Number and Algebra: Number knowledge, Level 3
AO1: Know basic multiplication and division facts. This means students will know basic multiplication facts are those that range from 0 x 0 = 0 to 9 x 9 = 81. The division basic facts are the inverse of the multiplication facts. So 6 x 4 = 24, 4 x 6 = 24, 24 ÷ 6 = 4 and 24 ÷ 4 = 6 are all basic facts. Students should commit their basic facts to memory as soon as they understand the meaning of the equations and can use number properties to work them out, 8 x 7 = 56 means "eight sets of seven" and can be worked out by doubling 4 x 7 = 28. Note that 56 ÷ 7 can mean "fifty-six shared among seven" or "how many sevens are in fifty-six".
AO2: Know counting sequences for whole numbers. This means students will know the forward number word sequence for whole numbers is the counting pattern of words and symbols, 0, 1, 2, 3, 4,... ∞ (infinity) while the backward sequence is the pattern 1000 000, 999 999, 999 998, 999 997, ...beginning with any whole number. At Level Three students should know these sequences in multiples of one, ten, e.g. 358, 348, 338,..., one hundred, e.g. 247, 347, 447,..., one thousand, etc. An important part of knowing these sequences is being able to name the number before and after a given number since this relates to taking an item off or putting an item onto an existing set, e.g. If a set contains 43 560 items, 43 559 items are left if one is removed and 43 561 items are in the set if one is added. This also applies to the sequence in tens, hundreds, thousands, etc. e.g. ten thousand removed from a set of 701 000 results in 691 000 objects left. At Level Three students should also have experience with counting sequences in tenths, e.g. 4.6, 4.7, 4.8, 4.9, 5 ,...
AO3: Know how many tenths, tens, hundreds, and thousands are in whole
numbers.
This means students will develop a multiplicative view of whole number place value involves more than
knowing the significance of the position of digits in a whole number, e.g. In 239 456 the 3 means three ten thousands.
Strategies for computation require a nested view of place value and understanding the scaling effect as digits move to the
right and left in place value. This means that nested in the thousands are hundreds, tens and ones in the same way that
nested in the tens are ones and tenths, e.g. 239 456 has 23 ten thousands, 2394 hundreds, and 23 945 tens, etc.
An understanding of nested place value is best demonstrated by calculations where place value units must be constructed
by combining or decomposing other place value units. For example, calculations like 2 004 - 700 = 
may require students to think of 1000 as ten hundreds. At Level Three students should connect the multiplicative value of the places, e.g.
one hundred thousand is ten times as much as ten thousand, and one hundred is the result of dividing one thousand by ten.
They should know the effect of multiplying and dividing by ten, 4200 is ten times more than 420 and 43 divided by ten is 4.3.
AO4: Know fractions and percentages in everyday use. This means students will understand the meaning of the digits in a fraction, how the fraction can be written in numerals and words, or said, and the relative order and size of fractions with common denominators (bottom numbers) or common numerators (top numbers). Fundamental concepts are that fractions are iterations (repeats) of a unit fraction, e.g. 3/5 = 1/5 + 1/5 + 1/5 and 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3. This means the numerator (top number) is a count and the denominator tells the size of the parts, e.g. in 5/3 there are five parts. The parts are thirds created by splitting one into three equal parts. This means that fractions can be greater than one, e.g. 4/3 = 1 1/3, and that fractions have a counting order if the denominators are the same, e.g. 1/3, 2/3, 3/3, 4/3,... The size of the denominator also affects the size of the parts being counted in a fraction. For example, thirds of the same whole are smaller than halves of the same whole. So fractions with common numerators have an order of size based on the size of the parts, e.g. 2/7 < 2/5 < 2/3 (< means “less than”). Students at Level Three should know simple common fraction-percentage relationships, including 1/2 = 50%, 1/4 = 25%, 1/10 = 10%, 1/5 = 20%, and use this knowledge to work out non-unit fractions as percentages, e.g. 3/4 = 75%.
