Begin here
This is the third part of a series about algebra. Part one, The Thing Called x, told where algebra comes from and what a letter is doing in a math problem. Part two, The Balance, taught two moves on a balance scale: restoring and balancing. This part has two halves. The first is a third move, sharing, for problems like 2x = 12. The second is the road from words to symbols, because on the test the problems arrive as words.
The scale is the same one, and it is meant to be touched. Under it, a line in plain words and a line in symbols say what the pans say, and both change as you work.
Tap Read at the top of any page to hear it read aloud, and Voice to choose the voice. On the scale pages, Hint says the question to ask, then the move by name, then does it in front of you.
The third move has a name too
Two moves came from al-Khwarizmi’s book: al-jabr, restoring, and al-muqabala, balancing. His book has a word for a third move as well. When a problem holds two squares, or three, he brings it back to one square, by sharing everything on both sides. The word is al-radd, returning: the problem is returned to its simplest form, one square.
Our version is smaller. When a problem holds two boxes, we bring it back to one box, by sharing everything on both sides. Here is the problem this part begins with.
Two boxes, the same thing twice, weigh 12. The 2 in 2x is a count, two of the thing: it is the hidden times sign from part one, 2 × x. There are no loose blocks to take away. To find what one box weighs, share.
Read the scale
Three scales. Each one says a sentence. Read the number in front of the variable as a count of boxes, and the loose blocks as blocks. Which sentence does each scale say?
Share it
Two boxes weigh 12, so one box weighs a fair share of 12: 12 shared into 2 equal parts, one part for each box. Tap Share, and the blocks go down to a tray, all of them in view, while the scale is held still. Tap the tray, or the pan, to deal one block into the next part, or tap Deal them all. When the parts are equal, keep one box and one part, on both sides at once, and the scale tells you what one box weighs.
The goal never changes: one box, by itself, on a level scale.
The problem:
Sharing 12 into 2 equal parts is written 12 ÷ 2 and read “twelve divided by two.” So the move, in the language of the test, is divide both sides by 2, and that is what the record writes: ÷ 2 under each side. The 2 you divide by is the 2 that counted the boxes. The check goes the other way: 2 × 6 = 12. Times is undone by divide, the way adding was undone by taking away.
The last block
Not every pile shares out evenly. Two boxes weigh 5. Deal, and one block is left in the tray: it cannot go to one box and not the other. Sharing still works. The last block is cut, and each part gets a half.
The problem:
Two and a half can be written three ways, and the test uses all three: as a mixed number, 2½; as a fraction, 5/2, five halves; as a decimal, 2.5. They are one number in three dresses.
Two moves, in order
Two boxes and 3 blocks weigh 11. Two moves are needed, and the order matters. Ask what is with the boxes that a single move can take away, and do that first. Then the boxes are alone on their pan, and sharing can begin. Try it the other way round if you like, and see what the blocks do.
The problem:
Which move?
Now nobody tells you which move it is. Read the problem, choose the move by name, and then make it on the scale. The question is always the same: what was done to the box, and what is the opposite?
The words for the signs
On the test, most problems arrive as words, and the words for the four signs are only a few. Here they are. Read them once, then practice: a phrase in words, and three sentences in symbols to choose from. The traps have names.
Less than, backwards
One kind of phrase runs backwards. “7 less than a number” starts from the number and takes 7 away, so it is x − 7, even though the 7 is said first. The same for “decreased by” and “the difference of.” They catch a lot of people, so this page is only about them.
Whole sentences
A whole sentence in words is a whole sentence in symbols: it has an equals sign in it, because it says that two things are the same. Read the sentence, choose the symbols, and then the scale appears with the problem loaded, for you to solve.
A question with a story
A test problem has a story and a question. The story hides an equation. Find it, solve it on the scale, and then answer the question in the words of the story, because the box counts something real.
The scale fades
Now the scale is gone, and only the sentence is left. The moves are the same, and they have their test names now: take away, give back, divide. Choose each move, and the record writes itself. If you need the scale back, tap Show me the scale.
You decide
Your turn. Three ways to work: with the scale, in your hands; with symbols only; or from words, where you find the symbols first. New problems as long as you want them. A star is for a first-try choice, or a solve with no tip of the scale and no wrong turn.
A look ahead
Here is a scale with a box on both pans. Two boxes and 3 blocks weigh one box and 7 blocks.
The moves you know are enough for it. A box can be taken from both sides, the way a block can, and then this is a problem you have solved before. Part four is about boxes on both sides, and about the fourth move, the opposite of sharing, for problems like “half of a number is 7.”