Mathematics 3

Ten of one place make one of the next place to the left, all the way up: 10 ones make a ten, 10 tens a hundred, 10 hundreds a thousand, 10 thousands a ten thousand, and 10 ten thousands a hundred thousand.

The places do not grow by adding 10 each time: each new place is 10 times the one before it.

A mix of coins is counted starting with the coins worth the most: 2 quarters, 1 dime and 3 nickels are counted 25, 50, 60, 65, 70, 75, so they are worth 75 cents.

Sort the coins by kind and start with the quarters, then the dimes, then the nickels.

Adding place by place works for three-digit numbers the same way as for two-digit ones: 352 + 214 is 300 + 200 = 500, 50 + 10 = 60 and 2 + 4 = 6, so 566.

Split each number by its places, add the hundreds, the tens and the ones, then put the parts together.

An estimate is enough when an exact sum is not needed: two classes collected 389 and 412 cans, which is about 400 + 400 = 800 cans in all.

Rounding each number to the nearest hundred makes the adding easy in your head.

Multiplication counts things in equal groups: 4 bags with 3 apples in each hold 12 apples, written 4 × 3 = 12 and read four times three.

One number tells how many groups there are and the other tells how many are in each group.

An equation says that two expressions name the same number: in 6 + 2 = 5 + 3 both sides are 8, and the equal sign joins them.

An expression is a number or a calculation, such as 6 + 2 or 4 × 3.

An equation is true only when both sides are the same number: 9 + 4 = 6 + 7 is true, 13 each, and 9 + 4 = 6 + 8 is false, because 13 is not 14.

Work out each side first, then compare.

A pan balance models an equation: 4 + 3 cubes on one pan and 5 + 2 cubes on the other keep the pans level, and the equation 4 + 3 = 5 + 2 is true.

Each pan holds one side of the equation.

The two sides of an equation can trade places: 7 = 3 + 4 says the same as 3 + 4 = 7, and 4 + 3 = 5 + 2 can be written 5 + 2 = 4 + 3.

An equation compares two sides, so it does not matter which side is written first.

Equations are written to show that a number and an expression, or two different expressions, are the same number: 24 = 6 × 4, and 6 × 4 = 30 − 6.

Pick a number and find different ways to make it: 24 is 6 × 4, 20 + 4, 30 − 6 and 48 ÷ 2.

A polygon is a closed flat shape with straight sides, named by its number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.

Count the sides and match the name: tri means three, quad means four, penta five, hexa six and octa eight.

A regular polygon has all sides equal and all angles equal: a square is regular, but a long rectangle is irregular, because its sides are not all equal.

Both parts must hold.

Polygons with the same number of sides share a name however different they look: a house-shaped pentagon and a pentagon with five equal sides are both pentagons, because each has 5 straight sides.

A polygon's group comes from its properties, not its looks, and the number of sides decides the name.

A polygon is classified as regular only after two checks, every side equal and every angle equal: a strip of paper marked to one side's length tests the sides, and a traced corner tests the angles.

The marked strip is laid along every other side, and a side longer or shorter fails.

Squares and rectangles are quadrilaterals too, because quadrilateral is the name for every polygon with 4 straight sides.

A square and a rectangle have their own names, and they also belong to the bigger group of quadrilaterals.

The metre is the basic unit of length in the metric system, and the other metric units of length are named as parts of it: the decimetre, the centimetre and the millimetre.

Every metric unit of length has metre in its name, and the front of the name tells which part of a metre it is.

A ruler, a metre stick and a measuring tape each show copies of one standard unit laid end to end from a starting point, the zero mark, so reading a tool is counting units from 0.

On a ruler the centimetre marks are equal steps, and the number at each mark counts the steps from 0.

Referents for a centimetre and a metre are lengths you know well: the width of a finger is about 1 cm, and one big step is about 1 m.

A referent is checked once against a ruler or a metre stick, then used anywhere without a tool.

An angle is formed where two arms meet at a common end point, the vertex, and the arms can be line segments, with two ends, or rays, which go on forever in one direction.

A line segment is a straight piece with two end points, like a stick.

The prefix of a metric unit tells how many of it make a metre: deci means 10 decimetres in a metre, centi means 100 centimetres in a metre, and milli means 1000 millimetres in a metre.

The more parts a metre is cut into, the smaller each part is, so the millimetre is the smallest of the three and the decimetre the biggest.

A sequence is a list of numbers, called terms, written in a certain order: in 5, 10, 15, 20 the first term is 5 and the fourth term is 20.

The order matters: 20, 15, 10, 5 uses the same numbers but is a different sequence.

Ordinal numbers name positions in a sequence: first, second, third, fourth and fifth, written 1st, 2nd, 3rd, 4th and 5th, so in 3, 6, 9, 12, 15 the 3rd term is 9.

Counting numbers tell how many, and ordinal numbers tell which place.

A table can show each term under its position: positions 1st, 2nd, 3rd and 4th above the terms 4, 8, 12 and 16 show that the 3rd term is 12.

Writing the positions in one row and the terms in the row below keeps them from getting mixed up, and reading down one column gives a position and its term.

The even numbers and the odd numbers are familiar sequences: 0, 2, 4, 6, 8 counts by 2s from 0, and 1, 3, 5, 7, 9 counts by 2s from 1.

Every whole number belongs to exactly one of the two sequences, so the evens and the odds take turns along the number line.

Other familiar sequences are the counting numbers 1, 2, 3, 4, skip counts such as 5, 10, 15, 20, and countdowns such as 10, 9, 8, 7, which go down.

Recognizing a familiar sequence means seeing its pattern right away, without working it out.

Clocks are standard tools for measuring time, and the basic unit of time is the second, about as long as it takes to say one thousand one.

Minutes and hours are built out of seconds.

A day is two 12-hour cycles: the a.m. hours run from midnight to noon, and the p.m. hours run from noon to midnight.

A clock face shows only 12 hours, so every time on it happens twice a day, once in each cycle.

Clocks count time in sixties: 60 seconds make a minute and 60 minutes make an hour, so one second is $\frac{1}{60}$ of a minute and one minute is $\frac{1}{60}$ of an hour.

Time does not go by tens like place value or money: after 59 seconds a new minute begins, and after 59 minutes a new hour begins.

A time of day tells how long it has been since 12:00: 3:20 p.m. is 3 hours and 20 minutes after noon, and 8:00 a.m. is 8 hours after midnight.

m.

An analog clock shows seconds, minutes and hours together: the thin second hand goes all the way around in 1 minute, the long minute hand in 1 hour, and the short hour hand in 12 hours.

The 60 small marks around the face count seconds for the second hand and minutes for the minute hand.

A statistical question is answered by collecting data that varies: how many pets does each classmate have? gets different answers, while how many legs does a dog have? has only one.

A statistical question needs many answers before it can be answered, because people or things differ.

Data collected for a statistical question give a possible answer and show how strong it is: if 11 of 24 classmates chose blue as their favourite colour, blue is the most common answer, though most chose other colours.

, but the counts show how sure it is: 11 of 24 is less than half, so 13 classmates chose something else.

A statistical question for a class is one the people being asked can answer: how many letters are in your first name? works, but how long were you when you were born? does not, because few people know.

Before collecting, the question is tested: can everyone asked answer it, and will the answers vary?

A set of data belongs to one time and one place: a survey of favourite sports in one Grade 3 class in winter may come out differently in another school, or in summer.

Data describe the people or things that were counted, when and where they were counted.

A good statistical question says clearly who is asked and what is counted: how many hours did each student in our class sleep last night? is clearer than how much do kids sleep?

A clear question makes the data easy to collect and to compare, because it says who is asked, what is counted and when.