This unit supports students developing an understanding of four-digit numbers and how to operate with them.
Specific Learning Outcomes Session One
Session Two
Sessions Three and Four
Session Five
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. While numbers are all around us in the environment the meaning of digits in those numbers 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. And so the system continues to represent very large numbers. The place values, one, ten, one hundred, one thousand, etc., are powers of ten. To represent all the numbers we could ever want we use just ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The word for digits also comes from our fingers. We don’t need a new number to represent ten because we think of it as one hand, 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 places so the reader knows the values represented by of 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 2 753 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, 2 000. Renaming a number flexibly is important. 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 2 610 + 4 390 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, 4 200 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. Place value was devised to count large numbers of objects and represent the set of objects using a succinct set of numerals. It is appropriate that students learn to use place value as a means to estimate the number of objects in a set and count the number exactly. Specific Teaching PointsSession One Writing four digit numbers in expanded form is a traditional place value task that can help students to calculate mentally and in written form. For example, 7 249 can be expanded to 7 000 + 200 + 40 + 9. It is also important for students to recognise which digits change when a given number is added to or subtracted from an existing number. For example, if 60 is added to 3 420 only the tens digit will change. However, if 60 is subtracted from 3 420 both the hundreds and tens digits will change since one hundred must be decomposed to create tens. Session Two The theme of digit change is more complex when students must determine the number to add, or subtract from, a given number to change it into a target number. For example, to change 567 into 607 the ones digit is invariant. The tens digit changes by adding four tens (40), which also causes the renaming of ten tens as one hundred. Session Three Renaming numbers in multiple ways is foundational to flexible calculation. The combining and decomposing of place value units means that a given number can have many names. For example, 456 can be represented as 4 hundreds, 5 tens and 6 ones. It is also 45 tens and 6 ones or just 456 ones. Some ways of renaming are particularly helpful. Decomposition of place value units is useful for subtraction. In the calculation 456 – 183 = ? treating 456 as 3 hundreds, 15 tens and 6 ones makes the subtraction very easy. Session Four The fundamental law of place value is that if ten units of one place are formed then a unit of the next highest place are created. For example, when 247 and 562 are combined ten tens are formed which creates another unit of one hundred. Similarly if a place value unit is partitioned it forms ten units of the place value next smaller to it. To calculate 819 – 30 = ? a one hundred unit can be decomposed to form ten tens, renaming the number as 7 hundreds, 11 tens and 9 ones. These rules for composition and decomposition of units can result in strangely appearing changes to numbers in counting sequences. For example, 970, 980, 990, 1 000, 1010, … and 3 004, 3 003, 3002, 3001, 3 000, 2999, 2998, … Session Five Nested place value is the idea that place value units are included in other place value units, for example, tens are within hundreds, and hundreds are within thousands. In real life an article costing $280 might need to be paid for in $20 notes. Knowing how many tens are in $280 (28 tens) is very helpful.
Students would benefit from working through these three easier units on place value prior to this one: Session OneIn this session students consolidate their understanding of three digit whole numbers and progress to four digit numbers. They learn to write numbers in expanded and compact form and apply place value to challenges in which they must change a given number to a target number. Whole Class Introduction
Independent challenge for students For this game, students work in teams of three. Each group needs a set of arrow cards. The game can be played with a full set of thousands, hundreds, tens and ones cards or the challenge can be reduced by playing with just hundreds, tens and ones cards. The instructions here are for the easier game. For each round the cards are shuffled and dealt equally so each player has three hundreds cards, three tens, and three ones cards. For each round there is a target clue (See PowerPoint), such as “Between 300 and 600” or “Within 50 less or greater than 700” or “Rounds to 300”. The players build numbers to satisfy the target. They can only build three numbers so the maximum number of points is three each round. No card can be used twice in a round. Reflection
Session TwoIn this session students explore renaming of three numbers in multiple ways. Flexibility in renaming is essential to mental and written calculation with any set of numbers, but is particularly important with whole numbers that form the basis of other sets, such as integers and rational numbers. Whole Class Introduction Calculator change games are a useful way for students to apply their understanding of place value. The difficulty of a particular change is a combination of the number of digits that need to change and the re-unitising of place value units required. For example, changing 435 to 495 is relatively easy since only the ten digit changes by six units of ten (60) and no re-unitising is required. However changing 435 to 395 is considerably more difficult as both the tens and hundreds digits change and one hundred needs to be re-unitised as ten tens.
Independent challenge for students Copymaster Three provides a series of calculator change challenges that increase markedly in difficulty. Allow the students access to calculators and photocopied Place Value People models (see Copymaster One) so they can experiment with different operations. You may decide to let students work in pairs or as individuals, for assessment purposes. Look for students to:
Reflection It is worthwhile to work through some of the challenges students find difficult, as a class, to support students in validating their ideas. Ideally, choose examples where re-unitising is required. For example: In this case both the tens and hundreds digits change. The operation is subtraction since the target number is less than the starting number. So to find the required operation, one hundred must be re-unitised, into ten tens. Actually it is easiest to see 217 as twenty-one tens and seven ones and find the answer using the difference between 21 and 18 (3 tens). 217 can be renamed as ‘one hundred and eleventy-seven’ and written as 211 7. This renaming can be shown by making 217 with Place Value People on the place value mat, cutting on hundred into ten tens, and moving those tens into their correct place. Then three tens (30) can be removed to make 187. Session ThreeIn this session the theme of renaming is continued. Students learn that whole numbers may have many different names depending on how place values are partitioned and combined.
Independent challenge for students Provide the students with copies of Cover Cathy Crocodile (3 Digit Version). You will need a coloured laminated set of boards and cards for every group of five students in your class. The activity can be set as a co-operative challenge to cover all the crocodiles or as a competitive game. Student will need to play the game several times to gain fluency in renaming three digit numbers. Reflection Take particular cards from the Cover Cathy Crocodile set and ask what numbers might be covered with that card. For example, the card 6 hundreds □ tens and 9 ones can be used to cover the sequence of numbers; 609, 619, 629, 639, …, 699, 709, 719, … Session FourIn this session students extend the number system to four digit numbers. They use Place Value People models to show how ten hundreds combine to form one thousand and the inverse operation of partitioning one thousand into ten hundreds. Whole Class Introduction Show the students a ‘book’ of ten hundreds stapled in the top left corner. How many people are in this collection? What clues do you need to find that out? Students should ask how many hundreds are in the collection (ten) and give one thousand as the total number of people. Use thousands together with other Place Value People units to form the quantities and ask the students to write numbers and words for those quantities. It is important to include zeros as place holders in some of the numbers. Independent challenge for students For this activity students should work in pairs or threes. Copymaster Four has a set of questions based on the number of residents living in Whangamata township. Students need to build the numbers using Place Value People and use the model to check their answers at each change in the number of residents. Insist that students record their answers to each question and justify how they know the number is correct. Early finishers can make up some other scenarios to change the population of the town. Look for your students to:
Reflection
Session FiveIn this session students apply their understanding of four digit numbers. They use Place Value People models to show how many hundreds, tens and ones are ‘nested’ in a four digit whole number. Whole Class Introduction
Independent challenge for students Copymaster Five has problems about identifying the number of tens and hundreds in a given number. The problems increase in difficulty. Some students will need Place Value Materials to attempt the problems but many will be able to solve the problems using symbols with understanding. Ask your students to check their answers using division, e.g. “How many tens are in 8 095?” can be checked with 8 095 ÷ 10 = 809.5 Reflection
Hello parents and caregivers Our next mathematics unit is based on Place Value. So we will be extending our knowledge of three digit numbers to include four digit numbers, like 7 403 and 2 579. We will represent four digit numbers in different ways to get an idea of the quantities the numbers represent. We will also rename the four digit numbers in flexible ways so that we can apply renaming to the operations, particularly addition and subtraction. |