Download Maths Skills for A-level Biologists

A minimum of 10% of the marks in
the exams involve maths.
• Answer the questions on the sheet.
• You may use a calculator.
• No cheating – work in silence!
1.
2.
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4.
5.
6.
7.
8.
29
68
8cm3
a) 24.6
b) 1.45
c) 0.0000100
Mrs Eves – 24, Miss Jesusanmi – 8, Mr Walker – 4
5/
6
160
29.2
• At least higher tier GCSE maths standard.
• This means that simple calculations do not
count towards the 10%.
• The specification lists all the skills you
need.
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Use of standard form
Ratios, fractions and percentages
Power, exponential and logarithmic functions
Significant figures
Mean, median, mode, range, standard deviation
Charts and graphs (including logarithmic scales)
Probabilities
Calculations of rate from results/graphs
Use of equations
Statistical tests
Uncertainties and percentage errors
Calculations of circumference, surface area and volume
• You should be able to give answers to an ‘appropriate number
of significant figures’. Use figures in the question to help you.
• Starting on the left, count from the first digit that is not a 0.
• When you get to the required number of sig. fig., look at the
next digit. If it’s 0 to 4, round down. If it’s 5 to 9, round up.
E.g. 0.048484848… is 0.05 to 1 sig. fig.
or 0.048 to 2 sig. fig.
or 0.0485 to 3 sig. fig.
Try the examples on the sheet – 3 minutes!
Number 1 sig.fig.
2 sig.fig.
3 sig.fig.
124507
100000
120000
125000
25.364
30
25
25.4
0.02565
0.03
0.026
0.0257
6.7909
7
6.8
6.79
Calculators use the standard order of operations:
Brackets
Indices (or Orders)
Division
Multiplication
Addition
Subtraction
This means your calculator might not do the sum you want.
To get it right, press = after each part of the calculation (or make
use of the brackets on your calculator!).
For the sum…
2+4
2
You might type in…
2+4÷2=
Which would give you…
4
Because your calculator ‘does’ BIDMAS.
If you type…
2+4=
Then…
÷2=
You’ll get the right answer, which is…
3
Or, you could type…
(2+4)÷2=
Try the examples on the sheet – 3 minutes!
1. 22
2. 33
3. 9
• Mean = (sum of values)/(number of values)
• Median – put numbers in order and select ‘middle’ value
• Mode = most frequently occurring value
• Range = lowest and highest values (or highest – lowest)
Try the example on the sheet – 5 minutes!
Mean = 19.9
Median = 19
Mode = 17
Range = 13 to 30 (or 17)
• For a sample of data, standard deviation gives an indication of
the spread of data around the mean.
Both curves
show a
normal
distribution.
• For now, if the standard deviations of two samples ‘overlap’,
there is no significant difference between them.
• We’ll learn how to calculate standard deviation later in the
year.
What information does the table above
give us?
• Overlap between ranges (mean ± s.d.) for
Pudelpointers suggests no significant difference in
heritability between tracking and searching.
• No overlap between ranges (mean ± s.d.) for Large
Munsterlanders therefore these is a significant
difference in the heritability of the tracking and
searching.
• Tracking is influenced by genetic factors more than
searching in Munsterlanders.
Powers – multiply the number by itself the number of times
stated in the power,
e.g. 23 = 2 x 2 x 2 = 8
This includes standard form,
e.g. 5.2 x 103 = 5.2 x 10 x 10 x 10 = 5200
You should be able to convert numbers to and from
standard form.
Includes a number (from 1 to <10) multiplied by
10x (where x is any whole number, including
negative numbers).
You must retain the sig. fig.
Try the examples on the sheet.
1. 243000 = 2.43 x 105
2. 0.00123 = 1.23 x 10-3
3. 469100000 = 4.691 x 108
4. 0.00001050 = 1.050 x 10-5
Exponential functions – also involve powers,
e.g. 2x = 2 multiplied by itself x times
A single bacterium is allowed to grow and reproduce for
24 hours. If it can divide by binary fission once every
20 minutes, how many cells will be present after 24
hours?
Answer:
There will be (24 x 60) / 20 divisions
= 72 divisions
The total number of cells will be 272 = 4.722 x 1021
We can use a logarithmic function to make it easier to accurately plot
graphs of data that covers several orders of magnitude (i.e. very
small numbers and very big numbers).
We generally use log10 in Biology.
With large numbers, we use standard form. This involves
10x, where x is called the exponent.
Log10 y = x
means
10x= y
So, we can use the log function to work out the exponent
of a number.
pH is a measure of the relative concentration of H+ ions in
a solution. It is calculated as follows:
pH = -log10[H+]
The contents of the stomach are found to have a H+ ion
concentration of 0.0001moldm-3. What is the pH?
pH = -log10[0.0001] = 4
If the pH of the blood is 7, what is the concentration of H+ ions?
pH = -log10[H+], so H+ conc. = 10-pH
H+ conc. = 10-pH = 10-7 = 0.0000001moldm-3
pH is a measure of the relative concentration of H+
ions in a solution. It is calculated as follows:
pH = -log10[H+]
Remember:
H+ conc. = 10-pH
Try the examples on the sheet.
pH = -log10[H+]
1. pH 3
2. pH 3.60
3. pH 8.33
1. 1.0 x 10-14 moldm3
2. 3.16 x 10-7 moldm3
3. 5.62 x 10-9 moldm3
H+ conc. = 10-pH
• Select appropriate graph for your data:
• Bar chart
• Histogram
• Scatter graph/line graph
• Remember:
• Label the axes
• Include units
• Continuous scales (must allow for accurate plotting – no multiples
of 3!)
• Best fit line is either straight (ruler!) OR curved (smooth!)
You can look for correlation (positive or negative) and
direct/inverse proportion.
For the next few slides, see if you can
explain what each graph shows.
Why has a particular type of graph
been chosen?
Bar chart with standard
deviation bars – which groups
are significantly different?
Positive correlation
– as the male head
length increases,
the female head
length increases.
X-axis labelled in
powers of 10, i.e.
10-2, 10-1, 100, 101,
102…
The Institute of Biology says:
‘Graphs should be joined dot to dot with straight lines when the
intermediate values can not be reliably predicted’.
You will usually use a line of best fit.
What information does the graph tell us?
a) The heights of all the students in a class
b) The eye colours of all the students in a class
c) The number of bacterial cells growing in a test tube
Be ready to explain your answers!
3 minutes!
a) The heights of all the students in a class
Histogram
b) The eye colours of all the students in a class
Bar chart
c) The number of bacterial cells growing in a test tube
Line graph
• Rate of reaction =
1
.
time taken for reaction
• OR: Rate of reaction = quantity of product formed .
time taken for reaction
• Rate of movement =
distance moved
.
time taken to move that distance
…and so on.
From a graph,
rate = gradient of line
Maximum rate = maximum gradient of line
1. 1.4 (2 marks)
1 mark for 4.2 ÷ 3
2.
You should be able to simplify a ratio,
e.g. 8:2 is the same as 4:1
144:12 is the same as 12:1 etc.
In Biology, we usually express ratios in the form x:1, where x is
any number (not necessarily an integer), or just as x, imagining the
‘:1’ part.
Allows us to make comparisons more easily.
• Need to understand how to interpret/use fractions.
e.g. finding ½ of a number
using formulae
Nothing more difficult than GCSE!
Answer the questions – 5 minutes!
1. 206.25
2. 37/60
3. 3/10
4. 12/5 or 2 2/5
• Calculate a percentage.
• Interpret percentages in tables and graphs.
• Calculate percentage increase/decrease:
Percentage increase = final value – original value X 100
original value
How would you calculate percentage decrease?
Answer the questions – 10 minutes!
1. 69%
2. 37 marks
3. 84.4%
Image length = real length x magnification
Pulmonary ventilation = tidal volume x ventilation rate
Cardiac output = heart rate x stroke volume
Need to recall, rearrange and use these.
Index of diversity =
N (N – 1)
Σ n (n – 1)
Where N = total number of organisms of all species, and n = total
number of organisms of each species.
Need to be able to use this to calculate index of diversity.
• You should be able to visualise 3D shapes from 2D diagrams.
• Given formulae, calculate:
• Circumference and area of circles
• Surface areas and volumes of regular blocks and cylinders
Circumference of a circle = 2πr or πd
Area of a circle = πr2
Surface area of a shape = sum of areas of all sides
Volume of a regular shape (prism) = area of cross-section x length
Try the calculations – 5 minutes!
• If a neurone has a mean diameter of 0.1mm and a length of
25cm, calculate the approximate volume of the neurone.
1. a) 25.1cm
b) 50.3cm2
2. Area of cross-section = 7.85x10-3mm2
Volume = 1.963mm3
Calculations of % errors
Calculations of standard deviation
Statistics:
• X2 (Chi-squared) – tests for differences between observed and
expected results
• Spearman’s rank correlation coefficient – tests for correlation
between two measurements from the same sample
• Student’s t-test – tests for differences between mean values