This unit requires students to work with the number of ones, tens, hundreds and thousands in four-digit whole numbers to improve their understanding of them.
Understanding place value is crucial if students are to develop the estimation and calculation skills necessary to become numerate adults. Our number system is sophisticated, though it may not look it. Numerals exist all around us in the environment. The meaning of digits, and the quantities they represent, can be challenging to understand. 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 one, ten, one hundred, one thousand and so on are powers of ten. Therefore, 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 numbers in our base-10 system. We don’t need a new number to represent ten because we think of it as one group of ten. Similarly, when we add one to 999, we write 1000 and do not need a separate 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’, (as in 6 + 0 = 6) and as a placeholder ( as in 7040). The term 'placeholder' means a number occupies a place, or several places, and describes the values represented by the other digits. For instance, the zero in 7040 acts as a placeholder in the hundreds and ones places.
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 which means that it represents seven hundred. Two is in the thousands column which means that it represents 2 units of one thousand, called 2000.
Understanding the nested nature of place value becomes very important as students learn to operate on whole numbers and extend their knowledge to 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 ten ones combine to form a unit of ten, ten tens combine to form a unit of one hundred, and ten hundreds combine to form a unit of one thousand. For example, the answer to 2610 + 4390 could be represented at 2000 + 4000 + 1000 = 7000, because 610 and 390 combine to form one 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. 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 opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The context used for this unit is people which should be engaging for all students. You might want to relate problems to people throughout the world or in regions of Aotearoa, and places relevant to your students, to provide more real-life settings. Encourage students to be creative by accepting a variety of strategies, and asking students to create their own problems for each other to solve. Ideas and strategies can be recorded on digital devices to easily share students thinking with their peers.
Te reo Māori vocabulary terms such as uara tū (place value), poro-tahi (ones block), poro-tekau (tens block), poro-rau (hundreds block) and whakarea (multiply) could be introduced in this unit and used throughout other mathematical learning.
This unit builds on the units at level 2 that explore place value of whole numbers to 1000:
You may want to revisit those units or, at least use some of the independent tasks. It is expected that students will develop an appropriate repertoire of basic multiplication facts- either prior to, or during, this unit.
Introducing Place Value People as a model
What number am I?
Copymaster 2 contains “What number am I?” challenges for the students to solve. The clues involve place value understanding and students are expected to use the Place Value People model to solve the problems if they need to. Look to see if your students:
Wish-upon-a-digit
A digital form of this game is called Wishball. It can be found at this link.
Students will need copies of Copymaster 1 (Place Value People) and Copymaster 3 (Scoresheets) to play. They will also need a way to randomly generate digits. This could be digit cards to draw from, a 10-sided dice, or an online random number generator.
The goal of the game is to use place value number operations to get from the starting number to the target number in as few turns as possible.
To set up the game:
For each turn:
When it will finish the game you have one opportunity to choose the digit you want (for example, if you were up to 467 with a target of 667, you can choose to have a 2, and add 200 to finish the game).
In the following sessions students are expected to apply multiplication and division to place value. They learn how many tens and hundreds are nested in whole numbers to four digits, and the effect of multiplying or dividing a whole number by ten.
Lucy’s Number Trick
In this session students investigate how many tens are in a whole number to four places.
In this session students put the concept of nested place value to work by solving problems. Most of the problems involve addition and subtraction, but a multiplicative view of place value is essential to the development of fluent strategies. Consider framing these problems in contexts relevant to your students and local area (e.g. number of students at our school each year, number of visitors to the library).
Dear family and whānau,
This week we have been exploring place value with whole numbers. We looked at the number of tens and hundreds in numbers like 4762 and used that knowledge to solve addition and subtraction problems. To help us we used a model called Place Value People so we could see what happened to the numbers we worked with. Discuss this strategy with your child. They can share how they used it to solve addition and subtraction problems.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/building-multiplicative-view-whole-number-place-value at 8:32pm on the 26th February 2024