Understanding mathematical terms and notation

Letters

Students need to be able to read and understand the various ways that letters are used in mathematics (MacGregor & Stacey, 1997).

Interpreting letters correctly

Various conventions need to be understood so that students can use this mathematical language and interpret letters correctly in context. These should be taught explicitly to students during the building knowledge phase of the teaching and learning cycle. A general convention that students should be aware of relates to italicised and non-italicised letters:

variables are italicised (e.g. x, y, z)
trigonometric functions are not italicised (e.g. sin, cos)
units of measurement are not italicised (e.g. km, min, cm)

Note that letters can be used as abbreviations of words, but this is not the same as the use of pronumerals in algebra.

Teachers can also provide opportunities for students to discuss various systems of mathematical notation. For example, teachers could:

provide students with a list of statements where letters are used (with or without a context)
ask students to identify what the letters might represent
have students provide justifications for their answers and challenge one another's interpretations.

Examples using and interpreting letters correctly

2m might indicate 2 metres in a measurement context
But in algebra it might represent the shorthand notation for 2 × m (where m is a pronumeral. i.e., the letter is standing in place of a number).

sin(x) does not involve multiplication at all.

In Roman numerals, IX means 1 before 10 (i.e., 9) while XI means 1 after 10 (i.e., 11).

y = mx + b
The equation indicates a linear relationship between the variables x and y, and m and b are called parameters.

Sample teacher prompts for introducing algebra

The examples below suggest some question prompts a teacher can use to introduce algebra to Year 7 students (VCMNA251).

A piece of material of length 8m is used to make costumes.
What does m represent here? How do you know?

2m + 3m equals 5m
What does m represent here? How do you know?

The area of a tile, A, is found by multiplying the length by the width.
What does A represent here? How do you know?

a for number of apples
Why might a have been chosen? Could we use another letter?

2(l+w)
What does this formula represent?

2x compared with IX
What do these two notations symbolise? How are they different? How do you know?
Discuss that 2x is an example of ax representing the product of a and x, compared with Roman numerals IX

Number types

Students need to be able to read and understand the notations for various numbers in mathematics:

fractions
decimals
positive and negative numbers (including integers)
percentages
surds.

Teachers can assist students with understanding the meaning of the notation if they read the notation "for meaning" rather than "spelling out" the symbols. To do this, teachers should provide opportunities for students to link words and symbols.

Two examples are shown below that show how teachers can support students to understand number types in a Year 7 class. There are multiple connections with the Victorian Curriculum (e.g. VCMNA239, VCMNA241, VCMNA242).

Using worded cards to match symbols and words

Teacher provides a set of worded cards with a range of mathematical terms and then another set of cards with symbols.
Students select one of the worded cards.
Students match as many of the symbol cards as they can.
Students may work in pairs or small groups to encourage discussion, joint decision making, and justification.

Examples using worded cards to match symbols and words

Brentwood Secondary College — Example of matching symbols and words

²⁄₃This symbol could be beside the following word cards: proper fraction, positive number, rational number, two divided by three, the result of two things shared between three people, reciprocal of ³⁄₂, numerator 2 and denominator 3

³⁄₂The symbol could be beside the following word cards: improper fraction, positive number, rational number, three divided by two, the result of three things shared between two people, reciprocal of ²⁄₃, numerator 3 and denominator 2

^-12⁄₅
The symbol could be beside improper fraction, negative number, rational number.

Using discussion to reflect on notation

Ask students: "Is there another way of reading this notation?"

The following table provides examples of:

various number types
the common reading (often like "spelling" the symbol)
alternatives which focus more on meaning.

Teachers need to ensure that their students understand the notation before they rely only on the common reading.

Examples of various ways of reading number notation

Example
Numbers	Common reading	Alternatives for reading to add meaning
0.4	point four	1.4 tenths
0.56	point five six	5 tenths and 6 hundredths 56 hundredths
3.12	three point one two (Note: three point twelve is incorrect)	3 and 1 tenth and 2 hundredths 3 and 12 hundredths
³/₅	three over five	3 fifths (i.e., 3 lots of 1 fifth) the result when 3 is shared between 5
⁷/₅	seven over five	7 fifths (i.e., 7 lots of 1 fifth). This is easier to see as 5 fifths and 2 fifths (i.e. 1 ²/₅) the result when 7 is shared between 5
-4	minus 4	negative 4 the number which is 4 units smaller than 0 below 0 on a vertical number line left of 0 on a horizontal number line
√2	root 2	square root 2 a number that when squared gives 2 side length of a square with area 2

Equivalent numbers

Students need to be able to read and understand the different words that are used to indicate equivalence in problems. They also need to understand how this equivalence should be represented:

simplify or reduce ³⁄₆ means to write as ¹⁄₂, i.e. to express as an equivalent number within the same number type (equivalent fractions)
convert ³⁄₆ to a percentage means to write as 50%, i.e. to express as an equivalent number but using a different number type.

Teachers should provide opportunities for students to clarify the terminology associated with naming and writing equivalent numbers. The following verbs are all terms that indicate that students should find an equivalent number:

reduce
simplify
convert
re-express
rename.

Any lesson needs to emphasise that these are all variations of expressing the same idea: writing the same number using different symbols.

The number line is a useful model to use as it emphasises the size of the number (by its position on the line) rather than the symbols used; for example, it can be used to show that ³⁄₆ is in the same position as ¹⁄₂, 0.5 and 50%.

Families of equivalent fractions

Introducing students to the term family of equivalent fractions is useful as it emphasises sets of equivalent fractions.

For the fraction ²⁄₃, the family of equivalent fractions is {²⁄₃, ³⁄₆, ⁶⁄₉, ⁸⁄₁₂, …}.

Examples of classroom strategies

An example of this strategy in a Year 7 class would be to engage students in the construction of a string number line (VCMNA242, VCMNA247).

Scenario

Provide students with a length of string, strips of paper and clips (for attaching the paper strips to the string).

Developing a number line

Ask students to write numbers on the paper strips and to decide where to attach to the string, starting with 0, 1 and 2.

Note that strips of paper are used as they hang down underneath the string and multiple symbols can be written down the one strip to show different ways of writing the same number (i.e., they are located at the same point on the number line).

Ask students to show where ¹⁄₂, ²⁄₂, ³⁄₂, ⁴⁄₂ would be on the number line:

¹/₂ needs a new strip

²/₂ would be written on the same strip as 1

³/₂ needs a new strip and could also be written as 1¹/₂

⁴/₂ would be written on the same strip as 2

Repeat for other fractions, this time for quarters up to 2 (i.e. ¹⁄₄, ²⁄₄ … ⁸⁄₄)

Continue until students have noted various fractions on the one strip (so same place on number line):

Proper fraction example: ¹/₂, ²/₄, ⁴/₈, etc. (“equivalent fractions, fraction family”)

Improper fractions & mixed number examples: 1¹/₂, ³/₂, ⁶/₄, 1²/₄

Extending the activity

This activity can be extended to include percentages and decimals:

¹/₂, 50%, 0.5, 0.50, 0.500 are on the same strip (so same place on number line)

³/₂, 1¹/₂, 150%, 1.5, 1.50, 1.500

This activity can also be extended to include negative integers and negative rational numbers:

^-¹/₂, ^-2/₄, ^-3/₆, -0.5, -0.50, -0.500

^-3/₂, –1¹/₂, -1.5, -1.50, -1.500

Discussing fraction simplification

Discuss with students that "simplifying" a fraction (such as ⁴⁄₈) means identifying the equivalent fraction that uses the smallest denominator, usually the first number they wrote on the paper strip (here ¹⁄₂).

Solution

Ask students, "Which is bigger: ³⁄₆ or ⁴⁄₈?"

The correct answer is "they are the same" but some students say ⁴⁄₈ as they are distracted by the larger numerator and larger denominator.