Multiplication and Division of Fractions and Worksheet - DOC by vcp13468


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									     MATHEMATICS / UNIT PLANNER                            –      VELS Level      3-4      Term / Year        2007         Year Level       3-6

     Dimension:           Number
     Focus:               Multiplication facts
       1.25                Drawing of diagrams to show sharing of up to 20 items

       1.5                 Counting by 2s, 5s and 10s from 0 to a given target …

       2.0          ... Students skip count by 2s, 4s and 5s from 0 to 100 starting from any natural number.
                    Students describe and calculate simple multiplication as repeated addition, such as 3 × 5 = 5 + 5 + 5; and division as
                    sharing, such as 8 shared between 4.

                    They use commutative and associative properties of addition and multiplication in mental computation (for example, 3 + 4
                    = 4 + 3 and 3 + 4 + 5 can be done as 7 + 5 or 3 + 9).

       2.25                Use of money as a model for grouping and unpacking lots of 10s

                           Use of written number sentences such as 20 † 4 = 5 to summarise sharing (partition) and „how many?‟ (quotition)

       2.5                 Automatic recall of number facts from 2, 5 and 10 multiplication tables

       2.75                Representation of multiplication as a rectangular array and as the area of a rectangle

                           Use of fact families to solve division problems, for example 5 × 7 = 35, 35 ÷ 7 = 5

       3.0          ... Students compute with numbers up to 30 using all four operations.
                    They provide automatic recall of multiplication facts up to 10 × 10.

                    They devise and use written methods for: whole number problems of addition and subtraction involving numbers up to
                    999; multiplication by single digits (using recall of multiplication tables) and multiples and powers of ten (for example, 5 ×
                    100, 5 × 70 ); division by a single-digit divisor (based on inverse relations in multiplication tables).

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       3.25              Appropriate selection and use of mental and written algorithms to add, subtract, multiply and divide (by single digits)
                          natural numbers

                         Multiplication of fractions by fractions through use of the rectangle area model (grid)

       3.75              Multiplication by increasing and decreasing by a factor of two; for example, 24 × 16 = 48 × 8 = 96 × 4 = 192 × 2 =
                          384 × 1 = 384

                         Recognition that multiplication can either enlarge or reduce the magnitude of a number (multiplication by fractions or

                         Use of inverse relationship between multiplication and division to validate calculations

       4.0          ... Students explain and use mental and written algorithms for the addition, subtraction, multiplication and division of
       Standard     natural numbers (positive whole numbers).

                    They add, subtract and multiply fractions and decimals (to two decimal places) …

       4.25              Use of index notation to represent repeated multiplication

                         Division of fractions using multiplication by the inverse

       4.75              Addition, multiplication and division of integers

       5.75              Division and multiplication of numbers in index form, including application to scientific notation


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                                                                  Background teacher knowledge

     Success depends on students having automatic recall of multiplication facts for all numbers from 0 to 10.

     Earlier, students have automatic recall of simple number facts but do not recall all the multiplication facts up to 10 × 10. The challenge is filling
     the gaps for the more „unusual‟ facts and developing speed and reliability.

     Students can make very good use of a calculator as a support initially, but their mathematical progress will be hindered if they do not have
     fluent recall of multiplication facts.

                                                                      Detecting lack of recall
     It is easy to identify when students do not have fluent recall of multiplication facts by direct questioning, but it is also evident in many other
     circumstances. For example, students without fluent recall of multiplication facts will find it difficult to carry out any computations that include
     multiplication facts (e.g. long multiplication), find factors of numbers, put fractions in lowest terms or work out areas etc.

     When students find it difficult to automatically recall their tables.

     One of the brightest maths students that I ever taught always had trouble recalling all his maths tables. He did however have excellent
     strategies to work out the answers efficiently. He would often say I don‟t know the answer to a table such as 8 x 6 but I know 4 x 6 is 24 so 8 x
     6 is double 24 which is 48. In some respects this type of thinking is of more value than the student who can automatically give you the answer
     yet have poor understanding of using the relationships between numbers. As part of your presentation of tables activities consider including
     activities that invite children to use clever strategies to get answers.

     Using simpler related tables as a scaffold.

     Children often know their 1, 2, 5 and 10 times tables but may find many of the others tricky. If they can put the commutative property of
     multiplication to practice they know 64 of the 100 tables. Add to this square numbers, favourites and number others that they know because
     they Try giving students the opportunity to use the tables that they know as stepping stones to reach tables above or below that they don‟t
     know. Imagine a window cleaner of a ten story building. He sets his scaffolds on the 2nd, 5th

                                                                       Assessment tasks

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     Examples of the types of tasks that would be illustrative of multiplication concepts, aligned from the Mathematics Online Interview:

            Question 32 - Multiplication tasks

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                                                                     Teaching Strategies
     Learning mathematics involves some memorisation. There will be some drill-and-practice, but it should be on a basis of strong understanding of
     mathematical principles. Memorisation need not be a chore. As with other memory intensive tasks (e.g. spelling), set clear goals that are not
     too overwhelming, enlist the help of parents, develop students‟ metacognition and make it fun with plenty of encouragement.

                          Focus:                                               Activity                                           Assessment /
                         Materials                                                                                                 Reflection
         one                             Set realistic goals

                                           Expect students to learn their tables, but in stages. Set realistic goals so that
                                           students build competence in stages without ever being overwhelmed. Start with
                                           the easiest tables

                                           2×, 5×, 10×,

                                           then move onto

                                           3×, 4×, 9×

                                           and then

                                           6×, 8×, 7×.

                                           Don't forget 1× and 0×.

                                           Make links where possible: e.g. relate 5× to reading the minute hand on a clock,
                                           7× to days in weeks etc.

         two                             Stress number patterns and properties

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                                           Students need first to learn multiplication tables with understanding. This means
                                           that they can work out the answers in a variety of ways, such as:

                                                4 × 6 is equal to 6 + 6 + 6 + 6 (i.e. I can add if needed, but it takes too

                                                4 × 6 = 3 × 6 + 6 (i.e. I can work out an unknown result by building on a
                                                 known near result)

                                                4 × 6 = 6 × 4 (i.e. if I know one I already know the other)

                                                4 × 6 = 2 × 12 (i.e. I can convert one I don‟t know to one I do know)

                                                known characteristics, such as 4 x 6 will be an even number (so it is not

                                           NOTE: Fluency follows understanding, and is no substitute for it. If students
                                           learn multiplication tables with understanding and are aware of number patterns,
                                           then it is not a big step to attain fluency.

                                           Using number patterns and number properties makes learning tables easier. For
                                           example, because of the commutative law, when it comes time to learn the 8×
                                           and 7× tables, the only new facts are 8×8 = 64, 8×7 = 56 and 7×7 = 49, because
                                           all the other facts are in previously learned tables. Teachers will have
                                           experienced that these are the three facts less likely to be known.

                                           Examples of patterns include the sequences of digits: eg for the 9s, the ones
                                           digits are 9, 8, 7, 6, etc, whilst the tens digits are 0, 1, 2, 3, etc. The sum of the
                                           two digits is 9 (e.g. 5 × 9 = 45 and 4 + 5 = 9)

                                           It is helpful to many students to see the patterns visually on a hundreds chart.

         three                           Enlist help of parents

                                           Time for individual practice at home is essential. Enlist help from parents to
                                           monitor their children‟s practice and test them. Remind parents of the importance
                                           of encouragement and also the importance of feedback - students should not
                                           practise mistakes.

                                           One piece of useful advice is to use simple flashcards, small scale, with only the
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                                           facts that are being targeted. Don‟t take much time on practising facts that are
                                           already known, but focus on missing skills.

                                           Another useful strategy is to use the multiplication grid up to 10 × 10. See how
                                           quickly the student can find the facts on the grid. This reinforces correct
                                                         1        2    3    4    5     6     7     8     9    10

                                                        11        12   13   14   15   16    17    18    19    20

                                                        21        22   23   24   25   26    27    28    29    30

                                                        31        32   33   34   35   36    37    38    39    40

                                                        41        42   43   44   45   46    47    48    49    50

                                                        51        52   53   54   55   56    57    58    59    60

                                                        61        62   63   64   65   66    67    68    69    70

                                                        71        72   73   74   75   76    77    78    79    80

                                                        81        82   83   84   85   86    87    88    89    90

                                                        91        92   93   94   95   96    97    98    99   100

         four                            Strengthen students' metacognition

                                           Pay attention to strengthening students‟ metacognition. Teach students some
                                           basic strategies for memorising that can be applied to many school subjects
                                           including spelling etc, such as “Copy-cover-write-check”. Ensure that students
                                           know that if they do not frequently check their answers, they may practise

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         five                            : Use games with a chance element

                                         There are many classroom games to make practice of tables enjoyable (e.g.
                                         adaptations of Bingo, dominoes, card games, Multo (from Maths300 ), commercial
                                         computer programs). Again, make sure that there is a process for challenging
                                         wrong answers, so they are not practised.

                                         If using competitions, include an element of chance, so that the winners are not
                                         predictable, and do not score only on speed of response. Avoid comparisons and
                                         simplistic „league tables‟ or „tables races‟ where the slower children are easily
                                         discouraged. Make sure children get credit when they achieve individually
                                         appropriate goals.

                                         Challenge students to record and beat their 'personal best', rather than focusing
                                         on relative performance. Praise, and even rewards, for achievements works
                                         wonders. Encouragement is critical, particularly for those who are slower to

                                         Accuracy is clearly far more important than speed, at any time. Do not over-
                                         emphasise speed of response and do not only play games that require
                                         speed to win. However, reasonable speed is a goal of attaining fluent

        Activity     A challenge that Land Grab
          six        requires children
                     to trace the      Aim – To fill more squares on the grid page than your opponent.
                     outline of their
                     tables on a grid  The game.-
                     sheet.             Players take it in turn to roll the two dice or turn over two cards. They then
                                       multiply the numbers together referring to the tables reference sheet if necessary.
                     Materials -       They select any vacant rectangle on the grid sheet that matches their numbers,
                     Tables            trace around the outline in their texta colour and write in the table.
                     sheet,. Thin      This continues for a set time limit or until the page is full. Children may use other
                     textas or         factors for an answer if they cannot make the square or rectangle rolled. For
                     highlighters, 2   example. Sam rolls a 6 and a 3, but there is no rectangle with those lengths left.
                     ten sided dice or He may use a 9 x 2 rectangle or an 18 x 1 rectangle if available. At the end of the
                     a set of 11 cards game children tally all their small squares and compare their tally to that of their
                     numbered 0 –      opponent to see who the winner is.
                     10, game board Wwhen students cannot fit a rectangle in because there is no rectangle left that
                     [ 1cm square      size they may break it up in sections [ ie the distributive property ] For eg aroll of

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                     grid page A4]       6x8 can be split into 3x8 and 3x8.

        Lesson                           Embedding them in activities and games
                                         An effective way to learn multiplication tables is by not making them the focus of
                                         the activity, but a necessary requirement. Games such as snakes and ladders
                                         using two dice that can be added , subtracted, multiplied or divided to avoid the
                                         snakes and catch the ladders, help children make decisions and use the number
                                         Below are a couple of activities that have the use of tables embedded in them.
                                         Children will need to have access to a 10 x 10 tables grid for reference.
                                         [As children continually refer to this grid during activities they are using their tables
                                         for a purpose rather than simply reciting them in unrelated activities.

        Activity                         Making Square numbers

                                         Materials - Set of counters or tiles / maths books.
                                         This activity and the next one invite children to make their tables using counters or
                                         tiles. Many children have never seen their multiplication tables. What is 6 x 4?
                                         While we know the answer is 24 what does it look like.

                                         Ask children to investigate making square numbers using tiles or counters and
                                         locate them on the tables reference sheet. Is there a pattern? For example nine is
                                         a square number because it spatially makes a square.

                                         As students investigate and make square numbers with their tiles, ask them to
                                         draw the squares in their books and write the corresponding multiplication and
                                         division facts.

                                         Making Rectangles

                                         Similar activity to the one above. The difference being that every time the children
                                         make a rectangle with the tiles, they use the same number of tiles to investigate
                                         the possibility of making other rectangles. For example the rectangle made with
                                         24 tiles can be 6 x 4, 8 x3, 12 x 2, and 24 x1. They then draw and record their
                                         findings as they did with the previous activity.

                                         As this diagram shows, our tables are a set of arrays that form rectangles and
                                         squares. If we look at the diagram from the front we see 6 groups of 4, yet from
                                         the side we see 4 groups of 6. Hence the understanding of the commutative
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                                         property of multiplication becomes visual and not an abstract image that we often
                                         teach by telling the children, „ just switch the number around.‟

        Activity                         Doubling and Halving
                                         Children enjoy this type of activity, which involves doubling one side and halving the other side.
                                                                      4 x 14
                                                                      2 x 28
                                                                      1 x 56

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