Around it, or across it: perimeter and area

I can make two numbers from one card.

Two side lengths produce two different answers, and the words decide which one is wanted.

The child sorts four questions to the right number and names the word in each that decided it.

First lesson. Hold up a card and say it is four inches long and six inches wide, then ask how big it is. Take every answer and write them all up.

Zack gives the hall pass two numbers on one page and then names only one thing about it; the area of that card is nowhere in the book.

Pairs run yarn round the edge of one card, cut it where it meets and read its length along a ruler; then cover a second identical card with one-inch squares and count them, writing the two numbers side by side.

Two numbers from one card, each written with the unit it belongs to.

Each pair sorts four questions about the card to one of the two numbers, writing beside each the one word from the question that decided it, then writes why one number is inches and the other square inches.

I can find the perimeter by going round.

The trip round the outside meets each pair of sides twice, which is why two sides of four and two of six make twenty and not ten.

Every side of the card carries a number, not just two.

Yesterday one of your two numbers came from the yarn. Today you find that number without the yarn.

Zack says the card is four inches long and six wide and names its perimeter in the same breath, and the book never shows the working.

Each student cuts a four-by-six card, writes a number on every side, then lays the ruler along one side at a time turning the corners and keeping a running count aloud until the finger returns to where it started.

A card with all four sides labelled and the running count written as a four-term sum.

Students write the four sides as one sum and again the short way, circle the line that shows more clearly why four and six each appear twice, and read the circled line to a partner.

I can find the area by tiling and then by multiplying.

The two numbers in the area equation are a number of rows and a number in each row, so the multiplication is a shortcut for something already on the desk.

The child counts by rows rather than by single squares and can say which number in the equation was which.

Yesterday you went round the edge. Today you cover the face, and the number that comes out is a different kind of number.

The book's third demonstration hands over the area rule and a page to use it on: five inches by eight, forty square inches.

Pairs lay one-inch squares onto a five-by-eight rectangle until nothing overhangs, write the two edge numbers, count by rows touching a whole row on each number word, and write the multiplication underneath.

A tiled rectangle with its two edge numbers and the equation written beneath it.

Students take a three-by-seven rectangle with no squares supplied, find its area by writing the multiplication alone, then lay one row of squares along it to check the row really holds seven.

I can compute two areas and compare them.

The same pan is two numbers in a sentence, a rectangle of squares and a product, and all three are one thing.

The child produces each area twice, by counting and by multiplying, and the two agree.

You can tile and you can multiply. Today two rectangles arrive as four numbers in a sentence and never get drawn for you.

Two loaf pans are given with their lengths and widths, and the book supplies the rule that turns them into an area.

Pairs draw both pans on grid paper at one square per inch, count the squares in each row by row and write the number, then multiply the sides and write that underneath.

Two drawn and labelled rectangles, each carrying a counted area and a multiplied area.

Pairs read the book's sentence about surface area and time, point at the larger rectangle and say which pan finishes first, then lay one rectangle over the other, mark and count the squares that stick out, and say that number both ways round.

I can build a field four times as large.

Four times as large is four copies, and what is extra is what the new field has that the old one did not.

The extra ground is shaded on the drawing and the number written matches the shading.

You can compute an area from its sides. Today one area is built from another, and the phrase says how.

The book gives last year's field as twenty yards by thirty, says the new field is four times as large, and asks how much extra ground a defender covers.

Pairs draw last year's field at one square per yard and count it by rows, then lay four copies of that rectangle side by side and count the squares, writing both areas inside the shapes.

Two drawings, the larger built from four copies of the smaller, with both areas written inside.

Pairs take last year's area off the new one, shade the extra on the drawing, then split the shaded extra down the middle and count one half, saying which number answers the book's question and which was a step on the way.

I can tell a scaling phrase from an adding one.

The numbers on the page are plain; what has to be extracted is which operation the sentence names.

Two labelled rectangles stand on one sheet, one at 2400 square yards and one at 604.

Yesterday four copies made the new field. Today the same sentence arrives with two words changed and the field comes out almost the same size as before.

This book never writes an equation; every operation it wants is carried in a phrase.

Pairs build the six-hundred-square-yard field, lay four copies together and count, underlining the phrase that told them to; then take the second strip, in which four times as large has become four more square yards, and draw what it asks for beside the first.

Two labelled rectangles on one sheet, each headed with the phrasing that produced it.

Each student writes one sentence about their own two drawings using the book's phrase correctly, and a partner checks it against the drawings rather than against the number.

I can say a difference both ways, and notice when a relation runs the other way.

Two sentences describe one arrangement, and a second relation on the same pair can point in the opposite direction.

Four sentences are written about one pair of rectangles, two about area and two about time, with the comparison word flipping between them.

You have built and compared areas. Today the same two pans give you a pair of sentences about area and a pair about time that do not point the same way.

The book gives two pans as four numbers, states that greater surface area usually means less baking time, and never prints the relation between them.

Pairs draw both pans, count the squares in each and check against the product, lay one rectangle over the other and count the squares that stick out, then write one sentence starting from each pan about that same difference.

Two overlaid rectangles with the difference visible, and four written sentences about them.

Students write the pair of time sentences for the same two pans and mark which comparison word had to flip against the area word, then swap papers and check a partner's four sentences against the rectangles.