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predict the temperature at the end of the Ist minute. The experiment is
timed (we used a volunteer timekeeper to shout out 5-4-3-2-1 at the end of
each minute). At the end of the 1st minute, the actual temperature is logged
on the worksheet and a prediction is made for the end of the 2nd minute.
Now, some students start to look at the rate of cooling as an indicator to sup-
port their estimate for the end of the 2nd minute. Also, students are asked
to estimate the temperature at the end of the 5th minute. The experiment
continues in this way for 10 minutes, each time students make future esti-
mates. After 5 minutes they estimate for end of the 6th minute and the end
of the 10th minute. By now, students are taking close account of the rate of
change and using this to make better and better estimates for each succes-
sive minute. There is an aura of quiet competitiveness and satisfaction when
estimates are close or even perfect. (Reading are taken to one decimal
place). After 10 minutes, they estimate for 30 minutes, 120 minutes and
24 hours. Again the requirement to explain the basis for the estimation is
emphasized. This last part requires students to share their mechanisms for
estimation and discuss how they expect the temperature to change after the
experiment has ended, that is, into the never-to-be-known. It was gratifying
that students happily watched a pizza warming in a microwave oven for
2 minutes then watched it cooling for 10 with rapt attention! — Figure 32 predict the temperature at the end of the Ist minute. The experiment is timed (we used a volunteer timekeeper to shout out 5-4-3-2-1 at the end of each minute). At the end of the 1st minute, the actual temperature is logged on the worksheet and a prediction is made for the end of the 2nd minute. Now, some students start to look at the rate of cooling as an indicator to sup- port their estimate for the end of the 2nd minute. Also, students are asked to estimate the temperature at the end of the 5th minute. The experiment continues in this way for 10 minutes, each time students make future esti- mates. After 5 minutes they estimate for end of the 6th minute and the end of the 10th minute. By now, students are taking close account of the rate of change and using this to make better and better estimates for each succes- sive minute. There is an aura of quiet competitiveness and satisfaction when estimates are close or even perfect. (Reading are taken to one decimal place). After 10 minutes, they estimate for 30 minutes, 120 minutes and 24 hours. Again the requirement to explain the basis for the estimation is emphasized. This last part requires students to share their mechanisms for estimation and discuss how they expect the temperature to change after the experiment has ended, that is, into the never-to-be-known. It was gratifying that students happily watched a pizza warming in a microwave oven for 2 minutes then watched it cooling for 10 with rapt attention!

Related Figures (73)

Braswell et al. 2001; NCES 2003; NCES 2008

FIGURE 2.1. An Expression and a Number Line

Dragging a point as shown in Figure 2.2 allows students to investigate the
difference between expressions and equations, laying the foundation for
thinking about an equation as the equality of two expressions, and build-
ing on their developing sense of a solution for a more general linear equa-
tion in one variable. Probing questions push students to look at these
relationships and to reason about the variables, the coefficients and the
output. — Dragging a point as shown in Figure 2.2 allows students to investigate the difference between expressions and equations, laying the foundation for thinking about an equation as the equality of two expressions, and build- ing on their developing sense of a solution for a more general linear equa- tion in one variable. Probing questions push students to look at these relationships and to reason about the variables, the coefficients and the output.

FIGURE 2.4 Balanced Systems
FIGURE 2.3. What Is a Solution — FIGURE 2.4 Balanced Systems FIGURE 2.3. What Is a Solution

FiGurE 5.1 Sand Dune in Morocco (photo by Fabio Cologna)
The 40th anniversary of the launch of the British Journal of Educational
Technology (BJET) took place in 2009. Also in 2009, coincidentally, ATM
instigated MT? (Mathematics Teaching interactive), an online web-based
publication. The first issue of BJET (in 1970) was most concerned with the
availability of audio-visual resources (such as 16mm film projectors), whereas
the first issue of MT7 contained, among other things, an article which
examines questions for which the answer is ‘It’s a parabola’ (see Figure 5.1

for an example). — FiGurE 5.1 Sand Dune in Morocco (photo by Fabio Cologna) The 40th anniversary of the launch of the British Journal of Educational Technology (BJET) took place in 2009. Also in 2009, coincidentally, ATM instigated MT? (Mathematics Teaching interactive), an online web-based publication. The first issue of BJET (in 1970) was most concerned with the availability of audio-visual resources (such as 16mm film projectors), whereas the first issue of MT7 contained, among other things, an article which examines questions for which the answer is ‘It’s a parabola’ (see Figure 5.1 for an example).

the framework illustrated in Figure 5.2.

In the teacher-demonstration approach, the teacher engages students in
discussing an on-screen geometric construction and may ask questions
about the objects on the screen to get the learners to explain what they
might expect would happen if some parts of the configuration were moved
or changed. This approach was found to allow teachers with little experi-
ence of using technology in the classroom to experiment with the tech-
nology with relatively small risk of things going wrong. In addition, this
kind of use requires less change in the classroom setting and needs fewer
resources than either organizing classes into a computer room or using a
class set of laptops in the regular classroom. The second approach (in
Figure 5.2) entails teachers providing previously created interactive files for
their learners. With such teacher-created files, students can experiment
with dynamic objects. This provides clear boundaries for learners and time
is not spent setting up the tasks; rather, learners can spend time exploring
the mathematics that is central to each task. No doubt there is quite some
teacher control over the material, but the approach can bring in opportun-
ities for creative thinking and problem solving by learners. The third
approach (in Figure 5.2) involves learners creating their own files, perhaps — the framework illustrated in Figure 5.2. In the teacher-demonstration approach, the teacher engages students in discussing an on-screen geometric construction and may ask questions about the objects on the screen to get the learners to explain what they might expect would happen if some parts of the configuration were moved or changed. This approach was found to allow teachers with little experi- ence of using technology in the classroom to experiment with the tech- nology with relatively small risk of things going wrong. In addition, this kind of use requires less change in the classroom setting and needs fewer resources than either organizing classes into a computer room or using a class set of laptops in the regular classroom. The second approach (in Figure 5.2) entails teachers providing previously created interactive files for their learners. With such teacher-created files, students can experiment with dynamic objects. This provides clear boundaries for learners and time is not spent setting up the tasks; rather, learners can spend time exploring the mathematics that is central to each task. No doubt there is quite some teacher control over the material, but the approach can bring in opportun- ities for creative thinking and problem solving by learners. The third approach (in Figure 5.2) involves learners creating their own files, perhaps

FiGuRE 6.1 Framework Showing Impacts of Digital Resources Used to Support
Learning and Teaching of Mathematics
Cognitive elements — FiGuRE 6.1 Framework Showing Impacts of Digital Resources Used to Support Learning and Teaching of Mathematics Cognitive elements

We implemented two approaches to study changes attributable to the inter-
vention: (1) comparing the student performances between the treatment
and control groups first before the intervention (8th grade TAKS) and then
after the intervention (9th grade TAKS for Algebra Items) (2) comparing
the paired student performances before and after the intervention for the
control group and then the treatment group. — We implemented two approaches to study changes attributable to the inter- vention: (1) comparing the student performances between the treatment and control groups first before the intervention (8th grade TAKS) and then after the intervention (9th grade TAKS for Algebra Items) (2) comparing the paired student performances before and after the intervention for the control group and then the treatment group.

FicurE 8.1 Using Cabri for Easy Providing of Data

FIGURE 8.3 The second lateral face as rotated from the previous one
fIGURE 8.2 A lateral face as rotated from the initial square — FIGURE 8.3 The second lateral face as rotated from the previous one fIGURE 8.2 A lateral face as rotated from the initial square

Students are given a plane and its equation in Cabri 3D (Figure 8.6). They
can observe that, when moving a point defining the plane, this latter varies
as well as its equation. A point Pwith displayed coordinates not belonging
to the plane is given. Then the plane and all its points are hidden and
students are asked to move point Pin order to put it as close as possible to
the hidden plane (Figure 8.7). Of course students must solve the task with
the plane remaining hidden. — Students are given a plane and its equation in Cabri 3D (Figure 8.6). They can observe that, when moving a point defining the plane, this latter varies as well as its equation. A point Pwith displayed coordinates not belonging to the plane is given. Then the plane and all its points are hidden and students are asked to move point Pin order to put it as close as possible to the hidden plane (Figure 8.7). Of course students must solve the task with the plane remaining hidden.

The principal phenomenon of interest is the wide range of freedom of
the points Pand Pcand yet the point D remains invariant. This can be expe-
rienced using dynamic geometry software in which Pand Pccan be dragged
about the plane while the point D fails to move, and that this holds for any
relative positions of A, B, and C (when Cis the mid-point of AB, the line Pa
of the choice of Pand the choice of Pe.
a a, >, >

Take any three points A, Band Con a line, (for the time being take Cnot
0 be the mid-point of AB). Now let Pbe any point in the plane not on the
ine ABC. Denote by Peany point other than Por Con the line PC. Form the
ines A Pc and B Pc and where they cross the lines PB and PA respectively,
Jenote the points by Paand Pb. Finally, let D be the point where the line Pa
Pb meets the line through ABC. The amazing fact is that D is independent
of the choice of Pand the choice of Pc. — The principal phenomenon of interest is the wide range of freedom of the points Pand Pcand yet the point D remains invariant. This can be expe- rienced using dynamic geometry software in which Pand Pccan be dragged about the plane while the point D fails to move, and that this holds for any relative positions of A, B, and C (when Cis the mid-point of AB, the line Pa of the choice of Pand the choice of Pe. a a, >, > Take any three points A, Band Con a line, (for the time being take Cnot 0 be the mid-point of AB). Now let Pbe any point in the plane not on the ine ABC. Denote by Peany point other than Por Con the line PC. Form the ines A Pc and B Pc and where they cross the lines PB and PA respectively, Jenote the points by Paand Pb. Finally, let D be the point where the line Pa Pb meets the line through ABC. The amazing fact is that D is independent of the choice of Pand the choice of Pc.

with 6 ‘hours’ and one with 10, together with a button that advances both
simultaneously. The second figure shows the result of 7 clicks

An immediate question is how many clicks of the button will be required to
get both the dials back to the 0 position again, and what relative positions
are possible for the two dials. This question in some sense abstracts numer-
ous situations in which two things are happening at different rates, such as
two wheels of different diameters on a vehicle, or two people walking at dif
ferent rates or using different pace-lengths and so on.
ferent rates or using different pace-lengths and so on. — with 6 ‘hours’ and one with 10, together with a button that advances both simultaneously. The second figure shows the result of 7 clicks An immediate question is how many clicks of the button will be required to get both the dials back to the 0 position again, and what relative positions are possible for the two dials. This question in some sense abstracts numer- ous situations in which two things are happening at different rates, such as two wheels of different diameters on a vehicle, or two people walking at dif ferent rates or using different pace-lengths and so on. ferent rates or using different pace-lengths and so on.

Polydials is software still under development, but it illustrates some of the
potential as well as some of the obstacles of pedagogic software. Available to
the user are dials with various alterable whole numbers of ‘hours’ marked on
them. Also available are buttons used to advance the hands by a single ‘hour
of one or more clocks simultaneously. Pictured below is a pair of clocks, one — Polydials is software still under development, but it illustrates some of the potential as well as some of the obstacles of pedagogic software. Available to the user are dials with various alterable whole numbers of ‘hours’ marked on them. Also available are buttons used to advance the hands by a single ‘hour of one or more clocks simultaneously. Pictured below is a pair of clocks, one

computing and memory capacity doubles every 18 months. Any fan of
digital photography will have seen an almost 100-fold increase in memory
card capacity at a given price point over the past 9 years from 64Mb to now
4Gb and above as standard. The increase in internet bandwidth driven by
the same underlying microprocessor and optical physics is actually increas-
ing at a rate that is twice as fast as Moore’s Law, with bandwidth at a price
point doubling every 9 months. We have seen connection speeds increase
from 56 kbps dial-up less than 10 years ago to speeds for 40 mbps projected
for WIMAX connections and 4G mobile connections over the next 12 to 18
months. The implications of this rapidly increasing capacity are far reach-
ing with computing technology moving from being scarce to becoming
abundant resources which are ‘too cheap to meter’. We are only part way
through this dramatic and fundamental transformation in widespread
access to computing power.

Such developments in technology are producing an increase in availabil- — computing and memory capacity doubles every 18 months. Any fan of digital photography will have seen an almost 100-fold increase in memory card capacity at a given price point over the past 9 years from 64Mb to now 4Gb and above as standard. The increase in internet bandwidth driven by the same underlying microprocessor and optical physics is actually increas- ing at a rate that is twice as fast as Moore’s Law, with bandwidth at a price point doubling every 9 months. We have seen connection speeds increase from 56 kbps dial-up less than 10 years ago to speeds for 40 mbps projected for WIMAX connections and 4G mobile connections over the next 12 to 18 months. The implications of this rapidly increasing capacity are far reach- ing with computing technology moving from being scarce to becoming abundant resources which are ‘too cheap to meter’. We are only part way through this dramatic and fundamental transformation in widespread access to computing power. Such developments in technology are producing an increase in availabil-

Using the graphical calculator or graphing software enables the pupils to
generate and explore graphs of linear functions quickly thus enabling them
to focus on the key properties of the functions and how they can be used to
create geometrical shapes with particular properties. — Using the graphical calculator or graphing software enables the pupils to generate and explore graphs of linear functions quickly thus enabling them to focus on the key properties of the functions and how they can be used to create geometrical shapes with particular properties.

The variety and degree of interactivity available on-line is increasing steadily.
Whereas a few years ago educational sites may have offered little more than
text, now there is a myriad of interactivity utilizing technologies such as
Flash, Java, Javascript, PHP, forums, conferencing and video. Recently the
browser Firefox, in version 3.5, has enabled the embedding of video directly
into a web page with out the need for any special display software. The web
is rapidly approaching the point where full interactivity is accessible to all.
So, what is available to the advanced level mathematics student? Mathemat-
ics seems to me to be ideally suited to the web. The student can investigate
graphs, geometrical constructions, algebraic manipulation, logical think-
ing, calculus, mechanics, statistical diagrams, simulations, video, and so on.
Some of these interactivities have been born and bred solely on the web,
interactive geometry being a good example. On the one hand proprietary
software such as Cabri Geometre, Geometer’s Sketchpad or Autograph
have developed their own on-line versions; on the other hand new packages
have developed their own on-line versions; on the other hand new packages — The variety and degree of interactivity available on-line is increasing steadily. Whereas a few years ago educational sites may have offered little more than text, now there is a myriad of interactivity utilizing technologies such as Flash, Java, Javascript, PHP, forums, conferencing and video. Recently the browser Firefox, in version 3.5, has enabled the embedding of video directly into a web page with out the need for any special display software. The web is rapidly approaching the point where full interactivity is accessible to all. So, what is available to the advanced level mathematics student? Mathemat- ics seems to me to be ideally suited to the web. The student can investigate graphs, geometrical constructions, algebraic manipulation, logical think- ing, calculus, mechanics, statistical diagrams, simulations, video, and so on. Some of these interactivities have been born and bred solely on the web, interactive geometry being a good example. On the one hand proprietary software such as Cabri Geometre, Geometer’s Sketchpad or Autograph have developed their own on-line versions; on the other hand new packages have developed their own on-line versions; on the other hand new packages

extent the MyMaths content is designed to replace the teacher.

In the United States the site Explorelearning has for some time been
using Flash and Shockwave to develop interactive material for pupils in
grades 3 to 12, not only for mathematics but science too. Mathematics is
covered up to the level of college algebra and pre-calculus. Explorelearn-
ing call these interactive resources ‘gizmos’ and although the site is not
free, many of the 450 gizmos available can be viewed without payment for
a few minutes. — extent the MyMaths content is designed to replace the teacher. In the United States the site Explorelearning has for some time been using Flash and Shockwave to develop interactive material for pupils in grades 3 to 12, not only for mathematics but science too. Mathematics is covered up to the level of college algebra and pre-calculus. Explorelearn- ing call these interactive resources ‘gizmos’ and although the site is not free, many of the 450 gizmos available can be viewed without payment for a few minutes.

Step by step illustration of algebraic techniques:
FMSP and MyMaths offer a subscription within which the user chooses
which syllabus elements they want to use. MathsNetAlevel requires the
user to choose their syllabus at the outset. Once the user has logged on,
they must select the module they wish to work on and then choose a topic.
From there on a vast array of resources awaits. Here is a selection from the
three sites: — Step by step illustration of algebraic techniques: FMSP and MyMaths offer a subscription within which the user chooses which syllabus elements they want to use. MathsNetAlevel requires the user to choose their syllabus at the outset. Once the user has logged on, they must select the module they wish to work on and then choose a topic. From there on a vast array of resources awaits. Here is a selection from the three sites:

Investigate parabolas
Investigate polar graphs — Investigate parabolas Investigate polar graphs

per second. So we can edit the function to show f1(x) = 84 — (2.3/60)x.
Happily the software gives an immediate response, so testing different the-
ories for accommodating the minutes to second conversion can be done
quickly by test and check. This feature keeps the discussion on track and
avoids being sidelined by tricky numeracy issues. These can be returned to
later. Often the very beginning of the experiment has an uneven cooling
rate, so a little ‘tweaking’ of the model needs doing. Having seen the con-
struction of the model, students feel in control of the coefficients. They
can move it up or down a bit by changing the ‘starts at’ value and change
the steepness by varying the ‘per minute’ value. They are very impressed by
their capacity to make a near perfect fit. Looking at the graph of our best
fit function, we can see roughly when the temperature will be down at 48°,
though this may require extending the axes. However, the estimating
power of the function becomes clear immediately. Students who hand
drew their graphs immediately see the flexibility of the software. Fitting a
graph by eye is very useful as it reinforces the power of the function. Clearly
this function fits our data very well and so we can use it to find when the
value of this function is 48. That is 84 — (2.3/60) x = 48. Immediately stu-
dents recognize that this is an equation. Their knowledge of how to solve
it can now be brought to bear. Powerfully the software includes a computer
algebra system (CAS). Here we can state the equation. Then work on it in — per second. So we can edit the function to show f1(x) = 84 — (2.3/60)x. Happily the software gives an immediate response, so testing different the- ories for accommodating the minutes to second conversion can be done quickly by test and check. This feature keeps the discussion on track and avoids being sidelined by tricky numeracy issues. These can be returned to later. Often the very beginning of the experiment has an uneven cooling rate, so a little ‘tweaking’ of the model needs doing. Having seen the con- struction of the model, students feel in control of the coefficients. They can move it up or down a bit by changing the ‘starts at’ value and change the steepness by varying the ‘per minute’ value. They are very impressed by their capacity to make a near perfect fit. Looking at the graph of our best fit function, we can see roughly when the temperature will be down at 48°, though this may require extending the axes. However, the estimating power of the function becomes clear immediately. Students who hand drew their graphs immediately see the flexibility of the software. Fitting a graph by eye is very useful as it reinforces the power of the function. Clearly this function fits our data very well and so we can use it to find when the value of this function is 48. That is 84 — (2.3/60) x = 48. Immediately stu- dents recognize that this is an equation. Their knowledge of how to solve it can now be brought to bear. Powerfully the software includes a computer algebra system (CAS). Here we can state the equation. Then work on it in

of the graph of the function they are looking for. In our lessons we restricted
exploration to a palette of possibilities consisting of linear, quadratic, recip-
rocal and exponential function. However, the software can cope with other
interesting functions, which we have tried out in teacher sessions, notably,
piecewise linear functions. These naturally accord with the descriptions of
the variation students suggested that is, a certain rate of decrease over a
certain range, followed by a different rate of decrease over the next part of
the range. With a final linear function of f(x) = room temperature after a
certain period, this can be an excellent model. Clearly, a well constructed
exponential will also provide an excellent model. — of the graph of the function they are looking for. In our lessons we restricted exploration to a palette of possibilities consisting of linear, quadratic, recip- rocal and exponential function. However, the software can cope with other interesting functions, which we have tried out in teacher sessions, notably, piecewise linear functions. These naturally accord with the descriptions of the variation students suggested that is, a certain rate of decrease over a certain range, followed by a different rate of decrease over the next part of the range. With a final linear function of f(x) = room temperature after a certain period, this can be an excellent model. Clearly, a well constructed exponential will also provide an excellent model.

FiGurE 18.1 Screen Shot of Modellus Used to Model an Oscillation

FicurRE 18.2. Screen Shot of Modellus Used to Animate a Model

FIGURE 18.3. The Animation at a Time When the Displacement is Zero

FiGuRE 18.5 Showing Video Data for an Experiment with Data Logged with a
Motion Sensor — FiGuRE 18.5 Showing Video Data for an Experiment with Data Logged with a Motion Sensor

[GURE 18.7 The Video is Stopped Just at a Critical Point in the Experiment

FiGuRE 18.8 The Video Stopped When the String is Slack

FiGurRE 19.1 Typical Result for Speed of Sound Measurement Using Audacity

FicureE 19.2. Test Launch in a Classroom, Smoke Detectors Off!

(note the inversion due to the sensor being pushed rather than pulled)

FiGuRE 19.4 Profile of the Motion of Bloodhound SSC The two graphs shown in Figure 19.4 are the velocity (smooth) and accel eration curves. There is a great deal that students can identify with the events taking place during the run such as the opening of parachutes and ignition of the rocket. The real cross-curricular approach comes when stu: dents launch their model car and attempt through various means to analyse the motion and compare it to their predictions. ween om aes Neki |e =" — a ” 1 a ons = see 06 Ue

FiGurE 19.5 A Still from a Film with Chalk Marks

FIGURE 20.2. A TI-Nspire Projectile Model
FIGURE 20.1 An Excel Spreadsheet Projectile Model — FIGURE 20.2. A TI-Nspire Projectile Model FIGURE 20.1 An Excel Spreadsheet Projectile Model

FIGURE 20.3 Vernier’s Logger Pro 3 and WDSS

©TDA 2008. Reproduced with permission from the Training and Development Agency for Schools

FIGURE 20.6 Tracking the Speed of del Potro’s service
ormats may not be readable, so video conversion software can be useful.

There is now a new range of personal digital cameras capable of higl
juality video capture at speeds of 210 and 420 frames per second, and <
ower quality at 1000 fps. Figure 20.6 shows the capture of the service actio
»f Juan del Potro on his way to the final at the 2009 ATP World Tour tenn:
inals at the London O2 stadium last year. The video clip: ‘del potro.avi’ we
ecorded at 210 frames per second using a Casio Exilim EX-FH20 hig
peed camera with telephoto lens: www.casio.co.uk/products/Digital %2
cameras /Exilim % 20High % 20Speed/EX-FH20BKEDA/ Technical
specifications /.

net ee ee = Wm 7 ros +1. Cc. — FIGURE 20.6 Tracking the Speed of del Potro’s service ormats may not be readable, so video conversion software can be useful. There is now a new range of personal digital cameras capable of higl juality video capture at speeds of 210 and 420 frames per second, and < ower quality at 1000 fps. Figure 20.6 shows the capture of the service actio »f Juan del Potro on his way to the final at the 2009 ATP World Tour tenn: inals at the London O2 stadium last year. The video clip: ‘del potro.avi’ we ecorded at 210 frames per second using a Casio Exilim EX-FH20 hig peed camera with telephoto lens: www.casio.co.uk/products/Digital %2 cameras /Exilim % 20High % 20Speed/EX-FH20BKEDA/ Technical specifications /. net ee ee = Wm 7 ros +1. Cc.

compares well with the speed of 115 mph (or 184 kph) shown on the court
side display. You can download a video clip together with a Tracker file from
which you can estimate the speed of delivery of a ball bowled by a young
cricketer from: www.ncetm.org.uk/Default.aspx? page= 14&module=com&
mode=102&comcid=5265&comf=45648&comfb=1&comu=0. Figure 20.7
shows a more sophisticated example of mathematical modelling applied tc
data from Usain Bolt’s Gold Medal winning 100 m sprint at the 2008

Olympics. — compares well with the speed of 115 mph (or 184 kph) shown on the court side display. You can download a video clip together with a Tracker file from which you can estimate the speed of delivery of a ball bowled by a young cricketer from: www.ncetm.org.uk/Default.aspx? page= 14&module=com& mode=102&comcid=5265&comf=45648&comfb=1&comu=0. Figure 20.7 shows a more sophisticated example of mathematical modelling applied tc data from Usain Bolt’s Gold Medal winning 100 m sprint at the 2008 Olympics.

I begin by getting the students to plot a sine curve and its first derivative
with their calculators (or software) in degree mode. At first sight the deriva-
tive curve is almost invisible as it runs so close to the »axis. But, on closer
inspection, it is clear that there is a wave and that maybe it is a periodic
function with a very small amplitude.

Tr a . ae | . Ff rod . cr ce 1
ing can be achieved than from handing students unjustified results.
a cy ee > ae

Let us take the introduction to the differentiation of sine functions as an
-xample of the use of technology in the teaching of the SL course. There are
nany approaches to doing this ranging from a rigorous analytical one to
imply presenting the students with the result. This approach could utilize a
sDC or any graphing software. When students in the class are comfortable
vith technology, there are many possibilities for introducing topics that do
10t need the depth required in a rigorous approach. A better understand-
ng can be achieved than from handing students unjustified results. — I begin by getting the students to plot a sine curve and its first derivative with their calculators (or software) in degree mode. At first sight the deriva- tive curve is almost invisible as it runs so close to the »axis. But, on closer inspection, it is clear that there is a wave and that maybe it is a periodic function with a very small amplitude. Tr a . ae | . Ff rod . cr ce 1 ing can be achieved than from handing students unjustified results. a cy ee > ae Let us take the introduction to the differentiation of sine functions as an -xample of the use of technology in the teaching of the SL course. There are nany approaches to doing this ranging from a rigorous analytical one to imply presenting the students with the result. This approach could utilize a sDC or any graphing software. When students in the class are comfortable vith technology, there are many possibilities for introducing topics that do 10t need the depth required in a rigorous approach. A better understand- ng can be achieved than from handing students unjustified results.

In mathematics studies SL, students undertake a project worth 20 per cent
of their total mark. This could be from any area of mathematics. Access to
powerful software could result in students simply adopting an approach that
relied on, for example, a statistics package to do all the serious mathematical
work for them. It is possible, however, to enable students to make use of the
leads to the conjecture that it might be possible to find a period that makes
the derivative have an amplitude of one, the same as the original sine curve.

After some trial and error and adjustment of the scale of the x-axis, it is
discovered that a value of approximately 57 for the coefficient of x yields
the appropriate amplitude. The intercepts of the modified sine curve are-
approximately z and 27 and, of course, 57=(180/7). Thus the first derivative
of sin((180/7)x) can clearly be seen to be cos((i80/z)x) leading to the idea of
radian measure. — In mathematics studies SL, students undertake a project worth 20 per cent of their total mark. This could be from any area of mathematics. Access to powerful software could result in students simply adopting an approach that relied on, for example, a statistics package to do all the serious mathematical work for them. It is possible, however, to enable students to make use of the leads to the conjecture that it might be possible to find a period that makes the derivative have an amplitude of one, the same as the original sine curve. After some trial and error and adjustment of the scale of the x-axis, it is discovered that a value of approximately 57 for the coefficient of x yields the appropriate amplitude. The intercepts of the modified sine curve are- approximately z and 27 and, of course, 57=(180/7). Thus the first derivative of sin((180/7)x) can clearly be seen to be cos((i80/z)x) leading to the idea of radian measure.

An interesting extension to the original question was to look at influence
that replacing goal difference with goal average had from the 1976-1977
season in the English Football League onwards. Another list was used to
calculate the goal ratio (GR) and this was plotted against PTS. — An interesting extension to the original question was to look at influence that replacing goal difference with goal average had from the 1976-1977 season in the English Football League onwards. Another list was used to calculate the goal ratio (GR) and this was plotted against PTS.

calculate the goal ratio (GR) and this was plotted against PTS.

SSAA) RSIS SAERLSEAN AIREY REM SSATET RSS STREWSAGA A TURTESSISTS ESS ATMA ERSEOUS NR A SAINI te DANS Ia REESE ie ces

Initially I mostly used the computer software for demonstration purposes
at the front of the class. For example creating a function f1(x) and then
creating dependant functions such as f2(x) = f1(x+a) and using a slider tc
explore the effect that changing the value of a has on the graph. This is z
really powerful way of exploring the various transformations of functions.
as it becomes simple to look at a variety of values for aas well as a variety of
starting functions. — SSAA) RSIS SAERLSEAN AIREY REM SSATET RSS STREWSAGA A TURTESSISTS ESS ATMA ERSEOUS NR A SAINI te DANS Ia REESE ie ces Initially I mostly used the computer software for demonstration purposes at the front of the class. For example creating a function f1(x) and then creating dependant functions such as f2(x) = f1(x+a) and using a slider tc explore the effect that changing the value of a has on the graph. This is z really powerful way of exploring the various transformations of functions. as it becomes simple to look at a variety of values for aas well as a variety of starting functions.

More recently I have started getting the students to regularly use the hand:
held devices in class. We frequently use the TI-Nspires to get a feeling for <
problem graphically before diving into the algebra. This has really helped my
students get a more intuitive feel for the mathematics. Also, since the TI
Nspire has a document structure, it is possible for me to pre-create activities
and copy them for students to use on the hand-held devices. Some really
good examples of the kind of things that can be done with the TI-Nspire tc
enhance post 16 teaching can be found on the Nspiring Learning website. — More recently I have started getting the students to regularly use the hand: held devices in class. We frequently use the TI-Nspires to get a feeling for < problem graphically before diving into the algebra. This has really helped my students get a more intuitive feel for the mathematics. Also, since the TI Nspire has a document structure, it is possible for me to pre-create activities and copy them for students to use on the hand-held devices. Some really good examples of the kind of things that can be done with the TI-Nspire tc enhance post 16 teaching can be found on the Nspiring Learning website.

learn by their mistakes and for creative thinking, I am finding an increas-
ing number of children coming to me in Year 3, totally switched off and
closed to mathematics especially when doing open ended investigations.
I hoped I would be able to reverse some of these effects through using the
Pro-bots, which children view as a ‘toy’, but through which I hoped to con-
vey mathematical skills.
I chose nine mixed ability Year 3 children to work in groups of three with
a Pro-bot per group. Several of the children are very wary of mathematics
and are reticent in taking risks with their learning. Could this be the knock
on effect of ten years of the National Numeracy Strategy and a culture of
testing? Although our School has an ethos of encouraging children to
testing? Although our School has an ethos of encouraging children to — learn by their mistakes and for creative thinking, I am finding an increas- ing number of children coming to me in Year 3, totally switched off and closed to mathematics especially when doing open ended investigations. I hoped I would be able to reverse some of these effects through using the Pro-bots, which children view as a ‘toy’, but through which I hoped to con- vey mathematical skills. I chose nine mixed ability Year 3 children to work in groups of three with a Pro-bot per group. Several of the children are very wary of mathematics and are reticent in taking risks with their learning. Could this be the knock on effect of ten years of the National Numeracy Strategy and a culture of testing? Although our School has an ethos of encouraging children to testing? Although our School has an ethos of encouraging children to

FiGuRE 27.1 [DCH2(tns-T) Page 2]

a ahr retamriintiaiaas alamari arenes eeimaicriaitaaians | aaiasiaaciaaaaairticaiacataciniiaaiasceaaaanicaaas

our triangles and would be expected to obtain identical measurements.
They also acknowledged that, by attempting to ‘translate’ a familiar paper
nd pencil task to a technological environment, although the technology had
utomated the angle measuring process, they had not appreciated the power
f the technology. It would have been possible for them to drawn a single
riangle and, by measuring its angles, students could have explored many — a ahr retamriintiaiaas alamari arenes eeimaicriaitaaians | aaiasiaaciaaaaairticaiacataciniiaaiasceaaaanicaaas our triangles and would be expected to obtain identical measurements. They also acknowledged that, by attempting to ‘translate’ a familiar paper nd pencil task to a technological environment, although the technology had utomated the angle measuring process, they had not appreciated the power f the technology. It would have been possible for them to drawn a single riangle and, by measuring its angles, students could have explored many

FIGURE 27.3 A dynamic triangle