A graph with a key

I can build a graph one object to a box and read a row as a count.

On a one-to-one graph the length of a row is the count. Nothing else makes a row long, and that is why the graph can be read at a glance.

The child sorts the tub, lays one bear in each box of the right row, writes the number at the end of every row, and reads all six counts before meeting the book's printed numbers.

First lesson. Tip out a tub of coloured bears and ask which colour there are most of. Take the guesses, then sort them into rows.

The book says to sort a pile of coloured bears, lay one bear in each box of a labelled row and write the number at the end of every row — and then prints all six numbers.

Children sort their own tub by colour, lay exactly one bear in each box of its labelled row with no gaps, and write the count at the end of each of the six rows before anybody opens the book to the printed answers.

A six-row graph built from the child's own tub with all six counts written, then checked against the six the book prints.

Children lay the longest row and the shortest row alongside each other box for box, count the boxes that stick out past the short one, and say what that number is — a question the book asks and never answers.

I can build a chart from a list and check it back against the list.

A chart is a claim about a list, so it can be wrong. Checking it against the list is part of making it, and when two sources disagree the disagreement has to be reported rather than smoothed over.

The child builds the chart by the book's own method, reads the tallest column, and finds and states the count that the book's two pages disagree about.

Yesterday one bear made one box. Today one X makes one visit, and the list is long enough that somebody will lose count.

A child in the book teaches another how to make a bar chart in two sentences: the places go along the bottom, and an X goes above a place for every time it was visited.

Children write the places along the bottom, work down the list marking one X above the right place for every entry, then count each column and read the tallest.

A chart built from the list by the book's method, with every column counted and checked back against the list one entry at a time.

Two of the book's pages give the walking trail a different count. Children find both, say which one their own chart agrees with, and say what they would need in order to settle it.

I can draw a graph where one figure stands for ten, and write the key.

As soon as one mark stands for more than one thing, the graph needs a sentence saying how much. Without it the rows cannot be turned back into counts by anybody.

The child draws one figure for every ten votes across all four rows, writes the key underneath, and hands the graph to a pair who recover all four counts from the rows and the sentence alone.

The bear graph needed one box per bear. Ask what a graph would look like for forty votes, and let somebody start drawing forty of anything.

The book hands over one line of results — forty, thirty, twenty and ten votes across four columns — and one line telling how a pictograph shrinks.

Children write the four counts in order off the results line, count forty by tens raising a finger on each number word, and draw one stick figure for each finger raised, one figure per square, stopping when the fingers stop — then do the same for the other three rows, lining each row up under the one above.

A four-row pictograph at one figure to ten votes, with the whole graph's votes counted by tens and checked against one hundred.

Children write the key sentence at the bottom and read it to a pair who have not seen the graph, then let that pair use it to say how many votes each row holds — and if a wrong number comes back, fix the sentence rather than the answer.

I can survey my class and choose the largest key that leaves no half figures.

A key is not a preference. It has to divide every count in the data, so choosing one means testing candidates against all of them and keeping the largest that survives.

The child surveys the class, tests keys of one, two, five and ten against all four counts, keeps the largest that leaves no half figure, and says which count knocked the bigger ones out.

Yesterday every count was a multiple of ten. Ask whether a class of twenty-seven will be that obliging, and go and find out.

In the story the kids ask around, tally what they hear, and only then draw.

Children take the four columns to every person in the room marking exactly one tally each and saying the four choices the same way every time, count each tally group by fives, add the four numbers and check the total against the number of people asked — a total too big means somebody was marked twice.

Four counts from the class with the total reconciled, a key-testing card with the failed keys crossed out, and four rows drawn at the surviving key.

Children write their key sentence, hand the graph to another pair to read all four counts back, and when a number comes back wrong work out whether the row or the sentence is at fault.

I can turn rows back into counts and say both directions of a comparison.

A graph is worth having because it answers questions. Every answer starts by turning rows back into counts with the key, and a comparison is one fact that can be said two ways.

The child recovers all four counts using the key, writes both directions of two comparisons, adds the four counts to one hundred, and defends a front-page choice with two counts inside the sentence.

Two graphs have been built and read back by strangers. Today the reading back is yours, and it has arithmetic in it.

The results line sets four counts against each other and the characters start arguing at once: one column takes the front page, one is waved off with the word only.

Children use the key to turn each row of figures back into votes — counting the figures first and then counting by tens — and write the four numbers without looking back at the results line, then lay two rows directly under each other and count the figures that stick out past the shorter one.

Four counts recovered and checked, both directions of two comparisons written, and the four counts added to one hundred.

Children say which column they would print on the front page with at least two counts in the sentence, answer a partner who must also use counts, and finish by saying how many of the hundred did not pick their column.

I can say both things about a piece of a circle graph: what fraction it is and how many children that is.

On a circle graph a piece stands for a share of everybody. The yellow half is not one child and not ten pieces — it is ten children, and only the size of the class makes that true.

The child fills the circle's pieces with cubes to match the stated counts, writes the count in each piece, and says for each piece both what fraction of the circle it is and how many children that is.

Every graph so far has had rows. Show a circle cut into a half and two quarters and ask how many children the half stands for. Take the answers, then count the cubes.

The book states the class size, the count of bus riders and the fraction in three separate sentences and leaves the reader to put them together.

Children count twenty cubes into the paper circle and say what one cube stands for, move cubes into the yellow half until it holds the bus riders' count, fill the blue and green quarters from what is left, and write the count in each piece.

A quartered circle with all three pieces filled and labelled with counts, and both facts said aloud about every piece.

Children find one half and one fourth of three other class sizes and write both counts for each, then say which class size makes a quarter that cannot be shared out evenly.

I can answer questions about a finished graph and say whether my answer is a category, a count or a fraction.

A graph gets read for other people's questions, and the questions are not all of one kind. Some want a name, some want a number, and one may have no single answer at all.

The child answers all four questions naming the pieces compared, identifies the question with no winner, gives the fraction answer both ways, and sorts the four answers by kind.

Yesterday the pieces got counted. Today somebody else's four questions arrive, and one of them cannot be answered the way it is asked.

The book asks four questions about its own graph and answers none of them.

Children answer the three category questions from the graph and write beside each answer which pieces they compared, then find the one that asks which of two answers more children gave and write the honest answer, because on this graph neither did.

Four answers with the compared pieces named, the no-winner question identified, and the fraction answer given as a fraction and then as a count.

Children write beside each of their four answers whether it is a category, a count or a fraction, and say which question was the odd one out and why.

I can survey my class, build a circle graph, and give it a title and labels.

A graph is finished when somebody who was not in the room can read it. That takes a title saying what it is about and a label on every piece, and it takes honesty about a count that does not fit a quarter.

The child tallies a three-answer survey, checks the total against the class, colours the pieces, writes a title and a label on every piece, and has another pair read the graph back.

The circle graph has been counted and read. Today the class makes one of its own, and it has to survive being handed to strangers.

The book shows the whole procedure in six short sentences: ask a question, collect the answers, count the class, draw a circle, colour the pieces, and put on a title and labels.

Children agree one question with three possible answers, ask every child in the class marking a tally as each answer is given, add the three counts and compare the total with the number of children — finding what was missed before anything is drawn.

A coloured circle graph of the class with a title and a label on every piece, and a note on the back about any count that does not fit a half or a quarter.

Children hand the graph to another pair to read back to them, and where a count does not fit a half or a quarter they say so on the back rather than forcing the piece to fit.