# Place value with whole numbers

Purpose

This unit builds students’ understanding of place value, extending to 6 digit whole numbers.

Achievement Objectives
NA3-3: Know counting sequences for whole numbers.
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
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, etc. The system continues, giving us the capacity represent very large quantities. The place values, one, ten, one hundred, one thousand, etc., are powers of ten. That means that the place immediately to the left of a given place represents units that are tens time 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 we could ever need. 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’, e.g. 6 + 0 = 6, and as a place holder, e.g. 7040. Place holder means it occupies a place, or several places, so the reader knows the values represented by the other digits. In 7040 zero is acting as a place holder in the hundreds and tens 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 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.

This unit is linked to stages 6 and 7 of the Number Framework

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 modelling numbers (arrow cards, Multibase Arithmetic Blocks, place value houses)
• using the digital learning object Modeling Numbers: 6-digit numbers to provide support with reading and writing numbers.

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.

Required Resource Materials
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.

Activity 1

1. Begin by distributing a set of 8 place value variation cards (Copymaster 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.
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:
(1065).
Have them discuss the zero in the number. Then ask them to remove or ‘zap’ another digit in 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 a place. Have the students make another of the 4-digit numbers and repeat this.

Activity 2

1. Make MAB materials available to the students.
Have them play Fish for 1000 (Copymaster 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.
2. Discuss the fact that numbers can be composed (and decomposed) in different ways.

Activity 3

1. Show the students 1,000 dots (Copymaster 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.
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, whilst 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.

Activity 4

1. Make sets of Copymaster 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 to each pair of students. Emphasise that the number of dots is ten times more 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 the dots that is important, not the size of the dots.

Activity 5

1. Show students the learning object Modeling Numbers: 6-digit numbers. Explain that it provides a model for representing numbers using place value equipment. Clicking on the arrow above a place adds one unit in that place to the model of the number, and clicking on the arrow below a place removes one unit from that place in the model.
2. Make the numbers 6, 7, 8, 9 and 10 (by clicking on the arrow above the ones place to add “one” at a time). Zoom in using the magnifying icon so students can see what is happening. Ask the students what they think will happen when you add another “one” (to make 11), then make the number and watch what happens. Ask the students what they think will happen if you count backwards 11, 10, 9. Then watch the place value equipment change as you remove one at a time using the arrow below the ones place. Repeat this by zooming out and using the same procedure with the thousands, ten thousands and hundred thousands. Support students to understand that the hundreds, tens and ones structure is repeated, and to appreciate that a new place is needed to show more than 9 in any particular place. Note that clicking the right arrow at the bottom of the screen will show different representations of the number: using place value houses, in standard form or represented on a three bar abacus.
3. Have students work in pairs to model numbers on the learning object. They can click the die at the bottom left of the screen and a number will appear in words for the student to build using the place value equipment. They can click on the question mark symbol to check to see whether their model is correct.
4. Have students work in pairs to explore making their own number, saying it aloud and then checking whether they are reading the number correctly using the speaker icon.

Activity 6

Introduce the game 11,111 (Copymaster 4) and support students to play this game in groups of 2 to 4 students.