10 of these is 1 of those

10 is a very important number in our place value number system. This is why it is called a decimal place value system. When we group 10 of these to make 1 of those, we are transforming 10 individual ones into a larger unit that is ten times the value. This means that each position in our number system has a value that is 10 times the value of the previous position as we move from right to left. When we move from left to right each position has a value that is one-tenth the value of the previous position.

The units of hundreds, tens and ones form an important repeating pattern in our place value system. This allows for large numbers to be read in syllables of hundreds, tens and ones. Each time the hundreds, tens and ones pattern repeats, the value of each position is 1000 times greater in value.

One way to think about this pattern is as a series of houses. Each house comprises three occupants—hundreds, tens, and ones—and the number name given to a house is 1000 times the value of the previous house as we move from right to left. For example, there is:

• the ones house, containing hundreds of ones, tens of ones and ones.
• the thousands house, containing hundreds of thousands, tens of thousands, and thousands.
• the millions house, containing hundreds of millions, tens of millions, and millions.

This hundreds, tens, and ones repeating pattern unit simplifies reading large numbers—if a student is able to read a 3-digit number then they can use this pattern to read any number. For example, students may find the number 23 532 604 very hard to say when it is written as numbers. However, by separating the number into units of hundreds, tens and ones, it becomes easier to read and say.

Foundation

In Foundation, the emphasis is on unitary thinking. Students develop fluency with the sequence of ones and using one-to-one matching to count collections. The power of 10 in our number system can be emphasised at this early stage by introducing 5 and 10 as useful benchmark numbers and through representations such as 10-frames, stacks and bundles of 10 items, and rekenreks.

Year 1

In Year 1 students develop more efficient counting strategies, as the range of numbers extends to 120. They count collections by systematically grouping loose objects in tens, and learn that numbers can be represented using units of different sizes: tens and ones. The process of grouping 10 ones together to form 1 ten introduces the powerful idea that 10 of these is 1 of those.

Students formalise the idea that 10 of these is 1 of those as they progress from counting collections by ones and grouping them in tens to multi-unit thinking, where they recognise that 10 ones is 1 ten. This realisation is the foundation for understanding the base-10 property of our number system.

Students apply the powerful idea that 10 of these is 1 of those to partition 2-digit numbers in different ways, by systematically exchanging 1 ten for 10 ones to represent numbers flexibly. They shift from recognising that a 2-digit number like 45 has "4 tens and 5 ones" to understanding that it is equivalent to "3 tens and 15 ones", "2 tens and 25 ones", "1 ten and 35 ones" and "45 ones".

Year 2

In Year 2 students extend the powerful idea that 10 of these makes 1 of those as they learn to read, write, order, compare and represent numbers to 1000. The base-10 property of our number system emerges as an ongoing relationship where 10 ones make 1 ten and 10 tens make 1 hundred. Students rename and regroup numbers to develop knowledge and skills that are the foundation for calculation strategies. They apply tens-based relationships between hundreds and tens and tens and ones to rename numbers. For example, when subtracting 78 from 123 they might rename 123 as 11 tens and 13 ones.

Related sequences

Year 3

In Year 3 students extend their understanding of whole number place to thousands, tens of thousands, and hundreds of thousands. They reapply the powerful idea that 10 of these is 1 of those as they identify the base 10 relationship between each new position and the previous one. When viewing a number like 475 512 students recognise that the digit 5 in the thousands position has a value that is 10 times the value of the digit 5 in the hundreds position.

Students encounter the significance of 1000 in our place value system as they work with numbers with up to 6-digits. They extend the idea that 10 of these is 1 of those to recognise that 1000 of these is 1 of those, and meet the power and efficiency of the base 10 property. They recognise that moving a digit 3 positions to the left results in 1000 times the value, because 1000 is $10 \times 10 \times 10$.

Representing numbers in the thousands draws students’ attention to the repeating pattern of ones, tens and hundreds in the naming of numbers. Students move from working with 1s, 10s and 100s of ones, to reading, writing, ordering, interpreting and representing 1s, 10s and 100s of thousands. The meaning of ‘thousands’ shifts from being the name of one position to encompassing a group of three place value positions: ones of thousands, tens of thousands and hundreds of thousands.

Students use the repeating pattern of ones, tens and hundreds to name larger numbers. Students learn to say the number of hundreds, tens and ones and then the name of the grouping. For example, 456 489 is read as "456 thousand, 489". Students learn the convention of leaving a space or adding a comma between the ones of thousands and the hundreds position in numbers with more than four digits. They recognise that when the space in a number with more than 4 digits is omitted, e.g. 456489, it is more difficult to name the number. The convention of leaving a space makes the repeating pattern of hundreds, tens and ones, and place value groupings, more salient.

Year 4

In Year 4, students extend their understanding of the place value system to decimal tenths and hundredths. They reason that if 10 of these is 1 of those, then 1 of those is 10 of these as they unpack the place value system in reverse. Just as 1 hundred is 10 tens and 1 ten is 10 ones, 1 whole is 10 tenths and 1 tenth is 10 hundredths. Students apply the base-10 property to generalise the idea that the value represented by each position is one-tenth the value of the previous position moving left to right. They learn that the position to the right of the ones is the tenths position and the position to the right of the tenths is the hundredths position. Students identify the ones as the axis of symmetry in our place value system. They recognise that the repeating pattern unit of ones, tens, hundreds is reflected across the ones position to reveal ones, tenths and hundredths.

Year 5

In Year 5, students continue to use the repeating pattern of ones, tens and hundreds to read, record and interpret an increasing range of larger and smaller numbers. They extend the decimal place value system to thousandths. The powerful idea that 10 of these is 1 of those is reinforced as students subdivide materials to model tenths, hundredths and thousandths and explain the multiplicative relationship between them. They apply the idea that 1 of these is 10 of those to rename decimals. For example, to calculate $0.6 \div 10$ they rename 6 tenths as 60 hundredths. Students use the base-10 property and sliding place value to compare the place value of digits by determining numbers that are 10, 100 or 1000 times, as well as one-tenth, one-hundredth, or one-thousandth the value of the original number.

Year 6

In Year 6, the foundations for index notation and representing numbers using other bases are developed. Students learn that any number multiplied by itself is a square number which can be recorded using index notation. For example, $3 \times 3 = 3^2$. Students apply the powerful idea that 10 of these is 1 of those to multiply decimals by multiples of powers of 10. For example, to multiply 2.34 by 20 students might double 2.34 to get 4.68 and use the base-10 property to recognise that $10 \times 4.68$ is 46.8.

Year 7

In Year 7 students represent natural numbers in expanded notation using place value and powers of 10. They apply the base-10 property as they explain connections between place value and expanded notation, e.g. $3750 = 3 \times 10^3 + 7 \times 10^2 + 5 \times 10^1$.

Students develop a more profound grasp the powerful idea that 10 of these is 1 of those as they recognise that the sequence of 10, 100, 1000, 10 000, 100 000, 1 000 000 … is synonymous with $10^1,10^2,10^3,10^4,10^5,10^6,10^7,...$

and how the base 10 number system enables concise representation of quantities of any size. Students increase their awareness of other bases and generalise the properties of bases as they become familiar with sequences including powers of 2, 3, and 5.

Year 8

In Year 8 students further extend their understanding of the number system as they are introduced to irrational numbers and terminating and recurring decimals. They recognise that the real number system includes irrational numbers which can only be approximately located on the real number line. For example, they understand that the value of pi lies somewhere between 3.141 and 3.142.

Students establish and apply exponent laws for powers of 10 with positive integer exponents to identify:

• the product of power rule, e.g. $10^3 \times 10^3 = 10^6$ because $(10 \times 10 \times 10) \times (10 \times 10 \times 10) = 10^6$
• the quotient of powers rule, e.g. $10^6 \div 10^3 = 10^3$ because $\frac{10 \times 10 \times 10 \times 10 \times 10 \times 10}{10 \times 10 \times 10} = 10 \times 10 \times 10$

Year 9

In Year 9 students investigate the real number system more deeply by representing relationships between irrational numbers, rational numbers, integers, and natural numbers. They apply the properties of place value as they represent decimals in exponential form. For example, they represent $ 0.475 = \frac{4}{10} + \frac {7}{100} + \frac {5}{1000}$ as $0.475 = 4 \times 10^{-1} + 7 \times 10^{-2} + 5 \times 10^{-3}$, or represent decimals such as 0.000023 as $23 \times 10^-5$.

Year 10

In Year 10 students learn to recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations. They apply exponent laws involving products, quotients and powers of variables to expand, factorise and simplify expressions and solve equations algebraically.