Place value with whole numbers
This unit builds students’ understanding of place value, extending to 6 digit whole numbers.
About this resource
Specific learning outcomes:
- Understand the structure of 6 digit whole numbers.
- Read and write six digit whole numbers.
- Represent numbers up to 999 999 using place value equipment.
Place value with whole numbers
Achievement objectives
NA3-3: Know counting sequences for whole numbers.
Description of mathematics
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. Place values such as one, ten, one hundred, and 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 greater 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 connections 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 the 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.
Opportunities for adaptation and differentiation
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:
- reducing the number of digits students are working with
- providing open access to a variety of materials for representing numbers (Arrow cards, Multibase Arithmetic Blocks, Place value houses)
- using a variety of physical and digital materials to model the construction, addition, and decomposition of numbers
- creating flexible groups (mahi tahi model) so that students can support each other, share their thinking, and model ideas for each other.
This unit is focused 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.
Required materials
See Materials that come with this resource to download
- Place value houses (.pdf) (Trendsetter, Thousands and Millions)
- Place value whole numbers 1 (.pdf)
- Place value whole numbers 2 (.pdf)
- Place value whole numbers 3 (.pdf)
- Place value whole numbers 4 (.pdf)
- Arrow cards (.pdf)
- MAB (Multibase Arithmetic Blocks), 1, 10, 100, 1000, also known as Dienes blocks
Activity
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.
1.
Begin by distributing a set of 8 place value variation cards (Place value whole numbers 1) to pairs of students. Have them order their set of 4-digit numbers from largest to smallest, laying them out in front of them. Consider starting with smaller numbers (e.g. 2 or 3-digit) if this is appropriate to the needs of your students.
2.
Once the sets are ordered, have the students rotate in pairs to an adjacent display of ordered cards and check whether these are ordered correctly.
3.
Ask the students to explain what they had to think about as they ordered their cards. Record their place-value ideas on a class chart.
4.
Distribute sets of Arrow cards to the student pairs and have them make one of the numbers in their set:
Have them discuss the zero in the number. Then ask them to remove, or ‘zap, another digit from their number and record the value of the digit they are removing. For example: the 6 in the number 1065 is removed. In doing so, 60 is being removed or subtracted from the number. This is clearly shown by using Arrow cards.
Ask what they noticed when the 6 (60) was removed. (A zero appeared in its place).
Discuss zero as a placeholder. Zero adds no value to the number but has the important job of holding it. Have the students make another of the 4-digit numbers and repeat this.
1.
Make MAB materials available to the students.
Have them play Fish for 1000 (Place value whole numbers 2) in pairs or groups of four to consolidate their understanding of the composition of 4-digit numbers.
Students can model the numbers with materials as needed to support their understanding. They can also relate these to numbers in their own culture.
2.
Discuss the fact that numbers can be composed (and decomposed) in different ways.
1.
Show the students 1,000 dots (Place value whole numbers 3). Have students tell you what they see and record all of their ideas using words and numerals. For example: ten groups of one hundred, 1000, 10 x 100, one hundred groups of ten, 1000, 100 x 10. Recording these in other languages that are relevant to your learners (e.g. te reo Māori) will increase engagement in this task.
2.
Ask the students to explain to a buddy what 10 thousand would look like. Listen to their predictions. Have students attempt to draw this. Show the 10,000 dots image and ask if this is ten thousand. Repeat step 1 above.
Point out that when we write the number 1000, there is no space after the 1 and before the three zeros, whereas with 10 000, a space is used to separate the ten from the three zeros. Highlight the fact that 10 thousand is ten times bigger by highlighting one row (1000) and then counting the rows.
3.
Write 100 000 on the class chart. Have the students describe what 100,000 might look like. Listen for and record their ideas (ten times bigger). Show one hundred thousand and repeat step 1 above. Again, point out how the number is written with the space between the 100 and the 000.
Ask the students to explain why.
4.
Write a 6-digit number on the class Place value houses. Have students read it together, referring to the hundreds, tens, and ones headings in each of the houses. Repeat with several 6-digit numbers, including several examples with zeros as placeholders. Highlight the hundreds, tens, and ones structure of both houses and how this helps when reading bigger numbers.
1.
Make sets of Place value whole numbers 3 with 9 copies of each representation (9 ones, 9 tens, 9 hundreds, 9 thousands, 9 ten thousands, and 9 one hundred thousands). Have sufficient sets for students to have 1 set for each pair.
2.
Distribute Trendsetter and Thousands Place value houses and dot sets for each pair of students. Emphasise that the number of dots is ten times greater each time the number has an extra zero and as it shifts one more place to the left.
3.
Have them take turns writing a number with up to 6 digits in the place value house and having their partner read and make the number using the dot materials. Emphasise that it is the number of dots that is important, not the size of the dots.
Introduce the game 11,111 (Place value whole numbers 4) and support students to play this game in small groups (i.e. 2-4 students).
Home Link
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.
The quality of the images on this page may vary depending on the device you are using.