Flips, turns and matching halves

I can fold a figure and say whether the halves matched everywhere.

A figure is symmetrical when a fold closes it with nothing sticking out. The answer comes from the fold, and the crease is the record of it.

The child folds all twelve figures, sorts them by overhang rather than by appearance, writes both counts, and draws the crease on every figure that passed.

First lesson. Hold up two figures, one symmetrical and one not, and ask the class which is which. Take the answers, then say that nobody has tested anything yet and hand out the paper.

The book gives the test in one sentence — fold the shape exactly in half — and then says what to look at once it is folded: two halves that are mirror images.

Children fold each figure so the two halves land on each other, hold the folded pair up to the window and look along every edge, then sort the whole tray of twelve onto a matched mat and a not-matched mat with both counts written at the top.

Twelve figures sorted by the fold, both counts written, and one line drawn along the crease of every figure that passed.

Children trace one hand, cut it out and try to fold it into matching halves, then set it beside a tracing of the other hand and say in one sentence what is true of the pair that was not true of the single hand.

I can say how many lines of symmetry a figure has and show a creased copy for each one.

How many is a claim, and a claim needs evidence. A figure with several lines means several creased copies on the desk, each one refoldable by somebody else.

The child's count for each figure equals the number of creased copies they can lay out, and they lower the count when a partner's check finds a copy that does not close.

Yesterday every figure that passed got one line drawn on it. Ask whether one was all there was, and let somebody find a second fold on a figure from the matched mat.

The book sorts figures three ways by a count: no lines, one line, several lines.

Children take a fresh copy of a figure for every line they want to test, crease each copy along a different line, keep the ones that closed with no overhang and set the failures aside, then count the copies that closed.

Each figure in the tray its count earns — none, one or several — with the creased copies laid out in a row as the proof of that count.

Children lay one figure's creases in front of a partner, say the count and the tray it earned, and let the partner refold a copy chosen at random; a copy that fails drops the count by one and may move the figure.

I can fold corner to corner, and I can change my own answer and say what changed it.

A no that was reached without folding on the diagonal is not a finished answer. Withdrawing it is what the extra test is for.

The child writes yes over their own no on at least one figure and says which fold changed it and what they had not done when they first answered.

Both lessons so far folded up and down and across. Ask whether those are the only two ways to fold a rectangle of paper, and let somebody fold a corner over.

The children in the story declare the monster not symmetrical, and much later one of them takes it back: it is symmetrical along its diagonal.

Children fold each of eight figures top to bottom and left to right and write no on any figure where neither fold matched, then take that no pile and fold every figure in it corner to corner both ways, writing yes over their own no each time the halves close.

Eight figures folded three ways, with every changed answer written over the first one so both are still readable.

Children pick one figure that changed and say a sentence naming the answer they gave first, the fold that changed it, and what they had not yet done when they answered — then go back and fold the original yes figures corner to corner too, writing how many of the three folds matched.

I can turn a figure one full way round and count the times it looks the same.

A figure can look the same at several points in one turn. That count is a property of the figure, and it is read off a fixed arrow rather than guessed.

The child pins each figure, turns it one full way round against the arrow, and gets the same count twice — and checks the flower against the number the book states.

Everything so far has been decided by a crease. Put a pinwheel on a pin, spin it, and ask what test would catch what just happened.

The book turns from folding to spinning and hands over two counted cases: a flower with three matches in a rotation and a pinwheel with four.

Children pin the letter Z through the dot at its centre and turn it slowly while a partner watches the fixed arrow, calling stop each time it looks exactly as it did at the start, then do the same with the three-petal flower and the four-bladed pinwheel, writing each count.

Three figures pinned and turned, two counts written and the flower's count checked against the number the book gives.

Children write their two counts side by side, read the book's name for what they have been testing, and say one sentence for each figure that uses that name with the number they counted in it.

I can sort figures by which test they pass, and I name the test before I name the answer.

The fold test and the turn test are different questions. A figure can pass either one alone, so an answer without its test attached does not say anything.

The child sorts four figures into passed both, only the fold, only the turn and neither, and hands a partner a figure naming the test first and the answer second, with the evidence for each.

Two tests are now on the table. Ask what a figure could do that would make the two tests disagree, and keep the answers up where they can be checked against the sort.

The book's closing section says outright that this is a different kind of symmetry and gives its test: the figure looks the same after it has been turned.

Children push a split pin through the centre of a card figure, trace all the way round it, turn the figure on the pin without lifting it and check every edge against the tracing, then run that test on all four figures and add what they already know about each figure's folds.

Four figures in four places on the desk — passed both, only the fold, only the turn, passed neither — with the tracing kept for each turn result.

Children take a figure from a pile that passed only one test, say which test it passed and which it failed in that order, and show the evidence for both: the tracing for the turn, the crease for the fold. The partner spins it on the pin before agreeing.

I can say a figure is asymmetrical and put my finger on the part that breaks the match.

Most things that were not drawn to be symmetrical fail the test a little. Deciding what close is going to mean, and naming what broke the match, is the child's job rather than the figure's.

The group says what close means before it sorts, both counts add to ten, and three failures are described in sentences a partner can act on without being pointed at.

Every figure so far was drawn on purpose. Hand out photographs of animals and ask whether the test still works on something nobody designed.

Having taught two tests, the book is careful to say that living things mostly fail them a little, and gives a case that fails outright: the fiddler crab with one oversized claw.

Children agree out loud what close is going to mean for their group, fold each of the ten animal cards down its middle and watch the edges, sort them onto a close mat and a not-close mat, then find the fiddler crab and put a finger on the part that stops the halves matching.

Ten cards folded and sorted, both counts written and checked against ten, and the crab's claw named with what is different about the two of them.

Children take three cards from the not-close mat and for each one name the part that broke the match in a sentence that would let a partner find the same part without any pointing.

I can test the story's claim about a name and say where I stand with the number in my sentence.

A claim in a book is something to test, not something to accept. Two checks have to agree — every letter has to close, and the row has to read the same reflected — and a position is only worth stating with the count attached.

The child builds the names, folds every letter, counts the failures, stands the mirror on each row, takes a mat, and states a position containing the number of letters that failed.

The unit has tested figures, letters and animals. Read the story's two sentences about its own names and ask whether anyone has checked them.

The story claims two of its names are symmetrical in capital letters, and a third character shortens her name until it is.

Each pair spells the story's names in capital letter cards one name per row, folds every card down the middle and turns face down any letter whose halves do not close, counts the face-down cards in each row, then stands a mirror upright on the middle of each name and says whether the two checks agree.

Every name built and folded, the failures counted per row, and each child standing at one of two mats with a position said aloud that has the count inside it.

Each child restates the other mat's reason in their own words before answering it, then names one thing they could find on the cards that would send them across; a partner from the other mat hands them a row to check, and anyone who moves says which card or which reversal did it.