Equal groups, and what is left over

I can say how many groups and how many things.

One arrangement carries two numbers, and confusing them is the error the whole unit is against.

Every answer is a number and a word: four bags, twenty peppers.

First lesson. Fill four cups with five counters each in front of the class and ask how many there are. Take both answers and do not settle it.

Abuela's instruction is precise: put five peppers in each plastic bag, then say how many peppers there are.

Pairs fill four cups with exactly five counters each, then answer alternating questions — how many bags, how many peppers, how many in this one bag — with the number and the word every time.

Four filled cups and each child's answers, each one carrying a number and a word.

Pairs add a fifth cup and fill it, predict both new answers before counting, and say which number grew by one and which grew by five.

I can change the group size and see what moves.

The number of groups and the number in each group vary independently, and the total follows the one that changed.

The child writes two numbers for each row and can say which one changed between the rows.

Yesterday one arrangement had two numbers. Today the number of plates stays put and what is on them changes.

Agnes hands out the same thing to every scout three times over, so the book holds the number of groups steady and changes only the group size.

Pairs build nine plates of ten sticks and write the two numbers, then clear and rebuild the same nine plates with five counters each and write the two numbers again.

Two rows recorded, each as a pair of numbers, with the number that changed marked.

One partner builds a row of equal plates while the other looks away; the counter says how many plates, how many in all, and names the group size.

I can hear the word that fills every bag.

Each is what makes five repeat in every bag; without it the same sentence puts five peppers on the table and stops.

Two builds stand side by side, one holding twenty and one holding five, and the child names the word between them.

You have changed the group size on purpose. Today the sentence changes instead, by one word, and the total moves by fifteen.

Abuela's instructions are the story's working orders, and their quantities only come out right if the little words are read exactly.

Pairs build the sentence as written — four cups, five counters in each — and count by fives to twenty; then rebuild with the in place of each, giving one bag of five and three empty cups, and count again.

Two builds side by side with their totals, and each child's sentence naming the word and both numbers.

Each child arranges a small build of their own with the same count in every cup and says one true sentence about it using each, with both numbers in it.

I can count on by a whole group.

A count that stops short tells you something is missing, and you can say where it would have landed without starting over.

The child says the next number without recounting the plates, and says the two things that got them there.

Yesterday one word filled every bag. Today every bag is filled and the count stops one bag early.

Every time Agnes counts, the count stops one group early, and that is how she learns Albert is gone.

Pairs set nine plates of five, count by fives to forty-five as Agnes does, then hold five more counters in hand and say the number the count would reach without laying them down.

The stopped count written and the reached number written beside it, with the two numbers said aloud.

One partner builds a row of plates of ten and secretly removes one; the other counts by tens and says how many plates are missing and how many sticks with them.

I can choose a count that fits the pile.

A skip count only works when the pile splits into equal groups with nothing left over, and the fastest count is not always the one that fits.

The class writes a rule in its own words naming both when a count works and when it does not.

You can count on by a group. Today three different counts are tried on one pile and one of them will not land.

Pablo escalates through the book, calling twos fast, fives faster, and tens very fast.

Pairs count the same forty counters three times, regrouped by twos, by fives and by tens, tallying one mark per number word and recording the three word costs; then add five counters and try all three counts again.

A tally table of word costs for the three counts, and the record of which counts landed on forty-five and which stranded counters.

The class returns to its opening votes and completes a sentence frame aloud, then settles one shared wording and writes it under the tally table.

I can pair them up to tell odd from even.

A leftover is information: the counter with no partner is what names a count odd.

The child reports the leftover before saying odd or even.

Yesterday a count left five counters stranded. Today the groups are pairs and the leftover has a name.

The book asks how you can tell an even number from an odd one, and then counts fifty pages of ocean animals past the reader.

Pairs count nine counters for the crabs and push them two at a time into the boxes, count the filled boxes by twos and say what is sitting outside; then build twenty-one, twenty-three and twenty-seven the same way, reporting the leftover each time before saying odd or even.

A tally of which counts left somebody standing, with each build's leftover reported.

Pairs turn number cards one at a time and place each on the odd or even pile, saying how many pairs the number makes and whether anyone is left standing before it moves.

I can say what one more does.

Adding one closes the gap or opens it, so parity flips every time, and one build can be tested against four others.

The child writes a rule that holds for every build on their table, including the one that starts at zero.

Yesterday the leftover named the number. Today one counter is added and the leftover changes hands.

The odd run ends at forty-nine, one more animal is added, and the count lands on fifty, a number the book calls even.

Pairs build nine and pair it, add one counter and pair again, saying both numbers and exactly what changed; then run the same test from twenty-one, twenty-three and twenty-seven and keep all four results visible.

Four before-and-after results kept side by side, each said as a full sentence.

Pairs fill in a sentence strip with the rule their builds support, read it to another pair, then test it by starting from zero and adding one, changing the strip if the test breaks it.