4905
/www/nrich/html/content/id/4905/
1
2011-02-01T00:00:01
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You will need a piece of squared paper for this activity. If you
have none you can get a sheet <a href="/content/id/4905/1cmSqs.doc">here</a>.<br></br>
<br></br>
Write the <span style="font-style: italic;">units digits</span> of
the numbers in the two times table from $1 \times 2$ up to $10
\times 2$ in a line. (Leave some room at the top of the paper, and
some space to the left and right.)<br></br>
It should look like this:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="155" width="388" alt="units of two times table" src="TT1.jpg"></mdo:image></div>
<div style="text-align: left;">You might be able to see some
patterns in these numbers.</div>
<div style="text-align: left;">Now, do the same thing with the
multiples of three. Remember, just write the <span style="font-style: italic;">units digits</span>, this time directly
underneath the line of the two times table, like this:</div>
<div style="text-align: left;"></div>
<div style="text-align: center;"><mdo:image height="184" width="388" alt="units of two and three times tables" src="TT2.jpg"></mdo:image></div>
<div>Continue by writing in the four and five times tables in the
same way. Again, just using the units digits.</div>
<br></br>
<div style="text-align: center;"><mdo:image height="240" width="388" src="TT3.jpg" alt="units digits of two, three, four and five times tables"></mdo:image></div>
<div>Now look at the whole array of numbers you have created.</div>
<div>What patterns can you find?</div>
<div>Try to explain why the patterns occur.</div>
<div>What do you notice about these four sets of numbers?</div>
<div>Can you predict what would happen next if we wrote in the next
times table?</div>
<br></br>
<div style="text-align: center;">***************</div>
<br></br>
<div>Well, why not add in the tables of sixes, sevens, eights,
nines and finally tens?</div>
<div>After that, for the sake of completeness, we could put in the
table of ones and zeros.</div>
<div>Do you have a grid that looks like this?</div>
<br></br>
<div style="text-align: center;"><mdo:image height="370" width="370" src="TT4.jpg" alt="units digits of all tables from zero to ten"></mdo:image></div>
<div>What patterns are there here?</div>
<div>What about repeats?</div>
<div>Can you predict what you will find?</div>
<div style="text-align: left;">How might you record the repeats
that you find?</div>
<div style="text-align: left;">Each line could be written like
this:</div>
<div style="text-align: left;"></div>
<div style="text-align: center;"><mdo:image height="165" width="366" src="TT5.jpg" alt="units digits of twos and threes in a circular rotation"></mdo:image></div>
<div>But you will probably find some other ways which are just as
good.</div>
<div>You could try writing all the tables like that.</div>
<div>Are some tables the same or similar to others?</div>
<div>Does it matter which way the arrows go?</div>
<div>What can you discover about the pattern of repeats?</div>
<div>Can you predict what you will find out about 'pairs' of
tables?</div>
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<p class="editorial">Those who submitted solutions found that there
are many different and interesting patterns to be found in this
problem. In addition, some tried to find explanations for these
patterns, which was great. As well as being fun, this problem
should also help with learning the times tables, as a student from
Devagiri CMI Public School, Calicut in Kerala, India found out.</p>
<p class="editorial">Firstly, let's look at the symmetrical
patterns that a few of you found in the tables.</p>
<p class="editorial">The numbers along the diagonals of the full,
completed square show a pattern, as these people discovered: Katie
from Sir Jonathan North CC, Isabel and Eleanor from The Manor Prep,
Veronika, Miriam, Albin, Carolien and Erin from the Independent
Bonn International School, Nitya Andrew and Sydney from Ysgol Pen Y
Bryn School in Wales, and a student from Devagiri CMI Public
School, Calicut.</p>
<p class="editorial">Katie found that the diagonal lines follow the
rule of symmetry. Isabel and Eleanor described this:</p>
<div>The second half of the diagonal is repeated by the same as the
first half but backwards.</div>
<br></br>
<p class="editorial">Nitya, Andrew and Sydney mentioned the pattern
using numbers to illustrate:</p>
In the right top corner the diagonal downwards numbers (i.e. from
top right to bottom left) are $0, 9, 6, 1, 4$ and on the left hand
bottom corners the diagonal upwards numbers are also $0, 9, 6, 1,
4$. It is as if the numbers are reflected in a mirror.<br></br>
<br></br>
<p><span class="editorial">Veronika, Miriam, Albin, Carolien and
Erin from the Independent Bonn International School noticed
"symmetry" with the rows for three and seven. They pointed
out:</span></p>
Looking at the sevens, we found out that the numbers were the same
as the threes in reverse.<br></br>
<br></br>
<p><span class="editorial">The student from Devagiri CMI Public
School also found this.</span></p>
<p><span class="editorial">A few people also found a symmetrical
pattern when loooking at the five times tables. Here, the only
digits in the table are $5$ and $0$. These numbers form a
cross-like pattern, running vertically and horizontally through the
middle of the table. The repetition of "$5$" and "$0$" should help
with remembering the five times tables: the "member of the table"
always ends with either a $5$ or $0$ and these alternate. Also, it
can show you quickly if a number can be divided by five with no
remainder: if this is true, the number must end in a "$5$" or
"$0$"</span>.</p>
<p class="editorial">Sam from the Orchard Community Primary School
looked at patterns along the columns and rows:</p>
The same pattern occurs from the same digit on both the rows and
the columns.<br></br>
<br></br>
For example, look at the sequence of numbers for the two times
table, starting in the second row from number two: $2, 4, 6, 8, 0,
2, 4, 6, 8, 0$. Now look at the third column, starting from number
two. This is the same: $2, 4, 6, 8, 0, 2, 4, 6, 8, 0$. This is the
same if you start from the number three, or four, or five ...<br></br>
<br></br>
<p><span class="editorial">A student from North Walsham Junior
School found this too, as did Nitya, Andrew and Sydney from Ysgol
Pen Y Bryn.</span></p>
<p class="editorial">Luke from Tudhoe Grange found another pattern,
but this time it is along the edges of the square. He discovered
that in the first column and the last column (ignoring the columns
of zeros), there are all of the numbers from $0$ to $9$.
Furthermore, the numbers are in order: the sequence increases by
one unit going down on the left hand side, and decreases by one
unit when going down the right hand side. The same is seen for the
first and last rows (ignoring the rows of zeros).</p>
<p class="editorial">James from Thornton Dales and James from
Tudhoe Grange found a similar sort of pattern. James from Tudhoe
Grange describes this:</p>
The end result shows that the numbers go up in the first column in
$1$s then the next column in $2$s and so on. The pattern continues
on to $10$s.<br></br>
<br></br>
<p><span class="editorial">Francesca and Katie went on to
explain:</span></p>
The patterns occur because all you are doing is timesing the number
of the first column by the number of whichever number of row you
want to create the numbers for e.g if you want to find the number
for the second column in the third row you would times three by two
equalling six because you times the row by the column. By this
method you can create the whole table.<br></br>
<br></br>
<p><span class="editorial">Some people found patterns when looking
at odd and even numbers in the table. For example, George from
Summerswood Primary found that the first line (for the one times
table) has alternating odd and even numbers, whilst the second line
has all even numbers. For the three times table, the numbers are
alternating odd and even and for the four times table, all numbers
are even. This pattern repeats itself ...</span></p>
<p class="editorial">A student from Devagiri CMI Public School also
noticed this. He explains:</p>
In the Tables of Even numbers, the units are always even , while in
the Tables of Odd numbers, alternate units are odd and even.<br></br>
<br></br>
<p><span class="editorial">He noticed this general rule: when
multiplying two odd numbers, the units are always odd. You only get
an even number if you multiply by an even number. So, to get an
even number for an answer, an even number must be involved in the
multiplication.</span></p>
<p class="editorial">Using this rule he reasoned:</p>
In the times tables of even numbers all the answers are even
numbers, because an even number is always involved. In the times
tables of odd numbers, only half the answers are odd numbers and
the other half even numbers. This is because you only mulitply the
odd number by an even number half of the time.<br></br>
<br></br>
<p><span class="editorial">As we have seen, we can find patterns
just by looking at the numbers in the tables. Also, we can try to
find further patterns by adding, subtracting, multiplying and
dividing various sets of numbers.</span></p>
<p class="editorial">Luke from Tudhow Grange and a student from
Devagiri CMI Public School looked at patterns found when adding
numbers. The student from India found complements of ten in the
table (complements are numbers that go together in some way). He
noticed:</p>
For each of the times tables, the units place starts with its own
respective number and ends with its complement of $10$. For
example, let us consider the one times table. This table starts
with $1$, and ends in $9$, and $1+9= 10$. So, $1$ and $9$ are
complements of ten. This pattern extends further.<br></br>
<br></br>
Let's stay with the one times table. The second number in the row
($2$ for the one times table) and the second last in the sequence
($8$) are complements in $10$ since $2+8=10$. This is true for the
third and third last numbers ($3+7=10$). This is also the case for
the fourth and fourth last numbers ($4+6=10$). For the fifth
number, you add it to itself ($5+5=10$).<br></br>
<br></br>
<p><span class="editorial">This pattern is seen for all of the
times tables. Try this for yourself, by looking along the rows of
the different times tables. Now look at the columns. Do you see the
same complements of tens?</span></p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/4905">This problem</a> involves learners in making and proving conjectures using patterned numbers and arithmetic sequences. It is, incidentally, a very interesting way of revising multiplication tables! It is also very useful for getting learners to predict what they think they will find out.</div>
<br></br>
<div>This is a good class investigation in that it can be taken on a long way but patterns can be found at an early stage so those who work more slowly will be doing some discovering.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could introduce the investigation as suggested or, alternatively, from a standard $10 \times 10$ 'table-square' simply dropping the tens. However, this does remove some of the exploration and discovery.</div>
<br></br>
<div>After the initial introduction learners could work in pairs so that they are able to talk through their ideas with a partner. Plenty of squared and plain paper should be available. Squared paper can be found <a href="/content/id/4905/1cmSqs.doc">here</a>.</div>
<br></br>
<div>You could show the group the cycles of repeats which are a very useful way of recording but learners may find another better way! When making them, make sure that the arrows are put in because this matters.</div>
<div>They can be put in like this:</div>
<div style="text-align: center;"><mdo:image alt="cycle of repeats" height="174" src="TT6.png" width="170"></mdo:image></div>
<div><a class="pdflink" href="/content/id/4905/TwTens2.pdf">This sheet</a> could be helpful.</div>
<div>�</div>
<br></br>
<div>At the end of the lesson the group should come together to discuss their explorations and discoveries. The factors of $10$ and the complements in $10$ (the numbers that add to make $10$) should arise in interesting ways. When pressed, can they give satisfactory explanations?</div>
<br></br>
<h3>Key questions</h3>
<div>What do you think you will find out from doing this?</div>
<div>How often does it repeat?</div>
<div>Which digits are there in the line?</div>
<div>Have you looked along the rows and up and down the columns?</div>
<div>Have you looked at any of the diagonals?</div>
<div>Would it help, when you're finding repeats, to extend the rows to $11$ lots of the number, $12$ lots, $13$ lots?</div>
<div>How did you make each row in the first place?</div>
<br></br>
<h3>Possible extension</h3>
<div>After exploring the many patterns in the investigation given learners could try <a href="http://nrich.maths.org/2791">Diagonal Sums</a>.</div>
<div>Some ideas for more extensions can be found on <a class="pdflink" href="/content/id/4905/Extensions4905.doc">this sheet</a>.</div>
<br></br>
<h3>Possible support</h3>
<p>Suggest using a standard $10 \times 10$ 'table-square' to help with the tables. If even this is proving difficult, start by using a $10 \times 10$ 'table-square' [such as <a class="pdflink" href="/content/id/4905/TableSq.pdf">this one</a>] that can be written on and crossing out the tens figures. The resulting unit-numbers can then be transferred to a plain sheet of squared paper. <a class="pdflink" href="/content/id/4905/TwTens1.doc">This ready-made sheet</a> might help.<br></br>
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Try looking along rows, up and down columns and perhaps along
diagonals for patterns in the digits.<br></br>
When you're finding repeats, you might want to imagine extending
the rows to $11$ lots of the number, $12$ lots, $13$ lots
etc.<br></br>
When you're trying to explain the patterns, don't forget how you've
made each row in the first place!<br></br></mdoxml>
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Katie from Sir Jonathan North School made the following
remark:<br></br>
<br></br>
"In the rows of odd numbers, except $5$, you have every number from
$1$-$9$, and in the rows of even numbers you have the multiples of
$2$. This occurs again in the columns. For example, (ignore the
presence of the columns of zero's for a second) in the first
column, third column, seventh colum and ninth column, every number
from $1$ to $9$ is present."<br></br>
<br></br>
Why do you think the rows and columns are the same?<br></br>
<br></br>
A number of you remarked that the table is symmetrical across the
diagonals. Can you explain why this is?<br></br>
<br></br>
Katie's first remark is the most interesting - we get the numbers
$1$ - $9$ in any row that is not a multiple of $2$ or $5$. Does it
surprise you that these are the factors of $10$?<br></br>
<br></br>
<span style="font-weight: bold;">Extension</span><br></br>
<br></br>
Matthew from QEB correctly observed that behind this problem is an
important kind of arithmetic called 'modular' or 'clock'
arithmetic. A search for either of these terms on the NRICH website
should yield some interesting problems and articles on this topic.
For an article aimed at secondary school pupils click <a href="http://nrich.maths.org/public/viewer.php?obj_id=4350">here</a>
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Tables Without Tens
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Making and proving conjectures
Numbers and the Number System
Patterned numbers
Sequences, Functions and Graphs
Arithmetic sequence