Building a multiplicative view of whole number place value

Purpose

This unit requires students to work with the number of ones, tens, hundreds and thousands in four-digit whole numbers.

Achievement Objectives
NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
Specific Learning Outcomes
  • Recognise the importance of zero as a place holder in whole numbers.
  • Recognise and apply understanding of the base ten system repeated naming pattern of thousands, hundreds, tens and ones.
  • Appreciate that tens are nested in hundreds and thousands and use that knowledge to rename whole numbers.
  • Rename whole numbers to make the addition and subtraction of collections of tens or hundreds easier.
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 very sophisticated though it may not look it. Numerals are all around us in the environment but the meaning of digits and the quantities they represent are 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, 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.

Opportunities for Adaptation and Differentiation

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:

  • Providing physical materials so that students can anticipate actions and justify their solutions. Place Value People is a representation that is accessible, allows for combining and partitioning units, and can be used with whole numbers up to 4 digits.
  • Connecting symbols with quantities, especially as the quantities are combined (addition), separated (subtraction). Support students to generalise properties that always occur, e.g. ten units of a place value combine to make one unit of next highest place value.
  • Using important mathematical vocabulary to discuss concepts, in particular words for the places (ones, tens, hundreds, etc.) and words associated with operations, e.g. addends and sum for addition.
  • Encouraging students to collaborate in small groups and to share, and justify, their ideas.
  • Using technology, especially calculators, in predictive, pattern-based ways, e.g. What will happen to 234 if you multiply it by ten?..divide it by ten?
  • Altering the complexity of the whole numbers and operations that are used. You might refer to the Level 2 place value units for some students.

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 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.

Required Resource Materials
  • Calculators,
  • Place Value People (made from Copymaster One)
  • Scissors
  • Stapler
  • Digit cards or ten-sided dice.
  • Copymasters One, Two, Three, Four, Five, and Six
  • PowerPoints One, Two, and Three
Activity

Session 1

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 also expected that students will learn a good repertoire of basic multiplication facts either prior to the unit or in the course of it.

Introducing Place Value People as a model

  1. Begin the lesson using PowerPoint One. The first three slides show numbers of people icons. All three collections contain 120 icons. The purpose of the class task is to establish that grouping objects helps us to count the number of objects efficiently and that tens-based grouping is the normal convention (though it was not always historically).
    What is an efficient way to count the number of people?
  2. Slide One: There are five blocks of 4 x 6 = 24 people. Students may approach the task additively, e.g. 6 + 6 = 12, 12 + 12 = 24 to find the number in each block. 24 + 24 = 48, etc. to establish the total. A multiplicative approach might be 4 x 6 = 24, 5 x 20 = 100, 5 x 4 = 12 so there are 120 people.
  3. Slide Two: There are three blocks of four tens. The use of place value units makes it much easier to count the 120 people. Look for students to apply multiplicative structure, such as, 4 x 10 = 40, 3 x 40 = 120. Discuss how 3 x 4 = 12 is used to find 3 x 40 = 120.
  4. Slide Three: The people icons are distributed randomly. Give the students a short time to count the icons since allowing them to continue is non-productive. The point is that counting in ones is time consuming and error prone. The animation creates collections of ten icons that might be used to estimate the total number.
  5. Show the students Copymaster One (place value people). Discuss that the 100 sheet is composed of ten rows of ten people or four groupings of 5 x 5 = 25 people. Slide Four of PowerPoint One illustrates how the number 365 can be made efficiently using the Copymaster. Slide Five poses some interesting questions about 365.
    • Question One: Within each 100 are ten tens. 300 is comprised of 30 tens. Since 60 is six tens there are 36 tens altogether. Some students may say that the other five is half of a ten or point five. That is correct. The answers of 36 tens and 36.5 tens are correct.
    • Question Two: In each ten there are two fives. An efficient way is to double the number of tens to get the number of fives, i.e. 2 x 36 = 72 (add the final five to get 73) or 2 x 36.5 = 73.
    • Question Three: To make 365 into 400, five are needed to make 370 and another 3 tens to make 400. 400 plus 100 equals 500. In total 100 + 30 + 5 needs to be added.

What number am I?

Copymaster Two 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:

  • Use a systematic approach to eliminating possible numbers as they work through the clues
  • Show a sense of quantity by making numbers with the Place Value People model
  • Demonstrate nested place value knowledge, e.g. 74 ten equal 740.

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 One (Place Value People) and Copymaster Three (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:

  1. Select three digits randomly to form a starting number and mark this on the number line at the top of their recording sheet.
  2. Select three digits randomly to form a target number and mark this on the number line as well.

For each turn:

  1. Select a digit randomly.
  2. Choose whether the digit represents ones, tens or hundreds (for example, a 3 could be worth 3, 30 or 300)
  3. Choose whether to add or subtract this number from the current running total and record this on the recording sheet.

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).

Session 2

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 effect of multiplying or dividing a whole number by ten.

Lucy’s Number Trick

  1. Play the first four slides of PowerPoint Two to introduce Lucy’s look for a pattern in the 11 x tables with two-digit whole numbers. Let students talk in pairs as each answer comes up.
    Is there a pattern?
    What is the pattern?
  2. While showing slides 5-7, invite your students to predict the answer to other 11 x facts. Prediction encourages them to conjecture on possible rules. Access to calculators will support students to check their predictions. After slide 7, discuss how the answer might be predicted. In general, for any two-digit number, 10a + b (written as ab), the answer has a ones digit of b. The tens digit is a + b and the hundreds digit is a. If a + b is less than ten the pattern is easier to spot. For example, 11 x 45 has an answer of 5 ones, 4 + 5 = 9 tens, and 4 hundreds (495).
  3. If a + b is greater than ten then some renaming occurs. Take 11 x 57 for example. The answer has 7 ones, 5 + 7 = 12 tens, and 5 hundreds. Since 12 tens are 120 another hundred is created. Therefore, the answer becomes 6 hundreds, 2 tens, and 7 ones (627).
  4. Noticing the pattern is important. The next step is to structure the pattern, that is, find out why it behaves the way it does. Challenge the students to use Place Value People (Copymaster One) to model some examples of the 11 x pattern. Slides 8 and 9 of PowerPoint Two illustrate 11 x 27. You might use the slides initially or allow students the opportunity to represent calculations themselves first. Essentially the pattern works because of the distributive property. Multiplication by eleven is the sum of multiplying by ten and one.
    Symbolically this property can be written as 11 x 27 = (10 x 27) + (1 x 27) = 270 + 27. The vertical algorithm on Slide Nine better illustrates how the digits in the second factor contribute hundreds, tens and ones to the product.
  5. What happens if the two digits in the second factor add to a number greater than nine?
  6. Students might recognise that if the sum of a + b > 9 (for any number 10a + b) there is more renaming required as tens are renamed to form hundreds. Slide Ten shows the calculation of 11 x 79 which is a good example.
  7. Challenge the students to practise their skills in calculating 11 x any 2-digit whole number. You might have a quiz to see how quickly they can use the digit pattern to find products. You might express the method as an algorithm (step by step procedure), such as:
  8. If time permits set related extension problems:
    1. Develop an algorithm for multiplying a three-digit whole number by 11.
    2. Develop an algorithm for multiplying a two-digit number by 101.
    3. Develop an algorithm for multiplying a two-digit whole number by 111.
    The solutions are:
    1. For any three-digit number, 100a + 10b + c (written as abc) the product when it is multiplied by 11 is a thousands, a + b hundreds, b + c tens, and c ones.
      For example, 345 x 11 = 3 795. Note that renaming is needed when a + b and/or b + c equal ten or more.
    2. For any two digit number, 10a + b (written as ab) the product is a thousands, b hundreds, a tens and b ones. No renaming will be needed.
      For example, 68 x 101 = 6 868.
    3. For any two-digit number, 10a + b (written as ab) the product when it is multiplied by 111 is a thousands, a + b hundreds, a + b tens, and b ones. Note the renaming needed if a + b > 9. For example 35 x 111 = 3 885.

Session 3

In this session students investigate how many tens are in a whole number to four places.

  1. Remind students about their learning from Session Two.
    What is the product of 10 x 26? How do you know?
    What is the product of 10 x 49? How do you know?
    What is the product of 100 x 83? How do you know?
  2. If necessary, use Place Value People to model the 10 x calculations (ten sets of the other factor). Discuss why modelling 100 x 83 might not be efficient and that thinking about pattern is more helpful.
    What is 83 composed (made up) of? (8 tens and 3 ones)
    What is 100 x 3? Why is the product the same as 3 x 100? (Commutative property)
    What is 100 x 8? So, what is the product of 100 x 80? Why? (Do students recognise the effect on the digits in the product of multiplying by ten?)
  3. Ask students to model the number 245 with Place Value People (Copymaster One).
    You need to make teams of ten for sports day. How many teams of ten can you make with 245 people?
  4. Ask students to anticipate the result of making tens then justify why they believe 24 tens exist in 245. Their explanations should include the fact that ten tens are nested in each hundred.
    How might we record what we have found in an equation?
  5. Let students offer possibilities like 24 × 10 = 250 or 240 ÷ 10 = 24. Discuss the meaning of the symbols, especially the division sign. In this case the process is measurement, i.e. How many tens make 240?
  6. Ask: What might we do with the extra five people?
    Some students might suggest that five is half of ten or that there are 24.5 tens in 245.
  7. Ask: How many tens can be made with …609 people? 444 people? 930 people?
  8. Do the students justify their answers by referring to the tens nested within the hundreds? Make 1 000 with Place Value People by attaching ten one hundred sheets together with a single staple.
    How many people are in this stack? (1000 since there are ten hundreds)
  9. Make the number 1 728 using Place Value people. Work out how teams of ten people can be made with that number.
  10. Look to see that students correctly model 1 728 with one thousand stack, seven hundreds sheets, two ten strips and eight singles.
    Do they unpack 100 tens in 100, 70 tens in 700, and 2 tens in 20?
    Do they name and justify that there are 172 tens in 1 728 (or 172.8)?
  11. Ask: How many teams of one hundred people could be made with 1 728 people?
  12. Look to see if your students recognise that ten hundreds exist in 1000, and another 7 hundreds exist in 700, making 17 hundreds altogether. If students give an answer of 17.28, recognise that this a very sophisticated answer since 0.28 means 28 hundredths. Record the operations of finding out how many tens and hundreds are in 1 728 as equations; 1720 ÷ 10 = 172 and 1700 ÷ 100 = 17.
    What patterns do you see in dividing by ten and one hundred?
  13. Provide your students with Copymaster Four that contains a set of related problems to work from. Look for students to develop fluency with renaming whole numbers.
  14. Mark the Copymaster together and discuss the strategies students used to solve the problems.

Session 4

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.

  1. Introduce Slides 1 and 2 of PowerPoint Three. You could use a large set of Place Value People to model the operations of 63 + 8 = 71 and 630 + 80 = 71 or simply play the animations and discuss what happens. The key idea is that students recognise the connection in both operations, that is 630 + 80 is the addition of 63 + 8 in units of ten.
    What is the sum of 6 300 and 800? How do you know? (63 + 8 = 71 in units of 100, so 7 100)
  2. Give your students other examples to calculate, with or without the support of Place Value People, depending on their fluency. Good problems are:450 + 70 =                  
    880 + 60 =                  
    390 + 90 =                  
    940 + 80 =  
  3. Slides Three and Four of PowerPoint Three deal with the same principle applied to subtraction. The examples of 82 – 7 = 75 and 820 -70 = 750 illustrate how a subtraction result with ones can be used to work out a result with units of ten. The principle works with units of any equal size, e.g. 8200 – 700 = 7 500 (units of 100).
  4. Give your students other examples to calculate. Check to see that they can demonstrate what is occurring with the quantities in the calculations, using Place Value People. Good examples might be:
    430 – 60 =                  
    910 – 80 =                  
    540 – 90 =                  
    820 – 70 =      
  5. Provide your students with Copymaster Five to work from in pairs. This worksheet consolidates and extends the concept of renaming a number to make calculations easier. Note that the principle is extended to four-digit whole numbers. Some students may need support, including modelling with Place Value People (Copymaster One).    

Session 5

  1. To consolidate the concept of nested place value, introduce and play the game Cover Cathy Crocodile 4 digits. The game can be made using Copymaster Six. Easier forms of the same game are available in the level two place value units (Cover Cathy Crocodile: 3 digits and  Cover Cathy Crocodile: 2 digits)
  2. The purpose of the game is to develop students’ fluency in renaming whole numbers up to four digits. To assess students’ ability to rename pose this question:
    Using thousands, hundreds, tens and ones, make up at least five names for 4768.
    For example, the first name might be 4 thousands, 7 hundreds, 6 tens, and 8 ones. The second name might start with 47 hundreds…
    Which name would be most useful to solve the problem 4768 + 900 =  ?
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Level Three