Place value to the thousands

I can say why counting a big pile one at a time does not work, using my own numbers.

Grouping is not a rule handed down. It is what people do when counting one at a time stops working, and you can watch it stop working.

The child records where each of the three attempts stopped and says what ended the first two using their own two numbers as the reason.

First lesson. Hold up a jar of beads and ask how long it would take to count. Write the class's guesses where they can be checked at the end.

The book tries three ways to count a crowd — a loose mill, a queue, and finally grouping — and rejects the first two in the characters' own words before the third works.

Each table takes about two hundred beads and three minutes and counts them three ways: once in a loose heap, once in a single line passed bead by bead, and once dropped into cups of ten, writing the number reached and where it stopped for each attempt.

Three attempts recorded with the number reached and the stopping point for each, and the two complaints matched against the two the characters make.

Tables fill cups to ten and set ten full cups on a tray, then hand their trays and cups to another table to read the total without emptying anything — and say what about the arrangement let it be read that fast.

I can rearrange a collection into groups of a new size and say what changed and what did not.

How many and how it was grouped are two different facts. A total does not tell you the grouping, so the size of the group is a choice somebody makes.

The child splits four tens into eight fives with nothing added or removed and says what changed — more groups, fewer in each — and that forty did not.

Yesterday cups of ten beat every other way of counting. Ask whether ten was the only size that would have worked, and hold the question open.

Forty bears are pulled off the pile into four groups of ten and the book says four times ten is forty; the same forty are then broken into eight groups of five and the book says to count by five to forty.

Children count forty bears off the pile into four hoops of ten, push the leftovers aside, then split each hoop in two so four hoops become eight without a bear entering or leaving, and count what has landed in each new group.

Both arrangements on the table with both multiplications written underneath, each number pointing at something in front of them.

Children touch each of the eight groups and count by fives to forty, say how many number words they said, and then answer out loud what changed when the total did not.

I can clip tens and bag hundreds until no ten is left to make, and read the three lanes.

Ten of one thing becomes one of the next thing. Done twice, it turns a pile that changes size every time you count it into three lanes you can read at a glance.

The child makes both exchanges without prompting, stops only when fewer than ten remain at each level, and reads the mat as bags, bundles and loose slips.

Cups of ten worked and ten was a choice. Today ten gets used twice, and the pile is far too big for cups.

Jim's collection outgrows counting twice — the number changes every time he counts and Everett loses his place among the bundles — and the repair is ten wrappers to a bundle and ten bundles to a bag.

Children try the loose count against a timer and disagree with the table beside them, then clip exactly ten slips into every bundle saying ten slips make one bundle as each clip closes, and drop exactly ten bundles into each bag saying ten bundles make one hundred as the bag fills.

About four hundred and fifty slips sitting as full bags, leftover bundles and loose slips in three labelled lanes, with each lane read aloud.

Children say which lane they would go to for exactly one hundred slips and which for exactly one, then say what would have to happen for a sixth bag to exist.

I can find how many of one place fit in the next and say the rule with a number in it.

The lanes are not an arbitrary row. Each one holds ten times what the lane on its right holds, and that same ten turns up at every step.

The child measures all three neighbouring pairs, gets ten each time, and states the rule in their own words with the number in it before the book's sentence is read.

Slips went into bundles and bundles into bags. Ask how many bundles a bag swallowed and how many slips a bundle swallowed, and whether that is a coincidence.

The book pitches four tents whose largest crowds are nine, ninety, nine hundred and nine thousand, and closes by saying each place is ten times bigger than the one to its right.

Children build each tent's largest crowd in cubes, then set one tent's crowd beside the next tent's and find how many of the smaller fit inside the larger, doing it for every neighbouring pair and saying whether the answer changed along the row.

Three measurements, all of them ten, and the rule said aloud with the number in it before the book's own sentence is read out.

Children take crowd cards and put each one in the tent that holds it without spilling, giving the tent's limit as the reason the next tent down is too small — then say what a tent to the left of the tiny tent would have to hold.

I can write the numeral a chart names, and fill a chart from a number.

Reading a chart and writing a number are one act done in opposite directions. Whichever end you start from, each column keeps its own meaning.

The child reads both of the book's charts aloud and as one number, fills an empty chart from named containers, and writes each numeral by hand.

The mat has held bags, bundles and slips. Show a chart with one digit per box and ask what has been left out and what has been kept.

The back of the book prints two charts and reads each one out loud, digit by digit and then as a whole number: four hundred thirty-seven from three boxes, two thousand one hundred fifty-six from four.

Children lay one digit tile in each box to match the book's three-box chart and read it left to right and then as a whole number, do the same with the four-box chart, and then fill an empty chart from the containers the character lists with nobody having given them a chart at all.

Four numerals written by hand, two from the book's charts and two from containers and a spoken count, each checked against what the character says.

The driveway count is read aloud in the book's order and never printed; children lay the four tiles, write the number on a blank card without looking at the mat, then turn the card over and say what would happen to the leftmost box if one more wrapper arrived.

I can build a number from its digits alone and say what each digit counts.

A digit is a count of cards of one colour. That is why moving a digit to another column changes the number without changing the digit.

The child rebuilds the count from the written digits with the sentence covered, then swaps two digits, rebuilds, and says which is larger and by how many cards.

Yesterday a chart was written as a numeral. Ask what the 8 in that numeral was actually counting, and take every answer before handing out the cards.

The book never writes a number in columns, but it says two of them out in full as groups of a thousand, groups of a hundred, groups of ten and singles, and prints the matching numeral both times.

Children set out that many cards of each kind for the count the book names, write one digit under each colour so the digit tells how many cards of that colour are there, then cover the sentence and lay the cards out again from their written digits alone.

The four-place count written as digits and as a sum of its four amounts, and rebuilt in cards from the digits with the book's sentence covered.

Children swap the digit in the hundreds column with the digit in the tens column, rebuild the cards to match, say the new number and by how many cards it differs, and say why the same two digits make two different numbers.

I can carry a trade up the mat until no lane holds ten, and say why one slip caused that many trades.

A single unit can set off a chain of exchanges or none at all. The chain is not in the unit; it is in what the other lanes were already holding.

The child completes every trade the count forces, saying what became what each time, then puts the same cube in a different count and says how many trades it causes there.

Every number so far has been built and read. Today one is pushed over an edge. Build nine of everything and ask what happens if one more wrapper turns up.

The book's last count is nine of everything — nine trash bags, nine grocery bags, nine bundles, nine loose — and Jim opens one more bar.

Children build nine thousand-cubes, nine flats, nine rods and nine units, read the four lanes in the book's order, add one unit cube, and trade ten units for a rod, then ten rods for a flat, then ten flats for a cube, saying what became what at every trade and reading the mat when it stops.

The mat carried from nine of everything to ten thousand, with every trade said aloud and the mat read again once no lane holds ten.

Children clear the mat, build six thousand three hundred fifty-four, put the same one cube in and say how many trades it causes now — then say what has to be true of a count before one cube can move four lanes.