Friday, July 10, 2009

A Morning Warmup for Middle and High Schoolers - No Calculators Please!

How many integers from -1001 ro 1001 inclusive are not equal to the cube of an integer?

Hint: This could be a real 'Thriller'!

Click Read more for comments...

1) Do you think daily exposure to these kinds of problems as early as 7th grade will improve student thinking, careful attention to details (reading!) and ultimately performance on assessments? I think you can guess my answer!

2) I've published many similar questions on my blog but I couldn't resist this tribute to MJ.

3) I strongly believe we must occasionally remove the calculator to force their thinking. The stronger student recognizes immediately that 1000 and -1000 are perfect cubes and that one does not need to count the cubes but rather the integers which are being cubed (aka, their cube roots). The student with less number sense and weaker basics will feel lost at first but eventually their minds will develop as well if challenged regularly.

4) I added some complications to this fairly common 'counting' problem, similar to many SAT problems. This type of question is also typical of 8th grade math contests. Where do you think the common errors would occur assuming the student has some idea of how to approach this? Is understanding the language the primary barrier or not?

5) Let me know if you use this in September to set the tone for the year!


Totally_clueless said...

Just to reinforce other concepts too, one could ask:
(1) How many numbers between -1001 and 1001 are cubes of real numbers?
(2) How many numbers between -1001 and 1001 are squares of real numbers?


watchmath said...

How about giving the following hint: 10^3=1000 but 11^3 is bigger than 1000.

mathmom said...

Nice problem, and I like TC's extensions (for high school -- we don't really do real number in middle school).

Predicted stumbling blocks:
1) off-by-one error on counting the numbers (all of them, or the valid cube roots)
2) missing the "not"
3) terminology -- not understanding what "not equal to the cube of an integer means" for any number of reasons.
4) forgetting how to multiply (cube) negative numbers

I probably wouldn't use it in September because the younger end of the middle school group will not know how to multiply negative numbers yet ;-) But I'll save it for later in the year (if I remember it!)

I sent the problem to my kids at camp (8yo and 13yo, both "math kids"). I'll let you know if they solve it -- I included a bribe ;-) The 13yo should have no problem with it, but there's a good 25% chance he misses the "not" or makes some other minor error. For the little guy, I included the following hints (not meant to be precise definitions, but to clarify the problem for him):

An integer is a positive or negative whole number, or zero (no fractions or decimals).

A number is the cube of an integer if it is equal to some integer cubed (to the power of 3).

3^3 means 3 to the power of 3 -- it's what we use when typing and can't make the power a little number up to the right

so 3^3 = 3 x 3 x 3 = 27. So 27 is the cube of an integer (it is the cube of 3).

The cube of a negative number is negative.

(-3)^3 = (-3) x (-3) x (-3) = -27. So -27 is the cube of an integer (it is the cube of -3).

Dave Marain said...

Agreed! TC's extensions lead to a nice discussion of domain and range for continuous functions although I would change the range to -1000 to 1000 inclusive. I might even restrict the range to the closed interval from -1 to 1. We can then analyze the problem both numerically and graphically and extend to all power functions of the form y = x^n where n is any positive real.

Mathmom, let me know how your sons respond. I don't think your older one will miss it at all! Clarifying the terminology is perfectly reasonable to allow your younger son to focus on the quantitative parts and the counting aspect.

I personally wouldn't give any hints about 1000 being 10^3. That's part of the base knowledge I'm assessing here and they should reason that 11^3 will be much greater than one more than 1000.

BTW, watchmath, thanks for the LaTeX info! I will need to try that out.

Anonymous said...

I like this.

I also like:

How many integers from -101 to 101 are not squares of integers?

I am not sure which is trickier.

What do you think?


Dave Marain said...

I like both versions and both require careful thinking! Try them out with students to see which produces more errors. My instinct is that the student who is able to solve one will also get the other!

mathmom said...


My little one wrote from camp with the correct answer. (No explanation -- I would expect him to be willing to give one verbally but not in writing at this point, and especially not in a letter from camp!)

No reply yet from the 13yo.

Eric Jablow said...

You know, I wouldn't make any references to Michael Jackson in any problem designed for middle schoolers.

Moordale said...

Is there a gramatical error in this question? I am having a hard time understanding what is been asked?

gemmabusarang said...

this is a real good warm up.. a real mind twisting...

Dave Marain said...

No grammatical error but I do see a silly typo:
Should be -1001 TO 1001 instead of "RO"!

mathmom said...

heh, turns out my little one did not solve the problem himself, but got a counselor to help him. ;-)