Elaborations on Level Four: Number and Algebra

The Ministry is migrating nzmaths content to Tāhurangi.           
Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz). 
When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024. 
e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

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 and knowledge

NA4-1: Use a range of multiplicative strategies when operating on whole numbers.

This means students will apply the properties of multiplication and division (commutative, distributive, associative and inverse) to number problems, particularly those requiring multiplication and division. Students should exercise critical choice in their method of calculation - mental, machine or paper - and recognise situations in which estimation should be used, including the checking of calculated answers. Strategies expected at Level Four include: using common factors and multiples, for example 37 + 41 + 40 + 38 = box. as 4 x 40 – 4, using the distributive property, for example 24 x 36 = 20 x 36 + 4 x 36, 9 x 78 = 9 x 80 – 9 x 2, or 276 ÷ 12 = 240 ÷ 12 + 36 ÷ 12, using the associative property, for example 12 x 33 = 4 x 99, or 216 ÷ 12 = 216 ÷ 2 ÷ 2 ÷ 3, and inverse operations (reversing), for example 354 ÷ 6 = box. as 6 x box. = 354. This objective also involves calculating powers, for example 43 = 4 x 4 x 4 = 64, and factorials, for example 4! = 1 x 2 x 3 x 4 = 24. Students should have strong mental strategies for operations on whole numbers but also accurately carry out standard written algorithms, particularly for multi-digit multiplication and division. Level Four corresponds to the Advanced Multiplicative stage of the number framework. Supporting teaching resources.

NA4-2: Understand addition and subtraction of fractions, decimals, and integers.

This means students will understand decimals as fractions, and be able to express decimals in fraction form and vice versa, for example 2.47 = 2 + 4 tenths + 7 hundredths (2 + 4/10 + 7/100 ), or 247 hundredths (247/100). They should solve addition and subtraction problems with decimals and with fractions (denominators must be related multiples), for example 13.2 – 5.79 = 7.41 and 3/4 + 7/8 = 13/8 = 1 5/8 by choosing appropriately from mental, machine and paper methods. Students should apply the strategies used for mental calculation with whole numbers to addition and subtraction of decimals, including standard place value, compensation after rounding, and applying inverse (reversing). Formal written algorithms for decimal addition and subtraction should be taught at Level Four after students have the place value knowledge required to understand them. Supporting teaching resources.

NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.

This means students will understand that finding a decimal or percentage of an amount involves finding a fraction of that amount, for example 40% of 56 = 0.4 x 56 = 4 x 5.6 = 22.4. They should be able to solve problems of the form a/b x c = d (a,b,c and d are whole numbers), where any one of the numbers is not known, for example 4/7 x box. = 24 (Four-sevenths of what number is twenty-four?) or box.% of 76 = 19 (What percentage of seventy-six is nineteen?). Students should be able to multiply fractions with understanding, for example 2/3 x 3/5 = box. as two-thirds of three-fifths, and use their multiplicative understanding of place value to solve multiplication and division problems with simple decimals, for example 1.6 x 0.4 = box. as 16 x 4 ÷ 100 = 0.64 and 24 ÷ 0.3 = box. as 24 ÷ 3 x 10 = 80. Supporting teaching resources.

NA4-4: Apply simple linear proportions, including ordering fractions.

This means students will solve problems involving linear proportions. “Linear proportion” is a term used to generalise situations that involve equivalent fractions. At Level Four students should be able to solve the following types of problems:

  • Comparing the size of two fractions, by converting them to equivalent fractions with a common denominator, or with reference to benchmark fractions, for example 2/3 > 4/9 because 2/3 is greater than one half while 4/9 is less, or because 2/3 = 6/9.
  • Finding equivalent ratios by either scaling up or down by a whole number multiplier, for example 2:5 is the same ratio as 8:20 (scaling up) or 12:18 is the same ratio as 2:3 (scaling down).
  • Finding equivalent rates by either scaling up or down with the same measurement units, for example 18km in 15mins is the same speed as 72km in 60mins.
  • Recognising when two “fraction of an amount” situations give equal or unequal answers, e.g 75% of $12 is the same as 25% of $36.
  • Recognising when sharing division situations give equal or unequal shares, for example three pizzas shared between five people is a smaller share than two pizzas shared between three people.
  • Finding how many measures of a fraction fit into one, for example A trip uses 2/5 of a tank of petrol. How many trips can be made on a full tank? (1 ÷ 2/5 = 5/2 = 2 1/2 ).
    Supporting teaching resources.

NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.

This means students will understand decimals and percentages as equivalent fractions, for example 3/8 = 375/1000 = 0.375 and 3/8 = 37.5/100 = 37.5%. They should know the fractions for halves, thirds, quarters, fifths, eighths, and tenths as decimals and percentages and be able to convert these decimals and percentages back to their simplest fraction form, for example 0.8 = 4/5. The fractions required also include those greater than one, for example 240% = 2.4 = 12/5. Supporting teaching resources.

NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.

This means students will use a mental number line that includes the relative size of integers and decimals to three places and the whole numbers they know from previous levels. They should be able to locate the position of integers and decimals to three places on a given number line with adherence to scale, particularly where tenths and hundredths divisions are given, for example

number line.

Knowing decimal place value involves more than knowing the significance of the position of digits in a whole number, for example in 24.671 the 7 means seven hundredths. 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 ones are tenths, hundredths and thousandths in the same way that nested in the hundreds are tens, ones, tenths, etc., for example 3.509 has 35.09 tenths, 350.9 hundredths, 3509 thousandths, etc... 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 4.2 – 2.68 = box. as the difference between 420 hundredths and 268 hundredths. Students should know the multiplicative relationship between place values, for example one hundredth equals ten divided by one thousand, and the effect of multiplying and dividing a given decimal by ten, one hundred, or one thousand, for example 30.4 divided by one hundred equals 0.304. Students should know the effect of adding and subtracting integers and be able to represent these operations on a number line, for example +3 - -2 = box. and +3 + +2 = box. have the same answer, +5. Supporting teaching resources.

Equations and expressions

NA4-7: Form and solve simple linear equations.

This means students will form and solve simple linear equations in the form y = mx + c, where x and y are related variables and where m is a whole number and c is an integer, for example q = 3p – c, or a + 5 = 4b. When the value of one variable is given the value of the other can be found by solving the equation, for example 3p – 6 = 18. Students should understand the equals sign as a statement of balance and know what operations to both sides of an equation preserve that balance, for example take off the same number from both sides. At Level Four students should be able to find the required value using both sensible estimation and improvement, and by formal methods of applying inverse operations, for example 3p – 6 = 18 so 3p = 24 (adding six to both sides) so p = 8 (dividing both sides by three). Supporting teaching resources.

Patterns and relationships

NA4-8: Generalise properties of multiplication and division with whole numbers.

This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to look at several examples, notice what changes (variables) and what does not, use appropriate mathematical terminology and symbols to describe the pattern, and apply the generalisation to other examples. At Level Four students should be able to describe and apply the properties of multiplication and division as these operations apply to whole numbers. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express generalisations using words and symbols, for example 4 x 6 = 24 so 24 ÷ 6 = 4 and 24 ÷ 4 = 6 (example) leading to a x b = c so c ÷ b = a and c ÷ a = b. This is the inverse relationship of multiplication and division. Supporting teaching resources.

NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.

This means students will describe the function rule for a linear relationship as well as recognise recursive relationships where more complex relationships are involved. For example, given the pattern of fish made with matchsticks and counters below, students should be able to represent the relationships in a table and graph and use these representations to predict the terms in the sequence:

 

fish pattern. fish graph.  

Counters

1

2

3

4

5

Matchsticks

8

14

20

26

32

Level Four students should be able to:

  • give linear rules connecting the variables  (for example, "the number of matchsticks is the six times the number of counters plus two", or "take one off the number of fish, multiply that number by six then add eight")
  • extend the graph or table of a linear relationship to predict further co-ordinate pairs, recognising that constant difference (add six in the fish pattern) is associated with points that lay on a line
  • use recursive methods to predict further members of a sequence where the relationship is non-linear.  For example, the sequence of triangular numbers: 

 

Number of rows

12345

...

10

Total of counters

1361015

...

55
 +2+3+4+5 

Recursive means finding what is added to or subtracted from one term to get the next. Supporting teaching resources.

Click to download a PDF of second-tier material relating to Level 4 Patterns and Relationships (282KB)