What a thing really costs

I can work a ledger and check it.

A ledger that does not balance means a line was dropped, and the job is to find which one rather than to accept the number.

The child's answer agrees with the book's last line, or the child finds and names the line they dropped.

First lesson. Show one ledger with its last line covered and ask what the covered number must be. Take the answers and cover it again.

This book prints the same table more than forty times, and every one of them balances: a price on top, costs beneath, and a last line that is the price minus the costs.

Pairs read a ledger aloud and name its three parts, add only the cost lines writing each running total in the margin, then subtract the circled total from the price and write what is left.

A worked ledger with running totals in the margin and the answer written at the bottom.

Pairs uncover the book's last line, set their number beside it, write agrees or add the column a second time to find the dropped line, and say aloud which line it was.

I can find the operation in a table with no signs.

There is no minus sign anywhere on the page; one word on its own line is the entire instruction.

Two answers are written for one table, one under each word, with the gap between them circled.

Yesterday you worked a ledger. Today you are going to look for the plus sign and find that there is not one.

Forty-three times this book prints a column of money with no operation sign anywhere in it, and what tells a reader to subtract is a single word.

Pairs hunt the card for any operation symbol and report there is none, find the one word that tells them what to do, work the table as that word says, then lift the word strip, lay the opposite one in its place and work the table again changing no digit.

A two-answer slip carrying both results and the gap between them circled.

Each student writes one sentence naming the word, the operation it called for and both answers, and a partner checks that the sentence carries two numbers and the name of the word.

I can add a column of decimals and test it against a budget.

A yes-or-no question about money is settled by a column of decimals with the points aligned and nothing dropped.

Two independent additions of the same column agree before anything is tested.

You can work a ledger somebody else laid out. Today you write the column yourself and there is a fixed amount it has to fit inside.

The book hands over a cart of five priced items, a stated budget and a tax rate, and asks a yes-or-no question the arithmetic settles.

Pairs copy the prices onto a cost card one per row with the points stacked, add from the bottom up, then trade cards and add the partner's column from the top down, reconciling any disagreement together.

A cost card with the total in the box, agreed by two independent additions.

Pairs add the bird seed on a sixth row, set the new total beside the budget and say which is larger, then multiply the total by eight hundredths, round to the nearest cent, add it on a seventh row and test again.

I can build a per-unit price and use it.

A price and a tax on separate sign lines make one per-pound price, and dividing a total by it says how much was bought.

The child says what the quotient counts, not just what it is.

You can add a column and test it. Today two of the numbers have to be joined before any dividing can start.

This is the book's own two-step calculation and the answer is a clue rather than an exercise.

Pairs write the price per pound and the tax per pound with the points stacked and add them, check the sum by a second route on a calculator, then lay play money in stacks of that amount until thirty dollars is laid out and count the stacks.

The per-pound price circled, and the division written as a sentence with the number of stacks counted.

Students label their answer twice, once saying six what and once saying what the thirty and the five each were, then read the book's line about a five-pound bag and say why the size of the purchase makes it a clue.

I can build a common unit before I compare.

Two prices that describe different amounts cannot be compared until both are expressed per the same thing.

The child says why the smaller number is not automatically the better buy, using the two unit words.

Yesterday you built a per-pound price. Today two prices arrive in different units and the smaller one is the worse buy.

The book sets a trap and then names it: one paint costs $18.95 and the other $35.96, and the cheaper-looking one is the worse buy.

Pairs write each price on a card with its unit underneath, fold a paper gallon strip into four equal parts labelled one quart each, divide the gallon price by four and write the answer on one part.

Two per-quart prices written on one line with the saving on a single quart between them.

Pairs buy eight quarts of the better brand as two gallon cans, multiply the gallon price by two, and write one sentence saying why they multiplied the gallon price and not the per-quart price they had just worked out.

I can find a half and a fourth of an amount.

A folded strip carries the relation between the half and the two fourths where an equation would not.

The child writes the half share two ways: as a half of the whole and as two of the quarter parts.

You have divided to find a unit price. Today an amount is divided into parts of different sizes and all three have to be visible at once.

The team's written budget divides its goal into a half and two fourths, and the book computes only one of those fourths.

Each student folds a strip standing for four hundred dollars in half and in half again, counts the parts, labels each with its amount by dividing, then colours the three budget shares onto the strip and writes the dollar amount inside each region.

A coloured strip with every part labelled and no part covered twice.

Students lay a second strip beside the coloured one folded once into halves, line them up, say how many fourths lie against one half, and write the half share both ways.

I can say what a percentage is acting on.

Changing what a rate acts on changes the answer more than changing the rate does.

Both bills are computed in full before either is judged.

You can take a part of an amount. Today two coupons compete and the bigger percentage loses.

Fifteen is the larger rate and it produces the larger bill, because the fine print aims it at one item and at the cheaper of the two.

Pairs add the two kit prices and circle the cart total, take a tenth of it by dividing into ten equal parts and subtract, then take fifteen hundredths of the cheaper item alone, round, subtract from that item and add the other kit back at full price.

Two computed bills side by side, each reached by its own method.

Pairs circle the smaller bill, say aloud why the bigger percentage did not win, and subtract what Nicole pays from the thirty dollars on her gift card.