Strand 1 Classroom Activity: How tall is a sixth class boy?
How tall is a sixth class boy?
Thanks to the students in Scoil Phadraig Naofa, Athy, Co. Kildare and their
teacher Willie O‟ Gorman for producing this sample activity.
Introduction:
Students often have difficulty looking at data sets in their entirety, focusing
instead on individual features of the data. They need to be encouraged to
consider the overall shape and distribution of the data set. This lesson activity
is an example of how a teacher led discussion in his class which resulted in the
students viewing the data set as a whole and encouraged them to look for
patterns and draw conclusions from the data. For second level the question
could be changed to “How tall is a first year student, junior cycle student etc.?”.
See the section titled “Extensions” for ways of extending the learning at post
primary level.
Curriculum Connections:
Representing and interpreting data
Fifth Class
Sixth Class
The student should be enabled to :
1
- Collect, organise and represent data using
pictograms, single and
multiple bar charts and
simple pie charts
- Read and interpret
pie charts and trend
graphs
pictograms, single and
multiple bar charts and
pie charts and trend
simple pie charts
graphs
- Compile and use simple data sets
Junior Cycle Post Primary
Students should be able to :
- explore different ways of collecting data
- design a plan and collect data
- work with different types of data in order
to clarify the problem at hand
- summarise data in diagrammatic form
including spreadsheets
- select appropriate graphical or numerical
methods to describe the sample
- use pie charts, bar charts, line plots,
histograms, stem and leaf plots and back
to back stem and leaf plots to display
data
- use a variety of summary statistics to
describe the data: central tendency - mean,
mode, median; variability - range, quartiles
- Explore and calculate averages of simple data sets and interquartiles
- draw conclusions from the graphical and
numerical summaries of data, recognising
- Use data sets to solve problems
assumptions and limitations
Strand 1 Classroom Activity: How tall is a sixth class boy?
Background:
On returning to school after our initial meeting with Rachel and the Project
Maths Group in Portlaoise, both Mr Breen and I were very excited about the
project and the mathematical idealism that it wished to promote.
Fortunately for us the educational philosophies and teaching methodologies of
active learning with the focus on the process more than the product are already
common place in the Primary School Curriculum.
Selecting suitable statistical topics that would provide us with enough natural
variation was our next challenge. After some discussion we agreed that both
groups would undertake different challenges, with Mr Breen‟s 5th class
endeavouring to solve the problem of „‟How Long is a Minute‟‟ and my 6th class
group trying to answer the „‟How Tall is a 6th Class Boy‟‟ question.
The Groups:
The 5th class group constituted of a group of 20 boys of similar mathematical
ability. The majority of these boys would be working comfortably within their
curriculum and wouldn‟t be experiencing huge difficulties or challenges with
their work.
The 6th class group was made up of 15 boys that find maths very difficult and
have ongoing problems with the subject.
The Activity:
STEP 1
The group were informed about their involvement in a mathematical project
initially without been given any other relevant details. This increased their
levels of interest and lead to a sense of anticipation and excitement
immediately.
When the boys were then told that as a group we were required to solve the
problem „‟How Tall is a 6th Class Boy‟‟ there was much confusion. This was to be
expected and was an excellent start to the process. Now through focused
2
Strand 1 Classroom Activity: How tall is a sixth class boy?
discussion we had to untangle this confusion.
At first the boys were very eager to find out „‟who wants to know this??‟‟ to
which I answered „‟the lady who is over the project‟‟
I was hoping to guide the discussion towards the idea of measuring different 6th
class boy‟s heights and deciding on how many to measure. I must admit that the
subsequent conversation which ensued was both pleasantly surprising and
inspiring at times.
Without much encouragement the boys almost immediately decided that we
needed to measure some heights. The majority of the boys initially believed
that we should measure the tallest and smallest. Unknowingly they were
introducing the mathematical concept of range and some of the boys started to
talk in terms of average without having the actual specific vocabulary
surrounding the concept.
As the conversation developed some of the boys began to talk in terms of
„‟measuring the two smallest and tallest‟‟. Now they were starting to grasp to
idea of expanding the data set. A certain boy introduced the idea of measuring
the heights of every 6th class boy an idea which was added to by another, by
saying „‟ why don‟t we get all the sixth class boys to line up from smallest to
tallest and pick the boy in the middle. This was a perfect example of how
concrete and practical maths is for these boys.
Agreement was reached by the group that we needed to measure more than 2
boy‟s heights yet not every boy‟s height. At this stage I intervened and
proposed that we measure 30.
Without having to introduce the concept of selecting from a random sample or
fair testing the boys independently came up with the idea of selecting 15 boys
from each 6th class and to do this by drawing names from a hat.
STEP 2
The boys had already covered the curricular area of height and were very
familiar with the process involved in measuring and recording a person‟s height,
therefore few, if any difficulties were experienced at this stage.
3
Strand 1 Classroom Activity: How tall is a sixth class boy?
They really enjoyed carrying out the practical side of the project and it gave
them a great sense of satisfaction and authority, feelings these boys would
rarely associate with maths.
Throughout the measuring process the mood of excitement and anticipation
really increased with huge surprise and awe been visible to see as they recorded
their friend‟s heights.
STEP 3
Once we had recorded all 30 heights our next lesson involved gathering all the
data together and using these actual results to try and solve our problem.
The boys were given the choice as to how they wanted to record the complete
30 pieces of data, after some short discussion it was agreed that it would be
recorded as 2 distinct sets, Ms X‟s and Ms Y‟s. This decision showed very
clearly the sense of ownership they had with the project and how personally
they were interpreting the work.
As the boys relayed the data for me to record on the whiteboard the
conversation surrounding the nature of the data continued. Without any
difficulty they were able to identify the outliers and the most regularly
occurring heights.
Discussion was then guided towards the idea of average and what exactly
average meant, to the majority of the boys average meant something in the
middle, obvious confusion between mean and median.
At this stage one of the boy‟s in the group calculated the „‟average‟‟ by
identifying the number which was exactly half way between minimum and
maximum recorded heights. When we had agreed upon this strategy as being an
effective manner in which to calculate the „‟middle/average‟‟ we decided to see
how many of our heights were less than the middle and how many were more
than the „‟middle‟‟. The results proved to be very interesting. Even though the
method we used to calculate the middle is a generally accepted method and one
which is very much promoted by your standard textbook we immediately
realised that a large majority of our heights were less than this so called
„‟middle‟‟. The healthy sense of confusion continued.
4
Strand 1 Classroom Activity: How tall is a sixth class boy?
STEP 4
We now recorded the heights of our second 6th class grouping and straight away
the boys started to notice the obvious differences and more importantly the
distinct similarities.
Our maximum and minimum had greatly decreased and increased and a cluster
effect started to occur around certain heights.
Before the lesson I had planned to conclude by
 Introducing the boy‟s to the idea of a line plot as a very effective method
of displaying this type of data.
 Enable each boy to create their own plot.
 And hopefully come to a conclusion of sorts.
In hindsight this may have been a little too ambitious on my behalf.
The boys struggled with the scale of the line plot and it took them a few
minutes to fully understand how we actually represented data on a line plot.
Issues and difficulties surrounding the specific language associated with
mathematics also started to become apparent.
Our lesson concluded at this stage with the majority of the boys now able to
comprehend how the line plot worked and how it would display a trend or in their
language „‟draw a picture of the information‟‟
Regarding our quest to answer the problem it was safe to say that the group
were now well capable of telling you „‟‟what height a 6th class boy wasn‟t‟‟ more so
than what he was. Agreement was reached between the group that we may
never be able to answer this question with 1 single answer so we decided that
over the couple of weeks we would try and find a solution set with 3/5 possible
conclusions.
STEP 5
We carried on from where we finished the previous day with our line plot and
extending the data set to include the heights of all 6th class boys, this delayed
us a little seen as it meant more measuring. Some members of the group seemed
very confident that we were now very capable of finding out „‟ How Tall is a 6th
Class Boy‟‟
5
Strand 1 Classroom Activity: How tall is a sixth class boy?
Considering that we were now expanding our data set and line plots to include
approximately 50 (all the sixth class boys in our school) different heights we
agreed that we would need new line plots. I also encouraged the idea of
reorganising our data into specific groupings i.e 130s 140s 150s etc, namely
sorting into tens. The boys agreed that this would be a good idea with one of
the lads commenting „‟ that‟s just a bit like the line plot‟‟. We also decided to
colour code our new enlarged line plots, with each ten been assigned a specific
colour.
With the data sorted in this manner the patterns started to become blatantly
obvious. Without using the specific mathematical vocabulary the lads
immediately recognised what statisticians refer to as mode, median, and range,
language unfortunately that these boys may never be comfortable with yet
concepts they can easily understand. Presenting the data in such a way also
made it much easier for the boys to construct and complete their new line plots.
With our extra information our data set had now expanded to include 47
heights. When this extra data was placed on our initial line plots it only
reaffirmed what many of our beliefs already were and confirmed many of the
patterns we had noticed.
For homework the boys were asked to write 5 sentences pertaining to the line
plots and data we had constructed and collected. I encouraged them to try and
focus on what we might call „‟conclusions‟‟.
To assist the boys in this task we formed a list of what we thought would be
relevant/helpful language and vocabulary.
Before we have even concluded the work the boys now want to know if we could
get a set of heights from another school to compare our heights and answers
with.
6
Strand 1 Classroom Activity: How tall is a sixth class boy?
STEP 6 – Conclusions
The boys found it difficult to form 5 conclusions but almost all the boys were
able to script 2/3 very satisfactory conclusions.
We discussed our ideas and then decided to reduce them down to 4
answers/conclusions.
After all our research into the height of a 6th class boy we concluded that

A typical 6th class boy is most likely to measure between 150cm-160cm.

It would be unusual for a 6th class boy to measure less than 139cm or
more than 175cm.

It‟s possible for a 6th class boy to measure less than 140cm and more than
174cm

In a typical 6th class the difference between the smallest and tallest boy
is most likely to be around 40cm.

If your asked to guess the height of a random 6th class boy say 158cm.
It was agreed that ATypical 6th Class was defined as a class with approximately
30 boys.
Step 7 - Comparisions.
The boys were now very eager to compare their heights with those of another
school and to assess how affective and appropriate their conclusions were.
The second set of data was gathered by a colleague in another primary school
and proved to be very interesting for both myself and the boys.
Initially we sorted the data from smallest to tallest and then decided to form a
second line plot to assist us in identifying comparisons and checking just how
appropriate our conclusions were.
Once again the subsequent conversation proved to be very interesting and
eventually we agreed that our initial observations proved to be very appropriate.
7
Strand 1 Classroom Activity: How tall is a sixth class boy?
(See both sets of data & line plots).
Step 8 – Formalising Language
Throughout the project I choose not to confuse the boys by using very
prescriptive mathematical language that one associates with statistics and data.
Instead I allowed concepts and ideas to develop very naturally and organically.
To my surprise a large majority of the boys had a natural understanding and
grasp of most of the mathematical concepts surrounding statistics without
knowing how to apply abstract formula or language.
Now I took the opportunity to introduce to them the formal language of data
Range
Mean
Median
Mode
Outliers
Both school’s mean/median were now calculated School A – Mean = 157cm & Median = 157cm, Mode = 158cm
Data Sets/ Tools + Resources
On the next few pages you will find the data sets collected from both schools.
They can be used to compare with the data sets collected by your class group.
You will also find resources for students who may need help organising their
work. Students should be encouraged to come up with their own strategies to
organise their work where possible as this forms a crucial part of the data
handling cycle.
8
Strand 1 Classroom Activity: How tall is a sixth class boy?
How tall is a Sixth Class Boy?
Data Set 1 (30 samples)
139cm, 145cm, 148cm, 149cm, 151cm, 152cm, 152cm, 153cm,
153cm, 153cm, 154cm, 154cm, 155cm, 155cm, 156cm,
157cm, 158cm, 158cm, 158cm, 159cm, 159cm, 162cm, 163cm,
165cm, 167cm,168cm 170cm, 170cm, 175cm.
Data Set 2 (30 samples)
160cm, 140cm, 155cm, 151cm, 151cm, 154cm, 137cm, 151cm, 156cm,
166cm, 150cm, 163cm, 147cm, 152cm, 147cm, 149cm, 144cm, 153cm,
169cm, 177cm, 154cm, 159cm, 161cm, 167cm, 164cm, 145cm, 148cm,
141cm, 146cm,150cm.
9
Strand 1 Classroom Activity: How tall is a sixth class boy?
How Tall is a 6th Class Boy?
Aim: To calculate the height of a 6th class boy?
Resources: ________________
________________
________________
_________________
_________________
_________________
Method :
1._________________________________________________
___________________________________________________
__________________________________________________
2._________________________________________________
___________________________________________________
___________________________________________________
3._________________________________________________
___________________________________________________
___________________________________________________
4._________________________________________________
___________________________________________________
___________________________________________________
5._________________________________________________
___________________________________________________
___________________________________________________
10
Strand 1 Classroom Activity: How tall is a sixth class boy?
Diagram
11
Strand 1 Classroom Activity: How tall is a sixth class boy?
Student work:
Students
organised data
on a line plot
with guidance
from the
teacher:
“Where should
we start?
Where should
we end?” etc.
Students
observed
patterns as they
plotted the data
points on the
line plot. They
could see how
the data
clustered
around certain
values.
12
Strand 1 Classroom Activity: How tall is a sixth class boy?
On completing
their line plot,
the students
organised the
data into
groups of ten,
allowing them
to look at the
data through a
different lens.
An easy
transition can
be made to
drawing a
histogram, and
now the
students see
the usefulness
of each type
of display, In
future data
analysis they
are equipped
to use
appropriate
graphical
displays to
analyse data.
13
Strand 1 Classroom Activity: How tall is a sixth class boy?
14
Strand 1 Classroom Activity: How tall is a sixth class boy?
15
Strand 1 Classroom Activity: How tall is a sixth class boy?
Notice how the students are confident to make
predictions based on their data. The students are also
beginning to develop concepts for hypothesis testing.
Asked “A boy walks in to the class and measures
182cm, how likely is it that he is a sixth class boy?”,
the students confidently replied very unlikely and
were able to justify their conclusion with reference
to their data.
16
Strand 1 Classroom Activity: How tall is a sixth class boy?
Reflections:
Throughout this process it was very noticeable and at times blatantly obvious
that many of the mathematical difficulties encountered by these boys are
specifically associated with language adversities.
Overall the project proved to be extremely interesting and exciting, for
everyone involved. It enabled the boys to work as mathematicians and explore
maths from a very natural and enjoyable perspective. It was a drastic move
away from what many would consider traditional mathematics, yet very
beneficial.
As we proceeded through the project I was always amazed to notice just how
much many of these kids knew naturally, from a sense of fairness when selecting
a group to actual practical methods of calculating the average of a set. The level
of interest which remained throughout was also refreshing and made working on
the project very easy. On reflection maybe creating interest in mathematics is
something that traditional methods and books fail to do.
Too often we as teachers may be guilty of making assumptions and having
unrealistic objectives, expectations which may be too advanced or even too
simplistic. When we take time to listen to children, give them a platform in
which to present/discuss mathematical ideas, one will be amazed as to how
innocent, imaginative, and quite often insightful their contributions can be.
I was very fortunate that I could assign a few lessons to the project. As the
boys took ownership of the project the opportunities for expansion and
integration across the Primary School Curriculum were glaringly obvious. From
being able to write up procedures to drawing artistic line plots.
One of the most interesting educational possibilities that was brought to my
attention by a student was to compare the heights of 6th class boys and girls.
Doing this would lead to extremely interesting conversations and work in the
area of science and gender comparisons. Unfortunately time and circumstances
contrived against us exploring this educational avenue.
In conclusion I believe all students know more about mathematics than we often
give them credit for. Unfortunately what they know may not fit neatly into what
a textbook or maths scheme expects or wants them to know.
17
Strand 1 Classroom Activity: How tall is a sixth class boy?
Extensions:
The way in which the teacher promoted rich mathematical discussions and
allowed the boys to encounter the confusions and then problem solve amongst
themselves, meant that the students had a strong sense of engagement with the
task. The key to encouraging students to become statistical investigators is to
pose the right questions and allow them to go through the processes of planning,
collecting data, organising data, analysing data and finally drawing up conclusions
from the data.
The lesson enabled the students to
-
Work with different types of data
-
Explore ways of collecting data
-
Generate data
-
Select a sample from a population
-
Recognise the importance of representativeness so as to avoid biased
samples.
-
Design a plan and collect data
-
Select appropriate graphical or numerical methods to describe the sample
-
Evaluate the effectiveness of different displays
-
Use appropriate displays to compare data sets.
-
Use a variety of summary statistics to describe data.
-
Recognise the existence of outliers.
-
Draw conclusions from graphical and numerical summaries of data,
recognising assumptions and limitations.
All of which are Junior Certificate learning outcomes! (Higher level learning
outcomes in bold).
18
Strand 1 Classroom Activity: How tall is a sixth class boy?
To extend the learning consider the following:
 Compare heights according to gender or age.
 The next activity could involve data which would not be
suitable to display in a line plot, so students are forced
to display data in a different way that will allow them
to see patterns.
 Analyse the shape of the different distributions,
including concepts of symmetry and skewness.
 In this activity the boys agreed that the typical height
of a sixth class boy is 150 – 160cm. Ask them to come
up with an argument that would convince someone that
the typical height is greater than 160cm. Students
then begin to see how statistics can be misused.
 Students could compare the relationship between
height and gender, height and age.
 Leaving cert students could be encouraged to provide
more in depth analysis, including standard deviation,
interquartile range etc.
[Type a quote from the document of the summary of an interesting point. You can position
the text box anywhere in the document. Use the Drawing Tools tab to change the
formatting of the pull quote text box.]
19