Begin here
This is the second part of a series about algebra. The first part, The Thing Called x, told where algebra comes from and what a letter is doing in a math problem. This part is about solving: finding the number in the box.
You will do it on a balance scale, with your own hands. Blocks go on and come off the pans. The scale tips or stays level. Under it, a sentence in symbols says what the pans say, and it changes as you work. By the end, the scale fades and the sentence is all you need.
Tap Read at the top of any page to hear it read aloud, and Voice to choose the voice.
The two moves have names
In part one you met two words from al-Khwarizmi’s book, written in Baghdad around the year 820. Al-jabr means to restore: to give back what was taken, so that a thing is whole again. Al-muqabala means to balance: to take the same amount from both sides, so that the two sides stay even and the problem gets simpler.
Restore and balance. In the beginning, that was all of algebra. In this part you will make both moves yourself.
There is no scale in al-Khwarizmi’s book. The scale is ours, a way to see what his words mean. But it fits them well. Here is one, holding a problem.
The box is the thing, x, with an unknown number inside. Each block weighs 1. The scale is level, so the box and 3 blocks weigh the same as 10 blocks. That is exactly what the sentence under the scale says: x + 3 = 10.
Read the scale
The sentence under a scale is a description of the pans. Before you move anything, make sure you can read one. Which sentence does each scale say?
Build it
Before you can solve a problem on the scale, you have to load it. The scale is locked while you load it, so it will not tip.
Build this one: . The box is already on the left. Tap the bin under a pan to give that pan a block. Tap a pan to take one back.
The balancing move
Now solve it. To solve is to get the box by itself, so that the scale tells you what it weighs. The box has 3 blocks with it. The opposite of adding 3 is taking 3 away.
Tap a pan to take one block off it, or drag a block off. Watch the scale, and watch the sentence. When the box stands alone on a level scale, tap the box to open it.
The record under the scale is your work, written the way it is written on paper. You did not write it; your hands did.
Both sides at once
You felt what happens when one side changes and the other does not. The rule is simple to say: whatever you do to one side, do to the other. So the move is really one move, made on both sides.
This problem needs more blocks taken away. Do it one pan at a time if you like, or use the new buttons under the scale, which take one block from both sides at once. The scale never tips.
The restoring move
This box is not whole. It came with 3 blocks missing, the dashed ones: the box, minus 3, weighs 7. To solve, make the box whole again. The opposite of taking 3 away is giving 3 back.
Give the box its 3 blocks, from the bin. To keep the scale level, give the right pan 3 blocks too. This is al-jabr: restoring.
Which move?
Now nobody tells you which move it is. Read the scale, decide, and then do it. The question to ask is always the same: what was done to the box, and what is the opposite?
From words to the scale
On the test, most problems arrive as words. The scale can hold words too. Read the sentence, build it on the scale, then solve it. The number you do not know is the box.
Turning words into symbols is a skill of its own. Part three is about it.
The scale fades
Now the scale is gone, and only the sentence is left. Nothing else has changed. The moves are the same moves, and the record is the same record. If you need the scale back, tap Show me the scale.
You decide
Your turn. Choose how to work: with the scale, in your hands, or with symbols only. New problems as long as you want them. A star is for a problem solved with no tip of the scale and no wrong turn.
A look ahead
Here is a problem this part did not solve. Two boxes, the same thing twice, weigh 10. The sentence says 2x = 10.
Taking blocks away will not help here; there are no blocks with the boxes. The move is a third one: sharing. Both pans are shared into two equal parts, and each box turns out to weigh 5. Part three is about that move, and about turning words into symbols.
Restore, balance, share. Those three moves solve most of the equations on the test.