“Students read, write, say, interpret and use numbers in common use, including whole numbers, fractions, decimals, percentages and negative numbers.” (Western Australian Minister for Education, 2013) Through their school years students work with a variety of number types. Learning begins with the natural numbers, progressing to whole numbers, integers and then rational numbers. We use these types of numbers in our day to day lives, to shop, cook, measure and communicate at home and professionally. As students enter their secondary school years, they are introduced to irrational numbers which may have specific uses in geometry, physics, biology or trigonometry. Students need the skills to be able to flexibly use all types of numbers to become numerate.
(Blankman, 2021)
Zero
Students need the opportunity to explore the idea of zero or that none are present. “Zero is one of the most important digits in the base- ten system, and purposeful conversations about it and its position on the number line are essential.” (Van de Walle, Karpe, & Bay-Wiliams, 2020) As students build their number sense zero becomes the dividing line between the positive and negative integers, the beginning of the whole numbers and to represent an “empty” place in the base ten number system. Zero can represent many things and students need an understanding of all facets of zero.
Natural Numbers, Whole Numbers and Integers
The first three types of numbers are linked and the ones most people are familiar with. We use these numbers easily to communicate about quantity, space and size.
Natural Numbers
The natural numbers are often referred to as cardinal numbers and are the first one’s children are introduced to. These include the set of positive whole numbers excluding zero. Children learn the natural numbers as they are introduced to verbal and object counting. The natural numbers begin at one and communicate how many are in a group. Natural Numbers are used to communicate about quantity. Students hear this type of number in their day to day lives as parents use them in day to day activities.
Whole Numbers
Whole numbers encompass the counting numbers along with zero. Whole numbers are generally regarded as positive numbers. Students work primarily with the whole number system in their elementary and middle school years.
Integers
Integers are comprised of the whole and natural numbers along with negative whole numbers. The negative numbers are those to the left of zero on the number line. Integers are commonly used to communicate about temperature, elevation, money and locations. Students are introduced to integers towards the end of their middle school or the beginning of their secondary school years, but will have been exposed to them for several years prior through media and day to day situations. It is impossible to communicate about a snow day with out using the negative integers. It is key for students to understand that the negative integers are a mirror of the positive integers that -2 is the same distance from 0 as positive 2. Negatives have the same magnitude as their equivalent positive integers.
Ordinal numbers
Ordinal numbers are used to communicate about order in a sequence. Children recognize first, second and third early and connect them to winning and losing. Ordinal numbers can communicate about order as well as ranking or preference.
Rational numbers
Rational numbers are used to represent numbers that are parts of wholes, fractions and decimals are the most common form of rational numbers.
Fractions
Fractions are so much more than pies and pizzas! Fractions serve several roles representing parts of wholes and groups. We encounter fractions daily in our kitchens, when sharing items, in sports and science. Hockey games are divided into thirds, football into quarters, soccer into halves. Standard recipes communicate measurements in fractions of cups and teaspoons. Fractions are one way of describing parts of a whole. Fractions can also represent parts of groups and are linked to ratios, percentages and decimals. We communicate about time in fractions referring to 30 minutes as half an hour or 15 minutes as a quarter hour. Fractions can be used as measurements communicating about small objects such as the width of a human hair. A key fraction understanding is the ability to connect equivalent fractions and convert between different equivalents.
Decimals
“People need to be able to interpret decimals for such varied needs as reading precise metric measures, calculating loans and mortgages, interpreting output on a calculator, and understanding sports statistics such as those at the Olympics, where winners and losers are separated by hundredths of a second. Decimals are critically important in many occupations: For nurses, pharmacists, and workers building airplanes, for example, precision affects the safety of the general public.” (Van de Walle, Karpe, & Bay-Wiliams, 2020) Decimals are an extension of the place value system yet pose their own challenges. They represent quantities that are greater than the number to the left of the decimal yet not quite the next whole number. Students encounter decimals daily in their environment, it is not possible to work with money without using decimals. Decimals are a form of rational numbers. Decimals can also be counter intuitive in their naming, the “pieces” in tenths are ten times the size of hundredths, when students expect that ten tens make 100. This tips the previously established place value system on its head as the names for decimal places form a mirror to the names on the left of the decimal.
Irrational numbers
Irrational numbers are one’s students will encounter later in their educational careers and are used less commonly. Examples of irrational numbers include constants like Pi as well as non-repeating decimals, and many roots.