# What could possibly be the point in this grade 8 math exercise?

My daughter has a math exercise to complete this week.

The numbers 1 to 100 are the answers. Using only numbers 1, 4, 8, and 9 she has to construct an equation to satisfy each answer. She can use addition, subtraction, multiplication, division and exponents (including roots).

That is the first part of the exercise. When she has answered as many questions as possible (100% is the target) she needs to write a few paragraphs on what techniques she used in order to come up with the correct response, including patterns noticed, and short-cuts taken.

Now while this may seem like a nice little brain exercise, I challenge anyone to sit and figure out ways to solve this in a pragmatic fashion. I looked at it last night and without simple, pure brute brain force there was no way I could easily come up with the desired equations. (You can only use use each number once in an equation.)

Let’s try 49.

9 X 4 = 36: 36 + 8 = 44: 44 + 1 = 45 - Nope.
9 + 1 = 10: 10 X 4 = 40: 40 + 8 = 48 - Nope.
(Square root 9) X 8 = 24: 24 X (Square root 4) = 48: 48 + 1 = 49 !!!

Now imagine having to do this for all 100 numbers!?!?! In grade 8?!?!?

Is there really a point to this?

Perhaps the point was to realize that if you do the brute-force thing for the values 1, 2, 3 and 5, you can then string those together to build all of the other answers? Except for the prime numbers, everything else can be factored down to multiples of 1, 2, 3 and 5, and multiplication can be reduced to repeated adding. Prime numbers can be created by summing (but not factoring) as well.

Does she have to use all those for each number, or does it simply say “can use” as you wrote?

49 = (8-1)(8-1)

Ah! Sorry. No, you must use each number once only; so 1, 4, 8, 9 is used in each response.

I’d guess the point is to get the student to think about using exponents and roots as “normal” math functions - since it appears that they are key to solving the complete set - and hence get more comfortable using them?

As for eighth grade - at seem like about the right time (if the exercise is going to be done at all) - much earlier and you probably haven’t done much with roots and exponents, any later and you’re into algebra and manipulating symbols (via roots and exponents, of course).

Joe

(8-1) * (9-√4)

Sure. Now solve for the other 99 answers. Sure looks like an exercise in tedium to me.

We did that in 7th grade, but we had to construct 1…100 using four 4s:

4 + 4 + 4 + 4 = 16
(4 * 4) + (4 * 4) = 32
4! + 4! + 4! + 4! = 96

etc.

IIRC there were two numbers that could not be constructed this way

1. 1
2. (9-8) +1
3. √9
4. 4
5. 4+1
6. 4+ (9-8) +1 (I just had to look back on how I got to 2 and copy that into here)
7. 8-1
8. 8
9. 9
10. 9+1
11. 9+ √4
12. 9+ √4 +1
13. 9+4
14. 9+4+1
15. √9*(4+1)
16. 8*√4
17. 9+8
18. 9*√4
19. 9+8+√4
20. 9+8+√4+1

I could go on. This took about 5 minutes to do. Granted, I’m not in 8th grade, but I found the project fun. I pretty much went with the first thing that worked in my head but there’s the beginings of a rudimentary pattern forming here as well that can be applied as numbers go higher.

I’m pretty sure you need to use all 4 numbers in the equation. I’ll check tonight. Otherwise you’re right, it’s a lot easier this way.

I figured you had to use all 4 numbers only once, which makes it harder. But you could start with 1[sup]489[/sup]= 1 and (8/4)*1[sup]9[/sup]=2, for example.

I feel ya. I’m having to get a math tutor for my 6th grader because she’s struggling and her father and I are no help at all. The last homework assignment I helped her with she got a D- on (something to do with Venn diagrams and factors - I read the chapter and I thought I understood it but apparently I was wrong).

My problem with math has always been, “Why?” Why am I doing this? What purpose does it serve? I can do what I call household and knitting math (balancing my checkbook, calculating how much carpet to buy, changing recipes, designing sweaters that fit, for example), but throw a string of random numbers that don’t stand for anything at me and I’m lost.

Sounds to me like this is a self-answering question. Once she’s written the few paragraphs, they should tell you what the point of this is.
But I think the point is to get a feel for numbers and operations and the various ways they can be combined, by playing around with them and experimenting with them.

I understand. I’m half venting about the amount of tedious homework, and half thinking maybe someone has been through this and it’s way simpler than I originally thought. I guess not. At this point, assuming an average of 5 minutes to answer one of the questions, then we’re talking 8 hours of homework for one subject alone. On the second week of school. In grade 8!

I can understand the point of the exercise (arithmetic is very important and kids don’t get enough practice to be quick at it in everyday life).

But having to do it in one week? That’s cruel for those not good at math.

(Note: this may not stop my husband and I from facing off to see who can do them all first. It sounds like fun to me.)

As I understand the assignment, there is no penalty for failing to complete it (you do as many as you can do, the goal is all of them). Some of the answers take far less than 5 minutes ( 1+4+8+9 = 24. done. 9+8+4-1=21. done ) while others will take far longer.

It may help not to attempt this one “ground up” (starting from the answer). She should manipulate the numbers every way she can think of, filling in the sheet with her results. Thus this also an excercise in strategic test taking. She should complete all the obvious ones, move on to the combinations which are more complex, and then try to see what the “missing” answers tell her. Are they all prime numbers? Are they all multiples of 5? etc.

I feel this is the entire point of the exercise and can be completed in, at most a couple of hours.

You can do them all from 1-100. It is 113 that is a problem. Here are some solutions.

Yeah, I would have interpreted “do as many of these as possible” to mean “do as many as you can before running out of time and/or patience,” not “do all the ones that are mathematically possible.”

Good point! Approached that way, some of them at least will take well under 5 minutes each.

Unfortunately, as her parent, I need to keep pushing her till Friday to do as many as possible. I can’t say, “Well you got half of them now, so you get the rest of the week off.” So, it’s going to be time consuming.

And believe me, she completely attacked the bottom-up principle to begin with. She ain’t dumb.

I do too! We’re dorks.

But really, it is a fun exercise in problem solving and becoming comfortable with creating your own formulas (which 8th graders just aren’t, but will soon need to be). The “easy” solutions should be plentiful, giving a solid B to anyone who gives it a half-assed attempt. Going the distance will require some effort.

I wonder if pairing the numbers up will be allowed - like using 14, 41, 18, 81, 19, etc.