All the way round, and what it covers

I can go all the way round a figure and say every side once.

A perimeter is a trip, not a formula. The figure decides how many numbers get added, because every side gets said once and the trip ends where it started.

The child walks both coops saying each side aloud, writes exactly as many numbers as the coop has sides, and can state the rule with the words around and every side in it.

First lesson. Tape a coop on the floor before the class arrives and ask how much fence it would take. Take the guesses, then walk it.

The book gives the rule twice at the back — the distance around a figure is called its perimeter, and you add the lengths of its sides to find it — and then acts it out four times with one roll of fence.

Children stand at one corner of the taped coop, walk a finger all the way round the tape and stop only on returning to the same corner, saying the length of each side aloud as they pass it while a partner taps the floor once for every side named — then write the four numbers in the order the finger passed them and read the strip as an adding sentence.

Two coops walked, the taped one and one built from a chain of twenty-four links, with four numbers and a total written for each.

Children tell a partner what they did both times using the words around and every side, then say the rule as an instruction another pair could carry out with a chain and a grid — and name the part of their rule that stops a side being counted twice.

I can add three, four or five side lengths and answer a claim about a figure with the sentence that settles it.

The number of sides is read off the figure before any adding starts. A claim about a total is answered by the adding sentence, not by remembering what the story said.

The child counts the sides of each figure before adding, writes the full adding sentence for all six rows, and gives a yes or no with the sentence as the reason.

Yesterday two coops both came to twenty-four. Show the six-figure chart and read the claim that every one of them does, then hand out the cards.

The chart at the back prints six figures as bare rows of side lengths, a claim that each has a perimeter of twenty-four feet, and two questions: is he right, and how do you know.

Children read the numbers on each figure card aloud and write how many sides that figure has before adding anything, then add each row and write the whole adding sentence in the book's own words, with plus between the numbers and equals before the total — all six rows, including the one with five numbers.

Six adding sentences with their totals, and a yes or no for each figure with the sentence pointed at as the reason.

Children add the five-number row again in a different order, say the new order aloud, say whether the total changed, and say which two numbers they put together to reach ten fastest.

I can build several different coops from one chain and say what changed and what did not.

A total round the outside says nothing about the shape. Two coops can spend exactly the same fence and not be the same coop, and only the list of sides tells them apart.

The child builds all four coops from a single chain, writes the adding sentence for each, and puts a number in every sentence about what changed.

Six figures all came to twenty-four yesterday. Ask whether they were all the same shape, and hand each pair one chain to settle it.

One roll of fence is carried to four places and laid out a different way in each, and the book prints the total after every laying: twenty-four feet, four times over.

Children lay the chain of twenty-four links as a coop nine by three and say the four numbers, then — without adding or removing a link — lay the same chain as eight by four, say the four numbers again, and say how many links were used both times.

Four coops built from one chain with an adding sentence recorded for each, including the three-sided coop whose sentence holds only three numbers.

The grandfather is told the coop is twenty-four feet round and nothing else. Children say aloud the two things he still cannot know and what they would have to tell him to let him build the right coop, using the numbers off their own record sheet.

I can make a measuring strip of equal lengths and measure with it.

A count of lengths means nothing unless every length is the same. The unit is a choice, and checking it against itself is part of measuring.

The child folds and marks eight equal lengths, checks two of them against each other before measuring, measures every side twice, and writes between 2 and 3 for a side that ends between knots.

The chain gave every coop its unit for free. Take the chain away and ask how a figure could be measured with a plain strip of paper.

Nef will not say how long to make each side, so Pythagoras finds an old piece of rope and ties knots in it, and the triangle that finally works has three lengths on one side, four on another and five on the longest.

Children fold a paper strip in half three times, mark every fold, and check two lengths by folding one onto the next — a strip whose lengths do not match is no good for measuring — then lay it along each side of the large triangle with a knot mark exactly at the corner and count whole lengths, measuring every side twice.

Three side counts for the triangle, recorded and with the largest circled, and every pair's three numbers the same if every strip was knotted the same way.

Children measure the rectangle with the same strip and record the counts, saying out loud when a side ends between two knots and writing between 2 and 3 rather than rounding it quietly to the nearer knot.

I can measure all four sides before I say what a figure is.

A figure's name is decided by measuring, not by how it looks from across the room. All four sides have to match, and a figure whose sides have not all been measured cannot be filed at all.

The child measures all four sides of every figure, writes each measure on the side it belongs to, and identifies the near-square by naming the two measures that gave it away.

Measuring has been trusted twice now. Hold up a near-square and ask the class what it is. Take the votes, then hand out the rulers.

A child in the book stops the others from calling something a square and states the rule they have to test it against: all the sides must be exactly the same length.

Children measure every side of each cardboard figure in whole inches and write each measure on the side they measured — all four even when the first two match — then file each figure on the square mat or the not-a-square mat, writing on the mat the measures that decided it.

Six figures measured all round with both mats filled, and the near-square that differs by one inch set beside a real square.

Children use the corner tester on all four corners of one figure from each mat and say what was the same about the corners of both figures even though one is a square and one is not.

I can build a square from rows of tiles and count the tiles by rows in both directions.

A covering is counted by rows rather than one tile at a time. Counting the same square both ways is what catches a row that came out the wrong length.

The child builds all three squares with every row the same length, counts each one across and down, and records each square as a side length, as rows with tiles in each, and as a total.

Everything so far has been measured round the outside. Put a square of tiles on the desk and ask what question the outside trip does not answer.

The book gives the exact tile counts from the courtyard: a square with three tiles on each side holds nine, one with four on each side holds sixteen, one with five on each side holds twenty-five.

Children lay a row of three tiles along the marked edge of the mat and add rows of the same length until the shape is as tall as it is wide, counting by rows with a touch per number, then build the four-square and the five-square the same way.

Three squares built and each recorded three ways — the tiles on one side, the rows and what each row holds, and the total.

Children count each square twice, once by the rows going across and once by the rows going down, saying both counts aloud — and when two counts of one square disagree, find the row that is the wrong length and fix it before recording anything.

I can add two counts and check the total against a separate count.

A check is a second route to the same number. Saying two totals agree is not the check; naming all three numbers and where each came from is.

The child adds the two smaller counts without looking at the third, counts the big square again separately, and decides whether the two numbers agree while naming nine, sixteen and twenty-five.

Yesterday every square got counted twice and the two counts had to agree. Today two different squares get added and checked against a third.

In the courtyard Pythagoras counts three arrangements and says what he found: the nine tiles in the red square plus the sixteen in the blue square equal twenty-five, and there are exactly twenty-five tiles in the big square.

Children rebuild the three squares, count each one by rows, then add the two smaller counts with the big square covered and write the sum before uncovering it and counting the big square again from the beginning.

Three counts and one sum, with the decision written down about whether the sum and the separate count agree.

Children say the check out loud with all three numbers in it and say where each number came from; agreeing without naming the three numbers does not count as having made the check.

I can cover a square with a smaller square and count the layings.

A covering can be counted in any unit that tiles the shape with no gaps and no overlaps. The count changes when the unit changes, and the shape does not.

The child cuts both squares to the stated sizes, measures the uncovered strip, counts how many layings of the small square cover the large one, and orders three bases while naming each gap.

Tiles have been one inch every time. Cut a square two and a half inches on a side and ask how many of those it takes to cover a five-inch square.

Two of the book's models have stated base squares: at least five inches on each side for the great pyramid, about two and a half inches on each side for the Nubian one.

Children measure and cut a five-inch square and a two-and-a-half-inch square, writing the side length on each as it is cut, then set the small square into one corner of the large one with two sides lined up exactly and measure the piece of the big side left uncovered.

The uncovered strip measured and named, the covering count reached by sliding the small square round, and the big square marked into that many parts.

Children add the step pyramid's four-inch square, put all three in order from widest to narrowest, and say each gap out loud — five to four is one inch, four to two and a half is one and a half inches.