Multiplication as equal groups

I can build equal groups and write what they say.

The word each is what makes the groups equal, and the two numbers in the equation do different jobs.

The child rings each number in the equation and touches the thing it counts.

First lesson. Put three mats out with two counters on each and write 2 × 3 = 6 beside them. Ask which number is the mats. Take the answers and leave it open.

Chapter three hands over the cleanest equal-groups picture in the book: three scarecrows, and every one of them gets the same parts.

Pairs set out three mats, put two counters on every one so no mat has more than another, count by twos across the mats one touch per mat, and write the equation beside the build.

Two builds with their equations, each number ringed and matched to what it counts.

Pairs clear their mats and build the mouths, three teeth on each of the three mats, count by threes and write the second equation under the first, then say what stayed the same and what changed.

I can show one quantity three ways.

Twenty-seven counters are nine groups of three and three groups of nine at the same time, and nothing has to move between the two sentences.

The child says both sentences about the same counters without touching them.

Yesterday two numbers did two jobs. Today the same counters swap which number does which, and the total does not move.

Peter does not guess the number he needs; he puts twenty-seven pebbles down in groups of three and the groups tell him nine.

Pairs count twenty-seven counters and push them into groups of three, line the groups up as nine rows of three and count by threes, then turn the whole arrangement a quarter turn and count by nines.

One array drawn on grid paper with a group of three ringed, and both equations written underneath.

Pairs hand the drawing to a partner and have them point to the group that was ringed, then say both sentences about the same counters without moving anything.

I can hear which operation the sentence wants.

The stick obeys words, and one of them turns three and nine into twenty-seven while the other turns them into twelve.

Both builds stand on the mat and the child names the count that settles which sentence the book meant.

You can read an array two ways. Today one word in a sentence decides whether you are multiplying at all.

The book says outright on its last page that the signal words are times and of, and in the river scene the whole repair rests on one of them.

Pairs set three counters for the stones still in the water, build the book's sentence across the top of the mat as three groups of nine counted to twenty-seven, and the swapped sentence underneath as three counters and nine more counted to twelve.

A two-part mat holding both results at once, with each count said aloud.

Pairs say which sentence puts the river back, using the boy's own number, and then swap a word in the chicken sentence and write one line telling Peter which version he can actually fix.

I can find the multiplier in a situation.

A multiplier does not always arrive as a number in an equation; here it comes out of two measured wingspans.

The child says how many of the small wingspan fit along the big one before any cookie is counted.

Yesterday a word set the operation. Today the number itself has to be found, and it is lying in two measurements.

Chapter four is the only place the book attaches its multiplier to a measurement: one foot of wings against four.

Pairs lay one cube for the small bird's wings and a strip of four beside it, say how many of the small span fit along the big one, then build the matching cookie piles for one, two and three cookies and say what the big bird's pile does each time.

The four-times column written out with the last two entries checked on the mats.

Pairs read the book's own statement about multiplying by four and say whether their piles agreed with it every time, pointing at the two piles that show it best.

I can build the times column as one repeated move.

A times column is not ten separate facts; it is one move made again and again, so a covered total can be recovered from its neighbours.

The child works out the covered total from the two beside it and says which two they used.

You have found a multiplier. Today one multiplier is used ten times running and the column that comes out has a shape.

The book prints its nine-times count twice as a bare column and once as a running list, and states in words what the list is doing.

Pairs snap bars of nine and lay them end to end down a strip, writing the running total at the end of each bar as it lands, and say each total aloud as it is written.

A strip carrying ten bars of nine with the running total written at the end of each.

Pairs cover one total in the middle with a card, work out what is under it using only the totals on either side, say which two they used and what they did with them, then lift the card.

I can check a product a second way.

A student who can reach one product by two independent routes has a way to catch an error in either.

Both routes are written on the strip and the child can say why they agree.

Yesterday the column let you recover a total. Today one product is reached twice by two different methods.

The book poses one unfinished problem and then instructs the reader to answer it twice, first by the finger procedure and then by adding five nines.

Pairs write a prediction, get the answer by curling down the fifth finger and reading the two counts, then build five equal piles of nine counters and add them with a running total said aloud.

A folded strip carrying the prediction, the procedure's answer and the adding answer side by side.

Pairs put the two answers side by side, say whether they match, and say in one sentence why the two routes give the same number, pointing at the five piles.

I can say the square products quickly.

Fluency here means the product arrives whole, with the array behind it for anything that does not.

Cards that were missed are rebuilt as squares of tiles and then answered again.

You have two ways to reach a product. Today a small set of products has to arrive without either of them.

On the voyage home Pythagoras states his rule in his own words: the length of a side, times itself, is the number of tiles in the whole square.

Pairs build a three-square and a four-square from tiles and count each by rows, say both products with eyes shut and check against the tiles, then sprint the nine square-product cards in mixed order, setting missed cards aside.

Two standing squares of tiles and a pile of missed cards.

Each missed card is rebuilt as a square of tiles, counted by rows, said aloud, and returned to the deck, and the miss pile is run once more.

I can say what times zero gives, and why.

Zero groups of anything leaves nothing on the table, and the rule is defended from the mats rather than recited.

The child predicts a product they never built and can say how they knew.

You can say your square products fast. Today a rule arrives that no amount of counting would ever give you.

The rhyme at the end states the rule flatly, and the last pages of the story spend themselves proving it.

Pairs set out three counts from the story on three mats, run the spell on each by making zero groups of that count, count what is left on the table and write the three equations one under the other.

Three mats and three written equations, with what all three answers have in common said aloud.

Without touching a counter, each child writes what one hundred times zero gives and then predicts a number their partner chooses, then defends the rule with the mats while the partner tries a number that might break it.