Mathematics 2

The counting order goes on past 100 in the same way, with a new hundred after each ninety-nine: 198, 199, 200, 201, and much later 998, 999, 1000.

Inside each hundred the tens and ones words come back in the same order you know: two hundred one, two hundred two, up to two hundred ninety-nine.

The sign < means less than and the sign > means greater than: 34 < 43 is read thirty-four is less than forty-three, and 43 > 34 is read forty-three is greater than thirty-four.

The two signs say the same comparison in two directions.

Skip counting by 10s can start at any number, and then only the tens change while the ones stay the same: 34, 44, 54, 64, 74.

Adding 10 adds one ten, so the ones digit never changes.

Adding facts with numbers to 10, and the subtraction facts that go with them, are recalled rather than counted: 8 + 6 = 14 and 14 − 6 = 8 come straight to mind.

Facts known by heart are the building blocks for bigger sums.

Adding and subtracting undo each other, so within 100 a difference is checked by adding and a sum by subtracting: 64 − 28 = 36 is right because 36 + 28 = 64, and 53 + 29 = 82 is right because 82 − 29 = 53.

For a difference, add the answer to the number that was taken away; you should get back the number you started with: 70 − 26 = 44 checks, since 44 + 26 = 70.

A side of a flat shape is a straight line segment, and a vertex is a point where two sides meet: a triangle has 3 sides and 3 vertices, and a square has 4 of each.

Sides and vertices are attributes, things about a shape that can be counted or checked.

A slide moves a shape in a straight line without turning it, so after the slide it points the same way it did before.

Slide a book across a desk: it moves to a new place, but its top is still at the top and it faces the same way.

Sides and vertices are counted from a marked start, going round the shape once: a triangle gives 3 sides and 3 vertices, and a square gives 4 and 4.

Put a small dot on one vertex.

A turn spins a shape around a point, like a wheel turning, so the shape faces a new way but stays in the same spot.

Pin a paper L at its corner and spin it a quarter turn: the L now lies on its side.

A circle has no sides and no vertices: its outline is one curved line, and a curved line is not a side.

A side has to be straight, a line segment from one vertex to the next.

Length is measured with units that have length themselves, all the same size: a book is 9 paper clips long when 9 same-size clips reach from one end to the other.

A unit is a length used again and again to measure.

A centimetre is a standard unit: every centimetre is the same length everywhere, so everyone who measures one crayon in centimetres gets the same number.

Hand spans and steps are different for different people, so two people can get two different counts for the same table.

Units are laid end to end from one end to the other with no gaps and no overlaps: gaps make the count too small, and overlaps make it too big.

Start the first unit exactly at one end of the object, put the next one touching it, and keep going to the other end.

A non-standard unit can give two people different counts for one length, while centimetres give everyone the same count: a table can be 7 of one person's hand spans and 9 of another's, but it is always the same number of centimetres.

Non-standard units are fine for comparing things measured by the same person with the same unit.

Tiling lays many copies of a unit end to end along a length, and iterating moves one copy along, marking where it ends each time.

Both ways measure the same length and give the same count.

A hundreds chart has ten numbers in each row across and ten in each column down, so one step right adds 1 and one step down adds 10: from 23, right is 24 and down is 33.

Rows run left to right like lines of writing, and columns stand up like pillars.

A pattern core can change more than one attribute at once: big red, small blue, big red, small blue changes size and colour together, and a copy must keep both.

Look at every attribute, colour, size, shape and position, because the core may repeat several at once.

Colouring a skip count shows its shape on a hundreds chart: counting by 10s from 4 colours one straight column, 4, 14, 24, 34, and counting by 5s from 0 colours two columns.

Shade every number the count says and look at the shape.

Pattern cores vary in how complex they are: red, blue is a simple core, while big red square, small red circle, big blue square, small blue circle is a core of four that changes in three ways.

A simple core changes one attribute between two elements.

A hundreds-chart pattern has a direction and a step: down a column adds 10, as in 7, 17, 27, and down a slant to the right adds 11, as in 12, 23, 34.

Name where the pattern runs, across, down or slanting, and how much each step adds.

Duration is how long something lasts, from its beginning to its end, which is a different question from when it starts: a song lasts just as long whether it is sung in the morning or at night.

Three questions about time are easy to mix up: when does it start, when does it end, and how long does it take.

Two durations are compared with words such as longer, shorter and about the same: a movie is longer than a song, and a sneeze is shorter than brushing your teeth.

Start two activities at the same moment and see which finishes first: the one still going lasts longer.

Happening first says nothing about lasting longest: breakfast comes before school, but school lasts far longer.

Order and duration are different.

Three or more events are ordered by duration from shortest to longest: a blink, brushing your teeth and a soccer game.

Line the events up by how long each lasts.

The wait until one event can be longer or shorter than the wait until another: on October 20, the wait until November 3 is longer than the wait until October 28.

Waits are durations too, from today until the event.

A survey question for an investigation offers a few clear choices and every person picks one: 'Which recess game do you like best: tag, soccer or skipping?' gives answers that can be counted.

Data can be collected by asking questions.

A question for an investigation asks something the class wants to know and can answer by asking each other: 'How do you get to school: walk, bus, car or bike?'.

Good investigation questions come from real wonderings: which snack to have at the class party, which game to play at recess, how most people get to school.

First-hand data is data collected by the person who uses it: when you ask your classmates and count their answers yourself, the data is first-hand.

Collecting first-hand data means you ask the questions, you hear the answers and you record them.

One mark is made for each answer, with nobody missed and nobody counted twice: 24 people asked must give 24 marks in all.

Go down the class list or around the circle so nobody is skipped or asked twice.

Tally marks are grouped in fives, four standing strokes with a fifth across them, so a row is counted by 5s: two bundles and one more stroke is 5, 10, 11.

The fifth mark goes across the four before it, making a bundle of five.