Question

Dear Mitch,

I believe you said that you would post the whole answer to the Christmas Balance Scale problem before the students leave at the end of the day tomorrow.  I know it's cutting it close, but could you just explain how to solve that balancing one?

Sincerely,

Jose B.

Southern part of the United States

Answer

Dear Jose,

First I would just like to reiterate that this problem is probably the most challenging question we have posted so far in our one year and one week of launched existence.   As I've mentioned before, it is CERTAINLY considered beyond the 'normal level' of challenge that one usually encounters before junior or senior year of high school. Even then, it is considered a serious challenge.

Since you already have the pictures, which can still be pulled up from the q&a 'archive' (the one dated November 29th, as you point out) and can still be printed right off the site, it's easy to make visual the changes in each step.

Substituting S for each Snowman, T for each Christmas tree, C for each Candy Cane, P for each 'Present', and an equal sign for each balancing point, we get the following three equations:

(a) T + P = S

(b) T = P + C

(c) 2S = 3C

And the question asks: How many P's = ONE T?

You asked for a hint, so here are the first two steps:

1. In order to make the two left sides of equations (a) and (b) equal, add one P to each side of equation (b).
2. This means that one S = 2P + C. OR, 2S = 4P+2C.

Equation (c) shows us that 2 S also equals 3C.

So, 4 P + 2 C = 3 C.

Now if we 'take away' two candy canes from each side, we can see that the weight of 4 presents equals the weight of one candy cane

Now, we can exchange the one candy cane in scale b with four presents, and then we can balance the Christmas tree in scale d with five presents

Even after you read this, it is highly likely to seem like a lot of words describing switches that you don't really get or which make no sense.  But trust me, you will gradually come to understand each  manipulation if you simply take index cards, cut them in half vertically, (the lines pre-printed are to be thought of as horizontal lines and should be held that way so the perpendicular cut makes them more square-like than they would be if you cut them along a horizontal. (This just seems to be a size and shape that students find convenient and helpful for this type of 'hands-on algebra.

Next make a supply of cars with one letter on each card.  There are only 4 relevant letters: C, for Candy Cane, P for Present, T for Christmas tree, and S for Snowman.   So, just to give yourself th4 freedom to practice manipulating variables, make ten C cards, 10 S cards, ten T cards, and ten P cards.

Next set up four makeshift visuals of balance scales on a horizontally oriented blackboard or dry erase board (in other words set the board flat on a desk or table or on the floor; vertically, gravity poses a nuisance. and put

These makeshift visuals for the scales is just something to help you arrange them properly and have your idea clearly observable to assess whether you are moving in the right direction or not. So each 'scale' they can be two paper plates sitting side by side and you put your cards on the plates, or you can simply draw four simple scales, which in math is typically a horizontal line with an isosceles or equilateral triangle underneath it, positioned with a vertex touching the horizontal at the midpoint, i.e., like a balance scale!

As much as possible, try to set it up the way it appears at the bottom of the question that introduces it on this website. (In other words, four vertically arranged).  Finally (and this is not trivial), somehow label the top one A, the second one B, third, C, and the fourth?  I bet you can guess!

Then using your new toy, follow each step we recommend here.  It will work. And then, after a few more tries, it will REALLY WORK in the quiet but strong part of your mind.  Good luck.

Hope this helped.

Stay Balanced,

Happy Holidays,

Mitch