1 7 Divided By 100
The Improving Mathematics Education in Schools (TIMES) Projection
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Multiplication and Division
Number and Algebra : Module 3
June 2011
Assumed Knowledge
Much of the building of understanding of early mathematics occurs concurrently, so a child can exist developing the basic ideas related to multiplication and division whilst besides investigating the place-value arrangement. All the same, in that location are some useful foundations necessary for multiplication and partition of whole numbers:
- Some experience with frontward and backwards skip-counting.
- Some experience doubling and halving pocket-sized numbers.
(run into F-iv Module Counting and Place Value and F-4 Module Addition and Subtraction)
Motivation
One way of thinking of multiplication is equally repeated addition. Multiplicative situations arise when finding a full of a number of collections or measurements of equal size. Arrays are a good way to illustrate this. Some partitioning problems arise when we try to suspension upwards a quantity into groups of equal size and when we try to undo multiplications.
Multiplication answers questions such every bit:
- ane
- Judy brought 3 boxes of chocolates. Each box contained half dozen chocolates. How many chocolates did Judy take?
- 2
- Henry has iii rolls of wire. Each gyre is 4m long. What is the total length of wire that Henry has?
Division answers questions such as:
- ane
- How many apples will each friend become if four friends share 12 apples equally
betwixt them? - 2
- If xx pens are shared between seven children how many does each child receive, and how many are left over?
Addition is a useful strategy for calculating 'how many' when two or more collections of objects are combined. When there are many collections of the same size, addition is not the nigh efficient means of calculating the total number of objects. For instance, it is much quicker to calculate half dozen × 27 past multiplication than by repeated addition.
Fluency with multiplication reduces the cognitive load in learning afterwards topics such every bit sectionalization. The natural geometric model of multiplication as rectangular area leads to applications in measurement. As such, multiplication provides an early link between arithmetic and geometry.
Fluency with division is essential in many later topics and division is central to the calculations of ratios, proportions, percentages and slopes. Division with balance is a fundamental idea in electronic security and cryptography.
Content
Multiplication and partitioning are related arithmetic operations and arise out of everyday experiences. For example, if every member of a family of 7 people eats 5 biscuits, we can calculate 7 × 5 to piece of work out how many biscuits are eaten altogether or nosotros can count by 'fives', counting one group of five for each person. In many situations children will use their hands for multiples of five.
For whole numbers, multiplication is equivalent to repeated addition and is often introduced using repeated add-on activities. Information technology is important, though that children run across multiplication equally much more than repeated addition.
If nosotros had 35 biscuits and wanted to share them equally amongst the family of 7, we would use sharing to distribute the biscuits as into 7 groups.
Nosotros tin write down statements showing these situations:
Introducing vocabulary and symbols
There is a smashing deal of vocabulary related to the concepts of multiplication and division. For instance,
Some of these words are used imprecisely exterior of mathematics. For example, nosotros might say that a kid is the product of her environment or we insist that children 'share' their toys fifty-fifty though we do not always look them to share every bit with anybody.
It is important that children are exposed to a diversity of different terms that apply in multiplication and division situations and that the terms are used accurately. Oft information technology is desirable to emphasise one term more others when introducing concepts, however a flexibility with terminology is to be aimed for.
Looking at where words come from gives us some indication of what they hateful. The word 'multiply' was used in the mathematical sense from the late fourteenth century and comes from the Latin multi meaning 'many' and plicare meaning 'folds' giving multiplicare - 'having many folds', which means 'many times greater in number'. The term 'manyfold' in English language is antiquated but nosotros nevertheless employ item instances such as 'twofold' or threefold'.
The give-and-take 'separate' was used in mathematics from the early on 15th century. It comes from the Latin, dividere meaning 'to force apart, cleave or distribute'. Interestingly, the word widow has the aforementioned etymological root, which can be understood in the sense that a widow is a adult female forced apart from her husband.
Use of the word 'product'
The production of 2 numbers is the result when they are multiplied. So the product of
iii and 4 is the multiplication 3 × 4 and is equal to 12.
Information technology is important that we utilize the vocabulary related to multiplication and sectionalisation correctly. Many years ago we were told to 'practice our sums' and this could apply to whatever calculation using any of the operations. This is an inaccurate use of the discussion 'sum'. Finding the 'sum' of 2 or more than numbers ways to add together them together. Teachers should have care not to use the word 'sum' for anything but addition.
The symbols × and ÷
The × symbol for multiplication has been in use since 1631. It was called for religious reasons to represent the cross. We read the argument 3 × 4 equally 'three multiplied by 4'.
In some countries a middle dot is used so 3 × 4 is written equally 3.4. In algebra it is common to non utilise a symbol for multiplication at all. Then, a × b is written as ab.
The division symbol ÷ is known as the obelus. It was first used to signify division in 1659. We read the statement 12 ÷ 3 as '12 divided by iii'. Another mode to write division in school arithmetic is to use the note 
'3 goes into 12'.
Mathematicians almost never use the ÷ symbol for segmentation. Instead they use fraction notation. The writing of a fraction is really another mode to write division. So 12 ÷ iv is equivalent to writing 
One time students are becoming fluent with the concepts of multiplication and division then the symbolic annotation, × for multiplication and ÷ for division, can be introduced. Initially, the ideas will exist explored through a conversation, then written in words, followed by a combination of words and numerals and finally using numerals and symbols. At each step, when the kid is ready, the use of symbols can reflect the kid's ability to bargain with abstruse concepts.
MODELLING MULTIPLICATION
Modelling multiplication by arrays
Rectangular arrays can exist used to model multiplication. For example, iii × v is illustrated by
We call 15 the product of three and 5, and we call iii and v factors of 15.
Past looking at the rows of the array we see that
By looking at the columns of the array we also run into that
This illustrates 3 × five = 5 × three. We say that multiplication is commutative.
Arrays are useful because they can be used with very small as well as very large numbers, and also with fractions and decimals.
CLASSROOM Action
Children can model multiplication using counters, blocks, shells or any materials that are available and arranging them in arrays.
- 1
- Children construct arrays using a variety of materials.
- ii
- Take a digital photograph.
- 3
- Draw the multiplication using words, words and numbers and finally words
and symbols.
Modelling multiplication by skip-counting and on the number line
Skip-counting, such as reciting 3, six, ix, 15,..., is ane of the earliest introductions to repeated addition and hence to multiplication. This can exist illustrated on a number line as shown for 3 × 5 = fifteen below.
3 × 5 = 15
On the number line, the fact that three + 3 + 3 + iii + iii = 5 + five + five is non then obvious; the previous prototype shows 5 + v + 5, whereas 3 + 3 + 3 + three + 3 looks quite unlike.
Skip-counting is important considering it helps children acquire their multiplication tables.
Modelling multiplication by expanse
Replacing objects in an assortment by unit squares provides a natural transition to the area model of multiplication. This is illustrated below for 3 × five.
At this stage, we are only using unit squares instead of counters or stars. We tin can also use the expanse model of multiplication later on for multiplication of fractions.
Backdrop of Multiplication
Ane of the advantages of the array and expanse approach is that properties of multiplication are more credible.
Commutativity
Every bit discussed above, turning the iii × 5 assortment on its side illustrates that 3 × 5 = 5 × iii because the full number of objects in the assortment does not alter.
Nosotros saw this before past looking at the rows and columns separately, but we can also do this by turning the rectangle on its side. The area of the rectangle does non change.
Associativity
Some other important property of multiplication is associativity, which says that
We can demonstrate this with the numbers two, iii and 4:
Associativity of multiplication ensures that the expression a × b × c is unambiguous.
Any-order property
We usually don't teach young children associativity of multiplication explicitly when introducing multiplication. Instead, nosotros teach the any-order holding of multiplication, which is a outcome of the commutative and associative properties.
Any-order property of multiplication
A listing of numbers can be multiplied together in whatever order to give the product of the numbers.
The whatever-club belongings of multiplication is analogous to the any- social club holding of addition. Both associativity and commutativity are nontrivial observations; notation that subtraction and division are neither commutative nor associative. One time we are familiar with the arithmetic operations we tend to take both associativity and commutativity of multiplication for granted, only as nosotros do for addition. Every so often, information technology is worth reflecting that commutativity and associativity combine to requite the important and powerful whatsoever-order properties for add-on and multiplication.
Multiplying 3 whole numbers corresponds geometrically to calculating the number of unit cubes in (or volume of) a rectangular prism. The whatever-gild holding of multiplication means that we can summate this volume by multiplying the lengths of the sides in whatever order. The order of the adding corresponds to slicing the volume up in different ways.
 | |  |
| 5 × 2 = 2 × 5 | (5 × 2) × 3 = (two × 5) × iii | |
 |  | |
| 3 × two = 2 × 3 | (3 × 2) × 5 = (ii × 3) × 5 | |
 |  | |
| 5 × 3 = iii × 5 | (5 × 3) × 2 = (3 × 5) × two |
We can apply this to the multiplication of 3 or more numbers, it doesn't matter in which club we practice this.
Distributivity of Multiplication over Improver
The equation three × (2 + 4) = (3 × 2) + (three × 4) is an case of the distributivity of multiplication over addition. With arrays, this corresponds to the post-obit diagram.
With areas information technology corresponds to the diagram beneath.
Multiplication is as well distributive over subtraction.
For case 7 × (ten − 2) = 7 × 10 − 7 × 2.
We use the distributive property to enable u.s.a. to reduce multiplication problems to a combination of familiar multiples. For case,
and
EXERCISE ane
Use the distributive police force to carry out the following multiplications.
a 
The outcome of multiplying by one
When any number is multiplied by 1, the number is unchanged. For example,
Nosotros telephone call one the multiplicative identity. It is important to accept this conversation with immature children in very elementary terms, using lots of examples in the early stages of developing understanding about multiplication.
Nothing is the identity element for add-on. When nothing is added to a set there is no effect on the number of objects in that ready. For example,
This is true for all addition. Hence, nosotros phone call zippo the identity element for add-on of
whole numbers.
The effect of multiplying by zero
When any number is multiplied by nada the upshot is cipher. Situations showing the result of multiplying past zero can exist acted out with children using concrete objects.
For example,
If I have 5 baskets with iii apples in each I have five × iii = 15 apples in total.
However, if I take 5 baskets with 0 apples in each, the result is 5 × 0 = 0 apples in total.
Learning the multiplication table.
Fluency with multiplication tables is essential for further mathematics and in everyday life. For a while it was considered unnecessary to learn multiplications tables by memory, but it is a corking help to be fluent with tables in many areas of mathematics.
If students can add together a single-digit number to a two-digit number, they can at least reconstruct their tables even if they have not yet developed fluency. Information technology is therefore essential to ensure that students can add fluently earlier they begin to learn their 'tables'.
We strongly recommend that students learn their multiplication facts up to 12 × 12. This is primarily because the 12 times table is essential for time calculations — at that place are 12 months in a yr, 24 hours in a twenty-four hour period, and 60 minutes in an hour. Familiarity with dozens is useful in everyday life because packaging in 3 × 4 arrays is so much more convenient than in 2 × five arrays. In add-on, the 12 × 12 table has many patterns that can exist constructively exploited in pre-algebra exercises.
A straightforward approach to learning the tables is to recite each row, either by centre or by skip-counting. However, students likewise need to be able to recall individual facts without resorting to the entire table.
Looking at the 12 × 12 multiplication table gives the impression that there are 144 facts to be learnt.
However, in that location are several techniques that can be used to reduce the number of facts to exist learnt.
- The commutativity of multiplication (8 × 3 = iii × 8) immediately reduces this number to 78.
- The 1 and 10 times tables are straightforward and their mastery reduces the number of facts to be learnt to 55.
- The 2 and 5 times tables are the easiest to learn and their mastery further reduces the number of facts to be learnt to 36.
- The 9 and 11 times tables are the adjacent easiest to skip-count because nine and 11 differ from 10 past i. This reduces the number of facts to 21. Children may notice the decreasing ones digit and increasing tens digit in the nine times table. They may also be intrigued past the fact that the sum of the digits of a multiple of 9 is always 9.
- The squares are useful and tin exist learnt merely as ane might learn a times table.
This reduces the number of terms to be learnt to 15.
Whatsoever techniques are used, the aim should be fluency.
MODELLING Segmentation
Division e'er involves splitting something into a number of equal parts, but there are many contrasting situations that can all be described by 'division'. Earlier introducing the standard algorithm for segmentation, it is worthwhile discussing some of these situations under the headings:
- Sectionalisation without remainder,
- Segmentation with residue.
Sectionalization without remainder
Here is a simple model of the division 24 ÷ 8.
Question: If I pack 24 apples into boxes, each with viii apples, how many boxes will at that place be?
We tin can visualise the packing process by laying out the 24 apples successively in rows of eight, as in the diagrams below.
The 3 rows in the last array use up all 24 apples, so there will exist 3 total boxes, with no apples left over. The result is written in mathematical symbols equally
The number 24 is called the dividend ('that which is to exist divided'). The number 8 is called the divisor ('that which divides'). The number 3 is chosen the quotient, (from the Latin quotiens significant 'how many times').
Modelling partitioning by skip-counting and on the number line
Partitioning without balance tin exist visualized as skip-counting.
On the number line we count in 8s until we reach 24.
EXERCISE 2
a
b
c
Using arrays to testify division without residuum is the inverse of multiplication
The rectangular array that nosotros produced when we modeled 24 ÷ 8 is exactly the same array that we would depict for the multiplication three × eight = 24.
In our example:
- The statement 24 = 8 × three means 'three boxes, each with 8 apples, is 24 apples', and
- The statement 24 ÷ 8 = 3 means '24 apples brand up 3 boxes, each with 8 apples'.
Sectionalisation without residuum is the changed process of multiplication. 
The multiplication statement 24 = viii × 3 tin can in plow be reversed to give a second division argument
which answers the question, 'What is 24 divided past 3?
This corresponds to rotating the array past 90°, and regarding information technology as made up of 8 rows of 3. It answers the question, 'If I pack 24 apples into boxes each property 3 apples, how many boxes will exist required?'
And so the division statement 24 ÷ 8 = 3 now has four equivalent forms:
EXERCISE 3
For each partition statement, write downward the corresponding multiplication statements, and the other corresponding sectionalization statement.
a
c
Two models of sectionalisation without remainder
children's questions oft concern pairs of
situations similar to those described hither.
If we have 24 balloons to share equally, in that location
are two ways we tin can share them.
The first way is past asking 'How many groups?'
For instance, if we have 24 balloons and we give
8 balloons each to a number of children, how many
children get viii balloons?
If we separate 24 balloons into groups of 8, then 3 children get eight balloons each.
We say '24 divided by eight is 3'. This is written equally 24 ÷ 8 = iii.
Nosotros tin see this from the array:
The 2nd style is by asking 'How many in each group?' For example, if we share
24 balloons amidst eight children, how many balloons does each child receive? We want
to brand viii equal groups. We practice this past handing out one balloon to each kid. This uses
8 balloons. Then we practise the same again.
We can exercise this 3 times, so each child gets 3 balloons.
Over again, nosotros can see this from the multiplication assortment:
So dividing 24 past 8 is the same as request 'Which number do I multiply 8 by to get 24?'
For each division problem, there is usually an associated problem modelling the same partition statement. The 'balloons' example higher up shows how two problems can have the same division argument. One problem with balloons is the associate of the other.
Exercise 4
Write downwardly in symbols the division statement, with its respond, for each problem below. Then write downward in words the associated problem:
a
b
c
d
Sectionalisation with residual
We volition now use apples to model 29 ÷ 8.
Question: If I pack 29 apples into boxes, each with 8 apples, how many boxes will there be?
As before, we can visualise the packing process by laying out the 29 apples successively in rows of 8:
We tin lay out 3 full rows, but the last row only has five apples, so there will exist three full boxes and 5 apples left over. The effect is written as
dividend divisor quotient balance
The number 5 is called the rest because there are v apples left over. The remainder is always a whole number less than the divisor.
Equally with sectionalisation without balance, skip-counting is the footing of this process:
Nosotros locate 29 between successive multiples 24 = 8 × iii and 32 = eight × 4 of the divisor viii. Then we subtract to discover the remainder 29 − 24 = 5.
We could also have answered the question above by maxim, 'There volition be four boxes, but the concluding box will exist three apples brusk.'
This corresponds to counting backwards from 32 rather than forwards from 24, and the corresponding mathematical statement would be
It is non normal practice at schoolhouse, nevertheless, to use negative remainders. Even when the question demands the interpretation corresponding to it, we will always maintain the usual school convention that the remainder is a whole number less than the divisor. Division without remainder can be regarded as division with balance 0. During the location process, we actually land exactly on a multiple instead of landing between ii of them. For example, 24 ÷ 8 = 3 residue 0, or more simply, 24÷ 8 = three, and nosotros say that
The respective multiplication and improver argument
The 29 apples in our instance were packed into iii total boxes of 8 apples, with 5 left over. We can write this as a sectionalisation, merely we tin likewise write it using a product and a sum,
So for sectionalization with rest there is a corresponding statement with a multiplication followed by an improver, which is more complicated than division without residuum.
Two models of division with remainder
As before, issues involving division with rest ordinarily have an associated problem modelling the aforementioned division statement. Standing with our example of
Question: How many bags of 8 apples can I make from 29 apples and how many are left over?
Question: I accept 29 apples and 8 boxes. How many apples should I put in each box so that in that location is an equal number of apples in each box and how many are leftover?
The following two associated questions model 63 ÷ x = six remainder iii.
Question: If I have 63 dollar coins, and x people to requite them to, how many coins does each person get if they are to each have the same number of coins? How many are left over?
Question: If I have 63 dollar coins, how many $x books can I purchase and how many dollars do I have left over?
EXERCISE v
Answer each question in words, then write downwards its the associated sectionalization problem and reply it.
- a
- How many 7-person rescue teams can be formed from xc people?
- b
- How many 5-seater cars are needed to transport 43 people, and how many spare seats are there?
Properties of Division
Order and brackets cannot be ignored
When multiplying 2 numbers, the guild is unimportant. For case,
When dividing numbers, however, the order is crucial. For example,
To visualise this calculation, twenty people living in 4 homes ways each abode has on average 5 people, whereas 4 people living in 20 homes means each habitation has on average 
Similarly when multiplying numbers, the use of brackets is unimportant. For case,
When dividing numbers, however, the apply of brackets is crucial. For example,
Division by nada
Before we used empty baskets of apples to illustrate that 5 × 0 = 0.
The same model tin can be used to illustrate why division by zero is undefined.
If nosotros have 10 apples to be shared as among v baskets each basket will accept
x ÷ five = 2 apples in each.
If the 10 apples are shared equally between 10 baskets, each basket has 10 ÷ 10 = 1
apples in each.
If x apples are shared between twenty baskets, each basket volition have 
What happens if we try to share 10 apples between 0 baskets? This cannot be washed.
| If | | ten ÷ 0 = a 1 |
| 10 ≠ a × 0. |
This action is meaningless, then we say that ten ÷ 0 is undefined.
Nosotros must always be conscientious to chronicle this to children accurately so that they understand that:
- 10 ÷ 0 is Not equal to ane and
- x ÷ 0 is Non equal to 0
but ten ÷ 0 is not defined.
Dividing by 4, 8, 16, . . .
Considering four = 2 × 2 and 8 = 2 × 2 × 2, we can separate past four and 8, and past all powers of 2, by successive halving.
To split up by 4, halve and halve again. For case, to divide 628 by 4,
To divide by 8, halve, halve, and halve once again. For example, to divide 976 past 8,
Multiplication Algorithm
An algorithm works most efficiently if it uses a small number of strategies that apply in all situations. And so algorithms exercise not resort to techniques, such as the use of near-doubles, that are efficient for a few cases just useless in the majority of cases.
The standard algorithm will not help you to multiply 2 single-digit numbers. It is essential that students are fluent with the multiplication of two unmarried-digit numbers and with adding numbers to 20 earlier embarking on any formal algorithm.
The distributive holding is at the center of our multiplication algorithm because it enables us to summate products one column at a time and and so add the results together. Information technology should be reinforced arithmetically, geometrically and algorithmically.
For example, arithmetically nosotros have 6 × 14 = 6 × ten + 6 × iv, geometrically we come across the same phenomenon,
and algorithmically nosotros implement this in the post-obit calculation.
| ane | 4 | ||
| × | 6 | ||
| 2 | four | ||
| half dozen | 0 | + | |
| 8 | four |
Once this bones property is understood, we can proceed to the contracted algorithm.
Introducing the algorithm using materials
Initially when children are doing multiplication they will act out situations using blocks. Eventually the numbers they want to multiply will become too large for this to exist an efficient ways of solving multiplicative bug. However base-10 materials or bundles of icy-pole sticks can be used to introduce the more efficient method - the algorithm.
If we desire to multiply 6 past 14 nosotros make vi groups of 14 (or xiv groups of 6):
Collect the 'tens' together and collect the 'ones' together.
This gives 6 'tens' and 24 'ones'.
And so make as many tens from the loose ones. There should never be more than than ix single ones when representing whatever number with Base-10 blocks.
This gives half dozen 'tens' + 2 'tens' + 4 'ones'.
We add the tens to get
14 × half-dozen = 10 × half dozen + iv × 6 = sixty + 20 + iv = 84
Eventually we should start recording what is existence done with the blocks using the multiplication algorithm vertical format. Eventually the back up of using the blocks tin can exist dropped and students can complete the algorithm without concrete materials.
Multiplying by a single digit
First nosotros contract the adding by keeping track of deport digits and incorporating the addition equally we go. The previous calculation shortens as either
depending on where the carry digits are recorded.
Intendance should be taken even at this early stage because of the mixture of multiplication and add-on. Note too that the exact location and size of the acquit digit is not essential to the process and varies across cultures.
Multiplying past a single-digit multiple of a power of 10
The next observation is that multiplying by a unmarried-digit multiple of 10 is no harder than multiplying past a single digit provided we continue track of place value. And so, to observe the number of seconds in 14 minutes nosotros calculate
fourteen × 60 = fourteen × six × 10 = 840
and implement it algorithmically as
| 1 | four | ||
| × | 6 | 0 | |
| eight | 4 | 0 |
Similarly, nosotros tin proceed runway of higher powers of x by using place value to our advantage. So
becomes
| 1 | 4 | |||
| × | half dozen | 0 | 0 | |
| 8 | iv | 0 | 0 |
For students who have met the underlying observation as role of their mental arithmetic exercises the just novelty at this point is how to lay out these calculations.
Multiplying by a two-digit number
The adjacent cognitive jump happens when nosotros apply distributivity to multiply two two-digit numbers together. This is implemented as two products of the types mentioned above. For example,
is used in the 2-footstep calculation beneath.
| 7 | 4 | ||
| × | vi | iii | |
| ii | 2 | two | |
| 4 | 4 | 4 | 0 |
| 4 | six | 6 | 2 |
This corresponds to the surface area decomposition illustrated below.
In the early stages, it is worth concurrently developing the arithmetic, geometric and algorithmic perspectives illustrated higher up.
Unpacking each line in the long multiplication adding using distributivity explicitly, as in
| 7 | four | ||
| × | vi | 3 | |
| 1 | ii | ||
| 2 | 1 | 0 | |
| 2 | four | 0 | |
| 4 | 2 | 0 | 0 |
| iv | six | 6 | two |
corresponds to the surface area decomposition
It is not efficient to do this extended long multiplication in order to calculate products in general, but it can be used to highlight the multiple use of distributivity in the process. The area model illustration used in this case reappears later as a geometric estimation of calculations in algebra.
The standard division algorithm
There is only ane standard segmentation algorithm, despite its unlike appearances. The algorithm tin can exist set out as a 'long division' calculation to show all the steps, or as a 'brusque division' algorithm where just the carries are shown, or with no written working at all.
Setting the calculation out as a long sectionalization
We could set the work out as follows:
5 × xxx = 150, then decrease 150 from 193
5 × 8 = xl, so subtract twoscore from 43
The standard 'long segmentation' setting-out, however,
allows identify value to work for the states even more efficiently,
past working merely with the digits that are required for
each item division. At each step another digit is
required − this is usually chosen 'bringing down the adjacent digit'.
5 × 4 = 20, then decrease 20 from 21.
Bring down the 9, and divide 19 by 5.
5 × iii = 15, then decrease 15 from 19.
Bring downward the 3, and split 43 past 5.
v × viii = 40, then subtract forty from 43.
Hence 2193 ÷ v = 138 residuum 3.
(Never forget to gather the calculation upward into a determination.)
The placing of the digits in the top line is crucial. The first step is '5 into 21 goes 4', and the digit four is placed higher up the digit 1 in 21.
Setting the calculation out equally a short division
Once the steps have been mastered, many people are comfortable doing each multiplication/subtraction stride mentally and writing downwardly only the comport. The calculation and so looks like this:
 | | Nosotros say, | '5 into 21 goes iv, residue 1'. |
| '5 into nineteen goes 3, remainder 4'. | |||
| '5 into 43 goes 8, remainder 3'. |
Zeroes in the dividend and in the steps
Zeroes will cause no problems provided that all the digits are kept strictly in their correct columns. This same principle is central to all algorithms that rely on identify value.
The instance to the right shows the long sectionalisation and short sectionalisation calculations for
We twice had to bring down
the digit 0, and 2 of the divisions resulted in a caliber of 0.
Using the computer for sectionalisation with remainder
People oftentimes say that division is hands done on the figurer. Segmentation with remainder, however, requires some common sense to sort out the answer.
Example
Use the estimator to convert 317 minutes to hours and minutes.
Solution
Nosotros can encounter that
With a estimator using the sectionalisation key: Enter 350 ÷ threescore, and the answer is five.833333… hours. So decrease 5 to become 0.833333…, and multiply by 60 to catechumen to l minutes, giving the answer 5 hours and l minutes.
Calculator assistance may be extremely useful with larger numbers, but experience with long sectionalization is essential to interpret the calculator display This phenomenon is common to many similar situations in mathematics.
Links Forward
The first application of multiplication that students are probable to come across is segmentation. When computing a division, we are constantly calculating multiples of the divisor, and lack of fluency with multiplication is a significant handicap in this process. The material in this module lays the foundation for multiplication, and so division, of fractions and decimals.
Other applications of multiplication include percentages and consumer arithmetic. For example, nosotros summate the price of an item inclusive of GST past calculating one.ane times its pre-GST cost.
A familiarity with multiplication and the expression of numbers every bit products of factors paves the way for one of the major theorems in mathematics.
The Key Theorem of Arithmetic states that every whole number bigger than 1 can be written every bit a product of prime number numbers and such an expression is unique up to the lodge in which the factors are written.
For example, 24 = 23 × iii and xx = 22 × five.
The Key Theorem of Arithmetic has far-reaching consequences and applications in reckoner science, coding, and public-key cryptography.
Last, but non least, a strong grounding in arithmetic sets a student up for success in algebra.
The division algorithm uses multiplication and subtraction. As such, partitioning demands that we synthesise a lot of prior knowledge. This is what makes division challenging, and for many students information technology is their first taste of multi-layered processes. The ability to reverberate on what you know, and implement it inside a new, higher-level process is one of the generic mathematical skills that division helps to develop.
The implementation of the sectionalization algorithm is typically a multi- step process, and as such it helps to develop skills that are invaluable when students move on to algebra. The link to factors is too critical in later years.
History
The product of two numbers is the same no matter how you summate it or how you write your respond. Just equally the history of number is really all about the development of numerals, the history of multiplication and division is mainly the history of the processes people have used to perform calculations. The development of the Hindu-Standard arabic place-value note enabled the implementation of efficient algorithms for arithmetic and was probably the master reason for the popularity and fast adoption of the notation.
The earliest recorded example of a partition implemented algorithmically is a Sunzi sectionalisation dating from 400AD in China. Essentially the same process reappeared in the volume of al Kwarizmi in 825AD and the modernistic-twenty-four hours equivalent is known every bit Galley division. It is, in essence, equivalent to modernistic-day long segmentation. Nevertheless, it is a wonderful example of how notation can make an enormous difference. Galley division is hard to follow and leaves the page a mess compared to the modern layout.
The layout of the long division algorithm varies betwixt cultures.
Throughout history there have been many different methods to solve problems involving multiplication. Some of them are nonetheless in use in different parts of the world and are of involvement to teachers and students every bit alternative strategies or considering of the mathematical challenge involved in learning them.
Italian or lattice method
Another technique, known as the Italian or lattice method is essentially an implementation of the extended version of the standard algorithm but in a dissimilar layout. The method is very sometime and might have been the one widely adopted if it had not been difficult to impress. Information technology appears to have outset appeared in Republic of india, only soon appeared in works by the Chinese and by the Arabs. From the Arabs it found its way beyond to Italian republic and can be plant many Italian manuscripts of the 14th and 15th centuries.
The multiplication 34 × 27 is illustrated hither.
34 × 27 = 918
In the pinnacle right rectangle 4 × 2 is calculated. The digit 8 is placed in the bottom triangle and 0 in the acme triangle.
Then 3 × ii is calculated and the effect entered as shown.
In the lesser right rectangle four × seven is calculated. The digit 8 is placed in the bottom triangle and the digit 2 in the top triangle. The result of three × 7 is also recorded in this way.
The greenish diagonal contains the units.
The blue diagonal contains the tens.
The orange diagonal contains the hundreds.
The digits are now summed along each diagonal starting from the right and each
result recorded every bit shown. Note that at that place is a 'carry' from the 'tens diagonal' to the 'hundreds diagonal'
References
A History of Mathematics: An Introduction, third Edition, Victor J. Katz, Addison-Wesley, (2008)
History of Mathematics, D. Eastward. Smith, Dover publications New York, (1958)
Knowing and Teaching Simple Mathematics: teachers' understanding of fundamental mathematics in China and the The states. Liping Ma, Mahwah, Northward.J.: Lawrence Erlbaum Associates, (1999)
History of Mathematics, Carl B. Boyer (revised by Uta C. Merzbach), John Wiley and Sons, Inc., (1991)
ANSWERS TO EXERCISES
Exercise one
| a | 9 × 32 | = 9 × 30 + 9 × 2 |
| = 270 + 18 | ||
| = 2888 |
| b | 31 × eight | = 30 × 8 + i × viii |
| = 240 + 8 | ||
| = 248 |
| c | 102 × eight | = 100 × 8 + 2 × viii |
| = 800 + xvi | ||
| = 816 |
Exercise two
- a
- 3, 6, nine, 12, 15, xviii, 21, 24, 27, 30, 33, 36, 39, 42. Hence 42 ÷ 3 = 14.
- b
- 11, 22, 33, 44, 55. Hence 55 ÷ 11 = v.
- c
- 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000. yard ÷ 100 = 10.
Practise 3
- a
- 8 = 2 × 4 and 8 ÷ 4 = 2.
- b
- 56 ÷ viii = vii and 56 ÷ 7 = 8.
- c
- 81 = nine × 9 and 81 ÷ 9 = ix. Because the divisor and the quotient are the same, the multiplication statement becomes a statement about squaring, and the other corresponding division statement is the same as the original statement.
Exercise iv
- a
- 24÷ four = half-dozen. If 24 children are divided into groups of four, how many groups are there?
- b
- xx ÷ 2 = 10. If 20 metres of textile is divided into ii equal pieces, how long is
- c
- 160 ÷ x =16. If 160 books are bundled into packages of 10 each, how many packages are there?
- d
- 35÷seven = 5. If a 35-day period is divided into 7 equal periods, how long is each period?
Exercise 5
- a
- Twelve 7-person rescue teams can be formed, with 6 people to spare. How many people will exist in seven equal groups formed from xc people? At that place will be 12 people in each group, with half-dozen left over.
- b
- Nine v-seater cars are needed, and there volition be two spare seats. How many people will exist in 5 equal groups formed from 43 people? In that location volition be 8 groups, with three people left over.
Exercise 6
- a
- 246 ÷ 4 = (246 ÷ 2) ÷2 = 123 ÷ 2 = 61
- b
- 368 ÷ viii = ((368 ÷ii) ÷ 2) ÷ 2 = (184 ÷ 2) ÷2 = 92 ÷ two= 46.
- c
- 163 ÷ 8 = ((163 ÷ 2) ÷ 2)÷2 =81
÷ two = 40
÷ two = 20
.
| d | 12 048 ÷ 16 | = (((12 048 ÷ 2) ÷ 2 ÷ 2) ÷ ii = ((6024 ÷ 2) ÷ 2) ÷ 2 |
| = (3012 ÷ 2) ÷ two = 1506 ÷ 2 = 752. |
The Improving Mathematics Education in Schools (TIMES) Projection 2009-2011 was funded by the Australian Government Section of Education, Employment and Workplace Relations.
The views expressed here are those of the author and exercise non necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.
© The University of Melbourne on behalf of the International Eye of Excellence for Didactics in Mathematics (Water ice-EM), the education partition of the Australian Mathematical Sciences Establish (AMSI), 2010 (except where otherwise indicated). This work is licensed nether the Artistic Eatables Attribution-NonCommercial-NoDerivs 3.0 Unported License.
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1 7 Divided By 100,
Source: http://amsi.org.au/teacher_modules/multiplication_and_division.html
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