One amount, many names: fractions, decimals and percents

I can add amounts that share a bottom number.

When the parts are the same size they can simply be counted together, and the bottom number stays put on both sides of the plus sign.

The bottom number is unchanged in every line the child writes.

First lesson. Lay two one-third strips together and ask what they make. Take the answers and write none of them down yet.

The book doubles four recipes and writes every doubled amount as a full sentence, so the arithmetic arrives spelled out in words.

Pairs lay one half strip on the cup mat and push a second beside it, then do the same with two thirds and with two quarters, saying what the pieces cover each time before writing anything.

A recording sheet with four lines, each written as amount plus the same amount again equals the answer.

Pairs trade sheets, read each other's four lines aloud, and rebuild with strips wherever the two answers differ, marking the line they fixed.

I can take one part away.

Taking a like part away leaves parts of the same size, so the bottom number is the same in all three places on the line.

Every difference the child writes keeps the bottom number.

Yesterday you doubled a pot. Today you halve one, and the pieces come off rather than going on.

Halving a pot is where this book does its clearest subtraction, and it spells every difference out in words.

Pairs lay two one-third strips for the peanut paste, slide one away out of reach and say what is left while touching it, then do the same with quarters and with eighths.

Three written differences, each carrying the starting amount, what was slid away, and what is left.

Pairs read their three lines aloud to each other and rebuild whichever line they disagree on, sliding the piece away again while the partner watches.

I can rename an amount without changing it.

Renaming changes what an amount is called and leaves how much is there alone.

The child builds each amount before writing its second name, never after.

You can add and take away like parts. Today one amount is going to get a second name and stay exactly the same size.

One line in this book carries three names for the same butter, and it renames two further amounts elsewhere.

Pairs read the butter line and say its three names in order, then cover one half strip exactly with quarter strips adding one at a time until nothing sticks out, and cover a quarter strip with eighths the same way.

A sheet carrying each amount with its second name on the line underneath, each built before it was written.

Pairs fill the eighth spoon twice into the quarter spoon and say whether the amount of red pepper changed, then say which name they would use if the only spoon left were the small one.

I can say what a dot adds.

How much a dot adds depends on what it sits beside, which is why the rule names the original value rather than a number.

The child refuses to answer with a single number and says why.

You can rename an amount. Today an amount is increased by half of itself, and the half is different every time.

The book states one rule and then works it three times with different starting values.

Pairs fold the two-beat strip end to end, crease and cut it to make the half, lay the half against the end of the whole strip and count the beat marks along the joined length; then run the same fold, cut and join for the four-beat and the one-beat strips.

Three joined lengths with the starting length, the half and the finished length said for each.

Pairs tell another pair one sentence naming a starting length, the half cut from it and the finished length, then answer the question how much does a dot add — and a single number is the wrong answer.

I can choose parts that add to a required total.

The total is fixed and the parts are chosen, so two different sets of cards can reach the same four beats.

Two measures both come to four using different cards, and the child names the cards rather than pointing.

Yesterday one amount grew by half of itself. Today a total is fixed in advance and you choose what fills it.

The book adds up two measures from the same song, uses a different number of notes in each, and reaches four both times.

Pairs check the book's two adding-up lines by laying cards for one of them and counting the total, then fill an empty measure box with cards so the beats come to exactly four, taking a card out and recounting if they land past.

Two measures built with different sets of cards, each counted aloud to four.

Pairs swap one measure with another pair, count theirs from left to right saying a number word only when a whole beat finishes, and write the sentence naming their own cards in order ending with four beats.

I can write one amount three ways.

An amount of cents is a fraction of a dollar, a decimal and a percent at the same time, and the coins prove all three describe one pile.

The child writes all three forms for a box the book never answers, with nothing to copy.

You can rename a fraction. Today an amount of money gets three names at once, and two of them look like different kinds of number.

The book counts two boxes of coins for the reader and answers both, then sets out a third and walks away.

Pairs lay out eighty-nine cents and write it in all three forms, build the book's first two boxes and write each three ways before reading the book's answer, then meet the third box and write all three forms themselves.

A three-blank recording strip for the third box, completed with nothing printed to copy.

Pairs hunt the next page for the sentence that settles the third box, find that two of the three fractions are already known, and write the sentence beside their answer as the reason they were right.

I can say what a digit is worth from where it sits.

The same digit written in two places names two different amounts, and the piles settle it.

The child's sentence names a position, not only a size.

Yesterday one amount had three names. Today one digit has two values and only its place says which.

The book makes a flat assertion and leaves it standing: ninety cents and nine cents are not the same.

Pairs read the assertion, build ninety cents and nine cents as two separate piles, and take turns finishing a spoken sentence about the two piles that names the pile and the position.

Two labelled piles with a spoken position sentence for each.

Students lay out what each of the three nines in the lamb's price is worth, order the three from largest to smallest, and write one sentence that would convince somebody who thinks the two piles are the same amount.

I can say which number goes inside.

The whole instruction lives in one small word, and the book's own answers rule out the reading that gives quotients above one.

Both columns are worked in full before either is judged.

You know the three names of a hundredths amount. Today the bottom number is not a hundred, and the procedure sentence has to be read very carefully.

The book states how to turn a fraction into a decimal in one sentence, and reversing the two nouns produces 2, 1.33 and 1.25 instead of 0.5, 0.75 and 0.8.

Pairs read the procedure sentence aloud twice, circle the word by and draw an arrow to the noun it attaches to, then work each of the three divisions as the sentence reads and again reversed, filling both columns.

Two full columns of quotients with the procedure sentence marked above them.

Students build half a dollar in coins, read the book's sentence about the coins being a fraction of a dollar, and say which column can possibly be an amount of a dollar, marking the ruled-out column and writing what the word decided.