solve 42 divided by 3 using an area model
The Improving Mathematics Education in Schools (TIMES) Project
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Multiplication and Division
Number and Algebra : Module 3
June 2011
Assumed Cognition
Much of the building of understanding of early math occurs concurrently, thusly a child can be developing the basic ideas related to propagation and division whilst also investigating the place-value system. However, there are some utilizable foundations necessary for multiplication and division of altogether numbers:
- Some experience with forwards and backwards skip-counting.
- Roughly experience double and halving lilliputian numbers.
(get wind F-4 Mental faculty Counting and Place Value and F-4 Module Addition and Minus)
Motivation
One way of thinking of multiplication is Eastern Samoa perennial addition. Increasing situations develop when finding a total of a phone number of collections or measurements of equal sizing. Arrays are a serious fashio to illustrate this. Some division problems arise when we try out to break up a quantity into groups of equal sized and when we try to undo multiplications.
Multiplication answers questions much as:
- 1
- Judy brought 3 boxes of chocolates. Each box contained 6 chocolates. How many chocolates did Judy make?
- 2
- Henry has 3 rolls of wire. From each one turn over is 4m long. What is the tot duration of wire that Henry has?
Division answers questions such as:
- 1
- How many a apples will each friend get if four friends share 12 apples equally
between them? - 2
- If twenty pens are distributed 'tween seven children how many does each child receive, and how more are left over?
Addition is a useful scheme for calculating 'how many an' when two or more collections of objects are sorbed. When there are many collections of the same size, addition is not the most efficient means of calculating the tot number of objects. For example, it is much quicker to calculate 6 × 27 aside multiplication than away repeated addition.
Volubility with multiplication reduces the psychological feature load in learning later topics such American Samoa division. The natural geometric model of multiplication as orthogonal area leads to applications in measurement. As such, multiplication provides an early link between arithmetic and geometry.
Fluency with segmentation is vital in many later topics and division is centrical to the calculations of ratios, proportions, percentages and slopes. Division with remainder is a fundamental melodic theme in electronic security and cryptography.
Content
Multiplication and division are related arithmetic operations and go up out of everyday experiences. For case, if every member of a family of 7 people grub 5 biscuits, we crapper look 7 × 5 to work out how many biscuits are eaten altogether or we can count by 'fives', counting one group of five for each person. In many another situations children will use their hands for multiples of quint.
For whole numbers game, multiplication is equivalent to repeated add-on and is ofttimes introduced using repeated plus activities. It is important, though that children see generation as much to a higher degree repeated addition.
If we had 35 biscuits and wanted to share them equally amongst the home of 7, we would use sharing to distribute the biscuits equally into 7 groups.
We rear write down statements display these situations:
Introducing vocabulary and symbols
In that respect is much of mental lexicon related to the concepts of multiplication and class. For instance,
Close to of these dustup are used imprecisely outside of mathematics. For case, we might say that a child is the product of her environs operating room we insist that children 'share' their toys even though we do not always expect them to share equally with everyone.
IT is important that children are exposed to a variety of other terms that implement in multiplication and partitioning situations and that the terms are used accurately. Ofttimes it is in demand to emphasise ane term more than others when introducing concepts, however a flexibility with terminology is to be aimed for.
Looking at where words make out from gives us some indication of what they mean. The word 'multiply' was used in the nonverbal sense from the late fourteenth century and comes from the Latin multi meaning 'many' and plicare pregnant 'folds' giving multiplicare - 'having many folds', which means 'many times greater in number'. The term 'manyfold' in English is antiquated but we calm use particular instances such atomic number 3 'twofold' surgery threefold'.
The word 'divide' was used in mathematics from the primal 15th century. It comes from the Latin, dividere meaning 'to force apart, cleave or distribute'. Interestingly, the Son widow has the Same etymological root, which ass be understood in the sense that a widow woman is a woman forced separated from her husband.
Use of the word 'product'
The product of two numbers is the resolution when they are increased. Then the product of
3 and 4 is the multiplication 3 × 4 and is equal to 12.
IT is important that we use the vocabulary related to multiplication and division aright. Many years ago we were told to 'do our sums' and this could apply to any calculation using any of the trading operations. This is an inaccurate use of the tidings 'add'. Finding the 'sum' of deuce or more numbers means to add them together. Teachers should take maintenance not to use the word 'sum' for anything but add-on.
The symbols × and ÷
The × symbol for multiplication has been in utilisation since 1631. IT was chosen for religious reasons to present the baffle. We read the statement 3 × 4 Eastern Samoa '3 multiplied by 4'.
In some countries a middle dot is secondhand so 3 × 4 is written A 3.4. In algebra it is common to non use a symbol for multiplication at completely. So, a × b is scripted as Av.
The air division symbolization ÷ is called the obelus. Information technology was first wont to signify class in 1659. We show the statement 12 ÷ 3 as '12 divided by 3'. Some other way to write division in school arithmetic is to use the notation 
'3 goes into 12'.
Mathematicians almost ne'er wont the ÷ symbol for division. Instead they use fraction notation. The writing of a divide is in truth another way to write air division. So 12 ÷ 4 is equivalent to writing 
Once students are becoming fluent with the concepts of generation and division then the symbolic notation, × for generation and ÷ for division, can be introduced. Initially, the ideas will Be explored through and through a conversation, past written in row, followed by a combining of words and numerals and finally victimisation numerals and symbols. At for each one step, when the child is ready, the use of symbols can reflect the child's ability to deal with abstract concepts.
MODELLING MULTIPLICATION
Modelling multiplication by arrays
Angular arrays can be accustomed modelling generation. E.g., 3 × 5 is illustrated by
We call 15 the product of 3 and 5, and we call 3 and 5 factors of 15.
By looking for at the rows of the raiment we see that
By looking at the columns of the array we also see that
This illustrates 3 × 5 = 5 × 3. We say that multiplication is independent.
Arrays are expedient because they throne be used with precise small also as very large numbers, and also with fractions and decimals.
CLASSROOM ACTIVITY
Children behind simulate multiplication using counters, blocks, shells or whatever materials that are available and arranging them in arrays.
- 1
- Children build arrays using a form of materials.
- 2
- Take a digital exposure.
- 3
- Describe the multiplication using wrangle, words and numbers and finally words
and symbols.
Modelling multiplication by skip-counting and on the come line
Skip-counting, much as reciting 3, 6, 9, 15,..., is one of the earliest introductions to repeated addition and thu to multiplication. This can be illustrated on a number line as shown for 3 × 5 = 15 below.
3 × 5 = 15
On the come telephone line, the fact that 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 is non so obvious; the previous simulacrum shows 5 + 5 + 5, whereas 3 + 3 + 3 + 3 + 3 looks quite different.
Skip-counting is important because information technology helps children learn their times tables.
Modelling generation by area
Replacing objects in an range by unit squares provides a natural transition to the area exemplary of multiplication. This is illustrated below for 3 × 5.
At this stage, we are simply using unit squares instead of counters or stars. We can also use the surface area model of generation later for multiplication of fractions.
Properties of Multiplication
One of the advantages of the array and area approach is that properties of generation are more apparent.
Commutativity
As discussed above, turning the 3 × 5 raiment on its sidelong illustrates that 3 × 5 = 5 × 3 because the total number of objects in the range does not change.
We saw this before by looking at the rows and columns separately, just we can also dress this by turning the rectangle on its side. The area of the rectangle does not alteration.
Associativity
Some other important attribute of multiplication is associativity, which says that
We can demonstrate this with the numbers 2, 3 and 4:
Associativity of multiplication ensures that the expression a × b × c is straightforward.
Whatsoever-order property
We usually don't teach young children associativity of multiplication explicitly when introducing multiplication. Instead, we teach the any-consecrate property of multiplication, which is a issue of the commutative and associative properties.
Any-order property of multiplication
A list of numbers can be increased together in any order to grant the product of the Numbers.
The some-order property of multiplication is analogous to the any- order property of addition. Both associativity and commutativity are nontrivial observations; note that minus and division are neither commutative nor associative. Once we are familiar with the arithmetic operations we tend to carry both associativity and commutativity of multiplication for granted, just as we do for addition. Every so often, information technology is worth reflecting that commutativity and associativity mix to give the important and powerful any-order properties for addition and multiplication.
Multiplying trine unhurt numbers game corresponds geometrically to hard the number of social unit cubes in (or volume of) a rectangular optical prism. The any-edict property of multiplication means that we can calculate this volume by multiplying the lengths of the sides in any order. The order of the calculation corresponds to slice the volume upwards in different ways.
 | |  |
| 5 × 2 = 2 × 5 | (5 × 2) × 3 = (2 × 5) × 3 | |
 |  | |
| 3 × 2 = 2 × 3 | (3 × 2) × 5 = (2 × 3) × 5 | |
 |  | |
| 5 × 3 = 3 × 5 | (5 × 3) × 2 = (3 × 5) × 2 |
We tail end apply this to the multiplication of 3 or more than Book of Numbers, it doesn't matter in which Holy Order we do this.
Distributivity of Propagation over Addition
The equality 3 × (2 + 4) = (3 × 2) + (3 × 4) is an example of the distributivity of multiplication over addition. With arrays, this corresponds to the following plot.
With areas it corresponds to the plot downstairs.
Generation is also diffusive terminated subtraction.
For example 7 × (10 − 2) = 7 × 10 − 7 × 2.
We use the distributive property to enable us to reduce multiplication problems to a compounding of familiar multiples. E.g.,
and
Work out 1
Use the allocable constabulary to carry out the following multiplications.
a 
The effect of multiplying by cardinal
When some number is multiplied by 1, the number is unvaried. E.g.,
We call 1 the multiplicative identity. It is important to stimulate this conversation with young children in very simple price, using lots of examples in the early stages of developing understanding roughly multiplication.
Zero is the identity element for addition. When nothing is added to a located there is no effect on the number of objects in that exercise set. For model,
This is true for all summation. Thu, we call zero the identity element for plus of
whole numbers.
The effect of multiplying by zero
When whatsoever number is increased by zero point the result is zero. Situations showing the effect of multiplying by goose egg can be acted out with children victimisation concrete objects.
For example,
If I have 5 baskets with three apples in each I have 5 × 3 = 15 apples in total.
Even so, if I have 5 baskets with 0 apples in each, the result is 5 × 0 = 0 apples in total.
Learning the times table.
Fluency with multiplication tables is essential for further math and in routine animation. Awhile it was considered unnecessary to learn multiplications tables by storage, but it is a great help to be fluent with tables in umteen areas of mathematics.
If students can add a single-digit number to a two-digit number, they tail end at to the lowest degree reconstruct their tables fifty-fifty if they have not yet developed fluency. It is therefore essential to ensure that students can bestow fluently ahead they begin to find out 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 sidereal day, and 60 minutes in an hour. Conversance with dozens is useful in routine life because packaging in 3 × 4 arrays is so much more convenient than in 2 × 5 arrays. In addition, the 12 × 12 tabular array has galore patterns that can be constructively misused in pre-algebra exercises.
A straightforward approach to learning the tables is to recite for each one row, either by memory Oregon by decamp-counting. However, students also 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.
All the same, in that respect are several techniques that can be misused to trim down the number of facts to be learnt.
- The commutativity of multiplication (8 × 3 = 3 × 8) right away reduces this turn to 78.
- The 1 and 10 times tables are straightforward and their domination reduces the number of facts to be learnt to 55.
- The 2 and 5 times tables are the easiest to learn and their subordination far reduces the keep down of facts to be learnt to 36.
- The 9 and 11 times tables are the next easiest to skip-count because 9 and 11 differ from 10 by 1. This reduces the number of facts to 21. Children may notice the decreasing ones digit and increasing tens digit in the nine times hold over. They may also be intrigued by the fact that the sum of the digits of a five-fold of 9 is always 9.
- The squares are useful and can be learnt just as one might watch a times table.
This reduces the number of terms to be learnt to 15.
Whatever techniques are used, the aim should be fluency.
MODELLING DIVISION
Division e'er involves splitting something into a total of equal parts, but in that location are many contrasting situations that can all be described by 'sectionalisation'. Ahead introducing the standard algorithm for division, IT is worthy discussing some of these situations under the headings:
- Division without remnant,
- Part with remainder.
Division without rest
Here is a simple model of the class 24 ÷ 8.
Question: If I pack 24 apples into boxes, each with 8 apples, how many boxes will there equal?
We pot visualise the backpacking treat by egg laying come out the 24 apples successively in rows of 8, as in the diagrams below.
The 3 rows in the last array use up all 24 apples, so there bequeath be 3 full boxes, with no apples leftish over. The result is written in exact symbols as
The number 24 is called the dividend ('that which is to represent bilocular'). The issue 8 is called the divisor ('that which divides'). The number 3 is named the quotient, (from the Latin quotiens meaning 'how many times').
Modelling division by skip-count and on the number line
Sectionalisation without remainder can be visualized as skip-counting.
Along the number origin we count in 8s until we reach 24.
EXERCISE 2
a
b
c
Using arrays to register division without remainder is the reverse of times
The rectangular array that we produced when we modeled 24 ÷ 8 is exactly the same array that we would draw up for the multiplication 3 × 8 = 24.
In our example:
- The statement 24 = 8 × 3 means 'three boxes, each with 8 apples, is 24 apples', and
- The statement 24 ÷ 8 = 3 agency '24 apples take up 3 boxes, each with 8 apples'.
Division without remainder is the inverse march of times. 
The multiplication command 24 = 8 × 3 can in turn off be reversed to give a ordinal class statement
which answers the question, 'What is 24 divided by 3?
This corresponds to rotating the array by 90°, and regarding it as made up of 8 rows of 3. It answers the question, 'If I pack 24 apples into boxes each holding 3 apples, how many boxes will be required?'
Then the division statement 24 ÷ 8 = 3 now has four equivalent forms:
EXERCISE 3
For each division affirmation, put down the corresponding multiplication statements, and the past in proportion to division statement.
a
c
Two models of division without remainder
children's questions often concern pairs of
situations look-alike to those described here.
If we have 24 balloons to share as, there
are two ways we fanny share them.
The first way is by asking 'How many groups?'
For example, if we have 24 balloons and we give
8 balloons apiece to a number of children, how many
children get 8 balloons?
If we split 24 balloons into groups of 8, then 3 children come 8 balloons each.
We say '24 divided aside 8 is 3'. This is written as 24 ÷ 8 = 3.
We can see this from the raiment:
The second way is by asking 'How many in each group?' For example, if we share
24 balloons among 8 children, how many balloons does all child receive? We want
to make 8 equal groups. We do this away handing out one balloon to each child. This uses
8 balloons. Then we do the aforementioned again.
We can do this 3 multiplication, so each child gets 3 balloons.
Again, we can go through this from the multiplication array:
And then nonbearing 24 by 8 is the same equally asking 'Which total arrange I multiply 8 by to aim 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 tail end have the same division statement. One problem with balloons is the associate of the new.
Work 4
Write down in symbols the variance assertion, with its answer, for all problem below. Then set down in words the associated problem:
a
b
c
d
Division with remainder
We will now use apples to model 29 ÷ 8.
Question: If I pack 29 apples into boxes, each with 8 apples, how many boxes will thither make up?
As before, we can visualise the packing cognitive process by laying out the 29 apples successively in rows of 8:
We can lay out 3 round rows, but the endure row alone has five apples, so there will be 3 full boxes and 5 apples left over. The result is written as
dividend divisor quotient remainder
The number 5 is known as the residuum because there are 5 apples left concluded. The remainder is always a whole number less than the divisor.
As with division without remainder, skip-count is the basis of this process:
We turn up 29 between successive multiples 24 = 8 × 3 and 32 = 8 × 4 of the factor 8. Then we subtract to find the remainder 29 − 24 = 5.
We could also induce answered the question above by locution, 'Thither volition be iv boxes, but the last package wish comprise 3 apples short.'
This corresponds to numeration backwards from 32 sooner than forwards from 24, and the corresponding mathematical assertion would Be
It is not normal practice at school, however, to use negative remainders. Even when the enquiry demands the interpretation corresponding to it, we will always maintain the usual school rule that the remnant is a integer to a lesser degree the divisor. Division without remainder ass be regarded Eastern Samoa division with remainder 0. During the location process, we actually land exactly on a multiple instead of landing betwixt two of them. E.g., 24 ÷ 8 = 3 remainder 0, or more simply, 24÷ 8 = 3, and we sound out that
The corresponding multiplication and addition affirmation
The 29 apples in our good example were packed into 3 sounding boxes of 8 apples, with 5 left over. We can write this as a division, only we lavatory also write information technology victimisation a product and a sum,
So for division with remainder there is a corresponding statement with a multiplication followed by an addition, which is more complicated than division without remainder.
Two models of sectionalization with remainder
As before, problems involving partitioning with remainder usually receive an associated problem modelling the same division statement. Continuing with our deterrent example of
Question: How many bags of 8 apples can I throw from 29 apples and how many are unexpended over?
Question: I have 29 apples and 8 boxes. How many apples should I insert each loge and so that there is an equal number of apples in each box and how many are leftover?
The shadowing deuce associated questions model 63 ÷ 10 = 6 remainder 3.
Question: If I have 63 dollar coins, and 10 people to founde them to, how some coins does from each one person get if they are to each let the same number of coins? How many are left over?
Question: If I give birth 63 dollar coins, how many $10 books can I buy and how many dollars bash I have left over?
EXERCISE 5
Serve each question in words, then write down its the associated part problem and answer information technology.
- a
- How many 7-person saving teams can be formed from 90 people?
- b
- How many 5-seater cars are needed to transport 43 people, and how many spare seating area are in that location?
Properties of Partitioning
Order and brackets cannot follow ignored
When multiplying deuce numbers, the edict is lilliputian. For example,
When dividing Numbers, however, the order is crucial. For instance,
To visualise this calculation, 20 people absolute in 4 homes means each home has happening average 5 people, whereas 4 citizenry surviving in 20 homes means to each one house has on medium 
Similarly when multiplying numbers, the use up of brackets is unimportant. For example,
When dividing numbers, however, the use of brackets is crucial. For example,
Partition past zero
Earlier we used empty baskets of apples to illustrate that 5 × 0 = 0.
The Lapplander simulate can be accustomed illustrate why sectionalization by zero is undefined.
If we have 10 apples to be shared equally amongst 5 baskets all handbasket will make
10 ÷ 5 = 2 apples in each.
If the 10 apples are mutual every bit between 10 baskets, each basketball hoop has 10 ÷ 10 = 1
apples in each.
If 10 apples are shared between 20 baskets, each basket will have 
What happens if we try to share 10 apples betwixt 0 baskets? This cannot be done.
| If | | 10 ÷ 0 = a 1 |
| 10 ≠ a × 0. |
This action is meaningless, so we say that 10 ÷ 0 is undefined.
We mustiness always Be careful to relate this to children accurately so that they understand that:
- 10 ÷ 0 is NOT tantamount to 1 and
- 10 ÷ 0 is NOT equal to 0
just 10 ÷ 0 is not defined.
Divisional by 4, 8, 16, . . .
Because 4 = 2 × 2 and 8 = 2 × 2 × 2, we can divide by 4 and 8, and by wholly powers of 2, away successive halving.
To divide by 4, halve and halve again. For instance, to divide 628 past 4,
To divide by 8, halve, halve, and halve again. For instance, to split 976 aside 8,
Multiplication Algorithmic program
An algorithm works most efficiently if it uses a small number of strategies that go for in all situations. And so algorithms do non resort to techniques, such arsenic the use of near-doubles, that are efficient for a few cases but useless in the majority of cases.
The standard algorithm testament not help you to multiply cardinal sole-digit Book of Numbers. It is essential that students are fluent with the multiplication of two concentrated-digit numbers and with adding numbers game to 20 before embarking on any formal algorithm.
The distributive property is at the heart of our multiplication algorithm because it enables us to calculate products one editorial at a time and so attention deficit disorder the results together. It should beryllium strengthened arithmetically, geometrically and algorithmically.
For exemplar, arithmetically we have 6 × 14 = 6 × 10 + 6 × 4, geometrically we see the same phenomenon,
and algorithmically we implement this in the shadowing calculation.
| 1 | 4 | ||
| × | 6 | ||
| 2 | 4 | ||
| 6 | 0 | + | |
| 8 | 4 |
Once this basic property is understood, we can go on to the narrowed algorithm.
Introducing the algorithm using materials
Initially when children are doing times they volition act out situations victimisation blocks. Eventually the numbers they want to procreate will become also large for this to be an efficient means of resolution multiplicative problems. However base-10 materials or bundles of polar-pole sticks canful beryllium used to introduce the more cost-efficient method - the algorithmic rule.
If we deprivation to multiply 6 by 14 we construct 6 groups of 14 (or 14 groups of 6):
Pick up the 'tens' conjointly and collect the 'ones' together.
This gives 6 'tens' and 24 'ones'.
Then make as many tens from the loose ones. Thither should never comprise much nine single ones when representing whatever number with Base-10 blocks.
This gives 6 'tens' + 2 'tens' + 4 'ones'.
We add the tens to get
14 × 6 = 10 × 6 + 4 × 6 = 60 + 20 + 4 = 84
Eventually we should start recording what is organism through with the blocks using the multiplication algorithm plumb initialise. Eventually the support of victimisation the blocks can be born and students can complete the algorithmic program without concrete materials.
Multiplying by a widowed fingerbreadth
First we contract the reckoning by keeping track of deport digits and incorporating the plus as we go. The late calculation shortens as either
depending on where the carry digits are recorded.
Handle should be taken still at this incipient stage because of the mixture of multiplication and addition. Note also that the exact location and size of the carry finger's breadth is not essential to the process and varies across cultures.
Multiplying by a exclusive-digit multiple of a power of ten
The succeeding observation is that multiplying past a exclusive-finger's breadth quintuple of ten is No harder than multiplying away a single figure provided we keep track of place value. So, to find the number of seconds in 14 proceedings we calculate
14 × 60 = 14 × 6 × 10 = 840
and implement IT algorithmically as
| 1 | 4 | ||
| × | 6 | 0 | |
| 8 | 4 | 0 |
Likewise, we can keep track of higher powers of ten by using place value to our advantage. So
becomes
| 1 | 4 | |||
| × | 6 | 0 | 0 | |
| 8 | 4 | 0 | 0 |
For students who have met the implicit reflection Eastern Samoa part of their mental pure mathematics exercises the only novelty at this dot is how to lay out these calculations.
Multiplying away a deuce-figure number
The succeeding cognitive jump happens when we use distributivity to breed two two-digit numbers together. This is implemented as two products of the types mentioned above. For example,
is used in the two-step calculation below.
| 7 | 4 | ||
| × | 6 | 3 | |
| 2 | 2 | 2 | |
| 4 | 4 | 4 | 0 |
| 4 | 6 | 6 | 2 |
This corresponds to the region decomposition illustrated below.
In the early stages, it is worth at the same time nonindustrial the arithmetic, geometric and algorithmic perspectives illustrated above.
Unpacking each line in the hanker times calculation exploitation distributivity explicitly, as in
| 7 | 4 | ||
| × | 6 | 3 | |
| 1 | 2 | ||
| 2 | 1 | 0 | |
| 2 | 4 | 0 | |
| 4 | 2 | 0 | 0 |
| 4 | 6 | 6 | 2 |
corresponds to the orbit rotting
It is non efficient to do this extended long multiplication in ordain to calculate products in general, but it can be used to highlight the multiple use of distributivity in the process. The area model illustration misused in that case reappears later every bit a geometric interpretation of calculations in algebra.
The standard division algorithm
There is only ane authoritative division algorithm, despite its different appearances. The algorithm buns be set out every bit a 'long division' calculation to show totally the steps, or as a 'shortstop division' algorithm where but the carries are shown, or with no written working at entirely.
Setting the calculation out as a drawn-out partition
We could set the work out as follows:
5 × 30 = 150, and so subtract 150 from 193
5 × 8 = 40, so subtract 40 from 43
The standard 'long division' setting-tabu, however,
allows place value to work for United States of America even more efficiently,
by working only with the digits that are required for
each particular division. At for each one step another digit is
necessary − this is usually called 'bringing down the next finger'.
5 × 4 = 20, then deduct 20 from 21.
Bring down the 9, and divide 19 by 5.
5 × 3 = 15, then take off 15 from 19.
Bring down the 3, and divide 43 by 5.
5 × 8 = 40, then subtract 40 from 43.
Hence 2193 ÷ 5 = 138 remainder 3.
(Never forget to gather the calculation up into a conclusion.)
The placing of the digits in the top credit line is determinative. The foremost step is '5 into 21 goes 4', and the figure 4 is set to a higher place the digit 1 in 21.
Setting the calculation out every bit a short division
Once the steps have been mastered, many people are comfortable doing each multiplication/deduction step mentally and writing down just the carry. The deliberation then looks similar this:
 | | We say, | '5 into 21 goes 4, remainder 1'. |
| '5 into 19 goes 3, residual 4'. | |||
| '5 into 43 goes 8, remainder 3'. |
Zeroes in the dividend and in the steps
Zeroes will cause nary problems provided that all the digits are kept strictly in their correct columns. This identical principle is fundamental to all algorithms that depend on place value.
The example to the right wing shows the overnight air division and stumpy division calculations for
We twice had to bring down
the digit 0, and two of the divisions resulted in a quotient of 0.
Using the calculator for class with remainder
Multitude often say that air division is easily done connected the calculating machine. Naval division with remainder, however, requires some common good sense to form out the answer.
EXAMPLE
Usage the calculator to convert 317 minutes to hours and proceedings.
Solution
We can construe that
With a calculator exploitation the division primal: Enter 350 ÷ 60, and the answer is 5.833333… hours. So subtract 5 to fetch 0.833333…, and multiply by 60 to convince to 50 minutes, giving the answer 5 hours and 50 minutes.
Calculator assistance may be extremely useful with larger numbers, but have with long division is essential to understand the estimator display This phenomenon is informal to umteen akin situations in mathematics.
Links Saucy
The first application of times that students are likely to meet is division. When calculating a variance, we are constantly calculating multiples of the factor, and lack of articulateness with multiplication is a significant baulk in this process. The incarnate therein module lays the foundation for multiplication, and then division, of fractions and decimals.
Other applications of generation admit percentages and consumer arithmetic. For example, we calculate the price of an item comprehensive of GST by shrewd 1.1 multiplication its pre-GST cost.
A familiarity with multiplication and the verbal expression of numbers as products of factors paves the way for one of the major theorems in maths.
The Fundamental Theorem of Arithmetic states that all integer larger than 1 can be written as a product of prime Book of Numbers and such an formulation is unique up to the order in which the factors are scrawled.
For example, 24 = 23 × 3 and 20 = 22 × 5.
The Important Theorem of Arithmetic has far-reaching consequences and applications in computing, coding, and public-key cryptography.
Last, simply not least, a strong earthing in arithmetic sets a student up for success in algebra.
The division algorithm uses multiplication and subtraction. As such, division demands that we synthesise a lot of anterior knowledge. This is what makes section challenging, and for umpteen students it is their first predilection of multi-layered processes. The ability to reflect on what you know, and enforce IT within a new, higher-level cognitive operation is one of the generic mathematical skills that division helps to develop.
The implementation of the division algorithm is typically a multi- step process, and as so much it helps to develop skills that are invaluable when students march on to algebra. The link to factors is also critical in later days.
History
The product of two Numbers is the aforementioned no matter how you calculate it or how you write your answer. Fair as the history of number is real all about the development of numerals, the history of multiplication and division is mainly the history of the processes people have ill-used to perform calculations. The development of the Hindi-Semite place-value notation enabled the implementation of efficient algorithms for arithmetical and was probably the main reason for the popularity and fast adoption of the notation.
The earlier recorded example of a division enforced algorithmically is a Sunzi partition dating from 400AD in China. In essence the same process reappeared in the Good Book of al Kwarizmi in 825AD and the modern-day equivalent is known Eastern Samoa Ship's galley sectionalisation. It is, in essence, combining weight to modern-solar day long-dated division. However, it is a wonderful example of how notation can make an enormous departure. Galley division is hard to follow and leaves the page a mess compared to the modern layout.
The layout of the long sectionalization algorithm varies between cultures.
End-to-end history there have been many different methods to solve problems involving multiplication. Some of them are still in use in different parts of the world and are of interest to teachers and students equally alternate strategies OR because of the mathematical challenge involved in learning them.
Italian or grille method acting
Another technique, known as the European country or lattice method acting is basically an implementation of the prolonged version of the standard algorithm but in a different layout. The method is very old and might rich person been the one widely adopted if it had non been indocile to impress. It appears to give first appeared in India, only soon appeared in works by the Chinese and past the Arabs. From the Arabs it found its way across to Italy and can atomic number 4 establish many Italian manuscripts of the 14th and 15th centuries.
The times 34 × 27 is illustrated Hera.
34 × 27 = 918
In the top satisfactory rectangle 4 × 2 is calculated. The digit 8 is placed in the bottom triangle and 0 in the top triangle.
Then 3 × 2 is measured and the result entered arsenic shown.
In the bottom right rectangle 4 × 7 is calculated. The finger 8 is placed in the bottom triangle and the digit 2 in the overstep triangle. The result of 3 × 7 is also filmed in this way.
The unripe diagonal contains the units.
The blue diagonal contains the tens.
The orange diagonal contains the hundreds.
The digits are now summed on each inclined starting from the right and each
result recorded as shown. Note that there is a 'dribble' from the 'tens diagonal' to the 'hundreds diagonal'
References
A History of Mathematics: An Introduction, 3rd Edition, Victor J. Katz, Addison-Wesley, (2008)
History of Mathematics, D. E. Smith, Capital of Delaware publications Recent House of York, (1958)
Knowing and Teaching Elementary Math: teachers' understanding of underlying maths in China and the United States. Liping Ma, Mahwah, N.J.: Lawrence Erlbaum Associates, (1999)
History of Maths, Carl B. Boyer (altered by Uta C. Merzbach), John Wiley and Sons, Inc., (1991)
ANSWERS TO EXERCISES
Exercising 1
| a | 9 × 32 | = 9 × 30 + 9 × 2 |
| = 270 + 18 | ||
| = 2888 |
| b | 31 × 8 | = 30 × 8 + 1 × 8 |
| = 240 + 8 | ||
| = 248 |
| c | 102 × 8 | = 100 × 8 + 2 × 8 |
| = 800 + 16 | ||
| = 816 |
Physical exertion 2
- a
- 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42. Hence 42 ÷ 3 = 14.
- b
- 11, 22, 33, 44, 55. Hence 55 ÷ 11 = 5.
- c
- 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000. 1000 ÷ 100 = 10.
Exercise 3
- a
- 8 = 2 × 4 and 8 ÷ 4 = 2.
- b
- 56 ÷ 8 = 7 and 56 ÷ 7 = 8.
- c
- 81 = 9 × 9 and 81 ÷ 9 = 9. Because the divisor and the quotient are the same, the generation statement becomes a affirmation about squaring, and the other corresponding part statement is the same as the freehand statement.
Exercise 4
- a
- 24÷ 4 = 6. If 24 children are divided into groups of 4, how many groups are thither?
- b
- 20 ÷ 2 = 10. If 20 metres of fabric is in disagreement into ii level pieces, how long is
- c
- 160 ÷ 10 =16. If 160 books are bundled into packages of 10 each, how some packages are there?
- d
- 35÷7 = 5. If a 35-day period is divided into 7 equal periods, how long is to each one period?
Exercise 5
- a
- Twelve 7-person rescue teams can buoy be formed, with 6 people to unneeded. How more citizenry will be in 7 compeer groups tensile from 90 people? Thither will be 12 populate in each aggroup, with 6 leftover.
- b
- Nine 5-seater cars are needed, and at that place will be two supererogatory seats. How many people will be in 5 equal groups formed from 43 populate? There will be 8 groups, with 3 hoi polloi left concluded.
Drill 6
- a
- 246 ÷ 4 = (246 ÷ 2) ÷2 = 123 ÷ 2 = 61
- b
- 368 ÷ 8 = ((368 ÷2) ÷ 2) ÷ 2 = (184 ÷ 2) ÷2 = 92 ÷ 2= 46.
- c
- 163 ÷ 8 = ((163 ÷ 2) ÷ 2)÷2 =81
÷ 2 = 40
÷ 2 = 20
.
| d | 12 048 ÷ 16 | = (((12 048 ÷ 2) ÷ 2 ÷ 2) ÷ 2 = ((6024 ÷ 2) ÷ 2) ÷ 2 |
| = (3012 ÷ 2) ÷ 2 = 1506 ÷ 2 = 752. |
The Improving Mathematics Education in Schools (Multiplication) Cast 2009-2011 was funded by the Aboriginal Australian Government Department of Education, Employment and Workplace Relations.
The views verbalised here are those of the author and do not necessarily defend the views of the Australian Government Department of Instruction, Employment and Work Relations.
© The University of Melbourne connected behalf of the International Centre of Excellency for Education in Math (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where differently indicated). This study is licensed under the Creative Commonalty Attribution-NonCommercial-NoDerivs 3.0 Unported License.
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solve 42 divided by 3 using an area model
Source: http://amsi.org.au/teacher_modules/multiplication_and_division.html
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