Developing an understanding of the meaning of place value is important in assisting students to move on from count-by-one methods for solving addition and subtraction. To be able to use efficient mental and written strategies to solve problems such as 54 – 38, students need to understand that 54 is both 50 + 4 and 40 + 14. That is, the meaning of the 5 in 54 is 5 tens. Partitioning 54 into 40 and 14 is the basis of using trading units when subtracting. This robust understanding of the meaning of place value is less common among students than is often imagined.
Students first learn to represent, say 16 tiles, by using a count. They then associate the word ‘sixteen’ with the quantity, as well as the numeral 16. However, writing or reading 16 need not mean “1 ten and 6 more”.
To see a sample of how some students reason about 16, watch these video clips.
What does 16 mean?
Using problems involving money and packs of gum
How students use groups of 10 when calculating can provide us with a clear sense of what they understand about place value.
Sarah gets 10 cents every day from her grandmother. How much money does Sarah have after 32 days?
Student 1 Part A
Student 1 Part B
Distinguishing between place value levels
One of the key features of a student’s place value understanding is how he or she works with groups of 10 when calculating. Do students use groups of 10 and can they coordinate counting in groups of 10?
Ten as a unit (PV1)
At Place Value Level 1, a student can coordinate counts of tens and ones when dealing with representations of quantity (i.e. visible units of tens and ones).
Tens and ones (PV2)
At Place Value Level 2, ten is both a collection of ones and a single unit of ten that can be counted, decomposed and exchanged for units of different value.
To assist students in solving problems involving multi-digit numbers, units of 10 can be represented in different forms, such as strips of 10 dots or even packets of chewing gum, as in the following tasks.
Student 8 Part A
Student 8 Part B
Understanding hundreds as collections of units of ten
As a student’s understanding of place value grows beyond coordinating exchanges of groups of ten in 2-digit numbers, the importance of the multiplicative relationship between units of ten and hundreds increases.
Hundreds, tens and ones (PV3)
At Place Value Level 3, each hundred is a collection of units of ten that can be counted, regrouped and exchanged for units of different value. For example, 621 can be thought of as 62 groups of ten and one as well as 6 hundreds, 2 tens and one. The exchange of units is reversible
The reversible exchange of units means that not only can the number of tens in 621 be determined without counting by ten but so can the number equivalent to 62 tens and one.
Pieces and Packets
Pieces and Packets - Reversing
A lot of gum: Confusing Pieces and Packets
The confusion between the size of the units and the number of units is sometimes evident in the explanations of high performing students.
A high performing Year 3 student confusing units