that I’ve been. Times I’ve needed the algebra I learned in high school and college? Never. Ever. Just as many times as I’ve needed to diagram a sentence to parse whether its subject-verb agreement needs correction. (And, yes, I’m very aware that this piece is littered with fragments. Intentionally so.)

One can always force external superfluous learning upon tasks, but that in no way renders the learning necessary.

Quite obviously, students who plan to go on in math, science or engineering absolutely must learn introductory algebra in h.s. The question is whether anyone else should.

I have a Ph.D. in Applied Mathematics from a three letter school in Cambridge, MA. So, I should be on the side of “everyone should learn algebra.” And, yet, I’m much closer to Mr. Hacker’s point of view.

There are a half dozen mathematical things that are more important for citizenship than general algebra: being comfortable estimating and working with both very large and very small numbers; understanding a few of the basic and fundamental ideas in probability; having a feeling for the meaning of the statistical results from, for example, polls; understanding simple rate equations; understanding compound interest.

I think you can gain a facility with all of the above without a semester long hs course in algebra. For example, it’s enough when learning about rate equations to learn how to identify the rate (some quantity per some unit time) ; to know what else your given and what you’re looking for (i.e., given time and looking for a quantity or given a quantity and looking for a time) and then using some simple rules to help you figure out if you divide or multiply what you’re given with the rate.

I personally think it’s possible to provide students with a much better grounding of the arithmetical competency and intuition that they’ll need to be citizens than we currently do by really rethinking what and how math is taught in HS.

For those who claim that algebra teaches students abstract thinking, there are much better vehicles for that. If that is the goal, then we should teach them the basics of formal systems: introductory logic and computer programming.

Anyone going into STEM-type fields absolutely must know algebra, and I’d argue that others should as well. As a computer scientist, I use algebra every single day that I write a line of code.

Do people really feel better in a world like this:

1. Numbers go in
2. Magic happens (algebra!)
3. Truth (but not understanding) comes out

Some people never use what they learn in algebra, music, art, PE, history, geography, and especially french and latin classes. Should we drop all of those too? Some people get extremely frustrated in all of those classes. There can be found bad PE teachers, bad french teachers, bad history teachers. Should we just drop them all?

Algebra is so fundamentally important in STEM and that’s where all the jobs are right now. So if you want your kids to have a better chance at getting a job, teach them algebra. Try Kahn academy if your algebra teacher [word removed].

Algebra is generalized computational math. If I buy two cookies for fifty cents each, how much will 10 cost? 100? or X? Algebra represents critical and representational thinking. If we stop trying to teach our students to think critically, and apply knowledge of one situation to better understand another, we are in big trouble. The real problem is that for the most part, we do not teach math properly in the U.S. Algebra is taught as disconnected memorized formulas after computational math is taught the same way. A we wonder why students struggle?

Math makes logical sense. Really! And not just to those of us who “get it”. Math is one way to explain a sense of logic and order that already exists in our universe. Not some extraneous subject we teach to children to torture them. There is significant and exciting research being done today on this very thing and we must change the way we teach math, from Kindergarten through College.


I was not a good math student. In college I was introduced to Algebra. My world and my thinking changed.
I was able to understand concepts of life. I cannot understand how it happened, but my thoughts and thinking changed. I believe Algebra should be taught in elementary school to students who are ready according to Piaget’s theory.

I graduated from the University of Illinois with a degree in liberal arts. I was able to choose either a science or a math class in college and I chose science, specifically geology, because it interested me; and, I knew math would be impossioble for me to pass. I graduated with honors and have a master’s degree plus many more hours. I teach high school English.

Today I understand algerba, but when I was younger it was only a burden. I found a career without algebra. We are doing a disservice to other kinds of thinkers by insisting that everyone succeeds in math. If I were the same student I was years ago, required to take algebra in order to graduate from college, it is unlikely I would graduate. I would not be a teacher. I would be a college drop-out.

Enough already! Everyone does not need college algebra in order to be successful in a career!

The old slick “It teaches you how to think” doesn’t go over so well with struggling high school students.

When they hit my office with failing grades in algebra, I always told them honestly: “No one, and I mean no one, has ever asked me in my adult life to solve for X. But almost every day I have to stop and figure out how to solve a problem whose solution doesn’t occur to me right away, in a situation that may be a bit frustrating. The home for acquiring this skill in high school is called math class.”

I was a student sure I couldn’t “do” math. I also never was taught to think logically. I was a A student of all things liberal arts, but floundered in science and math.
Math is about logic and thinking and processes and following directions and making connections and all of it needs to be DEMYSTIFIED so that all the kids can learn. Algebra? I tell my students they were doing it in first grade when they filled in the box that came after 2+3=……I tell them ratio is all about oatmeal that isn’t too thin and isn’t too thick…and that they need to understand the question before the can even hope to solve the problem.
Math and Science teach thinking skills and logical thinking, order and arguement. It isn’t really about the content. It is about the process.

….and one should proofread before submitting anything! :)