How many times as much

I can build a length exactly twice another.

Exactly is what turns a comparison into a measuring job, and the proof is that the short strip lays along the long one twice with no gap and no overhang.

The second laying finishes exactly at the far end, and the child can say by how much it misses if it does not.

First lesson. Hold up two strips and say one is twice the other. Ask how anybody could check, and take the answers before any measuring.

Pythagoras will not tune the pipes by ear; he gives one instruction with a multiplier in it, and the word exactly is what makes it a measuring job.

Pairs read the instruction and say what would count as following it, then build a second strip exactly twice the short one without a ruler, by laying the short strip down, marking where it ends, and laying it again from that mark.

Two strips, the long one built from two layings of the short, with the proof performed in front of a partner.

Each pair says one sentence to another pair naming both strips and using the words exactly twice as long, and the other pair checks it by laying the short strip along the long one while they watch.

I can say a comparison two different ways.

One pair of heights carries a factor and a difference at the same time, and both are true.

Both sentences are written with their equations underneath, and the child circles the one whose words match the book's.

Yesterday exactly twice was proved by laying. Today the same laying gives a second number that has nothing to do with the factor.

The book puts two heights side by side and never lets them go: Ray at twenty feet, Jack at four, and every problem comes out of that pair.

Groups tear tape at four feet and at twenty and label them, then lay four-foot lengths head to tail along the twenty-foot strip counting aloud until it is covered.

Two written sentences with their equations, one multiplicative and one additive, with the book's own wording circled.

Students take the two hoop heights, sixty feet and twelve, and answer how many of Jack's hoops stack up to Ray's, writing the equation and saying what they notice about the number.

I can say which phrase my measurement earns.

Two comparisons of the same kind between the same two objects, and only one of them is allowed the word exactly.

Each chosen phrase is written with what the strip actually did beside it.

You have said a comparison two ways. Today one sentence in the book contains two comparisons and gives them different words.

Pythagoras compares the two sets of pipes in one sentence and uses two different comparisons in it: the widths are exactly twice, the lengths are more than twice.

Pairs mark the small pipe's width on a narrow strip and lay the marked strip across the large pipe twice, saying whether it finishes exactly, falls short or overhangs; then do the same for the two lengths.

A record card with the phrase each measurement earned and, beside each, what the strip did.

Each pair says its two chosen phrases to another pair with the strips on the table, and the listening pair may not agree until they have laid the marked strip themselves.

I can catch a reversal that keeps the wrong word.

Three times, one third and two hundred sixty more are three true sentences about one pair, and only one word has to change when the direction does.

The child says which word has to change in the deliberately wrong reversal, and changes it.

You can choose the phrase a measurement earns. Today the phrase is right and the direction is wrong, which is harder to see.

One sentence carries the highest monthly distance and asks for the lowest, and the answer sits directly after it.

Pairs write the two months and their miles on strips, build both in base-ten pieces in rows that start level, and find how many times the November row fits inside the September row by laying copies of it end to end.

Three spoken sentences about one pair — three times, one third, and two hundred sixty more — with the rows pointed at for each.

A partner says one of the three sentences with the two months swapped but the comparison word left alone; the pair decides whether it is still true, says which word has to change, and changes it.

I can find the multiplier and build what it demands.

A multiplier can be recovered from two numbers about one thing and then applied to something else.

The tape is on the wall at the right height and the child can say how many times their own height it is.

Yesterday you found a factor between two given numbers. Today the factor is the thing you have to find first, and then use.

Ray gives two numbers about himself and Jack reads a rule out of them.

Pairs write Ray's height and hoop height on a card, test whether three really carries twenty to sixty by skip counting on a number line and writing the equation, then compute three times four and mark that height on the wall with the tape measure and a strip of tape.

A strip of painter's tape on the wall at the computed height, with the equation written beside the card.

Each pair picks a whole number of feet for a new player, works out that player's hoop height, and says the finished sentence to another pair, who check the arithmetic out loud.

I can use a multiplier to check a claim.

A comparison is a tool: once the factor is known it can be run on a second measurement to test something somebody said.

The child writes the product before reading the claim and then marks agree or disagree.

You can find a multiplier and build with it. Today somebody has already shouted an answer and your multiplier is going to check it.

Jack measures two shadows and then shouts a height for the beanstalk that the book never works out on the page.

Pairs find the two shadow measurements, write two times blank equals twenty and find the missing factor, check it against Ray's stated height, then multiply forty-one by ten showing it as forty times ten plus one times ten.

The missing factor written and checked, and the computed height written before the claim is read.

Each student answers in writing what height the same ten would give if the shadow had measured forty rather than forty-one, and how far off that is from the claim, then reads the sentence to a partner.

I can supply the sentence the book never writes.

Forty and eighty say nothing about each other until somebody says twice, and the fold is what produces it.

The child says both directions of the relation about strips that do not move between the two sentences.

You have run a multiplier and used it to test a claim. Today two numbers sit side by side in a book that never relates them at all.

This book compares constantly and quantifies both sides every time, and never once says a relation.

Pairs build two bar strips in ten-dollar units for the two jean prices, fold the longer onto the shorter to test whether it covers exactly twice, and count the units left standing to find the difference.

Two strips with the fold performed, and the factor and the difference both recorded.

Each student says and writes the relation in both directions using words the book never uses, and a partner checks that the strips did not move between the two sentences.