Finding an unknown

I can total both sides of a sentence and say whether it balances.

The equal sign says the two sides name the same amount. It does not say that a single number follows, which is why a true sentence can have an operation on both sides.

The child builds both sides of each sentence, totals each side without touching the other, rules it balanced or not, and for a broken one says which side is heavier and by how many.

First lesson. Write four plus one equals three plus two on the board and ask whether it is allowed. Take the answers, then hand out the mats.

The book opens by making the equal sign a physical claim rather than an instruction: a seesaw with a pumpkin on it, and then a run of sentences whose two sides are each computed and compared.

Children put four counters on the left of the mat and one beyond the plus, then three and two on the far side of the equal sign, count each side aloud and say the total — then build five minus one equals three plus one, starting with five and taking one away on the left, and saying both totals before ruling anything.

All four of the book's printed sentences built and ruled, including the two with an operation on both sides, with both totals said aloud before each ruling.

Children sort a deck of sentence cards into a balanced pile and a not balanced pile, say for every broken card which side is heavier and by how many counters, and then say what they would add to one side to make it balance.

I can say what the letter stands for and write the sentence without solving it.

Writing the sentence is a separate job from solving it. It turns on deciding which number in the story is the amount that left and which is the amount still there.

The child names what the letter stands for, sorts the story's two numbers into the one beside the letter and the one after the equal sign, writes both sentences without solving either, and says why one takes a minus and the other a plus.

Yesterday every sentence was already written. Today the sentence has to be made, and nobody is allowed to solve anything.

Between the story and the arithmetic the book performs a translation it names out loud: it says what the letter stands for, then writes the sentence.

Children read the raven story and finish the line R is the number of, write it on a card and check it against the book's own line, then say which of the story's two numbers is the amount that left and which is the amount still there and put each on a card.

Both sentences written from the stories with nothing pre-printed and neither one solved, read back to a partner with an account of where the numbers sit.

Children say why the ravens got a minus and the bats got a plus using what happened in each story, then take the kitten story and say what its letter would stand for without writing the sentence.

I can undo an operation on both sides to find the unknown.

The unknown is found by stripping the known numbers off its side — adding back what was subtracted, taking off what was added — and whatever is done to one side has to be done to the other or the claim stops being true.

The child undoes both of the book's additive cases with counters, checks each answer by rebuilding the original sentence, and can show what goes wrong when only one side is touched.

The sentences are written and unsolved. Ask how to get the letter by itself, and take the suggestions before touching the counters.

The book works four haunted-house problems and each one runs the same procedure: name the unknown with a letter, write the sentence, then do one thing to both sides to strip the number away from the letter.

Children put the letter card face down on the left of the mat and build the sentence round it, then — six ravens having flown off — put six counters back on the left and six on the right as well, count the right side and say that number, then undo the bats' plus twelve by taking twelve off each side.

Both additive problems solved on the mat and each answer checked by putting the numbers back and watching the original total return.

A partner deliberately works one side only and leaves the other alone; children say what goes wrong and show it on the mat, using the words both sides and still balanced.

I can build an array, write its multiplication, and predict the next case before I draw it.

Seven lines each way do not make seven rows. The count comes from the bands the lines cut, and the drawing, the labelled rows and the equation are three views of one arrangement.

The child draws seven lines each way, numbers the bands, writes eight times eight equals sixty-four, checks by counting in eights, and predicts the six-line case with a reason before drawing it.

The unknowns so far have had letters. Read the two sentences about how the board was drawn and ask how many squares that makes — the book never says.

Jack draws seven lines one way and seven lines the other way and paints the squares, and the book never says how many squares that makes.

Children rule seven evenly spaced lines across a sheet and seven more square to them, number the bands down the left edge and across the top, and say how many rows and columns the seven-and-seven lines made — anyone who wrote seven counts again with a finger in each band.

The grid drawn and numbered with eight times eight equals sixty-four written underneath and checked by counting the shaded squares in eights.

Before drawing anything, children write down how many squares six lines each way would make and why, then draw it in a fresh corner and count — saying whether the prediction held and what the count of lines has to do with the count of bands.

I can find a length the book never gives and name the sentences that make it certain.

A text can fix a number without stating it. Two sentences are enough — the total and the fact that the parts match — and the rest of the numbers on the page have nothing to do with it.

The child sorts the sentence cards into those that pin a number down and those that do not, finds the side, checks it by adding back four times, and names the two sentences that made the answer certain.

Yesterday a number came out of a structure. Today it comes out of two sentences, and half the sentences on the card are no help at all.

The book tells the reader the whole way round a softball diamond and that its four sides match, then asks how far apart to space the bases and never divides.

Children read the sentences off the cards, put every number they find on the diamond outline where it belongs, discover the question about the bases has no number to answer it and mark where the missing number goes, then sort the cards into a pile that pins a number down and a pile that does not.

A 240-strip cut into four equal pieces with one counted, the division written, the sixty added back four times and all four sides of the outline labelled.

Children write one sentence naming the two cards that made sixty certain, then try a fifth number on the outline and say what goes wrong; a partner checks that the sentence names cards rather than repeating the answer.

I can produce a measure the book never states and hand it in with the sentence that makes it the only answer.

A sentence can name one measure of a thing and not another. Reading it carefully enough to say which is which is what makes the missing one findable at all.

The child marks what the sentence fixes and what it does not, cuts the card to those dimensions, covers and counts to twenty-four by rows, and hands in the number with the sentence and the two deciding words underlined.

Sentences decided the answer yesterday. Today one sentence gives two measures and withholds a third, and the question is which is which.

The hall-pass sentence gives the card's two dimensions and never gives its area, and the number twenty-four appears nowhere in the book.

Pairs read the sentence aloud twice, underline every number in it and write beside each what it measures, then write underneath the one measure of the card the sentence names and the one it does not — touching no card yet — before cutting a card to the two dimensions they underlined.

The card covered corner to corner in one-inch squares with nothing overhanging, counted aloud by rows, and the count written with its unit and the matching multiplication underneath.

Children put the number with its unit at the top of a sheet and copy the book's sentence beneath it, underline the two words that gave the size of the card, and write one sentence saying why no other card could have been meant — then say to another pair which number the book printed and which they produced, both with their units.

I can find how many equal jobs close a gap, and say why a part one still counts as a whole one.

Some unknowns are reached by running the arithmetic backwards. The quotient is not always the answer: three and a half rooms is what the numbers say and four rooms is what gets cleaned.

The child computes the shortfall and labels it, takes out the raking, counts the rooms in fifty-cent jumps, and writes one sentence saying why a leftover of a few cents still costs a whole room.

Every unknown so far has come out exactly. Today the number lands between two answers and somebody has to choose.

The total is fixed by two book prices and the money in a pocket, and the number of fifty-cent jobs needed to close the gap is never worked out.

Children add the two book prices and subtract the money in hand, labelling what is left as short by so it is not read as money held, then lay two three-dollar cards for the promised raking, subtract that total, and jump up the number line in fifty-cent steps from zero, tallying one room per jump until they reach or pass what is still owed.

The shortfall worked in three labelled stages and a count of rooms reached by jumps rather than by division, with the jumps still visible.

The jumps have probably landed past what is owed rather than exactly on it. Children say aloud what the leftover means — whether part of a room can be cleaned — and write the number of rooms with one sentence saying why they rounded as they did.

I can recover a number a table leaves out and show the sentences that force it.

A rule plus a starting amount fixes every row, including the ones nobody printed. An answer is finished when the arithmetic and the sentences that force it arrive together.

The child states the rule in their own words, recovers day five, checks the five-day total against the figure in the text, finds the first day the pay passes a dollar, and hands in a card carrying both numbers, the arithmetic and the underlined sentences.

The unit has found lengths, areas and counts. Today it finds a row of a table, and the card will come back if the number arrives alone.

The offer sentence states a rule and four days of pay, and the book never prints day five or the day the wage first passes a dollar.

Partners read the offer sentence aloud twice and each says the rule back in their own words while the other checks it against the sentence, write the four given days into the recovery table, work out day five from day four using the rule, and add the five days in the running-total column to check against the figure the text gives.

Day five and the dollar crossing both recovered, with only the sentences that force the answer underlined — not every sentence that has a number in it.

Children keep doubling and comparing each day's pay against one hundred cents until one is bigger, write that day in the crossing box, and fill in the evidence card: both numbers, the arithmetic under each, and the underlined sentences copied below. A card with numbers and no sentences goes back, and so does one with sentences and no numbers.