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You are here: HomeLessons → Open problems

What does it mean to change a math problem from being CLOSED to being OPEN? Essentially, the problem changes from having just one correct answer to having several. I also give several examples of how to change problems.

From CLOSED math problems (with one answer)
to OPEN ones

First, I want you to watch this short video by Jo Boaler. Seriously. Take a few minutes. It's not long, and I think you will be glad you did!

Jo mentioned the simple task of finding the perimeter of a rectangle when its sides are given, and changing that into a problem where students are asked to give two different rectangles with a given perimeter.

That is changing the task from a "closed" format, or a simple performance task, into an "open" format and into a LEARNING/GROWING type of task. It essentially means that now the problem has many possible answers, instead of just one. And it was a very simple change that did it!

(This also helps students develop a growth mindset.)

As you are planning your math lessons, check if some of the tasks you will be giving to your children/students can be CHANGED into open problems.

You can OFTEN take a textbook problem and do this. And, I don't mean you'd need to do this to every problem in the book but do it to SOME so that your children/students can have the opportunity of seeing that math can be ADVENTUROUS — you can be CURIOUS in math class — and it can even be FUN! It's not just about spitting out correct answers to calculation problems!!




Examples of "opening up" math problems

  1. Calculate 25 × 42. (4th grade)

    Change to:

    Calculate 25 × 42 in two different ways. Compare your ways with those of your friend.

    I just need to break in on myself here, and show you one way to calculate this. 25 is special because it goes into 100 evenly. So, instead of 25 × 42, I calculate 100 × 42 in my head... which is 4,200. Now, the real answer is just 1/4 of that. And... to find 1/4 of any number is SO easy, all you need to do is halve it, and halve it again. So, 1/2 of 4,200 is 2,100, and 1/2 of that is 1,050. So that's the answer, and that is the quickest way I know to find it. But, there are other ways as well! The idea here is NOT to compete with the quickest way to calculate it, but to find several different ways, and let students/children compare the methods.


  2. Find 3 + 4 and 5 + 4 and 3 + 7 and 1 + 6 etc. etc. (1st grade)

    Change to:

    Find ways to make 7 with two numbers. Also, write them in a pattern!

    Of course, I use that in Math Mammoth (see Sums with 9 and similar lessons in 1st grade materials).

    EXTENSION. One way children may write the pattern is
    0 + 7
    1 + 6
    2 + 5
    3 + 4
    4 + 3
    5 + 2
    6 + 1
    7 + 0

    You could then ask if they could still continue their pattern... :^)

    YES, they will come to negative numbers!   :^)
    0 + 7
    1 + 6
    2 + 5
    3 + 4
    4 + 3
    5 + 2
    6 + 1
    7 + 0
    8 + (-1)
    9 + (-2)
    ...

  3. Find the volume of a rectangular prism with a height of 2.1 m, depth of 4.5 m, and width of 3.8 m. (6th grade).

    Change to:

    Make a word problem where one needs to calculate the volume of a rectangular prism in order to solve it. (One idea is to use a pool, for example.)

    I'm sure students will come up with a great variety of word problems! Now, here's an extension: What if the floor of the pool is gradually sloping? How do you find its volume then????


  4. Find 5-2 (8th grade)

    Again, making a pattern is the way to go, I feel.

    Change to:

    Write a pattern using positive, zero, and negative exponents, using your chosen base. Try different kinds of bases! (whole number, zero, one, negative, even fractional base...)

    I have written about negative exponents with a pattern before, so I'll just refer you to that page.

See also

Problem Solving: Opening up Problems
This article gives a great example of producing MANY open problems from the closed problem "How many 5p coins are needed to make 45p?", in terms of what is known (information given), what is unknown (what needs to be found out) and what restrictions are placed on its solutions.




By Maria Miller

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