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Proportionality and scatter |
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CONTENTS |
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The workout comprises a series of sides that guide you systematically through the topic concept by concept, skill by skill. The slides may be used with or without the support of a tutor. The methodology is based on problem-solving that advances in logical succession by concept and difficulty. The student is presented with a problem or series of questions, and the next slide presents the fully-worked solution. To use the material you must sign-in or create an account.
blacksacademy.net comprises a complete course in mathematics with resources that are comprehensive.
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SAMPLE FROM THE WORKOUT |
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SLIDE 1 - QUESITON 1 |
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SLIDE 2 - SOLUTION |
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SAMPLE FROM THE LIBRARY |
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QUESTION [difficulty 0.4] |
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SOLUTION |
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DEPENDENCIES |
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CONCEPTS |
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LEV. |
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Algebraic proportionality problem
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678.3 |
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Algebraic inverse proportionality problem
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678.8 |
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Scatterplot
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679.3 |
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Interpolation
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679.7 |
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Prediction
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679.7 |
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Extrapolation
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679.9 |
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Critique of extrapolation in science
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680.0 |
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Fallacy
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680.1 |
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Difference between line of best fit and actual value
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680.2 |
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Extending proportionality to non-linear functions
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680.8 |
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y = k f(x)
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680.8 |
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Graph of root x is reflection in y = x of graph of x^2
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690.6 |
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Rational indexes in proportionality relations
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690.6 |
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Combining proportionality relations
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690.9 |
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Graphing proportionality relations
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691.2 |
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RAW CONTENT OF THE WORKOUT |
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SLIDE 1
Consolidation
The extension of an elastic string is proportional to the force applied to stretch it. The natural length of an elastic string is the length it has when no force is applied to it.
An elastic string with natural length 1.2 m is stretched by a force of 45 N to a length of 1.35 m. Assuming the string remains elastic when a larger force is applied, what would be the length of the string when stretched by a force of 57 N?
SLIDE 2
The extension of an elastic string is proportional to the force applied to stretch it. The natural length of an elastic string is the length it has when no force is applied to it. An elastic string with natural length 1.2 m is stretched by a force of 45 N to a length of 1.35 m. Assuming the string remains elastic when a larger force is applied, what would be the length of the string when stretched by a force of 57 N?
Solution
Let the extension of the string be x and the force applied be F. The extension of an elastic string is proportional to the force applied to stretch it. ?
Make this into an equation by introducing the constant of proportionality ? .
The extension is the length of the string less the natural length.
?
Substitute the given values to find k.
? ?
Substitute for k ?
Find F when ?
The length of the string with this force is
SLIDE 3
If x bananas cost y dollars. How much does each banana cost? What is the cost of z bananas?
SLIDE 4
If x bananas cost y dollars. How much does each banana cost? What is the cost of z bananas?
Solution
Use ratios
SLIDE 3B
If x bananas cost £y. How much does each banana cost? What is the cost of z bananas?
SLIDE 4B
If x bananas cost £y. How much does each banana cost? What is the cost of z bananas?
Solution
Use ratios
SLIDE 5
If M apples cost Q pence per kilogram, what is the mass of apples that can be bought for P pence and how many apples do you get for this money?
SLIDE 6
If M apples cost Q pence per kilogram, what is the mass of apples that can be bought for P pence and how many apples do you get for this money?
Solution
You can buy apples for P pence and their mass is kg.
Note. This allows for the possibility of fractional apples; that is, half an apple and so on.
SLIDE 7
Consolidation – inverse proportionality
A variable Y is inversely proportional to another variable X. When , .
? Find X when
? Find Y when
SLIDE 8
A variable Y is inversely proportional to another variable X. When , .
?
?
SLIDE 9
X and Y vary directly. The table provides values of X and Y.
X Y
1.96 7
a 18
2.66 b
Find a and b.
SLIDE 10 Need to make it b in the second part of the answer
X and Y vary directly.
X Y
1.96 7
a 18
2.66 b
SLIDE 11
? ?
x y s t
4 12 6 14
6 18 8 20
8 24 10 26
Which of these two relationships is linear proportional and which is linear non-proportional?
Find the equations linking x and y, and s and t respectively?
SLIDE 12
? ?
x y s t
4 12 6 14
6 18 8 20
8 24 10 26
? is linear proportional. The equation is .
? is linear non-proportional. The equation is
SLIDE 13
Latitude / °N 90 80 70 60 50 40 30 20
90 - Lat / ° 0 10
Mean temp / °C -32 -25 -20 -9 -3 +4 +14 +25
A latitude of 90° corresponds to the North Pole. The table shows the relationship between mean surface temperature and latitude. Complete the table and make a scatterplot for the data.
SLIDE 14
Latitude / °N 90 80 70 60 50 40 30 20
90 - Lat / ° 0 10 20 30 40 50 60 70
Mean temp / °C -32 -25 -20 -9 -3 +4 +14 +25
? What kind of correlation is exhibited?
? Draw the line of best fit within the range of the datapoints.
SLIDE 15
The scatterplot exhibits positive correlation. Find the equation of the line of best fit shown in the diagram.
SLIDE 16
The gradient is . The intercept is . The equation of the line of best fit is , where T is the mean surface temperature and L is .
SLIDE 17
Interpolation
The equation of the line of best fit is , where T is the mean surface temperature and L is . To interpolate is to insert a point within the range of the data. Interpolation is used to make a prediction.
By interpolation predict the mean surface temperature of a town with latitude . (This corresponds to on the scale used in the diagram.)
SLIDE 18
The graphical method is shown above. By substituting into the equation , we also obtain .
SLIDE 19
Extrapolation
Extrapolation is to continue the linear relationship outside the range of the data. That is, to extend the straight line beyond the least or greatest datapoint.
By extrapolation predict the temperature for a location corresponding to on the horizontal axis.
SLIDE 20
The graphical method is shown above. By substituting into the equation , we also obtain .
Extrapolation is considered “risky” in science. Suggest a reason why.
SLIDE 21
Extrapolation is considered “risky” in science. There may be good reason why a linear relation does not continue outside the range of the experimental data on which it is based.
This question is based on experimental data. * In fact, the relationship does not continue linearly, and a maximum mean surface temperature of 27°C obtains. Thus, the prediction that 80° (10°N) corresponds to 31°C is a significant error. Continuing known mathematical relations into the unknown is an error of deduction. (Errors of deductions are known as fallacies.)
* http://www-das.uwyo.edu/~geerts/cwx/notes/chap16/geo_clim.html
SLIDE 22
The diagram shows data for ten of the largest world companies with the line of best fit.
? Predict the income of a company with sales of $220bn.
? For the company that has the greatest difference in income from the line of best fit, what was the difference between the income predicted by the line of best fit and the actual income?
SLIDE 23
? Predict the income of a company with sales of $220bn.
$51 bn
? For the company that has the greatest difference in income from the line of best fit, what was the difference between the income predicted by the line of best fit and the actual income?
SLIDE 24
The diagram shows the relationship between mass and density for a class of planetary body together with its line of best fit.
? What kind of correlation is shown?
? Find the equation of the line of best fit.
? Estimate the volume in cubic meters of the planetary body with mass .
SLIDE 25
? The correlation is negative.
? Intercept at 0.6. For the gradient, we must put the two axes in the same units. There are grams in one kilogram, so
Gradient
The equation is where is the density in and m is the mass in g.
? When mass is , density is 0.5
1 cubic meter is . Hence the volume of the planetary body is
SLIDE 26
Complete the following
graph function curve
line
… parabola
… hyperbola
… half parabola
… cubic curve
SLIDE 27
graph function curve
line
parabola
hyperbola
half parabola
cubic curve
SLIDE 28
Extending proportionality to non-linear functions
y is directly proportional to is written
We derive the equation
where k is the constant of proportionality.
Example
It is given that y is directly proportional to . When , . Find the equation . Find y when
Solution
When .
SLIDE 29
It is given that y is directly proportional to for . When , . Find the equation . Find y when . Find x when . Give your answers to 3 significant figures.
SLIDE 30
It is given that y is directly proportional to . When , . Find the equation . Find y when . Find x when . Give your answers to 3 significant figures.
Solution
When
When
SLIDE 31
It is given that y is directly proportional to for .
Sketch the graph corresponding to this relationship.
SLIDE 32
It is given that y is directly proportional to for .
The graph of is a parabola. Since , we have only the positive part of the parabola.
SLIDE 33
It is given that P is directly proportional to for . When , . Find the equation for the relationship, complete the following table, and plot the graph.
t 0 1 4 9 16 25
P 0 0.6
SLIDE 34
t 0 1 4 9 16 25
P 0 0.2 0.4 0.6 0.8 1.0
SLIDE 35
Sketch and for onto the same coordinate plane. Add onto the graph a sketch of the line . What is the relationship between the graphs of and ?
SLIDE 36
The graphs of both and , where are the positive parts of a parabola.
The graph of is the reflection of in the line .
SLIDE 37
It is given that Y is directly proportional to for . When , . Find the equation expressing this in index form. Find Y when .
SLIDE 38
It is given that Y is directly proportional to for . When , . Find the equation expressing this in index form. Find Y when .
Solution
When
SLIDE 39
Combining proportionality relations
SLIDE 40
y is inversely proportional to . When .
t is directly proportional to x. When .
Find the equation linking y in terms of x.
SLIDE 41
y is inversely proportional to . When .
t is directly proportional to x. When .
Find the equation linking y in terms of x.
Solution
y is inversely proportional to
t is directly proportional to x.
.
SLIDE 42
Water is being poured into a cone at a constant rate. At the volume is also . When the volume is . Which of the following graphs shows the relationship between V and t? Find the equation connecting V and t.
SLIDE 43
Water is being poured into a cone at a constant rate. At the volume is also . When the volume is . Which of the following graphs shows the relationship between V and t? Find the equation connecting V and t.
Solution
SLIDE 44
Water is being poured into a cone at a constant rate.
The depth of the water proportional to the cube root of the volume. When , .
Which of the following graphs shows the relationship between V and d? Find the equation connecting V and d.
SLIDE 45
Water is being poured into a cone at a constant rate.
The depth of the water proportional to the cube root of the volume. When , .
Which of the following graphs shows the relationship between V and d? Find the equation connecting V and d.
Solution
SLIDE 46
Water is being poured into a cone at a constant rate.
and
? Find the relation between depth and time.
? The capacity of the cone is reached at . Find the volume and depth of the cone when it is full.
SLIDE 47
Water is being poured into a cone at a constant rate.
and
? Find the relation between depth and time.
? The capacity of the cone is reached at . Find the volume and depth of the cone when it is full.
or
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