A rule that makes the next number

I can count a row of tens and say where the count stops.

A count by tens needs the ten to be a thing you can touch. Nine bundles reach ninety and the tenth is what reaches a hundred.

The child counts the row from either end and lands on the same last number both times.

First lesson. Ask the class to count a hundred sticks by ones and stop them. Ask what could be done to the sticks to make the count shorter.

Agnes asks every scout for ten twigs and counts the returning scouts by tens. Nine scouts get her to ninety and she stops.

Pairs bundle ten craft sticks with an elastic saying ten twigs, one scout as each is tied, build nine bundles, then touch each bundle and count aloud by tens with the book's printed line, stopping where Agnes stops.

Ten bundles counted to a hundred, and a written record of where the count stopped at ninety and how many scouts were still to come.

One partner lays out between four and ten bundles while the other looks away; the counting partner turns back, counts the row by tens and says the last number, and they swap until each has counted four rows.

I can count the same set by twos and by fives and say which took fewer words.

The size of the step decides how many number words a count costs. The set does not change; the counting does.

The child counts the same ten plates both ways and says which count took fewer number words, using the written numbers as the reason.

Yesterday the step was ten. Today the same ten groups get counted with a smaller step, and then a middle-sized one.

The book runs the same count twice with different group sizes: two cheesy puffs per scout and five pegs per tent.

Children set nine plates with two counters on each and count by twos touching one plate per number word, add the tenth plate to reach twenty, then clear the plates and count nine plates of five by fives with the printed line.

Three written numbers — where the twos stopped, where they finished, and where the fives stopped.

Children count the same ten plates once by twos and once by fives and say how many number words each count took, using the numbers on their paper to say which was quicker.

I can count by threes along the yards and say the feet.

A step can carry a unit. Each new yard adds three feet, so the count by threes is not an exercise — it is a conversion the book uses on the next page.

The child answers any yard from one to six from a called card before walking it, and only walks to check.

Counting by twos and fives moved along plates. Today the step moves along the floor, and each step is worth three feet.

The book hands over its own procedure in the back matter and then names the step: add three feet for each new yard. Its rows run one yard to six.

Children lay six one-yard ribbons end to end with no gaps, walk the line saying three, six, nine, twelve, fifteen, eighteen with one touch per number word, and start again at the first ribbon if the voice gets ahead of the foot.

Six ribbons walked in both directions with the matching number card laid on each, and the sixth ribbon's number checked against the book's own sentence.

A partner draws a yard card face down and calls it; the child says the feet before moving, then walks the ribbons to check, and any card that had to be walked to answer goes back in the pile.

I can make the next number by pushing together the two before it.

Not every rule is a fixed step. This one adds the two numbers standing next to each other, so the step grows as the row grows.

The child rebuilds a covered count by pushing the two piles immediately to its left, and says the two numbers used before saying the answer.

Three lessons of fixed steps. Ask what the step is in this row — one, one, two, three, five, eight — and let the class find that there is not one.

Eli writes down eight exhibits and then states, in his own words, the rule that made them: take any two numbers in order and add them, and you get the next one.

Pairs build the first six exhibits as piles of counters in Eli's order, then one partner covers a pile and the other pushes the two piles immediately to its left together, counts the result and says what the covered pile must be before the card is lifted.

Four rebuilt exhibits and four adding sentences written under the strip, ticked where they match Eli's printed ones.

The strip goes face down and each child states the rule cold in their own words; a partner names two numbers standing next to each other and asks what comes after, and the speaker must answer and say which two they added.

I can say the next number before I see it, and show the two numbers I used.

A rule is worth having because it lets you commit before the answer arrives. A number written down with its two addends can be told apart from a lucky guess.

The child writes both predictions before either page is read, with the two numbers used beside each, and does not rub out a wrong one.

Yesterday the covered pile was already on the strip. Today the next exhibit has not been read yet, and the number has to be written before it is.

Twice Eli says a number before he sees the animals, and twice the book confirms him: he predicts thirteen gorillas and the text answers exactly 13.

Before anyone reads on, every pair writes a number in the seventh box with the two numbers they added underneath it, and folds the strip so nobody can change it; then the gorilla page is read and the strips come up.

Two written and folded predictions, each with its addends, and the book's number written beside any that was wrong rather than replacing it.

Children tell a partner why they were willing to write a number down before seeing the animals; an answer has to name the rule and at least two counts from the strip, and I guessed is sent back.

I can test a rule along a row and say where it first goes wrong.

A rule is a claim about every place in the row. It is not beaten by an opinion but by one place where it produces a number the row contradicts.

The child circles the first disagreement for each failing rule and names both numbers that disagreed there.

The class now has a rule that works. Today two rules that also look reasonable get tested against the same numbers.

Amanda names three candidate rules aloud and discards two of them, and the book prints the counts she tests them against.

Children write the book's seven counts along a strip, then test the adding-one rule by writing what it produces underneath and circling the first place their answer and the book's count differ, saying both numbers aloud, and do the same for doubling.

A strip carrying three tested rules, with the first disagreement circled for each of the two that fail and every agreement marked for the one that holds.

Children tell a partner which rule works and how they know, naming the count where each of the other two first broke with both disagreeing numbers in the sentence, and say whether their partner named the same breaking place.

I can draw the rule and find the two earlier days inside my drawing.

One growth can be written two ways. The branching picture and the sum of two earlier days describe the same thing, and finding one inside the other is what makes the rule make sense.

The child extends the drawing one more day by the stated rule, counts the pairs, and can point at where the two earlier days show up in the new one.

Rules have been rows of numbers. Today the same rule is a picture that grows, and the numbers have to be found inside it.

Amanda draws her explanation in the dirt with letters, an A for adults and a B for babies, and the book prints both the drawing's first days and the rule that extends it.

Children draw the four days Amanda describes, labelling every pair A or B exactly as she does and writing the count under each day, then apply her rule to draw day five — every A pair drawing a new B pair beneath it and every B pair becoming an A.

A five-day drawing with every pair labelled and every day counted, checked against the book's own sentence for the fifth day.

Children point at their day-five drawing and tell a partner where the day-three pairs and the day-four pairs both show up in it, and say one sentence naming the two earlier days and the number they make.