This unit builds students’ understanding of place value, extending to 6 digit whole numbers.
Understanding place value is crucial if students are to develop the estimation and calculation skills necessary to become numerate adults. Our number system is based on groupings of ten. Ten ones form one ten, ten tens form one hundred, ten hundreds form one thousand, and so on. The system continues, giving us the capacity to represent very large quantities. The Place values such as one, ten, one hundred, one thousand are powers of ten. That means that the place immediately to the left of a given place represents units that are ten times more than the given place, e.g. thousands are ten times greater than hundreds.
Ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to represent all the numbers in a base ten number system. A new number is not needed to represent ten, because it can be thought of as one group of ten. Similarly, when one is added to 999, we write 1000. Therefore, we do not need a separate number symbol for one thousand. The position of the 1 in 1000 tells us about the value it represents. Zero has two uses in the number system, as the number for ‘none of something’, e.g. 6 + 0 (i.e. none of something) = 6, and as a placeholder, e.g. 7040. A number is a placeholder when it occupies a place, or several places. This allows for communication of the values represented by other digits. For example, in 7040 zero acts as a place holder in the hundreds and tens places. In turn, this communicated the value of the 7 and 4 digits.
Place value means that both the position of a digit, as well as the value of that digit, indicate what quantity it represents. In the number 2753, the position of the 7 is in the hundreds column, meaning it represents seven hundred. 2 is in the thousands column which means that it represents two units of one thousand, called 2000.
Understanding the nested nature of place value is necessary for students to operate on whole numbers and decimals. Nested means that the places are connected, e.g. within hundreds there are tens, within ones there are tenths. Renaming a number flexibly is an important application of nested place value.
In particular, it is vital that students understand that when ten ones are combined they form a unit of ten, when ten tens are combined they form a unit of one hundred, and when ten hundreds are combined they form a unit of one thousand. For example, the answer to 2610 + 4390 is 7 thousands since 610 and 390 combine to form another thousand. Similarly, when a unit of one thousand is ‘decomposed’ into ten hundreds, the number looks different but still represents the same quantity. For example, 4200 can be viewed as 4 thousands, and 2 hundreds, or 3 thousands and 12 hundreds, or 2 thousands and 22 hundreds, etc. Decomposing is used in subtraction problems such as 7200 – 4800 = □ where it is helpful to view 7200 as 6 thousands and 12 hundreds.
At Level 3 students need to develop a multiplicative view of place value that includes understanding the relative size of quantities represented by different numbers. A nested view of 230 as 23 tens allows multiplicative connection between 23 and 230. 230 is ten times larger than 23, and 23 is ten times smaller than 230. Such knowledge can be expressed with equations, 23 x 10 = 230, 10 x 23 = 230, 230 ÷ 10 = 23. Multiplication and division basic facts can be leveraged for harder calculations, 4 x 3 = 12 so 4 x 30 = 120 (ten times more). 30 x 4 = 120 as well. 12 ÷ 3 = 4 so 120 ÷ 30 = 4.
The learning activities in this unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. Ways to support students include:
This unit is focussed on the place value structure of whole numbers and as such is not set in a real world context. Learning to read and write numbers in Māori or other Pacific languages will support students’ developing understandings, because number names are derived from their place value structure in these languages. Numbers in te reo Māori can be used throughout this unit.
The activities in this unit could be taught in succession over a number of days to provide a concentrated focus on building place value knowledge. Alternatively, selected activities could be used to support place value understanding while students are working on solving number problems.
Activity 1
Activity 2
Activity 3
Activity 4
Activity 5
Activity 6
Introduce the game 11,111 (Copymaster 4) and support students to play this game in small groups (i.e. 2-4 students).
Dear parents and whānau,
In class we have been learning to understand very big numbers, their value and how to read them correctly. It is important for students to be able to read numbers up to a million and beyond, and understand their structure. It would be helpful if your child could practice this at home. Find examples of real, large numbers (e.g. in real estate or car sales) and practice reading these. Challenge them to find the highest and lowest prices advertised.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/place-value-whole-numbers at 9:08pm on the 26th February 2024