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# Elaborations on Level Two: Number and Algebra

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

NA2-1: Use simple additive strategies with whole numbers and fractions.

This means students will learn to treat whole numbers as units of ones that can be split and recombined to make calculations easier. Additive strategies are about a type of thinking not the operation of addition. So additive strategies can be applied to addition (for example, 47 + 38 is 50 + 40 – 5), subtraction (for example, 74 – 8 =  as 74 – 4 – 4 = ), multiplication (for example, 4 x 6 =  as 4 + 4 + 4 + 4 = , which is 8 + 8 = ), division (for example, 18 ÷ 3 = , as 5 + 5 + 5 = 15 so 6 + 6 + 6 = 18). Additive strategies may also be applied to finding fractions of sets particularly halves, thirds, quarters, fifths, eighths and tenths. Level Two corresponds to students being proficient at the Early Additive stage of the number framework. Supporting teaching resources.

### Number knowledge

NA2-2: Know forward and backward counting sequences with whole numbers to at least 1000.

This means students will know the forward number word sequence to 1000 is the counting pattern of words and symbols, 0, 1, 2, 3, 4...1000 while the backward sequence is the pattern 1000, 999, 998, 997 ... At level Two students should know these sequences in multiples of one ten (for example, 358, 348, 338...) and one hundred (for example, 247, 347, 447...) 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 800 items, 799 items are left if one is removed. This also applies to the sequence in tens and hundreds, for example, ten removed from a set of 503 results in 493 objects left. Supporting teaching resources.

NA2-3: Know the basic addition and subtraction facts.

This means students will know the basic addition facts from 0 + 0 = 0 to 9 + 9 = 18. So 4 + 1 = 5, 8 + 6 = 14, and 9 + 3 = 12 are all basic addition facts. The basic subtraction facts are the subtraction equivalent of the addition facts, so 5 – 1 = 4, 5 – 4 = 1, 12 -3 = 9 and 12 – 9 = 3 are all examples. It is important that students understand the commutative property of addition (for example, 4 + 7 = 7 + 4), and the inverse nature of addition and subtraction (for example, 6 + 7 = 13 so 13 – 7 = 6), as a foundation for more difficult problems, as well as a way to connect basic facts. Students also need to encounter the unknown in different positions within their basic facts, for example, 4 +  = 12 and   – 5 = 8. Supporting teaching resources.

NA2-4: Know how many ones, tens, and hundreds are in whole numbers to at least 1000.

This means students will develop an additive view of whole number place value by knowing the significance of the position of digits in a whole number, for example, in 456 the 5 means five tens. However, many strategies for computation require a nested view of place value. This means that nested in the hundreds are tens in the same way that nested in the hundreds and tens are ones, for example, 456 has 45 tens and 456 ones. An understanding of nested place value is best demonstrated by calculations where tens must be constructed from ones, hundreds constructed from tens, tens created from breaking hundreds and ones created from breaking tens. For example, calculations like 456 + 70 = , or 456 - = 396, show whether students can apply place value in this way. Supporting teaching resources.

NA2-5: Know simple fractions 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). Fundamental concepts are that fractions are iterations (repeats) of a unit fraction, for example, 3/4 = 1/4 + 1/4 + 1/4 and 4/3 = 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 4/3 there are four 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... Note that whole numbers can be written as fractions, for example = 1. Fractions in everyday usage include halves, thirds, quarters (fourths), fifths, eighths, and tenths. Supporting teaching resources.

### Equations and expressions

NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.

This means students will be able to 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 1000, and simple fractions. 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 and of drawing arrays to record simple multiplication and division strategies. Formal written algorithms for multi-digit addition and subtraction should not be taught at Level Two until students have the place value knowledge required to understand them. Supporting teaching resources.

### Patterns and relationships

NA2-7: Generalise that whole numbers can be partitioned in many ways.

Students at level two should understand that numbers are counts that can be split in ways that make the operations of addition, subtraction, multiplication and division easier. From Level One students understand that counting a set tells how many objects are in the set. Building on this thinking at Level Two is to realise that the count of a set can be partitioned and that the count of each subset tells how many objects are in that subset. Also required is understanding that partitions of a count can be recombined. For example, a count of ten can be partitioned into 1 and 9, 2 and 8, 3 and 7, etc. This objective also involves critical choice of partitioning. For example, 8 + 6 = can be solved by partitioning 6 into 1 and 5, 2 and 4, 3 and 3. Of these partitions 2 and 4 is the best strategic choice since it recombines into a “ten and...” fact, that is, 8 + 6 = 8 + 2 + 4 = 10 + 4. At Level Two students are expected to understand the strategic importance of using place value as a way to partition numbers. Students should apply their partitioning generalisation to many problem types including combining (27 + 9 = ), separating (105 - 19 = ), comparing (45 + , = 106), duplicating (8 x 5 = ) and sharing (20 ÷ 4 = ). Supporting teaching resources.

NA2-8: Find rules for the next member in a sequential pattern.

This means students will explore sequential patterns, either spatial, for example, , ... or numeric, for example, 1, 3, 5, 7 ... A pattern has consistency so further terms of it can be anticipated from those already known. In spatial patterns students should be able to identify the repeating element, for example, in the pattern above, and use this to predict the shape in a given ordinal position, for example, the next shape is , the eleventh shape will be . For simple number patterns students should identify the consistent “gap” between the terms (for example, 1, 3, 5, 7... two is added each time), and use this additive difference to find further terms. Students should also develop their concept of relations between variables using spatial patterns that can be represented using numeric tables of values, for example, for this pattern, how many squares make 7 crosses? Supporting teaching resources.

 1 cross5 squares 2 crosses10 squares 3 crosses15 squares

Click to download a PDF of second-tier material relating to Level 2 Patterns and Relationships (203KB)