Explain How To Check Multiplication Using Addition Or Division
The Improving Mathematics Education in Schools (TIMES) Project
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Multiplication and Sectionalisation
Number and Algebra : Module 3
June 2011
Causeless Knowledge
Much of the building of understanding of early on mathematics occurs concurrently, so a child can be developing the bones ideas related to multiplication and segmentation whilst also investigating the identify-value system. Nevertheless, in that location are some useful foundations necessary for multiplication and partitioning of whole numbers:
- Some experience with frontwards and backwards skip-counting.
- Some feel doubling and halving small numbers.
(come across F-4 Module Counting and Identify Value and F-4 Module Addition and Subtraction)
Motivation
One style of thinking of multiplication is as repeated addition. Multiplicative situations arise when finding a total of a number of collections or measurements of equal size. Arrays are a practiced way to illustrate this. Some division problems arise when nosotros attempt to break up a quantity into groups of equal size and when we effort to undo multiplications.
Multiplication answers questions such every bit:
- 1
- Judy brought 3 boxes of chocolates. Each box contained half-dozen chocolates. How many chocolates did Judy have?
- 2
- Henry has iii rolls of wire. Each coil is 4m long. What is the total length of wire that Henry has?
Division answers questions such every bit:
- i
- How many apples will each friend get if iv friends share 12 apples every bit
between them? - two
- If twenty 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 ii or more than collections of objects are combined. When at that place are many collections of the aforementioned size, add-on is not the almost efficient means of calculating the total number of objects. For example, it is much quicker to calculate vi × 27 by multiplication than past repeated addition.
Fluency with multiplication reduces the cognitive load in learning later topics such equally division. The natural geometric model of multiplication equally rectangular expanse leads to applications in measurement. As such, multiplication provides an early link betwixt arithmetic and geometry.
Fluency with partitioning is essential in many afterward topics and division is central to the calculations of ratios, proportions, percentages and slopes. Division with remainder is a fundamental thought in electronic security and cryptography.
Content
Multiplication and division are related arithmetic operations and arise out of everyday experiences. For example, if every member of a family of vii people eats 5 biscuits, we can summate 7 × 5 to work out how many biscuits are eaten birthday or nosotros can count by 'fives', counting one group of v for each person. In many situations children will utilize their easily for multiples of five.
For whole numbers, multiplication is equivalent to repeated add-on and is often introduced using repeated addition activities. It is of import, though that children see multiplication as much more than repeated improver.
If we 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.
We can write downwards statements showing these situations:
Introducing vocabulary and symbols
There is a great deal of vocabulary related to the concepts of multiplication and segmentation. For example,
Some of these words are used imprecisely exterior of mathematics. For case, we might say that a kid is the production of her environment or we insist that children 'share' their toys fifty-fifty though we do not always expect them to share equally with everyone.
Information technology is important that children are exposed to a variety of different terms that apply in multiplication and sectionalization situations and that the terms are used accurately. Often it 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 up from gives united states some indication of what they mean. The give-and-take '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 we even so use particular instances such as 'twofold' or threefold'.
The word 'divide' was used in mathematics from the early 15th century. It comes from the Latin, dividere meaning 'to force apart, carve or distribute'. Interestingly, the word widow has the same etymological root, which tin can be understood in the sense that a widow is a woman forced apart from her married man.
Utilize of the word 'product'
The production of two numbers is the result when they are multiplied. So the product of
3 and 4 is the multiplication 3 × four and is equal to 12.
It is of import that we utilize the vocabulary related to multiplication and partitioning correctly. Many years ago nosotros were told to 'practice our sums' and this could utilize to whatsoever calculation using any of the operations. This is an inaccurate use of the discussion 'sum'. Finding the 'sum' of 2 or more numbers ways to add together them together. Teachers should take intendance non to utilize the discussion 'sum' for annihilation only addition.
The symbols × and ÷
The × symbol for multiplication has been in utilize since 1631. It was chosen for religious reasons to stand for the cross. Nosotros read the statement 3 × 4 as '3 multiplied by 4'.
In some countries a middle dot is used so 3 × 4 is written as 3.4. In algebra it is common to not employ a symbol for multiplication at all. So, a × b is written as ab.
The sectionalisation symbol ÷ is known as the obelus. It was first used to signify segmentation in 1659. We read the statement 12 ÷ 3 equally '12 divided by 3'. Another style to write segmentation in school arithmetic is to utilize the notation 
'3 goes into 12'.
Mathematicians nigh never employ the ÷ symbol for partition. Instead they use fraction notation. The writing of a fraction is really another mode to write division. And then 12 ÷ 4 is equivalent to writing 
One time students are condign fluent with the concepts of multiplication and division then the symbolic annotation, × for multiplication and ÷ for sectionalisation, can be introduced. Initially, the ideas will be explored through a conversation, so written in words, followed by a combination of words and numerals and finally using numerals and symbols. At each step, when the child is ready, the use of symbols can reflect the child's power to bargain with abstract concepts.
MODELLING MULTIPLICATION
Modelling multiplication past arrays
Rectangular arrays can exist used to model multiplication. For instance, 3 × 5 is illustrated by
We call 15 the product of 3 and 5, and we call 3 and five factors of 15.
Past looking at the rows of the array we run across that
Past looking at the columns of the array we besides see that
This illustrates 3 × five = 5 × 3. Nosotros say that multiplication is commutative.
Arrays are useful because they tin can be used with very small also as very large numbers, and likewise with fractions and decimals.
CLASSROOM Activity
Children can model multiplication using counters, blocks, shells or whatsoever materials that are available and arranging them in arrays.
- 1
- Children construct arrays using a diversity of materials.
- 2
- Take a digital photograph.
- 3
- Describe 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 every bit reciting three, half dozen, 9, 15,..., is ane of the earliest introductions to repeated addition and hence to multiplication. This can exist illustrated on a number line every bit shown for iii × five = 15 below.
3 × 5 = 15
On the number line, the fact that 3 + 3 + 3 + iii + 3 = five + 5 + 5 is not so obvious; the previous epitome shows 5 + 5 + v, whereas 3 + 3 + 3 + three + three looks quite different.
Skip-counting is of import considering it helps children learn their multiplication tables.
Modelling multiplication by surface area
Replacing objects in an array 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 of measurement squares instead of counters or stars. Nosotros can as well employ the area model of multiplication afterward for multiplication of fractions.
Backdrop of Multiplication
Ane of the advantages of the array and expanse approach is that properties of multiplication are more than apparent.
Commutativity
As discussed above, turning the 3 × v array on its side illustrates that 3 × 5 = v × iii because the full number of objects in the assortment does not change.
We saw this before by looking at the rows and columns separately, simply we tin can also practice this past turning the rectangle on its side. The area of the rectangle does not change.
Associativity
Some other important holding of multiplication is associativity, which says that
We can demonstrate this with the numbers two, 3 and 4:
Associativity of multiplication ensures that the expression a × b × c is unambiguous.
Whatsoever-gild property
We usually don't teach young children associativity of multiplication explicitly when introducing multiplication. Instead, we teach the any-order belongings of multiplication, which is a event of the commutative and associative properties.
Any-lodge property of multiplication
A listing of numbers can exist multiplied together in whatever gild to give the product of the numbers.
The whatever-order property of multiplication is analogous to the any- gild property 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, it is worth reflecting that commutativity and associativity combine to requite the important and powerful whatsoever-order properties for add-on and multiplication.
Multiplying three whole numbers corresponds geometrically to calculating the number of unit cubes in (or book of) a rectangular prism. The any-guild belongings of multiplication ways that nosotros can summate this volume past multiplying the lengths of the sides in whatever order. The gild of the calculation corresponds to slicing the book upward in different ways.
 | |  |
| 5 × 2 = two × five | (v × ii) × three = (2 × 5) × iii | |
 |  | |
| 3 × ii = two × 3 | (3 × 2) × 5 = (two × 3) × five | |
 |  | |
| 5 × iii = iii × 5 | (5 × iii) × ii = (three × v) × ii |
We can utilise this to the multiplication of iii or more numbers, information technology doesn't thing in which social club we do this.
Distributivity of Multiplication over Addition
The equation 3 × (two + 4) = (three × two) + (3 × 4) is an instance of the distributivity of multiplication over improver. With arrays, this corresponds to the following diagram.
With areas information technology corresponds to the diagram below.
Multiplication is also distributive over subtraction.
For example vii × (10 − 2) = vii × x − 7 × 2.
We use the distributive belongings to enable us to reduce multiplication problems to a combination of familiar multiples. For example,
and
Exercise ane
Use the distributive law to carry out the following multiplications.
a 
The event of multiplying by 1
When whatsoever number is multiplied past one, the number is unchanged. For example,
Nosotros call 1 the multiplicative identity. It is important to have this conversation with young children in very elementary terms, using lots of examples in the early stages of developing understanding about multiplication.
Zero is the identity element for addition. When nothing is added to a gear up there is no consequence on the number of objects in that ready. For case,
This is true for all addition. Hence, we call zippo the identity element for addition of
whole numbers.
The effect of multiplying by zero
When any number is multiplied past naught the result is nothing. Situations showing the effect of multiplying by null can exist acted out with children using concrete objects.
For example,
If I accept 5 baskets with three apples in each I have 5 × 3 = 15 apples in total.
Yet, if I have v baskets with 0 apples in each, the effect is v × 0 = 0 apples in total.
Learning the multiplication table.
Fluency with multiplication tables is essential for farther mathematics and in everyday life. For a while information technology was considered unnecessary to learn multiplications tables by retentivity, but it is a great help to be fluent with tables in many areas of mathematics.
If students can add together a unmarried-digit number to a ii-digit number, they can at least reconstruct their tables even if they accept non yet adult fluency. It is therefore essential to ensure that students tin can add fluently earlier they brainstorm 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 — there are 12 months in a year, 24 hours in a day, and 60 minutes in an hour. Familiarity with dozens is useful in everyday life because packaging in 3 × 4 arrays is then much more convenient than in 2 × 5 arrays. In add-on, the 12 × 12 table has many patterns that can be 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 also need to exist able to remember private facts without resorting to the unabridged tabular array.
Looking at the 12 × 12 multiplication table gives the impression that at that place are 144 facts to be learnt.
Still, there are several techniques that can be used to reduce the number of facts to be learnt.
- The commutativity of multiplication (eight × 3 = iii × 8) immediately reduces this number to 78.
- The 1 and ten times tables are straightforward and their mastery reduces the number of facts to be learnt to 55.
- The ii and five times tables are the easiest to learn and their mastery further reduces the number of facts to exist learnt to 36.
- The ix and 11 times tables are the next easiest to skip-count because 9 and eleven differ from 10 past 1. This reduces the number of facts to 21. Children may observe the decreasing ones digit and increasing tens digit in the nine times table. They may likewise be intrigued by the fact that the sum of the digits of a multiple of 9 is always 9.
- The squares are useful and can exist learnt but equally one might acquire a times table.
This reduces the number of terms to be learnt to 15.
Whatsoever techniques are used, the aim should be fluency.
MODELLING DIVISION
Sectionalisation always involves splitting something into a number of equal parts, but there are many contrasting situations that can all be described by 'sectionalisation'. Before introducing the standard algorithm for partitioning, it is worthwhile discussing some of these situations under the headings:
- Division without balance,
- Division with remainder.
Partition without balance
Here is a simple model of the division 24 ÷ eight.
Question: If I pack 24 apples into boxes, each with 8 apples, how many boxes will at that place be?
Nosotros can visualise the packing process by laying out the 24 apples successively in rows of 8, equally in the diagrams below.
The iii rows in the last array use upwardly all 24 apples, and so there will be three 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 be divided'). The number 8 is called the divisor ('that which divides'). The number iii is called the quotient, (from the Latin quotiens pregnant 'how many times').
Modelling partition by skip-counting and on the number line
Sectionalization without remainder can exist visualized equally skip-counting.
On the number line we count in 8s until we reach 24.
Do two
a
b
c
Using arrays to show segmentation without remainder is the inverse of multiplication
The rectangular array that we produced when we modeled 24 ÷ 8 is exactly the same array that we would draw for the multiplication 3 × eight = 24.
In our example:
- The statement 24 = 8 × 3 means 'iii boxes, each with 8 apples, is 24 apples', and
- The statement 24 ÷ 8 = iii means '24 apples make upwardly 3 boxes, each with 8 apples'.
Division without remainder is the inverse process of multiplication. 
The multiplication statement 24 = 8 × three can in turn be reversed to give a 2d partitioning statement
which answers the question, 'What is 24 divided past 3?
This corresponds to rotating the array by 90°, and regarding information technology as fabricated upwardly of 8 rows of iii. Information technology answers the question, 'If I pack 24 apples into boxes each property 3 apples, how many boxes will be required?'
So the division statement 24 ÷ 8 = 3 now has four equivalent forms:
Practice iii
For each division statement, write down the respective multiplication statements, and the other corresponding division statement.
a
c
Two models of division without remainder
children's questions often concern pairs of
situations similar to those described here.
If nosotros accept 24 balloons to share every bit, there
are two ways nosotros can share them.
The first way is past request 'How many groups?'
For example, if we have 24 balloons and we requite
8 balloons each to a number of children, how many
children get 8 balloons?
If we split 24 balloons into groups of 8, then 3 children become 8 balloons each.
Nosotros say '24 divided by viii is three'. This is written as 24 ÷ eight = 3.
We tin encounter this from the array:
The second way is by asking 'How many in each grouping?' For example, if we share
24 balloons among 8 children, how many balloons does each kid receive? Nosotros want
to make viii equal groups. We practice this by handing out one airship to each child. This uses
8 balloons. Then we do the same again.
We tin do this three times, then each child gets 3 balloons.
Again, we can see this from the multiplication assortment:
So dividing 24 by 8 is the same as asking 'Which number practice I multiply 8 by to become 24?'
For each division trouble, there is usually an associated problem modelling the same division argument. The 'balloons' example above 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 partition statement, with its reply, for each problem below. Then write down in words the associated problem:
a
b
c
d
Sectionalisation with residual
We volition now use apples to model 29 ÷ viii.
Question: If I pack 29 apples into boxes, each with 8 apples, how many boxes volition at that place be?
As before, nosotros can visualise the packing process by laying out the 29 apples successively in rows of 8:
We can lay out three full rows, merely the final row simply has v apples, and so there volition exist 3 full boxes and 5 apples left over. The event is written as
dividend divisor quotient remainder
The number v is called the balance considering there are 5 apples left over. The residuum is ever a whole number less than the divisor.
Equally with sectionalisation without remainder, skip-counting is the basis of this procedure:
We locate 29 between successive multiples 24 = 8 × 3 and 32 = 8 × iv of the divisor eight. Then nosotros subtract to discover the remainder 29 − 24 = v.
We could also have answered the question above by proverb, 'In that location volition be four boxes, merely the concluding box will be iii apples short.'
This corresponds to counting backwards from 32 rather than forward from 24, and the corresponding mathematical argument would be
It is not normal do at school, however, to use negative remainders. Fifty-fifty when the question demands the estimation corresponding to information technology, we will always maintain the usual school convention that the remainder is a whole number less than the divisor. Partitioning without residual tin be regarded equally division with residual 0. During the location process, we actually state exactly on a multiple instead of landing between 2 of them. For instance, 24 ÷ 8 = iii remainder 0, or more simply, 24÷ 8 = 3, and nosotros say that
The corresponding multiplication and addition statement
The 29 apples in our example were packed into three full boxes of 8 apples, with 5 left over. Nosotros can write this every bit a division, but we can also write information technology using a product and a sum,
So for division with remainder there is a respective argument with a multiplication followed by an addition, which is more than complicated than division without residual.
Two models of division with remainder
Equally before, problems involving division with rest unremarkably accept an associated trouble modelling the aforementioned division statement. Standing with our case of
Question: How many numberless of 8 apples tin I make from 29 apples and how many are left over?
Question: I have 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 ÷ ten = half dozen remainder iii.
Question: If I have 63 dollar coins, and ten people to give them to, how many coins does each person go if they are to each take the aforementioned number of coins? How many are left over?
Question: If I have 63 dollar coins, how many $10 books can I buy and how many dollars practice I accept left over?
EXERCISE 5
Reply each question in words, then write down its the associated division trouble and answer it.
- a
- How many vii-person rescue teams tin can exist formed from 90 people?
- b
- How many 5-seater cars are needed to transport 43 people, and how many spare seats are in that location?
Properties of Division
Order and brackets cannot be ignored
When multiplying ii numbers, the order is unimportant. For example,
When dividing numbers, however, the order is crucial. For case,
To visualise this calculation, 20 people living in 4 homes means each home has on average 5 people, whereas 4 people living in 20 homes means each home has on average 
Similarly when multiplying numbers, the employ of brackets is unimportant. For case,
When dividing numbers, yet, the apply of brackets is crucial. For instance,
Division by zero
Earlier we used empty baskets of apples to illustrate that 5 × 0 = 0.
The same model can exist used to illustrate why partition by zero is undefined.
If we take 10 apples to be shared equally amidst 5 baskets each basket will have
10 ÷ five = 2 apples in each.
If the 10 apples are shared equally between ten baskets, each basket has 10 ÷ 10 = ane
apples in each.
If ten apples are shared between 20 baskets, each basket will have 
What happens if we effort to share ten apples between 0 baskets? This cannot be done.
| If | | 10 ÷ 0 = a 1 |
| 10 ≠ a × 0. |
This activeness is meaningless, so nosotros say that 10 ÷ 0 is undefined.
We must always be careful to relate this to children accurately so that they understand that:
- ten ÷ 0 is NOT equal to ane and
- 10 ÷ 0 is NOT equal to 0
but 10 ÷ 0 is not defined.
Dividing past 4, eight, xvi, . . .
Because 4 = 2 × 2 and viii = 2 × 2 × ii, we can divide by four and eight, and by all powers of 2, by successive halving.
To divide by 4, halve and halve again. For example, to divide 628 past four,
To divide by 8, halve, halve, and halve again. For example, to divide 976 by viii,
Multiplication Algorithm
An algorithm works nigh efficiently if it uses a minor number of strategies that apply in all situations. So algorithms exercise not resort to techniques, such as the use of most-doubles, that are efficient for a few cases but useless in the majority of cases.
The standard algorithm will not aid you to multiply 2 unmarried-digit numbers. It is essential that students are fluent with the multiplication of two single-digit numbers and with calculation numbers to twenty before embarking on whatsoever formal algorithm.
The distributive holding is at the eye of our multiplication algorithm considering it enables us to calculate products i column at a time and and then add the results together. It should be reinforced arithmetically, geometrically and algorithmically.
For example, arithmetically we take vi × 14 = 6 × ten + 6 × iv, geometrically we meet the same phenomenon,
and algorithmically nosotros implement this in the following calculation.
| ane | 4 | ||
| × | six | ||
| 2 | four | ||
| 6 | 0 | + | |
| viii | 4 |
Once this bones property is understood, we tin 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 as well large for this to be an efficient means of solving multiplicative problems. All the same base-10 materials or bundles of icy-pole sticks tin be used to introduce the more than efficient method - the algorithm.
If nosotros want to multiply vi by 14 we brand 6 groups of 14 (or xiv groups of six):
Collect the 'tens' together and collect the 'ones' together.
This gives half-dozen 'tens' and 24 'ones'.
Then brand as many tens from the loose ones. At that place should never exist more than nine single ones when representing any number with Base-10 blocks.
This gives 6 'tens' + 2 'tens' + iv 'ones'.
We add the tens to get
14 × 6 = 10 × 6 + iv × vi = 60 + 20 + 4 = 84
Eventually we should start recording what is being done with the blocks using the multiplication algorithm vertical format. Eventually the back up of using the blocks can be dropped and students can complete the algorithm without physical materials.
Multiplying by a single digit
First we contract the calculation by keeping track of carry digits and incorporating the addition as we go. The previous adding shortens as either
depending on where the carry digits are recorded.
Care should exist taken even at this early stage because of the mixture of multiplication and addition. Note also that the exact location and size of the carry digit is not essential to the process and varies beyond cultures.
Multiplying past a single-digit multiple of a power of ten
The next ascertainment is that multiplying past a single-digit multiple of ten is no harder than multiplying by a single digit provided we keep runway of place value. So, to notice the number of seconds in fourteen minutes nosotros calculate
14 × 60 = 14 × six × 10 = 840
and implement it algorithmically every bit
| one | 4 | ||
| × | half dozen | 0 | |
| 8 | iv | 0 |
Similarly, nosotros tin keep track of higher powers of 10 by using identify value to our advantage. Then
becomes
| ane | four | |||
| × | half-dozen | 0 | 0 | |
| 8 | 4 | 0 | 0 |
For students who have met the underlying observation as part of their mental arithmetic exercises the just novelty at this point is how to lay out these calculations.
Multiplying past a two-digit number
The side by side cognitive jump happens when nosotros employ 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 two-stride calculation below.
| 7 | 4 | ||
| × | 6 | 3 | |
| 2 | 2 | 2 | |
| 4 | 4 | 4 | 0 |
| 4 | half dozen | 6 | 2 |
This corresponds to the area decomposition illustrated beneath.
In the early stages, it is worth meantime developing the arithmetic, geometric and algorithmic perspectives illustrated above.
Unpacking each line in the long multiplication calculation using distributivity explicitly, as in
| vii | 4 | ||
| × | 6 | 3 | |
| 1 | 2 | ||
| 2 | 1 | 0 | |
| 2 | 4 | 0 | |
| 4 | 2 | 0 | 0 |
| 4 | 6 | 6 | 2 |
corresponds to the area decomposition
It is not efficient to do this extended long multiplication in order to calculate products in general, but information technology can be used to highlight the multiple use of distributivity in the process. The expanse model illustration used in this case reappears subsequently as a geometric interpretation of calculations in algebra.
The standard division algorithm
There is simply ane standard division algorithm, despite its different appearances. The algorithm can exist set out as a 'long division' calculation to bear witness all the steps, or as a 'curt partitioning' algorithm where just the carries are shown, or with no written working at all.
Setting the calculation out as a long division
We could set the work out every bit follows:
5 × 30 = 150, then decrease 150 from 193
v × 8 = 40, then subtract 40 from 43
The standard 'long division' setting-out, however,
allows identify value to work for united states fifty-fifty more efficiently,
by working merely with the digits that are required for
each particular division. At each step another digit is
required − this is usually called 'bringing downward the next digit'.
five × 4 = 20, then decrease xx from 21.
Bring downward the 9, and split 19 past 5.
5 × 3 = xv, and then subtract 15 from 19.
Bring downwardly the 3, and divide 43 by 5.
5 × 8 = 40, then subtract 40 from 43.
Hence 2193 ÷ v = 138 remainder iii.
(Never forget to gather the calculation up into a conclusion.)
The placing of the digits in the top line is crucial. The outset step is 'five into 21 goes iv', and the digit 4 is placed above the digit 1 in 21.
Setting the calculation out as a brusk division
One time the steps have been mastered, many people are comfortable doing each multiplication/subtraction step mentally and writing down simply the carry. The adding and so looks like this:
 | | We say, | 'five into 21 goes 4, remainder one'. |
| 'v into 19 goes 3, remainder 4'. | |||
| '5 into 43 goes viii, remainder iii'. |
Zeroes in the dividend and in the steps
Zeroes will cause no issues provided that all the digits are kept strictly in their right columns. This same principle is fundamental to all algorithms that rely on identify value.
The example to the right shows the long division and short division calculations for
We twice had to bring downwards
the digit 0, and two of the divisions resulted in a quotient of 0.
Using the calculator for division with residual
People often say that division is easily washed on the computer. Division with residual, however, requires some common sense to sort out the answer.
EXAMPLE
Use the calculator to convert 317 minutes to hours and minutes.
Solution
We can see that
With a calculator using the division fundamental: Enter 350 ÷ 60, and the answer is five.833333… hours. So subtract 5 to get 0.833333…, and multiply by 60 to convert to 50 minutes, giving the reply 5 hours and 50 minutes.
Estimator assistance may be extremely useful with larger numbers, merely experience with long division is essential to translate the estimator display This miracle is mutual to many similar situations in mathematics.
Links Forward
The first awarding of multiplication that students are probable to meet is division. When calculating a division, we are constantly calculating multiples of the divisor, and lack of fluency with multiplication is a pregnant handicap in this process. The material in this module lays the foundation for multiplication, and so sectionalization, of fractions and decimals.
Other applications of multiplication include percentages and consumer arithmetic. For instance, we calculate the cost of an item inclusive of GST by calculating 1.1 times its pre-GST cost.
A familiarity with multiplication and the expression of numbers equally products of factors paves the style for 1 of the major theorems in mathematics.
The Fundamental Theorem of Arithmetic states that every whole number bigger than ane tin be written equally a production of prime numbers and such an expression is unique up to the guild in which the factors are written.
For example, 24 = 23 × three and 20 = iiii × 5.
The Central Theorem of Arithmetic has far-reaching consequences and applications in computer science, coding, and public-key cryptography.
Last, merely non least, a strong grounding in arithmetics sets a student upwardly for success in algebra.
The division algorithm uses multiplication and subtraction. Equally such, partition demands that we synthesise a lot of prior cognition. This is what makes sectionalization challenging, and for many students it is their kickoff sense of taste of multi-layered processes. The ability to reflect on what you know, and implement it within a new, higher-level process is 1 of the generic mathematical skills that division helps to develop.
The implementation of the sectionalisation algorithm is typically a multi- step procedure, and as such it helps to develop skills that are invaluable when students move on to algebra. The link to factors is likewise critical in subsequently years.
History
The product of ii numbers is the same no thing how you calculate it or how you write your respond. Just as the history of number is really all near the development of numerals, the history of multiplication and sectionalisation is mainly the history of the processes people accept used to perform calculations. The development of the Hindu-Arabic place-value notation enabled the implementation of efficient algorithms for arithmetic and was probably the main reason for the popularity and fast adoption of the notation.
The earliest recorded example of a division implemented algorithmically is a Sunzi division dating from 400AD in People's republic of china. Essentially the same procedure reappeared in the book of al Kwarizmi in 825AD and the modern-day equivalent is known as Galley division. It is, in essence, equivalent to modern-day long segmentation. However, it is a wonderful example of how note tin can make an enormous divergence. 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 between cultures.
Throughout history at that place accept been many dissimilar methods to solve issues involving multiplication. Some of them are still in use in dissimilar parts of the world and are of interest to teachers and students every bit culling strategies or because of the mathematical challenge involved in learning them.
Italian or lattice method
Some other technique, known every bit the Italian or lattice method is essentially an implementation of the extended version of the standard algorithm but in a unlike layout. The method is very onetime and might have been the ane widely adopted if information technology had not been difficult to print. It appears to accept first appeared in India, but soon appeared in works by the Chinese and by the Arabs. From the Arabs it establish its way across to Italian republic and tin can be found many Italian manuscripts of the 14th and 15th centuries.
The multiplication 34 × 27 is illustrated hither.
34 × 27 = 918
In the top right rectangle 4 × ii is calculated. The digit 8 is placed in the lesser triangle and 0 in the height triangle.
Then iii × ii is calculated and the result entered as shown.
In the bottom right rectangle four × 7 is calculated. The digit 8 is placed in the lesser triangle and the digit 2 in the superlative triangle. The result of 3 × 7 is likewise recorded in this way.
The green diagonal contains the units.
The blueish diagonal contains the tens.
The orange diagonal contains the hundreds.
The digits are at present summed forth each diagonal starting from the right and each
result recorded as shown. Annotation that at that place is a 'bear' 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. E. Smith, Dover publications New York, (1958)
Knowing and Teaching Simple Mathematics: teachers' understanding of fundamental mathematics in China and the U.s.. Liping Ma, Mahwah, N.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 1
| a | 9 × 32 | = 9 × 30 + 9 × 2 |
| = 270 + 18 | ||
| = 2888 |
| b | 31 × 8 | = 30 × 8 + 1 × 8 |
| = 240 + viii | ||
| = 248 |
| c | 102 × 8 | = 100 × 8 + 2 × eight |
| = 800 + 16 | ||
| = 816 |
Exercise ii
- a
- 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42. Hence 42 ÷ iii = 14.
- b
- eleven, 22, 33, 44, 55. Hence 55 ÷ 11 = v.
- c
- 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000. 1000 ÷ 100 = 10.
Practice 3
- a
- 8 = 2 × 4 and 8 ÷ 4 = 2.
- b
- 56 ÷ 8 = vii and 56 ÷ vii = viii.
- c
- 81 = nine × 9 and 81 ÷ 9 = 9. Because the divisor and the quotient are the aforementioned, the multiplication statement becomes a statement almost squaring, and the other corresponding division argument is the same as the original statement.
Exercise four
- a
- 24÷ 4 = 6. If 24 children are divided into groups of 4, how many groups are there?
- b
- 20 ÷ 2 = x. If 20 metres of material is divided into two equal pieces, how long is
- c
- 160 ÷ x =16. If 160 books are bundled into packages of 10 each, how many packages are at that place?
- d
- 35÷7 = 5. If a 35-day flow is divided into 7 equal periods, how long is each period?
Do 5
- a
- Twelve 7-person rescue teams can be formed, with vi people to spare. How many people will be in seven equal groups formed from 90 people? In that location will be 12 people in each group, with 6 left over.
- b
- Nine 5-seater cars are needed, and there will be two spare seats. How many people will be in 5 equal groups formed from 43 people? There will be 8 groups, with three people left over.
Exercise 6
- a
- 246 ÷ 4 = (246 ÷ ii) ÷ii = 123 ÷ two = 61
- b
- 368 ÷ 8 = ((368 ÷2) ÷ 2) ÷ two = (184 ÷ 2) ÷two = 92 ÷ 2= 46.
- c
- 163 ÷ viii = ((163 ÷ 2) ÷ 2)÷2 =81
÷ two = twoscore
÷ 2 = 20
.
| d | 12 048 ÷ 16 | = (((12 048 ÷ 2) ÷ 2 ÷ 2) ÷ 2 = ((6024 ÷ 2) ÷ 2) ÷ 2 |
| = (3012 ÷ 2) ÷ two = 1506 ÷ 2 = 752. |
The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Authorities Department of Education, Employment and Workplace Relations.
The views expressed here are those of the author and exercise not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.
© The University of Melbourne on behalf of the International Centre of Excellence for Instruction in Mathematics (Water ice-EM), the teaching division of the Australian Mathematical Sciences Found (AMSI), 2010 (except where otherwise indicated). This work is licensed nether the Artistic Commons Attribution-NonCommercial-NoDerivs iii.0 Unported License.
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Explain How To Check Multiplication Using Addition Or Division,
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