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This unit presents a range of strategies for solving multiplication and division problems with multi-digit whole numbers.

Multiplication and Division

Students are encouraged to notice the structure of problems, and to anticipate which strategies might be best suited to solving them. This unit builds on the ideas presented in the Multiplication Smorgasbord session in Book 6: Teaching Multiplication and Division. The New Zealand Curriculum requires students to understand and use a range of mental, written and digital calculation strategies to multiply and divide multi-digit whole numbers.

Students at this stage partition and recombine numbers to simplify calculations and draw on their knowledge of multiplication facts and related division facts with factors up to ten. Understanding whole number place value underpins all strategies in this unit. The purpose of this session is to explore the range of strategies that your students already use to solve multiplication and division problems. This will enable you to evaluate which strategies need to be focused on in greater depth as well as identifying students in your group as "expert" in particular strategies. Problem 1 Copymaster Two : Vanessa bikes 38 kilometres each day for five days.

How many kilometres has she travelled by the end of the five days? Ask students to work out the answer in their head if they can and record their strategy on paper. Give the students an appropriate amount of thinking time. Then ask them to share their solutions with their learning partner. The following are possible responses:. As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the strategies that are elicited at this stage, recording them to reflect upon later in the unit perhaps in a modelling book.

Be aware that some students may elect to add rather than multiply. Problem 2 Copymaster Three : There are rowers entered in the eights rowing champs at the Maadi Cup, not including the drivers coxwains. That is 20 crews. There are 24 rowers left. Dividing by 8 is like dividing by 2 then 2 then 2 so half is 92 and half 92 is 46 and divide by 2 again leaves me 23 so the answer is That is two crews. Watch for repeated subtraction or partial use of multiplication, such as:. Ask students to reflect on the strategies that have been discussed in the session and evaluate which strategies that they personally need further work on, perhaps using thumb signals - thumbs up - confident and competent with the strategy, thumbs sideways - semi confident, thumbs down - not yet confident.

Use this information to plan for your subsequent teaching from the exploring section outlined below. Over the next two to three days, explore the following strategies, making explicit the strategy you are concentrating on as the teacher and the reason for using the selected strategy. The following questions are provided as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs.

How much money does the bike cost? The place value strategy involves multiplying the ones, and tens separately then combining the partial products. In the above problem the student might say the following: I multiplied 7 x 50 and got , then I multiplied 7 x 4 and got I added and 28 to get If the students do not seem to understand the partitioning concept, show the problems physically. Some students may find it useful to record and keep track of their thinking. An extension of the place value strategy involves the use of standard written form for multiplication.

Place value partitioning division Pisi has an after-school job at the market, bagging pawpaw into bags of 6. If there are pawpaw to be bagged, how many bags can he make? The long division written form will be familiar to most teachers. That leaves me with I can take away from that, which is 20 x 6. That leaves If I take another pawpaw away I get 24, which is 4 lots of 6. This thinking could be recorded as:. If the students do not seem to understand the partitioning concept, show the problems physically, e. An extension of the place value strategy involves the use of standard written form for division.

The rounding and compensating strategy involves rounding a number in a question to make it easier to solve. In the above question 48 can be rounded to 50 by adding 2. If the students do not seem to understand the tidy numbers concept, use place value equipment or a large dotty array to show the problems physically. Recording might look like this:.

View how to do this using Videos Seven and Eight. Note that the problems posed here are using a tidying up strategy rather than tidying down. If one of the factors is just over a tidy number such as 42 then standard place value partitioning tends to be a more useful strategy. Sarah uses nine bus tickets every week to travel around town. She wins tickets in a radio competition. How many weeks will the tickets last her? Rounding and compensating for division involves finding a number that is close to the dividend starting amount and working from that number to find an answer. For the question above, a student might say:.

If the students do not seem to understand the rounding and compensating concept, use place value materials, or a large dotty array, to represent the problems physically. Students may find it useful to record and keep track of their thinking, especially if they partially divide the dividend at first. Use the same context of buses to pose problems where rounding and compensating is a sensible strategy.

Proportional adjustment involves using knowledge of factors and multiples to create easier equations that have the same answer. Factors are proportionally adjusted to make one or both factors easier to work from. In the above problem the factors could be adjusted as follows: Or, using doubling and halving:.

If the students do not seem to understand the proportional adjustment concept, use a large dotty array to show the problems physically. In division, proportional adjustment involves changing both numbers in the equation by the same factor. Therefore, the numbers used to proportionally adjust the problem must be factors of both numbers in the equation dividend and divisor. Zero is the identity element for addition.

When nothing is added to a set there is no effect on the number of objects in that set. This is true for all addition. Hence, we call zero the identity element for addition of whole numbers. When any number is multiplied by zero the result is zero. Situations showing the effect of multiplying by zero can be acted out with children using concrete objects. 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 great help to be fluent with tables in many areas of mathematics.

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If students can add a single-digit number to a two-digit number, they can at least reconstruct their tables even if they have not yet developed fluency. 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. A straightforward approach to learning the tables is to recite each row, either by heart or by skip-counting. However, students also need to be able to recall individual facts without resorting to the entire table. However, there are several techniques that can be used to reduce the number of facts to be learnt.

Before introducing the standard algorithm for division, it is worthwhile discussing some of these situations under the headings:. Question: If I pack 24 apples into boxes, each with 8 apples, how many boxes will there be? We can visualise the packing process by laying 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 will be 3 full boxes, with no apples left over.

The result is written in mathematical symbols as. Using arrays to show division without remainder is the inverse of multiplication. Division without remainder is the inverse process of multiplication. For each division statement, write down the corresponding multiplication statements, and the other corresponding division statement. What happened in this example, and why?

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  7. If we have 24 balloons to share equally, there are two ways we can share them. For example, if we have 24 balloons and we give 8 balloons each to a number of children, how many children get 8 balloons? We want to make 8 equal groups. We do this by handing out one balloon to each child.

    This uses 8 balloons. Then we do the same again. For each division problem, there is usually an associated problem modelling the same division statement. One problem with balloons is the associate of the other. Write down in symbols the division statement, with its answer, for each problem below. Then write down in words the associated problem:. 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 We can lay out 3 full rows, but the last row only has five apples, so there will be 3 full boxes and 5 apples left over.

    The result is written as. The number 5 is called the remainder because there are 5 apples left over. The remainder is always a whole number less than the divisor. As with division without remainder, skip-counting is the basis of this process:. This corresponds to counting backwards from 32 rather than forwards from 24, and the corresponding mathematical statement would be. It is not normal practice at school, however, 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 remainder 0. During the location process, we actually land exactly on a multiple instead of landing between two of them. The 29 apples in our example were packed into 3 full boxes of 8 apples, with 5 left over.

    We can write this as a division, but we can also write it using 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. As before, problems involving division with remainder usually have an associated problem modelling the same division statement. Continuing 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 have 29 apples and 8 boxes.

    How many apples should I put in each box so that there is an equal number of apples in each box and how many are leftover? Question: If I have 63 dollar coins, and ten people to give 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?

    Multiply & divide with Grade 5 worksheets from K5

    Answer each question in words, then write down its the associated division problem and answer it. 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 of a person. Similarly when multiplying numbers, the use of brackets is unimportant. The same model can be used to illustrate why division by zero is undefined.

    If 10 apples are shared between 20 baskets, each basket will have an apple in each. What happens if we try to share 10 apples between 0 baskets? This cannot be done. We must always be careful to relate this to children accurately so that they understand that:. To divide by 8, halve, halve, and halve again. For example, to divide by 8,. An algorithm works most efficiently if it uses a small number of strategies that apply in all situations.

    So algorithms do not resort to techniques, such as the use of near-doubles, that are efficient for a few cases but useless in the majority of cases. The standard algorithm will not help you to multiply two single-digit numbers.

    Multiplication and Division :

    It is essential that students are fluent with the multiplication of two single-digit numbers and with adding numbers 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 column at a time and then add the results together. It should be reinforced arithmetically, geometrically and algorithmically. Once this basic property is understood, we can proceed to the contracted algorithm. 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 be an efficient means of solving multiplicative problems. However base materials or bundles of icy-pole sticks can be used to introduce the more efficient method - the algorithm. Then make as many tens from the loose ones. There should never be more than nine single ones when representing any number with Base blocks. Eventually we should start recording what is being done with the blocks using the multiplication algorithm vertical format.

    Eventually the support of using the blocks can be dropped and students can complete the algorithm without concrete materials. First we contract the calculation by keeping track of carry digits and incorporating the addition as we go. The previous calculation shortens as either. Care should be 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 across cultures.

    The next observation is that multiplying by a single-digit multiple of ten is no harder than multiplying by a single digit provided we keep track of place value. So, to find the number of seconds in 14 minutes we calculate. Similarly, we can keep track of higher powers of ten by using place value to our advantage. For students who have met the underlying observation as part of their mental arithmetic exercises the only novelty at this point is how to lay out these calculations.

    The next cognitive jump happens when we use distributivity to multiply two two-digit numbers together. This is implemented as two products of the types mentioned above. In the early stages, it is worth concurrently developing the arithmetic, geometric and algorithmic perspectives illustrated above. Unpacking each line in the long multiplication calculation using distributivity explicitly, as in.

    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 interpretation of calculations in algebra. There is only one standard division algorithm, despite its different appearances. Divide 21 by 5. Bring down the 9, and divide 19 by 5. Bring down the 3, and divide 43 by 5. Never forget to gather the calculation up into a conclusion.

    The placing of the digits in the top line is crucial. The calculation then looks like this:. Zeroes will cause no problems provided that all the digits are kept strictly in their correct columns. This same principle is fundamental to all algorithms that rely on place value. The example to the right shows the long division and short division calculations for.

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    We twice had to bring down the digit 0, and two of the divisions resulted in a quotient of 0. It is possible to extend the division algorithm to divide by numbers of more than one digit. See module, Division of Whole Numbers F to 4.