Elaborations on Level Three: Number and Algebra

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In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

Number strategies

NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.

This means students will use a range of mental strategies based on partitioning and combining to solve addition and subtraction problems with multi-digit whole numbers and simple decimals (tenths). These strategies include standard place value, for example 603 – 384 = box. as 60 – 38 tens less one (219), rounding and compensating, for example 923 – 587 = box. as 923 – 600 + 13 = box. , and reversing (applying inverse), for example 923 – 587 = box. as 587 + box. = 923. Students should also connect known multiplication facts to solve multiplication and division problems, for example 13 x 6 = box. as 10 x 6 + 3 x 6 = box. (distributive property), 14 x 9 = box. as 2 x (7 x 9) = box. (associative property) and 36 ÷ 9 = box. using 4 x 9 = 36 (inverse). This multiplicative understanding allows students at Level Three to find fractions of quantities, for example two-thirds of 24 as 24 ÷ 3 x 2 = 16, find simple equivalent fractions related to doubling and halving, for example 3/4 = 6/8 , to add and subtract fractions with the same denominators, for example 3/4 + 3/4 = 6/4 = 1 2/4, and to convert improper fractions to mixed numbers, for example 17/3 = 5 2/3. Students should know the decimals and percentage conversions of simple fractions (halves, quarters, fifths, tenths) and use these to solve simple percentage of amount problems, for example 50% is fifty out of one hundred. 50% is one half so 50% of 18 is 9 or five is half of ten. Level Three corresponds to the Advanced Additive stage of the number framework. Supporting teaching resources.

Number knowledge

NA3-2: 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". Supporting teaching resources.

NA3-3: 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, for example 358, 348, 338..., one hundred, for example 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, for example 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. for example 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, for example 4.6, 4.7, 4.8, 4.9, 5... Supporting teaching resources.

NA3-4: 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 that involves more than knowing the significance of the position of digits in a whole number, for example 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, for example 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 = box. may require students to think of 1000 as ten hundreds. At Level Three students should connect the multiplicative value of the places, for example 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. Supporting teaching resources.

NA3-5: 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, for example 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, for example 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, for example 4/3 = 1 1/3, and that fractions have a counting order if the denominators are the same, for example 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, for example 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, for example 3/4 = 75%. Supporting teaching resources.

Equations and expressions

NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.

This means students will use words, symbols and diagrams to explain their number strategies to others. Recording also allows students to think through solutions to problems and allows them to reduce their working memory load by storing information in written form. This is particularly important for the solving of complex, multi-step problems. Students should be able to write the numerals for whole numbers to 1 000 000 at least, simple fractions, percentages and decimals. They should also be able to write addition, subtraction, multiplication and division equations with understanding of the meaning of these operations and of the equals sign as meaning “equal to”. Similarly, they should know which operation to perform on a calculator if the numbers are beyond their mental range. Students should also be familiar with using empty number lines to record addition and subtraction strategies, arrays to record multiplication and division strategies, and strip diagrams or double number lines to solve problems with fractions and percentages. Formal written algorithms for multi-digit addition and subtraction should be taught at Level Three after students have the nested place value knowledge required to understand them. Supporting teaching resources.

Patterns and relationships

NA3-7: Generalise the properties of addition and subtraction with whole numbers.

This means students will generalise, which means to establish properties that hold for all instances. Generalisation begins with noticing patterns and relationships in a few specific instances, defining the variables involved, noticing the relationships between the variables, then using appropriate mathematical terminology and symbols to describe the relationships. At Level Three students develop many generalisations that allow them to perform mental strategies effectively. These generalisations include the commutative property of addition and multiplication, for example 7 x 8 = 8 x 7, the associative property of addition and multiplication, for example (2 x 3) x 4 = 2 x (3 x 4), the distributive property of multiplication, for example 8 x 7 = 8 x 5 + 8 x 2, the inverse relationships of addition and subtraction, and of multiplication and division, for example 6 x 7 = 42 so 42 ÷ 7 = 6, and identities for all four operations, for example 17 x 1 = 17, 17 ÷ 1 = 17. It is not expected that students use algebraic symbols to express these generalisations. However, students should be able to look for relationships across the equals sign in equations to determine missing numbers, for example 4 x 12 = box. x 6 without calculating 4 x 12. Supporting teaching resources.

NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.

This means students will recognise that a sequential pattern can be either spatial, for example spatial sequence. .., or numeric, for example 1, 3, 5, 7... A pattern has consistency so further terms of it can be anticipated from those already known. The focus in this thread is that students become increasingly sophisticated at describing the relationships between variables found in sequential patterns. With spatial patterns, students at Level Three should be able to identify the repeating element, for example spatial3. , and use simple multiplicative thinking to predict the shape in a given ordinal position, for example Every third shape is spatial diamond. so the thirtieth shape will be spatial diamond. so the thirty-second shape will be circle. With number patterns students should identify the consistent relationship between variables in simple multiple situations, for example 4, 8, 12, 16... are all multiples of four, or identify the additive “gap” between the terms, for example 4, 7, 10, 13... three is added each time. They should be able to describe these rules in their own words and use their rules to find further terms. Students also use tables, graphs, diagrams and word rules to find and describe relationships in patterns, for example

 

pegs.

Towels

Pegs

1

2

2

3

3

4

4

5

pegs graph.

“There is always one more peg than the number of towels. The first towel took two pegs.” Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 3 Patterns and Relationships (319KB)