Is She Cheating?
A Bar Chart
The Middle Number
Was Ann Cheating
Many Sixes
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Starting Snakes and Ladders
Some games start by throwing a six with a die. Snakes and
Ladders is one of them. Ludo is another. You cannot begin until
you throw a six.
Do you think this is.fair?
Would it be better for all the players to start together?
In what other ways could you start?
| a |
Write down two of them. |
Is She Cheating?
Ann, Brian and Charles are playing Ludo. Brian and Charles
have started but Ann has not. She says it takes too long to get a
six. Instead, she wants to throw a one to begin. Charles thinks
that this is cheating. He says that he had to get a six so Ann
must too.
| a |
Is it easier for Ann to get a one? |
Brian is not sure. He suggests two experiments. First they
will see how long it takes to get a six. Then they will do the
same for a one.
Ann throws the die until she gets a six. Charles records the
result of each throw. He then counts up how many throws it took
to get a six. They do this until they have thrown 25 sixes.
The results are shown in Table 1.
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Numbers showing on the die |
How many throws to get a six |
| 1st time |
1,3,2,4,5,1,6 |
7 |
| 2nd time |
2,2,4,1,6 |
5 |
| 3rd time |
6 |
1 |
| 4th time |
3,6 |
2 |
| 5th time |
1,1,4,5,6 |
5 |
| 6th time |
4,6 |
2 |
| 7th time |
1,4,5,3,3,3,2,4,3,5,2,6 |
12 |
| 8th time |
4,4,1,3,2,6 |
6 |
| 9th time |
3,4,6 |
3 |
| 10th time |
5,6 |
2 |
| 11th time |
3,1,5,6 |
4 |
| 12th time |
2,1,4,5,3,3,6 |
7 |
| 13th time |
6 |
1 |
| 14th time |
2,4,2,6 |
4 |
| 15th time |
5,4,3,1,2,3,4,5,1,2,6 |
11 |
| 16th time |
3,2,6 |
3 |
| 17th time |
2,3,2,4,5,1,4,6 |
8 |
| 18th time |
6 |
1 |
| 19th time |
5,3,6 |
3 |
| 20th time |
4,6 |
2 |
| 21st time |
3,2,4,1,5,5,2,1,2,3,3,5,2,1,6 |
15 |
| 22nd time |
4,2,1,3,5,3,4,2,2,6 |
10 |
| 23trd time |
5,1,2,1,3,3,1,6 |
8 |
| 24th time |
1,6 |
2 |
| 25th time |
6 |
1 |
Table 1 - Throwing a Six - Ann's Results for 25 Sixes.
You will need a die and page R1.
Work as two pairs. One pair is to find out how long it takes
to throw a six. The other pair is to find out how long it takes
to throw a one. One person from each pair throws the die, and the
other records the results in the appropriate table on page R1.
Exchange results with the pair working with you.
| |
c |
Use their results to complete the
last column only of the other table on page R1. |
| |
d |
What was the smallest number of
throws you needed:
to get a one,
to get a six? |
| |
e |
What was the largest number of
throws you needed:
to get a one,
to get a six? |
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f |
What is the smallest possible
number of throws you could take:
to get a one,
to get a six? |
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g |
Are you more likely to throw a six
after you have thrown a one
than after you have thrown a five? |
| * |
h |
Charles got a six
on his first throw. Ann took rive throws for her first six.
Who will take longer to get a six next time? |
| * |
i |
Is there a largest possible
number of throws it could take to get a six? |
A Bar Chart
It is difficult to interpret the figures in Table 1. A diagram
can make things simpler. Ann drew a bar chart of her results.
This is Figure 1.
She shaded in a square for each of her 25 results. For
example, it took seven throws to get a six the first time. She
shaded in the square above 7. The next time it took seven throws
to get a six, she shaded in the square above the first one.
Figure 1 - Throwing a six - Ann's results for 25 sixes
The bar chart shows that the six turned up on the second throw
of the dice five times. This was more often than any other number
of throws. In other words, the most common number of throws
needed to get a six was two. When there is a most common number,
it is called the MODE. For Ann, Brian and Charles the mode was
two throws.
You will need page R2.
| a |
Complete the bar chart on Figure
4 to show your results. |
Remember to give the chart a title. Check that you have shaded
25 squares. Compare your bar chart with that of the other pair
working with you.
| b |
What was the mode when throwing
a six? |
| c |
What was the mode when throwing
a one? |
The Middle Number
Ann and Charles threw a six 25 times. They wrote down the
number of throws it took to get a six in order of size. For
example, they got six on the first throw four times, so they
wrote down four 1's. Their results were:
| 1 |
1 |
1 |
1 |
2 |
2 |
2 |
2 |
2 |
3 |
3 |
3 |
4 |
4 |
5 |
5 |
6 |
7 |
7 |
8 |
8 |
10 |
11 |
12 |
15 |
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Look at the '4' which is arrowed. There are 12 results to the
left of it and 12 results to the right of it. It is halfway
through the results when thev are put in order of size. We say
that 4 is the MEDIAN. The median is the middle number.
| a |
Copy Ann and Charles's results
in order of size. Below them write your results from
throwing a six in order of
size. Below this, write your results from throwing a one
in order of size. |
| b |
What was your median number for
throwing a six?
What was your median number for throwing a one? |
Was Ann Cheating?
Remember, the median tells you the middle number of throws you
needed.
| a |
If throwing a six
is much harder than throwing a one,
will the median for six be:
bigger,
or smaller,
or about the same as the median for one? |
| b |
If it is equally hard to throw a
six or a one, will the median
for six be:
bigger,
or smaller,
or about the same as the median for one? |
Look at the two medians you got for throwing a six and for
throwing a one.
| c |
Do you think it is harder,
easier or about the same to throw a six
or to throw a one? Give your
reasons. |
| d |
Would it make any difference to
starting the game of Ludo if you had to throw a one
instead of a six? |
Many Sixes
You will need page R3.
In your experiment you threw a six (or a one) 25 times. This
takes a long time, but it is not very many sixes. It would be
fairer to do the experiment many times. One quick way of doing
this is to collect together the results of the whole class.
Your teacher will collect your results for throwing ones and
sixes.
| a |
Complete the bar charts of
Figures 5 and 6 on page R3. Your teacher will tell you
how to number the vertical axes. |
One will show the class results of throwing a six. The other
will show the class results of throwing a one.
Look at the class results for throwing a six.
| b |
What is the mode? |
| c |
What is the median? |
Look at the class results for throwing a one.
| b |
What is the mode? |
| c |
What is the median? |
Look at your answer to Was Ann
Cheating, Question c.
| f |
Do you still think the same?
Give a reason. |
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