Genuine Systems of Equations (Help!?)

Kogut and I are on retreat in lovely Maine, listening to pandora extensively, working on re-writing our curriculum to be awesome in all the ways we’ve talked about so far.

In some ways, it’s going really well. Kogut (see below) is taking over Unit 0 and planning how we will get to know the students and start teaching them all those vitally important skills of working together, persevering through frustration, discussing math, using various tools, listening to each other, taking breaks when needed, etc.

I’m working on Unit 1 (linear, then non-linear patterns), using a combination of “math is cool!” awesome visual patterns and “math is useful” real-life patterns. Mapping our foci over the course of the weeks (anything always goes, but sometimes we’ll require some tabling, and then some graphing). And thinking about a myriad of ridiculous/fun products students can make to communicate their understanding (flipbooks, videos, write-ups, self-help pamphlets, etc.).

But. We are both into the idea of having a year-long project (problems that students can return to over the course of the year, that use various tools from throughout the year and allow students to showcase sophistication in the math as they learn more). Systems of equations seems like a pretty ideal topic to both of us — you can use tables! equations! diagrams! graphs! technology! to solve systems. But. Finding non-contrived systems problems. I can’t stomach the problems that pretend we wouldn’t already know the answer, or that are just totally irrelevant.

So I googled “genuine systems of equations” and got this blog post by Dan Meyers. Which mostly served to make me feel more hopeless because … yeah, I agree. And I have such a hard time finding genuine/authentic systems of equations problems (yes, I know about cellphones).

Here are three sample problems that might be acceptable. Do you have suggestions? Or more problems? Keep in mind that we will not require that every problem be solved in a specific way just a variety (most problems: solve 3 ways; some: 2 or 4 ways). We will also use some of Dan’s 3-act system-y problems, but one of our goals is to relate as many problems as possible to Community Leadership …

Problem 1: (This is in the realm of math is cool, not actually useful. It is borderline to me.)

Burning Candles.  People are lighting candles at a vigil to honor those who died in the Bangladesh Garment District Factories Collapse.  There are two different types of candles: green and red to represent their flag.  The red candles are 6 inches tall and burn ½ an inch per hour.  The green candles are 7 inches tall and burn ¾ of an inch per hour.

(a)  A photographer wants to take a dramatic picture when all of the candles are the same height; when should she take the picture?

(b) The local fire department wants assurance there will be organizers with all of the candles throughout the night until all of the candles have been extinguished (or have burned completely).  If the vigil starts at 8 PM, when should the organization expect to have volunteers shifts end?

Problem 2: (I think this is ok … though maybe stupid because the answer to whether Consalvo ever raised as much as Conley will be answerable by September 24th.)

Fundraising.  Dan Conley entered the Boston mayoral race on April 3rd with $866,000.  This amount was about twice as much as his closest rival, Rob Consalvo.  Use the internet to determine a reasonable amount each candidate fundraised per week.

a) Use what you find to determine whether Consalvo had as much campaign money as Conley at any point in time between April 3rd and the Democratic Primary on September 24th.

b) What do you think about corporations or unions donating large amounts of money to campaigns?

Problem 3: (This is a somewhat simplified but fairly real scenario.)

Minimum Wage.  According to the US Census, the average cost of living for a one-parent, two-child family (FAMILY A) in Boston is $5,260 per month.  Assuming a family could get a childcare subsidy or arrange for free childcare, the average goes down to $3,769.  The cost of living for a two-parent, three-child family (FAMILY B) in Boston is $7,058.  If we assume one of the parents provides childcare, the average goes down to $5,006.  Assume the parent in Family A works a minimum wage job, $8.00/hour.  Assume one of the parents in Family B works a slightly above minimum wage job, taking home $9.50 per hour.

a)  After how many hours will each family “break even” (i.e. earn enough money to cover their expenses)?

b)  After how many hours will the families have the same amount of savings/debt?

c)  Read this:  What should the government do to help this situation?




3 thoughts on “Genuine Systems of Equations (Help!?)

  1. Ok, things are getting a little better. Here’s two problems I wrote today, both of which I’m happy with.

    Problem 4 (compares male and female intake of calories):
    Read the article, write and answer your own question.
    (I have a backup set of questions if necessary.)

    Problem 5:
    Girls’ Education in Afghanistan. According to the United Nations Girls’ Education Initiative, in Afghanistan in 2004, 18% of females aged 15-24 are literate, compared to 50% of males in that age range (
    Assume that the male literacy rate is increasing at a rate of 1.27% per year.
    UNGEI’s goal is that females and males will be equally literate (have achieved “educational parity”) by 2024 (20 years after their initial report).
    a) At what rate must female literacy increase to meet this goal?
    b) Using your rate for female literacy increase from part a, compare male and female literacy in Afghanistan from 2004 till 2024.
    c) What do you think will happen with literacy once males and females achieve educational parity?
    d) Find another article about girls education in Afghanistan.
    Write a soapbox with a claim about girls education, using mathematical and other facts as evidence.
    Write a plan for how you would work to increase girls’ education in Afghanistan.

  2. Love these questions, however I would start with the kind you can guess and check for AND can have answers that are found by drawing. Example in minute. When I hand the students the lunch cart price list and have them build a problem backwards, I get a lot of success with mixture problems. Not highly functional, but works.

    My favorite to start with again are not terribly “useful,” AND I promise you that the most surprising of your students will find the most creative solutions…the pet store and parking lot problems! Why? Because there are multiple entry points and folks can DRAW the solutions. It is really cool to watch esp. the 2nd language learners. (Bird and dog legs or motorcycle and car tires…can we add to this repretorie with other than 2’s and 4’s?)


    • Hi Amy,

      Thanks for your comment! We enjoy problems like the pet store/tires ones. This project is for the whole year, and the goal, indeed, is to have problems that students can solve in multiple ways (perhaps first using a diagram or table, then graph, then equations, for one example). So just to clarify that we’re not starting with these problems 🙂

      We think we might end up using one or two of the pet store type problems, even though it is one of the problems that makes me cringe because obviously if you can count the legs, you can count the animals, but present it as this is fun(ny), not useful and that’s ok!


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