Corners and degrees

I can test an angle against a right angle I folded and say which side of ninety it falls.

A right angle is not a matter of opinion. It is a corner a folded tester either fits or does not, and ninety degrees is its number.

The child folds a right-angle corner, tests every angle card against it, sorts them onto the two mats saying the card's number and then ninety, and states the sorting rule in a sentence containing ninety.

First lesson. Hold up two angle cards, one at 85 and one at 95 degrees, and ask which is the right angle. Take the votes, then fold the tester.

The story hands over four exact measures and one benchmark: a hallway that measured only 55, one that measured 120, the smallest tunnel angle at 40, and the right angle at 90 that a character keeps testing everything against.

Children fold a paper circle in half and in half again so the corner where the creases meet is a right angle, press it into a table corner and a book corner to feel where it fits exactly, then lay it into each angle card with one folded edge on one edge of the card and the points touching, and say whether the card opens narrower, the same, or wider.

Every card sorted onto the smaller-than and larger-than mats, with the card's number and then ninety said aloud before each card left a hand, and both mats counted.

Children state the one rule that sorted every card with the number ninety in the sentence, write it on the strip, circle the two words that decide the mat, and read it to another pair — fixing it if one of their cards breaks it.

I can test every corner of a figure I folded and say how many right angles it has.

A figure can be described by counting its right angles, and the count comes from testing every corner — including the one that looks least likely.

The child folds both ways, tests all the corners of each figure, records both counts, and says which corner of the triangle is the right angle and where it sits relative to the fold line.

Yesterday the tester sorted cards somebody else drew. Today it goes on paper you fold, where nobody has told you what the answer should be.

The book defines a right angle by its measure and then hands over a figure that is nothing but folded paper.

Children fold a scrap twice to build their own tester and keep only the desk and book corners it fits exactly, fold one square in half edge to edge and test all four corners of the shape in their hand, then fold a fresh square corner to corner and test all three corners of the triangle, checking the corner opposite the fold line last.

Two folds made and both figures entered on the chart with their shape names and their right-angle counts, each count said aloud before it was written.

Children put the rectangle, the right triangle and the untouched square side by side, test the untouched square and add it to the chart, then sort all three from most right angles to fewest and say which two tie and how the tester proved it.

I can measure an angle in whole degrees and say which move went wrong when two measures disagree.

Measuring is a procedure with three moves in a fixed order. A wrong measure is almost always one of the three slipping, and naming which one is how it gets fixed.

The child numbers the scale so that ninety lands a quarter of the way round and one hundred eighty opposite zero, measures four openings, and where a partner's number differs by more than five degrees names the move that slipped before changing anything.

The tester has answered every question so far with yes or no. Ask how much wider than ninety the 120 card is, and let the class find that the tester cannot say.

A character carries a circle with numbers around its edge and uses it exactly the way a protractor is used: centre on the vertex, zero on one ray, read where the second ray crosses.

Children write zero on the tick at the right edge of a blank circular scale and number their way round in tens, checking with a partner that ninety landed a quarter of the way round and one hundred eighty exactly opposite zero, then set the centre on the vertex, turn the scale until zero lies along the lower ray, and follow the other ray to the edge — saying centre, zero, read while their hands do it.

Four openings measured in whole degrees and recorded, with a pencil laid along any ray too short to reach the edge and no measure written that could not be lined up.

Children trade strips and remeasure a partner's four diagrams; wherever two numbers differ by more than five degrees both scales go on that diagram at once and the child says whether the centre slipped or zero slipped before either number is changed.

I can join two right angles, measure what I made, and say why it has to be one hundred eighty degrees.

Angles add. Two ninety-degree corners joined at a point make one hundred and eighty degrees, and that can be argued from the corners themselves once the number is known.

The child joins two right angles point to point, traces and measures the opening, then argues the result with the scale covered and answers a partner's question with a number in the answer.

Two lessons have measured single angles. Put two right-angle corners together on the desk and ask what has been made, before anybody measures it.

A character calls out one instruction in the riding ring — double the right angle to make a straight angle — and the book closes by giving both numbers, ninety and one hundred eighty.

Children set two right-angle corners so their points touch at one dot with one straight edge of each lying along the other, trace right around the outside opening, then measure it with the centre of the scale on the dot and write that measure above the two they started with.

The traced straight angle measured and all three numbers written as one sentence: ninety and ninety make one hundred eighty.

With the scale covered, children say why two right angles joined this way have to make a straight angle using only the corners in their hands, answer one question from a partner with a number in the answer, then fold the traced angle in half along the dot and measure one half to test whether every half of a straight angle is a right angle.

I can measure the angles that meet at a point, add them, and say whether the point closes.

Whether a point closes is decided by a total, not by how the arrangement looks. Three hundred and sixty degrees is what all the way round comes to.

The child measures every angle touching the dot, adds the measures, says what the total came to, and locates the gap on a prepared arrangement that falls short.

Yesterday two angles were added to make a straight angle. Ask what would happen if pieces were laid all the way round a point, and mark a dot.

A character states the rule in one sentence and the book then has the others use it: the internal angles of every shape meeting at a point add up to three hundred and sixty degrees.

Children measure every corner of a paper triangle and a paper rectangle and write each measure on the piece, checking the rectangle's four corners against each other, then mark a dot and lay pieces round it so their corners meet with no overlap, measure every angle touching the dot and add the measures.

One built point with every touching angle measured and the total written, and the book's own name for a closed point written beside it.

Children take the prepared arrangement card, measure the angles at its dot and add them, say how far the total is from a closed point, and put a finger on where the gap is.

I can find the missing angle at a point and cut a piece to that measure.

The missing angle is arithmetic: add what is there, take it from three hundred and sixty, and the difference is the measure of the piece still needed. The light behind the card is the check.

The child totals and subtracts for five prepared points, cuts a wedge to one of the differences, measures the wedge before placing it, and holds the card to the light.

Yesterday a point came out forty degrees short and somebody put a finger on the gap. Today the gap gets a number and then gets cut.

Once a closed point measures three hundred and sixty degrees, the arithmetic follows: add what is there, subtract from three hundred and sixty, and the difference is the piece still needed.

Children measure the angles at each of five prepared points and write all five totals before subtracting anything, then write three hundred and sixty at the head of the second column, subtract each total, and say what the number in the third column is the measure of.

Five missing angles found by subtraction, and one paper wedge cut to one of those measures and measured before it was placed.

Children lay the wedge into the gap and hold the card up to the light; if light shows they measure the wedge and the gap again and say which of the two was wrong.

I can count the right angles on a folded figure and record where I found each one.

A count of a physical figure is only as good as the account of where the things counted were. Two honest counts differ because they searched different places, and that is settled by testing a spot rather than recounting.

The child searches the outside edges and the places where folded paper meets, writes a note beside every right angle saying where it was, and settles a disagreement by testing the disputed spot.

The tester has been used on cards, on folds and on a scale. Hand out the finished frogs and ask how many right angles are on one.

The book asks how many right angles a half-built frog shows, answers ten, and then tells the reader to count again on the finished frog — warning them to include the edges as well as the places where folded paper meets.

Children write the book's own count at the top of the sheet labelled as the book's, then test the finished frog recording every right angle with a short note saying where it was, working round the outside edges first and then the places where folded paper meets, and writing the total only once every line is filled.

A recording sheet with a note beside every right angle found, and a total written last.

Children trade sheets with a partner whose total differs, find the first row where the sheets disagree, put the frog between them with a finger on the spot, decide together whether the tester fits there, and say which of them had searched a place the other had not.

I can find a total from two facts the text gives and say which sentence gave me each one.

A book can say enough without saying everything. Two stated facts — how many rectangles and how many right angles each carries — are enough for a total, and each factor has a sentence behind it.

The child folds and verifies the four rectangles, reaches the total by multiplying rather than counting singly, and names the sentence that supplied each factor before writing the product.

Yesterday a count had to be searched for. Today the count is not in the book at all, but two of its sentences are enough to produce it.

Three sentences about a tower section say how many pieces it takes, how many become rectangles and how many become triangles, and the total number of right angles is never printed.

Children underline every number in the three sentences and write how many pieces a section takes, how many become rectangles and how many become triangles, then fold four squares in half and test one rectangle's corners with the tester, checking the count of four against the book's sentence before continuing.

Four counters on each rectangle, the total reached without counting singly, the multiplication written beside it, and the sentence that gave each factor said aloud.

Pairs state in one sentence why the book could leave the total out and still have said enough, pointing at the two sentences as they speak.