PatternsARelationsApproachToAlgebra

PatternsARelationsApproachToAlgebra
Patterns
A Relations Approach
to Algebra
Junior Certificate Syllabus
Leaving Certificate Syllabus
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Contents
Chapter 1: Relationship to syllabuses,
introduction and unit learning outcomes . . . . . . . . . . . . . . . . . . 5
Chapter 2: Comparing Linear Functions . . . . . . . . . . . . . . . . . 11
Chapter 3: Proportional
and non proportional situations . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter 4: Non constant rates of
change-quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 5: Exponential growth. . . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 6: Inverse Proportion . . . . . . . . . . . . . . . . . . . . . . . . . 53
Chapter 7: Moving from linear to
quadratic to cubic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 8: Appendix containing
some notes and suggested solutions. . . . . . . . . . . . . . . . . . . . . 65
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Chapter 1: Relationship to
syllabuses, introduction and
unit learning outcomes
5
Patterns a Relations Approach to Algebra
Junior Certificate Mathematics: Draft Syllabus
Topic
Description of topic
Students learn about
Learning outcomes
Students should be able to
4.1 Generating
arithmetic
expressions
from
repeating
patterns
Patterns and the rules that govern
them; students construct an
understanding of a relationship as
that which involves a set of inputs, a
set of outputs and a correspondence
from each input to each output.
4.2 Representing
situations
with tables,
diagrams and
graphs
Relations derived from some kind of
• use tables, diagrams and graphs as tools for representing
context – familiar, everyday situations,
and analysing linear, quadratic and exponential patterns
imaginary contexts or arrangements
and relations (exponential relations limited to doubling
of tiles or blocks. Students look at
and tripling)
various patterns and make predictions
about what comes next.
• develop and use their own generalising strategies and
ideas and consider those of others
• use tables to represent a repeating-pattern situation
• generalise and explain patterns and relationships in words
and numbers
• write arithmetic expressions for particular terms in a
sequence
• present and interpret solutions, explaining and justifying
methods, inferences and reasoning
4.3 Finding
formulae
4.4 Examining
algebraic
relationships
Ways to express a general
relationship arising from a pattern or
context.
• find the underlying formula written in words from which
the data is derived (linear relations)
Features of a relationship and how
these features appear in the different
representations.
Constant rate of change: linear
relationships.
Non-constant rate of change:
quadratic relationships.
Proportional relationships.
• show that relations have features that can be represented
in a variety of ways
• find the underlying formula algebraically from which the
data is derived (linear, quadratic relations)
• distinguish those features that are especially useful to
identify and point out how those features appear in
different representations: in tables, graphs, physical
models, and formulas expressed in words, and
algebraically
• use the representations to reason about the situation
from which the relationship is derived and communicate
their thinking to others
• recognise that a distinguishing feature of quadratic
relations is the way change varies
• discuss rate of change and the y-intercept, consider how
these relate to the context from which the relationship is
derived, and identify how they can appear in a table, in a
graph and in a formula
• decide if two linear relations have a common value
(decide if two lines intersect and where the intersection
occurs)
• investigate relations of the form y = mx and y = mx + c
• recognise problems involving direct proportion and
identify the necessary information to solve them
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Patterns a Relations Approach to Algebra
Leaving Certificate Mathematics: Draft Syllabus
Students learn
about
3.1 Number
systems
Students working at FL should be
able to
In addition, students
working at OL should be
able to
• generalise and
explain patterns
• revisit the operations of addition,
and relationships
multiplication, subtraction and
in algebraic form
division in the following domains:
• recognise irrational numbers and
appreciate that R≠Q
• N of natural numbers
• Z of integers
• Q of rational numbers
• R of real numbers
and represent these numbers on
a number line
• appreciate that processes can
generate sequences of numbers
or objects
• investigate patterns among these
sequences
• use patterns to continue the
sequence
• recognise whether
a sequence
is arithmetic,
geometric or
neither
• find the sum to
n terms of an
arithmetic series
In addition, students working at
HL should be able to
• verify and justify
formulae from number
patterns
• investigate geometric
sequences and series
• prove by induction solve
problems involving finite
and infinite geometric
series including
applications such as
recurring decimals and
financial applications,
e.g. deriving the
formula for a mortgage
repayment
• derive the formula for
the sum to infinity of
geometric series by
considering the limit of a
sequence of partial sums
• generate rules/formulae from
those patterns develop decimals
as special equivalent fractions
strengthening the connection
between these numbers
and fraction and place value
understanding
• consolidate their understanding
of factors, multiples, prime
numbers in N
• express numbers in terms of their
prime factors
• appreciate the order of
operations, including brackets
• express non-zero positive rational
numbers in the form a x10n,
where n ∈ N and 1 ≤ a < 10 and
perform arithmetic operations on
numbers in this form
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Patterns a Relations Approach to Algebra
Strand 4: Algebra
This strand builds on the relations-based approach of Junior
Certificate where the five main objectives were
1. to make sense of letter symbols for numeric quantities
2. to emphasise relationship based algebra
3. to connect graphical and symbolic representations of algebraic
concepts
4. to use real life problems as vehicles to motivate the use of
algebra and algebraic thinking
5. to use appropriate graphing technologies (graphing
calculators, computer software) throughout the strand
activities.
Students build on their proficiency in moving among equations,
tables and graphs and become more adept at solving real world
problems.
Junior Certificate Draft Syllabus page 26,
Leaving Certificate Draft Syllabus page 28.
National Council for Curriculum and Assessment (2011), Mathematics Syllabus,
Junior Certificate [online] available: http://www.ncca.ie/en/Curriculum_and_
Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/
Junior_Cert_Maths_syllabus_for_examination_in_2014.pdf
National Council for Curriculum and Assessment (2011), Mathematics Syllabus,
Leaving Certificate [online] available: http://www.ncca.ie/en/Curriculum_and_
Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/
Leaving_Cert_Maths_syllabus_for_examination_in_2013.pdf
Accessed September 2011.
A functions based approach to algebra
The activities below focus on the important role that functions play
in algebra and characterise the opinion that algebraic thinking is the
capacity to represent quantitative situations so that relations among
variables become apparent. Emphasising common characteristics
helps us to think of functions as objects of study in and of themselves
and not just as rules that transform inputs into outputs. Students
examine functions derived from some kind of context e.g. familiar
everyday situations, imaginary contexts or arrangements of tiles or
blocks. The functions-based approach enables students to have a deep
understanding in which they can easily manoeuvre between equations,
graphs and tables. It promotes inquiry and builds on the learner’s prior
knowledge of mathematical ideas.
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Patterns a Relations Approach to Algebra
In this unit:
1. Students link between words, tables, graphs and formulae when
describing a function given in a real life context.
2. Students come to understand functions as a set of inputs and a set
of outputs and a correspondence from each input to each output.
3. Students identify what is varying and what is constant for a
function.
4. Students will understand the terms variable and constant,
dependent and independent variable and be able to identify them
in real life contexts.
5. Students will build functional rules from recursive formulae.
6. Students focus on linear functions first, identifying a constant rate
of change as that which characterises a linear function. Students
will identify this as constant change in the y - values for consecutive
x - values. They will identify this change in y per unit change in x as
the slope of the straight line graph.
7. Students will see that linear functions can have different ‘starting
values’ i.e. value of y when x = 0. They will identify this as the
y – intercept.
8. Students will see how the slope and y intercept relate to the
context from which the function is derived.
9. Students will explore a variety of features of a function
and examine how these features appear in the different
representations–
a. Is the function increasing, decreasing, or staying the same? If the
function is increasing the slope is positive, if decreasing the slope
is negative and if constant it has zero slope.
b.Is the function increasing/decreasing at a steady rate or is the
rate of change varying? If it is increasing at a constant rate this is
characteristic of a linear function.
10. Students will investigate real life contexts which lead to quadratic
functions where the rate of change is not constant but the rate of
change of the rate of change is constant. Students will identify this
as constant change of the changes in the y values for consecutive x
values
11. Students will examine instances of proportional and non
proportional relationships for linear functions. Students will know
the characteristics of a proportional relationship when it is graphed
– it is linear and passes through the origin (no ‘start-up’ value, i.e.
no y - intercept) and the ‘doubling strategy ‘[if x is doubled (or
increased by any multiple) then y is doubled (or increased by the
same multiple)].
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Patterns a Relations Approach to Algebra
12. Students investigate contexts which give rise to quadratic functions
through the use of words, tables, graphs and formulae.
13. When dealing with graphs of quadratic functions students should
contrast the quadratic with the linear functions as follows:
• the graphs are non linear
• the graphs are curved
• the changes are not constant
• the change of the changes is constant
• the highest power of the independent variable is 2
14. Students should investigate situations involving exponential
functions using words, tables, graphs and formulae and understand
that exponential functions are expressed in constant ratios between
successive outputs.
15. Students should see applications of exponential functions in their
everyday lives and appreciate the rapid rate of growth or decay
shown in exponential functions.
16. Students should discover that for cubic functions the ‘third
changes’ are constant.
17. Students will understand that x3 increases faster than x2 and the
implications of this in nature and in design.
18. Students will investigate situations where inverse proportion
applies and contrast these with linear, quadratic, and exponential
relationships. Students will see that the product of the variables is a
constant in these situations.
19. Students will interpret graphs in cases of ‘relations without
formulae’.
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Chapter 2: Comparing
Linear Functions
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Patterns a Relations Approach to Algebra
Student Activity – comparing
linear functions
(This activity may be introduced with students at any
stage - JCOL, LCFL, LCOL) GRAPH PAPER REQUIRED
The activities give students an opportunity to link this strand with
co-ordinate geometry and can be taken before or after it. Through
the study of patterns of change, they can get a more meaningful
understanding for such characteristics as slopes, intercepts,
equation of a line, etc.
In this activity, students
• Identify patterns and describe different situations using tables and
graphs, words and formulae
• Identify independent and dependent variables and constants
• Identify ‘start amount’ in the table and formula and as the
y-intercept of the graph
• Identify the rate of change of the dependent variable in the table,
graph (as the slope) and formula
• Identify linear relationships as having a constant change between
successive y values (outputs)
• Know that parallel lines have the same slope (same rate of change of
y with respect to x)
• Connect increasing functions with positive slope, decreasing
functions with negative slope and constant functions with slope of
zero. (It is important that students realise that tables, graphs and
algebraic formulae are just different ways to represent a situation.)
The following data shows the measured heights of 4 different
sunflowers on a particular day and the amount they grew in
centimetres each day afterwards.
Sunflower
Start height/cm
Growth per day/cm
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Patterns a Relations Approach to Algebra
Investigate how the height of each sunflower changes over time. Is
there a pattern to the growth? If you find a pattern can you use it to
predict how tall the sunflowers will be in say 30 days, or in any number
of days? What different representations could you use to help your
investigation?
Represent this information in a table similar to the one below, showing
the height of the sunflowers over a six day period for each situation (a),
(b), (c) and (d).
Time in days
0
Height in cm
Change
1. Where is the ‘start height’ for each plant seen in the tables?
2. Where is the amount the plant grows by each day seen in the
tables?
3. What do you notice about the change between successive outputs
for each the tables?
(4 and 5 may be omitted as the main aim of the activity is for
students to recognise the features of a linear relationship in a table
and in a graph.)
4. For each of the situations and tables (a), (b), (c), and (d) identify 2
values which stay the same and 2 values which vary.
Situation and Table
a
Varying
b
c
d
Staying the same
We call the values which vary ‘variables’ and the values which stay the
same ‘constants’.
We have identified 2 variables. Which variable depends on which?
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Patterns a Relations Approach to Algebra
We call one variable the dependent variable and we call the other
variable the independent variable.
5. Which is the independent variable and which is the dependent
variable in each case above?
Plotting graphs of the situations represented in table 1:
Using graph paper, on the first set of axes, graph situations a and b.
Which variable do you think should go on which axis? Discuss.
We usually plot the independent variable on the x-axis and the
dependent variable on the y-axis.
We can think of the x values as the inputs and the y values as the
outputs.
1. What observations can you make about graphs of a and b?
2. How is each observation seen in the situation?
3. How is each observation seen in the table?
4. Are the sunflowers a and b ever the same height? Explain.
y - intercept and slope
5. Where are the start heights seen in the graphs? We call these values the y - intercepts of the graphs.
6. Where are the growth rates seen in the graphs? (As the day increases by 1 what happens to the
height?)
We call this value the slope of the graph – the
change in y per unit change in x.
7. If the number of days increases by 3, what does the height increase
by?
What is the average increase in height per day?
Is this the same as the slope?
8. Can you come up with a way of calculating slope?
9. Going back to the sunflower example, what does the slope
correspond to?
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Patterns a Relations Approach to Algebra
Formulae in words and symbols: The following can be
applied to Sunflowers (a) and (b) JC Syllabus 4.3 – HL in
bold
4.3 Finding
Formulae
Ways to express a general
relationship arising from a
pattern or context
• finding the underlying
formula written in words
from which the data is
derived (linear relations)
• find the underlying formula
algebraically from which
the data is derived (linear,
quadratic relations)
Using the table and the graph (which are just different representations
of the same situation):
1. Describe in words the height of the plant in (a) on a particular day
(e.g. day 4) in terms of its ‘start height’ and its growth rate per day.
2. Describe in words the height h of the plant in (a) on any day d, in
terms of its ‘start height’ and the amount it grows by each day.
3. Write a formula for the height h of the plant in (a) on any day d, in
terms of its ‘start height’ and the amount it grows by each day.
4. Identify the variables (dependent and independent) and the
constants in the formula.
Formulae in words and symbols: Sunflower (b)
5. Where is the ‘start height’ seen in the formulae for a and b? Where is the amount grown each day seen in the formula for a and
b?
6. Identify where each variable and constant in the formulae appears
in the graphs.
Possible Homework or class group work: Investigate the pattern
of sunflower growth in situations b and c using tables, graphs and
formulae as you did for situations a and b.
Next class: Investigate the pattern of sunflower growth in situations c
and d using tables, graphs and formulae as you did for situations a and
b and b and c.
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Patterns a Relations Approach to Algebra
Intersecting graphs c and d
1. After how many days are the sunflowers in situations (c) and (d) the
same height?
2. Why do you think sunflower (c) overtakes sunflower (d) even
though sunflower (d) starts out with a greater ‘start height’ than
sunflower (c)?
Increasing functions and positive slopes
1. As the number of days increased what happened to the height of
the sunflower in each situation?
2. What sign has the slope of the graph in each situation?
Line with constant slope
Monica decided to plant a plastic sunflower whose height was 30cm.
Represent this information in a table, a graph and an algebraic formula
for day 0 to day 5.
1. What shape is the graph?
2. Contrast the situation with the previous sunflower situations. Why
is the graph this shape?
3. Where is the 30cm in the table? Where is the 30cm in the graph?
4. How much does the sunflower grow each day? Where is that in the
table? Where is it in the graph?
5. As the number of days increased what happened to the height of
the sunflower?
6. What is the slope of the graph?
7. What formula would describe the height of the sunflower?
8. Where is the 30cm in the formula?
9. What are the limitations to the model presented for sunflower
growth? It is important for students to realise early on that it is impossible
to mirror accurately many real life situations with a model and they
must always recognise the limitations of whatever model they use.
Line with a negative slope:
You have €40 in your money box on Sunday. You spend €5 on your
lunch each day for 5 consecutive days (take Monday as day 1).
As the variable on the x - axis increases what happens to the variable
on the y - axis?
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Patterns a Relations Approach to Algebra
Is sunflower growth a realistic situation for negative slope?
Can you think of other real life situations which would give rise to linear
graphs with negative slopes?
Summary of this Student Activity:
The height of each plant depends on how old it is. We have an input
(time in days) and an output (height of the plant) which depends on
the input. We say that the output (in this case height of the plant) is the
dependent variable and the input (time) is the independent variable.
When we use x and y, x is the independent variable and y is the
dependent variable.
We see that a, b and d have different starting heights and b and c have
the same starting heights. The starting height appears in the table as
the height on day 0 and in the graph as the y value when the x value
(number of days) is zero. In the formula it appears on its own as a
constant.
We see that there is a constant rate of change each day for the plant’s
height in each situation. In the table it appears as a constant change
for successive y outputs which characterises linear relationships. In the
graph it appears as the slope of the line – constant change in y for each
unit change in x. In the formula for a particular situation this constant
rate of change is multiplied by the independent variable, in this case
the number of days. When two plants have the same rate of change in
their growth per day their graphs are parallel, i.e. they have the same
slope.
When two plants have different growth rates per day the graphs will
intersect and the plant with the higher growth rate will eventually
overtake the plant with the lower growth rate even if the one with
lower growth rate starts at a greater height.
Student Activity – draw a graph from the
situation without a table of values JCOL
As in the previous activities, there are obvious links to co-ordinate
geometry and to functions/graphs.
It is important that students are confident at sketching/drawing linear
graphs without the need to formally set out a table, plot points and
then draw the graph.
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Patterns a Relations Approach to Algebra
Eventually, they should be able to use this knowledge/skill when
presented with a similar set of conditions.
(Discussion point: Is the linear relationship a continuous one or just
discrete values? Is joining the points always appropriate?)
Draw graphs of linear functions without making a table of
values, by comparing them to the graphs you have already
made.
1. On the same axes you used for Student Activity 1 (situations a
and b) sketch a graph of the following sunflower growth pattern:
starting height 5cm, grows 2cm each day.
Compare this graph to the graphs of situations a and b? Explain the
comparison.
2. On the same axes you used for Student Activity 1 (situations b
and c) sketch a graph of the following sunflower growth pattern:
starting height 6cm, grows 4cm each day.
Compare this graph to the graphs of situations b and c? Explain the
comparison.
3. On the same axes you used for Student Activity 1 (situations c
and d) sketch a graph of the following sunflower growth pattern:
starting height 6cm, grows 2cm each day
Compare this graph to the graphs of situations c and d? Explain the
comparison.
Situations a and b
Situations b and c
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a: y = 2x + 3
b: y = 2x + 6
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b: y = 2x + 6
c: y = 3x + 6
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Situations c and d
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c: y = 2x + 8
d: y = 3x + 6
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Chapter 3: Proportional
and non proportional
situations
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Patterns a Relations Approach to Algebra
Student Activity: Proportional and non
proportional situations JCOL
In the following examples students examine instances of
proportional and non proportional relationships for linear
functions. Students arrive at a knowledge of the characteristics
of a proportional relationship when it is graphed – it is linear
and passes through the origin, has no ‘start-up’ value, i.e. no
y-intercept and the ‘doubling strategy‘ works. In other words, if
x is doubled (or increased by any multiple) then y is doubled (or
increased by the same multiple).
Example 1: The following table shows a pattern of growth of a
fast growing plant.
Time in days
0
1
2
3
4
5
6
Height in cm
2
7
12
17
22
27
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1. What information does the table give about the pattern of growth?
2. Can you predict the height of the plant after 12 days? Can you do
this in different ways?
3. If the number of days is doubled will the height also double?
4. After 4 days is the height double what it was after 2 days? Explain
your answer.
After 6 days is the height double what it was after 3 days? Explain
your answer.
Did the ‘doubling strategy’ work?
5. What is the formula for the height of the plant in terms of the
number of days elapsed and the start up value (use words first,
then use symbols)?
6. Can you predict from the formula whether or not the ‘doubling
strategy’ will work?
When the ‘doubling strategy’ does not work we are dealing with a
non proportional situation. When the ‘doubling strategy’ works it is a
proportional situation. The variables are proportional to each other.
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Patterns a Relations Approach to Algebra
Example 2: This table represents a context where each floor in a
building has 5 rooms.
Number of floors
0
1
2
3
4
5
6
Total number of rooms
0
5
10
15
20
25
30
How many rooms in total would be in a building which has 12
floors?
1. Find 3 different ways to solve this problem.
2. If you double the number of floors what happens to
the total number of rooms in the building?
3. If you treble the number of floors what happens to the total
number of rooms in the building?
4. Does the ‘doubling strategy’ work?
5. Does the ‘trebling strategy’ work?
6. Are the variables proportional to each other?
7. What is the formula relating the number of rooms to the number
of floors (use words first, then use symbols for HL)?
8. Can you predict from the formula, whether or not the ‘doubling
strategy’ will work?
Plot graphs for the above 2 tables from example 1 and example 2
9. What is the same about the two graphs? What is different about
the two graphs?
10. For which graph will the ‘doubling strategy’ work? Explain.
11. Which graph represents a situation where the variables are
proportional to each other?
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Patterns a Relations Approach to Algebra
Example 3: In Home Economics students made
gingerbread men. They used 2 raisins for the eyes and 1
raisin for the nose on each one. How many raisins did they
use for 5 gingerbread men?
1. Represent this information in a table showing the number of raisins
used for 0 to 5 gingerbread men.
2. State the number of raisins used in words.
3. State the number of raisins used in symbols.
4. Plot a graph to model the situation.
5. Does the ‘doubling strategy’ work here? Explain from the graph,
table and symbols.
Example 4: Growing ‘worm’
The diagrams show a new born, 1 day old, and 2 day old ‘worm’ made
from triangles.
1. Draw up a table for the number of triangles used for a 0 to 5 day
old worm.
2. What is the relationship between the age of the worm and the
number of triangles used (use words first and then symbols)?
3. How many triangles in a 4 day old worm?
4. How many triangles in an 8 day old worm? Does the ‘doubling
strategy’ work?
5. Are the variables proportional to each other? Explain using the
table and formula.
6. Plot a graph to model the situation.
7. How does the graph show whether or not the variables are
proportional to each
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Patterns a Relations Approach to Algebra
Example 5: How many blue tiles are needed for n yellow tiles?
(JC HL and may be appropriate for OL also.)
This activity has great potential and can be used with students at any
stage. Students must be allowed to make the patterns themselves.
There is a great opportunity to reinforce the concept of equality when
students generalise the relationship differently. Students can make the
patterns themselves with blocks or draw them. By physically making the
pattern students will start to see the relationships.
1. Represent in a table the number of blue tiles needed for 0 to 5
yellow tiles.
2. How many blue tiles are needed for 4 yellow tiles? (accessible for
OL)
3. What is the relationship between the number of blue tiles and the
number of yellow tiles? Express this in words initially. (accessible for
OL)
4. Consider how other students express this relationship. Are all the
expressions equal? How can you verify this?
5. Does the doubling strategy work here?
6. If not why not?
7. Draw a graph showing the relationship between the number of
yellow and the number of blue tiles. Can you explain from the table
and the graph whether or not the ‘doubling strategy’ works?
8. Are the variables proportional to each other? Explain.
Class Discussion on proportional and non proportional
situations
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Patterns a Relations Approach to Algebra
1. When you look at these tables how can you tell if the variables are
in proportion?
2. Model the 4 situations represented in the tables by drawing graphs
for each table.
3. When you look at linear graphs where the variables are
proportional what do you notice about the graphs?
4. When you look at linear graphs where the variables are not
proportional what do you notice about the graphs?
5. Write a formula relating the variables for each of these tables.
6. In the formulae there is an amount you multiply by and an amount
you add. How are these quantities represented in the graphs?
7. Give a real life context for each of these tables.
Next activity on graph matching:
JC HL: Matching all the different representations simultaneously
JC OL: Matching representations in appropriate pairs
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Patterns a Relations Approach to Algebra
Match up the stories to the tables, graphs, and formulae and fill in
missing parts.
(This could form the basis for group or class discussion. Note these and other
examples are referred to on the next page. Label the axes of the graph.)
Story
Changing euro to
cent.
Table
No.
of
cars
€/w
1
450
2
500
3
550
4
600
5
650
Graph
y=100/x
m/
kg
y=100x
t/
mins
1
65
2
110
3
155
4
200
5
245
6
290
A cookbook
recommends
45 mins per
kg to cook a
turkey plus an
additional 20
minutes.
Henry has a
winning ticket
for a lottery for
a prize of €100.
The amount he
receives depends
on how many
others have
winning tickets.
Linear/
non linear,
proportional/non
proportional?
Justify
Y = 45x + 20
John works for
€12 per hour.
The graph and
table show how
much money
he earns for the
hours he has
worked.
Salesman gets
paid €400 per
week plus an
additional €50
for every car sold.
Formula
y=12x
h
A/€
1
12
2
24
3
36
4
48
5
60
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Linear and
proportional
because it’s a line
(constant change)
through (0,0)
25
Patterns a Relations Approach to Algebra
Story
Henry is one of
the winners of a
prize of €100.
The amount he
receives depends
on how many
others have
winning tickets.
A cookbook
recommends
45 minutes per
kg to cook a
turkey plus and
additional 20
minutes.
John works for
€12 per hour.
The graph and
table show how
much money
he earns for the
hours he has
worked.
Changing euro to
cent.
Table
Graph
n
€
1
100
2
50
3
33.33
4
25
5
20
6
16.67
m/k
g
t/k
mins
1
65
2
110
3
155
4
200
5
245
6
290
h
€
0
0
1
12
2
24
3
36
4
48
5
60
euro cent
0
0
1
100
2
200
3
300
4
400
5
500
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Formula
Linear/
non linear,
proportional/non
proportional?
Justify
y = 100/x
Can you
write this
formula
another way?
Not linear.
(This relationship
represents an
inverse proportion.
This is dealt with
later on in the
document.)
For each couple
(x,y) in this
situation what is
the value of the
product xy?
y = 20+45x
Linear but not
proportional
The graph is a line
but it does not
pass through (0, 0)
y = 12x
Relationship
is linear and
proportional as
its graph is a line
through (0,0).
y = 100x
Relationship
is linear and
proportional
because its graph
is a line through
(0,0). The table
shows constant
change for
successive outputs
where the input
x is a natural
number.
26
Patterns a Relations Approach to Algebra
Story
Fixed wage +
commission
Table
Cars
sold
€
0
400
1
450
2
500
3
550
4
600
5
650
© Project Maths Development Team 2011
Graph
Formula
y = 400+
50x
www.projectmaths.ie
Linear/
non linear,
proportional/non
proportional?
Justify
Linear but not
proportional.
The graph is a line
but it does not
pass through (0,0).
27
Patterns a Relations Approach to Algebra
Examples of proportional situations:
• Changing euro to cent – if you have no euro you have no cents, if
you double the number of euro you double the number of cent, if
you treble the number of euro you treble the number of cent
Number of cent = 100 x number of euro.
• Changing miles to kms.
• If you get paid by the hour the more hours you work the more
money you get in your wages.
• The circumference of a circle is proportional to its diameter (also to
its radius).
• If you drive (walk, run or cycle) at a constant speed the distance
travelled is proportional to the time taken.
Give 2 other examples of proportional situations.
Examples of non proportional:
• The recommended time it takes to cook a turkey is 45 minutes
per kilogram plus an additional 20 minutes: total cooking time in
minutes = 45 x number of kg + 20 minutes.
• You are a car salesman and you get paid a fixed wage of €400 per
week + €50 for every car sold.
• When you pay your ESB bill you pay a fixed standing charge plus a
fixed amount per unit of electricity used.
• The costs for a retailer selling shoes, for instance, are the costs of
shoes purchased plus fixed overhead costs (rent, heating, wages,
etc.)
Give 2 other examples of non proportional situations
What is the same and what is different about
the following relationships?
(i) To convert metres to inches the approximate conversion factor is 1
metre equals 39.37 inches.
(ii) To convert temperatures from degrees Celsius to degrees Fahrenheit
multiply the number of degrees Celsius by 1.8 (9/5) and add 32.
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Chapter 4: Non constant
rates of changequadratic functions
29
Patterns a Relations Approach to Algebra
Student Activity: Non constant rates of
change - quadratic functions JC OL
We saw that for linear relationships, which give straight line
graphs, there was a constant rate of change as seen by constant
change in the outputs for consecutive input values. Are all rates of
change constant? We will look for patterns in different situations
and investigate their rates of change. Students will investigate
contexts which give rise to quadratic functions through the use of
tables, graphs and formulae.
When dealing with graphs of quadratic functions students should
contrast the quadratic with the linear functions as follows:
• the graphs are non-linear
• the graphs are curved
• the rate of change is not constant
• the rate of change of the rate of change is constant i.e.
constant change of the changes
• the highest power of the independent variable is 2.
Example 1: Growing Squares Pattern. Draw the
next two patterns of growing squares.
We wish to investigate how the number of tiles in each pattern is
related to the side length of each square. Identify the independent and
dependent variables.
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Patterns a Relations Approach to Algebra
Complete the table
Side length of each square
Number of tiles to complete each square
1
1
2
4
3
9
4
5
6
7
8
9
10
1. How is the number of tiles used to make each square related to the
side length of the square? Write the answer in words and symbols.
What difference do you notice about this formula and the formulae
for linear relationships?
2. Looking at the table, is the relationship linear? Explain your answer.
3. Predict what a graph of this situation will look like. Will it be a
straight line? Explain your answer. Plot a graph to check your
prediction.
4. What shape is the graph? Is the rate of change of the number of
tiles constant as the side length of each square increases? Explain
using both the table and graph.
5. Why is the graph not a straight line? How can you recognise
whether or not a graph will be a straight line using a table?
Note: Students should come up with the last two columns in the
table below themselves when looking for a pattern in the outputs and
through questioning. It is not envisaged that they will be labelled as
below initially for them.
Side length
Number of tiles
1
2
3
4
5
6
7
8
9
1
4
9
Change
Change of changes
3
5
2
6. We saw that the changes in the table were not constant. Is there a pattern to them? Calculate the change of the changes.
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Patterns a Relations Approach to Algebra
7. When the change of the changes is constant we call this
relationship between variables a quadratic relationship.
8. Give 3 characteristics of a quadratic relationship from the ‘growing
squares’ example.
Example 2: Growing Rectangles JCOL
Complete the next two patterns in this sequence of rectangles.
Look at the pattern of growing rectangles. Make a table for the number
of tiles in each rectangle for rectangles of heights from 1 to 10. Make
observations about the values in the table.
What would a graph for this table look like? Will it be linear? How do
you know? Make a graph to check your prediction.
Complete the table
Height of rectangle
Width of rectangle
Number of tiles to make
each rectangle
1
2
2
2
3
3
4
5
6
7
h
1. Can you see 1 group of 2 (or 2 groups of 1) in the first diagram, 2
groups of 3 (or 3 groups of 2) in the second diagram, etc?
2. The last row of the table gives a formula for the number of tiles in a
rectangle of height h. Write down the formula in terms of the height of each rectangle.
Call this formula (i) Check that the formula works for values already calculated.
3. Will a graph for this table be linear? Explain using the table below.
Is the change of the changes constant?
What type of relationship is indicated by the pattern of the
changes? Explain
What do you notice about the change of the changes?
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Patterns a Relations Approach to Algebra
Height of rectangle
Number of tiles
1
2
3
4
5
6
7
2
6
Change
Change of changes
4
h
4. Predict what a graph of the situation will look like. Plot a graph to check your prediction.
5. What shape is the graph?
6. Does this relationship have the 3 characteristics identified in the last
example for a quadratic relationship? Explain.
Relate growing rectangles to growing squares JC OL (with
scaffolding)
Make the biggest square possible from each of the 5 rectangles you
have previously drawn. Shade the remaining area in each rectangle.
Draw the next two rectangles in the pattern below.
Complete the table for the
number of tiles in each
rectangle in terms of the
height of each rectangle.
Height
Number of tiles
Area of the rectangle = area of square
+ area of a rectangle
1
2
12+1
2
6
3
12
4
5
6
7
h
1. What is the formula for the number of tiles in a rectangle of height
h in the above sequence, using the pattern seen in the right hand
column with specific numbers? Call this formula (ii)
2. Show that the two formulae (i) and (ii) derived for the number
of tiles in the each rectangle are equivalent expressions using (a)
substitution and (b) the distributive law.
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Patterns a Relations Approach to Algebra
Example 3: Staircase Towers JC HL
Look at the staircases below. Make a table representing the relationship
between the total number of tiles and the number of towers. Make
observations about the values in the table. What would a graph look
like? Would it be linear? How do you know? Make a graph to check your
prediction.
Allow students to continue the pattern themselves and make their own
observations. The following questions could be used as prompt questions
if necessary.
1. How many tiles do you add on to make staircase 2 from staircase 1?
2. How many tiles do you add on to make staircase 3 from staircase 2?
3. How many tiles do you add on to make staircase 4 from staircase 3?
4. Complete the table below
Number of
towers
Number of tiles
1
1
2
3
3
6
4
5
6
7
8
9
10
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Patterns a Relations Approach to Algebra
5. Will a graph for this table be linear? Explain, using the table below.
Side length
Number of tiles
1
2
3
4
5
6
7
1
3
6
Change
Change of changes
2
3
1
6. Are the changes constant? Are the changes of the changes
constant?
7. Predict what a graph of the situation will look like. Plot a graph to check your prediction.
8. What shape is the graph?
We need to develop a formula for the number
of tiles required for the nth tower.
Compare the pattern for the ‘growing rectangles’ with that of the
staircase towers.
1. What is the formula for the number of tiles in the nth rectangle in
terms of n?
2. Hence what is the formula for the number of tiles in the nth
staircase in terms of n?
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Patterns a Relations Approach to Algebra
3. The staircase pattern showing the numbers 1, 3, 6, 10, ... could
also be shown using the above pattern. What do you think these
numbers are called? (Look at the shapes formed).
Contrast the rates of change for ‘growing
rectangles’ and ‘staircase towers’ – use tables
and graphs JC HL
Growing rectangles
Side length
Number of tiles
1
2
3
4
5
6
7
8
9
10
2
6
12
20
30
Change
Change of changes
4
2
2
6
8
10
Staircase Towers
Side length
Number of tiles
1
2
3
4
5
6
7
1
3
6
Change
Change of changes
2
3
1
Compare how the values of y increase for the same change in x in
‘growing rectangles’ and ‘staircase towers’. Focus on how the numbers
change in the first two columns
1. What is the same about the graphs and the tables for ‘growing
rectangles’ and ‘staircase towers’?
2. What is different about the graphs and the tables for ‘growing
rectangles’ and ‘staircase towers’?
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Patterns a Relations Approach to Algebra
3. How is the non-linear nature of the graph related to the tables and
the context?
4. What do the changes increase by for ‘growing rectangles’?
5. What do the changes increase by for ‘staircase towers’?
6. How is the difference in the answers to the last two questions
shown in the graphs for each situation?
Which situation shows the greatest change of the changes?
Example 4: The growth of a fantasy creature called Walkasaurus JC HL
The following table shows how Walkasaurus’s height changes with
time.
Age (years)
Height (cm)
0
1
1
2
2
4
3
7
4
11
5
16
6
22
1. Will a graph for this table be linear? Explain using a table. Note: The last two columns of the table need not be given to
students. Students should draw on the skills they have learned in
previous activities and should not need these prompts. The aim of
this approach is to ask questions that give students the opportunity
to think and investigate in order to make sense of the features of a
quadratic relationship.
Age (years)
Height (cm)
0
1
2
3
4
5
6
1
2
4
7
11
16
22
Change
Change of changes
1
2
1
2. Predict what a graph of the situation will look like. Plot a graph to check your prediction.
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Patterns a Relations Approach to Algebra
3. What shape is the graph? Compare the pattern of y values for Walkasaurus’s height with the
pattern of y values for the staircase towers.
4. How is the pattern similar?
5. How is the pattern different?
6. By comparing with the staircase towers pattern write a formula for
Walkasaurus’s height h in terms of his age d in years.
7. Using the same axes and scales plot the graphs for the staircase
towers and Walkasaurus’s height?
What is the same and what is different about the two graphs?
Example 5: Gravity and Quadratics JCHL
This question is set in context so that students will see the application
of quadratics in everyday life.
a. If a cent is dropped from a height of 45m, its height changes over
time according to the formula h = 45 – 4.9 t2, where t is measured
in seconds. Use mathematical tools (numerical analysis, tables,
graphs) to determine how long it will take for the cent to hit the
ground. What does the graph of height vs. time look like? What connections do you see between the graph and the table?
b. Suppose you wanted the cent to land after exactly 4 seconds. From
what height would you need to drop it? Explain your answer.
c. Suppose you dropped the cent from the top of the Eiffel Tower
(300m high). How long would it take to hit the ground? What does the graph of height vs. time look like?
What connections do you see between the graph and the table?
JCHL Linear and quadratic:
i
Investigate the relationship between length and width for a
rectangle of fixed perimeter
ii
Investigate how change in the dimensions of a rectangle of fixed
perimeter affects its area.
Given a rectangle with a fixed perimeter of 24 metres, if we increase
the length by 1m the width and area will vary accordingly. Use a table
of values to look at how the width and area vary as the length varies.
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Patterns a Relations Approach to Algebra
Length/m
Width /m
Area/m2
0
12
0
1
11
11
2
10
3
4
5
6
7
8
9
10
11
12
1. Predict the type of graph you expect to get if you plotted width
against length? Explain your answer using the values of the change between
successive outputs.
2. What is different about these changes and changes in previous
examples, e.g. sunflower growth? Plot the graph of width against length to confirm your predictions.
3. What can you say about the slope of this graph?
4. Explain what the slope means in the context of this situation.
5. Express the width in terms of the length and perimeter of the
rectangle.
6. What are the variables and the constants in the situation?
7. Does it matter which you decide to call the independent variable?
8. What do you notice about the values of area in the table – are they
increasing, decreasing or is there any pattern?
9. Does the area appear to have a maximum or minimum value? If it does what value is it?
10. Predict the type of graph you expect to get if you plotted area
against length. Explain your answer.
11. Plot the graph of area against length to confirm your predictions.
12. What shape is this graph?
13. Use the graph of area plotted against length, for a fixed perimeter
of 24, to find which length of rectangle gives the maximum area.
Does this agree with your conclusion from Q 9?
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Patterns a Relations Approach to Algebra
14. Express the area in terms of the length and perimeter of the
rectangle. Check that the formula works by substituting in values of
length from the table. Extension: What if the rectangle had a fixed area – what is
the relationship between the length and width? (See inverse
proportion).
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Chapter 5: Exponential
growth
41
Patterns a Relations Approach to Algebra
Student Activity: Exponential growth
JCOL LCOL, LCHL, – Exponential relations limited to doubling and
tripling
Students investigate situations involving exponential functions
using words, tables, graphs and formulae and understand that
exponential functions are expressed in constant ratios between
successive outputs.
Students should see applications of exponential functions in their
everyday lives and appreciate the rapid rate of growth and decay
shown in exponential functions.
Example 1: Pocket money story
Contrast adding 2 cent each day (multiplication as repeated
addition) with multiplying by 2 each day (exponentiation as repeated
multiplication).
I put this proposition for pocket money to my Dad at the beginning of
July.
“I just want you to give me pocket money for this month. All I want is
for you to give me 2c on the first day of the month, double that for the
second day, and double that again for the 3rd day ... and so on. On the
1st day I will get 2c, on the 2nd day 4c, on the 3rd day 8c and so on
until the end of the month. That is all I want.”
Was this a good deal for Dad or is it a good deal for me?
Make a table to show how much money I will get for the first 10 days
of the month.
Time/days
Money/c
0
2
1
4
2
3
4
5
6
7
8
9
10
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Patterns a Relations Approach to Algebra
1. Using the table would you expect a graph of this relationship to be
linear? Explain.
2. Using the table would you expect a graph of this relationship to be
quadratic? Explain.
3. Check changes, change of the changes, change of the change of
the changes etc.
4. What do you notice? Is there a pattern to the differences? If so
what is it?
5. Predict the type of graph you will get if you plot money in cent
against time in days.
6. Make a graph to check your prediction. What do you notice?
7. Can you come up with a formula for the amount of money you will
have after n days?
8. Using your calculator find out how much money you would get on
the 10th day, the 20th day, the 30th day, 31st day?
9. What are the variables in the situation? What is constant in this
situation?
10. Where is the factor of 2 (the doubling) in the table? Where is it in
the graph?
11. Contrast this situation with adding 2c every day. How much would
you have on day 31?
12. How would the formula change if your Dad trebled the amount of
money, starting by giving you 2c on the first day as above? Make a
table and come up with the formula for this new situation.
13. When people speak of ‘exponential growth’ in everyday terms,
what key idea are they trying to communicate?
y=abx
Final value = starting value multiplied by (growth factor) no. of intervals of time elapsed
(with the independent variable in the exponent).
This type of growth is called exponential growth. The variable is in the
exponent. There is a constant ratio between successive outputs. This
constant is called the growth factor. During each unit time interval
the amount present is multiplied by this factor. Can you think of other
examples of things which grow in this way e.g. double over a constant
period of time? (The growth factor does not have to be a ‘doubling’.)
Computing: Byte = 23 bits, kilobyte = 210 bytes, Megabyte = 220 bytes
Population growth, paper folding, cell division and growth, compound
interest – (repeated multiplication)
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Patterns a Relations Approach to Algebra
Exponential growth in legend–the origins of the
game of chess and the payment for its inventor
Legend tells us that the game of chess was
invented hundreds of years ago by the Grand
Vizier Sissa Ben Dahir for King Shirham of India.
The King loved the game so much that he offered
his Grand Vizier any reward he wanted. “Majesty,
give me one grain of wheat to go on the first
square of the chess board, two grains to place
on the second square, four grains to place on the third square, eight
on the fourth square and so on until all the squares on the board are
covered”. The King was astonished. “If that is all you wish for you
poor fool then you may have your wish”. When the 41st square was
reached, over one million million grains of rice were needed. Sissa’s
request required more than the kingdom’s entire wheat supply. In all,
the king owed Sissa more than 18,000,000,000,000,000,000 grains of
wheat which was worth more than his entire kingdom. Foolish king!
Square
Rice on each square
Pattern for rice on
the square
Rice on the board
1
1
1
1
2
2
1x2
1+2
3
4
1x2x2
1+2+4
4
8
5
Exponential growth and computing power
See the following link for more on this legend and Moore’s Law dealing
with the exponential growth of transistors in integrated circuits leading
to smaller and faster computers.
http://www.authorstream.com/Presentation/Arkwright26-16602-wheat1-Exponential-Growth-Moores-Law-seedcount-presentation-002Entertainment-ppt-powerpoint/
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Patterns a Relations Approach to Algebra
Example 2: How many ancestors do you have? JC OL
1. You have 2 parents, 4 grandparents, and eight great grandparents.
Going back further into your family tree, how many ancestors do
you have 5 generations ago, n generations ago? Make a table to
show this. (30 years on average per generation?)
2. What example is this the same as numerically? How does it differ
from it? (Think in terms of time intervals.)
Example 3: Paper folding JC OL
If I fold a sheet of paper in half once I have 2 sections. If I fold it in half
again I have 4 sections. What happens if I continue to fold the paper?
Make a table, plot a graph and come up with a formula to describe the
situation. Is it linear, quadratic, exponential or none of these?
Example 4: Growth of bacteria JC HL
1. If we start with 1 bacterium which by growth and cell division
becomes 2 bacteria after one hour – in other words the number of
bacteria doubles every hour – how many of these bacteria would
there be at the end of 1 day (24 hours)?
2. What type of growth is this – linear, quadratic, exponential, or none
of these?
3. Would this growth increase indefinitely? Explain. (Exponential
growth would not continue indefinitely – the bacteria would run
out of for example nutrients or space.)
4. Certain bacteria double in number every 20 minutes. Starting
with a single organism with a mass of 10-12g, and assuming
temperature and food conditions allowed it to grow exponentially
for 1 day, what would be the total mass produced?
5. What would be its total mass after 2 days? (The mass of the earth
is approximately 6 x 1024kg.)
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Patterns a Relations Approach to Algebra
Example 5: Compound interest and exponential growth LC OL
€650 is deposited in a fixed interest rate bank account. The amount
(final value) in the account at the end of each year is shown in the
following table.
End of year
Final value
1
2
3
4
5
€676.00 €703.04 €731.16 €760.41 €790.82
1. How can you tell from the table if the above relationship is linear,
quadratic or exponential? Explain.
2. If you plot a graph of final value against time what does the graph
look like for this limited range of times? How might plotting more
points help?
3. What would be the effect of increasing the interest rate? Make
a table showing the final values for the first five years using an
interest rate of 10% per annum compound interest. Plot a graph
for this data. Compare the graph to the graph produced for the
lower interest rate.
4. What formula expresses the final value after t years in the above
compound interest formula if the initial value is €650?
5. What is the danger of drawing a conclusion about a relationship
from a plot using a limited number of points?
6. From just the 2 data points below what are two possible formulae
which could relate the x and y values?
x
y
1
2
2
4
Population growth
The tendency of populations to grow at an exponential rate was
pointed out in 1798 by an English economist, Thomas Malthus, in
his book “An Essay on the Principle of Population”. He suggested
that unchecked exponential growth (of course people die and
war and famine still occur) would outstrip the supply of food and
other resources and would lead to disease and wars. He wrote that
“Population, when unchecked, increases in a geometrical ratio.
Subsistence increases only in arithmetic ratio”.
Of course population growth for any particular country depends on the
following factors: birth rate, death rate, immigration, emigration, wars,
famine, diseases and other possible factors.
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Patterns a Relations Approach to Algebra
Example 6: Growth of world’s population LC OL
1. The world’s population reached 6 billion in 1999. On 07/01/2009
the world’s population was 6,755,987,239. http://www.census.gov/
ipc/www/popclockworld.html
On 07/01/10 the world’s population was 6,830,586,985. (Write
each of these figures to 3 significant figures if you wish and use
scientific notation.) Approximately how many people more were in
the world one year after 07/01/09? What factor did the world’s population increase by in the year? This
is the growth factor (rounded to 2 decimal places).
2. Assuming this growth rate continues what do you multiply the
population in 2010 by to get the population in 2011?
3. Make a table of expected world population values from 2010 to
2015. Note how much the population increases by each year.
4. What type of growth is this?
5. Write a formula for the expected value of the world’s population n
years from 2010?
6. Using trial and error and the expected growth rate of 1% per
year, how many years from 2010 would you expect the world’s
population to have doubled?
7. Using trial and error and the expected growth rate of 2% per
year, how many years from 2010 would you expect the world’s
population to have doubled?
8. ( LC HL) Using logs work out the exact number of years for the
world’s population to double assuming a constant growth rate of
1% per year.
http://www.learner.org/interactives/dailymath/population.html
The rate of Earth’s population growth is slowing down. Throughout
the 1960s, the world’s population was growing at a rate of about
2% per year. By 1990, that rate was down to 1.5%, and by the year
2015, it’s expected to drop to 1%. Family planning initiatives, an
aging population, and the effects of diseases such as AIDS are some
of the factors behind this rate decrease.
Even at these very low rates of population growth, the numbers
are staggering. By 2015, despite a low expected 1% growth rate,
experts estimate there will be 7 billion people on the planet. By
2050, there may be as many as 10 billion people living on Earth. Can the planet support this population? When will we reach the limit of our resources?
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Patterns a Relations Approach to Algebra
Example 7: Population of Sylvania LC HL
The table shows the population in thousands of a small mythical
country for various years
Year
Population in
thousands
1825
200
1850
252
1875
318
1900
401
1925
504
1950
635
1975
800
1. Using the table above, check to see if the relationship between
time and population is linear, quadratic or exponential or none of
these? Explain.
2. Predict what a graph of the data will look like.
3. Plot a graph of this data. What does the graph look like?
4. About how long does it take for the population to double? Explain how you got the answer from the table. Explain how you got the answer from the graph.
5. What would you expect the population to be in 2000, 2050, and
3000? Explain how you got these answers from (i) the table (ii) the
graph
6. Here are 3 different formulae that can be used to calculate
population in this situation.
Explain what t, r, and n represent.
7. Explain how
fits the table and the situation and how it
can yield the correct answer.
fits the table and the situation and
8. Explain how
how it can yield the correct answer.
9. Explain how
fits the table and the situation and
how it can yield the correct answer.
10. Do you expect actual data values to match data values obtained
using the formula? Explain.
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Patterns a Relations Approach to Algebra
Example 8 LC HL Is the population growth shown in the graph below
exponential?
3000
World population in millions
2500
2000
1500
1000
500
Year
1650
1700
1750
1800
1850
1900
1950
1. Make a table and investigate if the growth is exponential – does
it always increase by the same factor for successive equal time
intervals?
Calculate the ratio between output values for successive equal time
intervals.
2. Calculate what constant growth factor over the first 100 years
(1650 to 1750) would have given the same result in that period. Use this growth factor for the next two intervals of 100 years.
Plot the points for 1850 and 1950.
What can you conclude about this growth pattern?
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Patterns a Relations Approach to Algebra
Exponential Decay: Compounded depreciation (an investment
decreases to 0.80 of its value at the end of every year) and the decay of
nuclear waste are examples of exponential decay. They involve repeated
multiplication but in this case the growth factor is less than 1.
Example 1 LC OL Alan is 4 metres away from a wall. He jumps towards
the wall and with each jump he halves the distance between himself
and the wall.
Draw up a table of values to show Alan’s distance from the wall with
each step (jump).
Step
Distance from
the wall/m
0
4
1
2
2
3
4
5
6
1. Looking at successive output values in the table, explain whether or
not the relationship is linear, quadratic, or exponential.
2. Plot a graph of Alan’s distance from the wall against the step
number towards the wall.
3. How is this graph different than the graph obtained for compound
interest?
4. How is this graph like the graph obtained for compound interest?
5. Will Alan’s distance from the wall ever be zero?
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Patterns a Relations Approach to Algebra
Example 2: LC OL During a visit to the hospital, Jane receives a dose of
radioactive medication which decays or loses its effectiveness at a rate
of 20% per hour.
1. If she receives 150mg of the medication initially, approximately how
many milligrams of the medication remain in her body after the first
6 hours?
2. Draw up a table showing the amount of radioactive medication in
her body for 1 up to 6 hours, increasing by 1 hour each time.
3. Plot a graph of mg of medication left in her body against hours
elapsed.
4. Is it possible to reduce the amount of radioactive medication in
Jane’s body to 0? Explain.
5. What are the implications of this for the decay of radioactive waste
material? (Some radioactive materials take more than 20,000 years
for half of it to decay. See mathematical tables for half lives of
radioactive elements.)
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Patterns a Relations Approach to Algebra
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Chapter 6: Inverse Proportion
53
Patterns a Relations Approach to Algebra
Student Activity: Inverse Proportion
For an input x and an output y, y is inversely proportional
to x, if there exists some constant k so that xy = k.
The constant k is called the constant of proportionality.
Example 1: Sharing a fixed amount of prize money
Imagine you won the lotto and the prize was €1,000,000. You don’t
know how many other people also had the winning numbers so you
are speculating on how much you will collect. If you are the only winner
then you collect €1,000,000. However, if there are 2 winners you
collect €500,000.
1. Make up a table showing number of winners and the
corresponding amount won per person.
2. Is it a linear relationship? Explain
3. Is it a quadratic relationship? Explain.
4. Plot a graph of prize collected per winner against number of
winners.
5. Describe the graph.
6. Is this like exponential growth/decay, i.e. is there a constant ratio
between successive outputs for equal changes of the independent
variable?
7. What formula relates winnings per person, number of winners and
total value of the prize?
8. There are variables and a constant in the equation. Identify the
variables and the constant.
9. Which variable is the independent variable?
10. Which variable is the dependent variable? (The resulting graph is called a hyperbola. In real life contexts
we are only considering positive values of the independent
variable.)
11. As the number of winners doubles what happens to the amount of
money each winner gets?
12. As the number of winners trebles what happens to the amount of
money each winner gets?
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Patterns a Relations Approach to Algebra
13. As the number of winners quadruples what happens to the amount
of money each winner gets?
14. When the number of winners decreases what happens to the
amount of money each person wins?
15. When the number of winners becomes very big what happens to
the amount of money each person receives
16. Will the graph ever touch the x axis? Explain.
17. Will the graph ever touch the y axis? Explain.
This type of relationship is called an inverse proportion. The product
of the variables is a constant.
• The more people who share a pizza the smaller the slice of pizza
each one receives. The size of the slice is inversely proportional to
the number of people sharing.
• Length varies inversely with width for a rectangle of given area. A = l x w.
• Depth of a fixed volume of liquid in different cylinders varies
inversely as the area of the base of the cylinder. V = Area of base x h
Example 2: Relationship between time spent travelling a fixed distance
and speed for the journey
1. Consider a situation where people travel between two points a
fixed distance apart. Assume that different people travel with
different constant speeds between the two points. What happens to the time taken to complete the journey between
the two points as the speed increases?
2. Is there a speed which will give a time of 0? Explain.
3. Is there a value of time corresponding to a speed of 0? Explain.
4. What do these two answers tell you about the graph?
5. Make up a table showing speed and the corresponding time taken.
(You can choose the distance between the two points and the
speed and the units used.)
6. Is it a linear relationship between speed and time? Explain.
7. Is it a quadratic relationship? Explain.
8. Plot a graph of time taken against speed.
9. Describe the graph.
10. Is this like exponential growth/decay – is there a constant ratio
between successive outputs for equal time intervals?
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Patterns a Relations Approach to Algebra
11. What formula relates the speed, time taken and the distance
between the two points?
12. There are variables and a constant in the equation. Identify the
variables and the constant
13. Which variable is the independent variable?
14. Which variable is the dependent variable? The resulting graph is called a hyperbola.
15. As the speed doubles what happens to the time taken for the
journey?
16. As the speed trebles what happens to the time taken for the
journey?
17. As the speed quadruples what happens to the time taken for the
journey?
18. When the speed has a low value what happens to the time taken
for the journey?
19. When the speed has a high value what happens to the time taken
for the journey?
20. Will the graph ever touch the x axis? Explain.
21. Will the graph ever touch the y axis? Explain.
22. Is this an example of an inverse proportion? Explain.
23. Give two examples of inverse proportions.
http://www.articlesforeducators.com/dir/mathematics/cat_and_mouse.
asp
If it takes 5 cats 5 days to catch 5 mice, how long will it take 3 cats to
catch 3 mice?
If a boy and a half can mow a lawn in a day and a half, how long will it
take 5 boys to mow 20 lawns?
Hint: Only vary 2 quantities at any one time – keep the third one
constant.
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Chapter 7: Moving from
linear to quadratic to cubic
functions
57
Patterns a Relations Approach to Algebra
Student Activity:Moving from linear to
quadratic to cubic functions. LC OL
The linear model, occurs when there is a constant rate of
change. When there is a consistent force for change, the
quadratic model often fits well. For example, gravity is a
constant force. Many movements under its influence are
essentially quadratic. Cubic models appear in locations such
as highway designs which require a smooth transition from
a straight line into a curve.
Example 1: Using a cube students investigate linear,
quadratic and cubic relationships
For a block with edge lengths of 1 unit, the perimeter of
the base is 4 units, the surface area is 6 square units and
the volume is 1 cubic unit. What would the values be for a block with
edge lengths of 2 units or 3 units or 34 units or n units?
1. Make tables for perimeter, for surface area and for volume as the
edge lengths of the block increase. Examine the tables to predict
the shape of the graph for each of the three relationships. Explain
your predictions. Make the graphs for perimeter vs. edge length,
surface area vs. edge length and volume vs. edge length and
compare them with your predictions.
Students complete a table for the perimeter of the base for cubes with
different edge lengths
Edge length /cm
Perimeter of
the base of the
cube /cm
1
2
3
4
5
6
7
1. Predict the shape of graph for the above relationship.
2. Explain your prediction.
3. Write a formula for the perimeter of the base in terms of the edge
length.
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Patterns a Relations Approach to Algebra
4. Plot a graph to show the above relationship.
5. Check if values for perimeter predicted by the formula agree with
values predicted by the graph.
Complete the table below for total surface area of the cube.
Edge length /cm
Surface area
of the cube /
cm2
1
2
3
4
5
6
7
1. Predict the shape of graph for the above relationship.
2. Explain your prediction.
3. Write a formula for the total surface area in terms of the edge
length.
4. Plot a graph to show the above relationship.
Complete the table below for volume of the cube
Edge length/cm
Volume/cm3
1
2
3
4
5
6
7
1
8
27
Change
7
19
Change of the
changes
12
Change in the
changes of
the changes
1
6. Predict the shape of graph for the above relationship.
7. Explain your prediction.
8. Write a formula for the perimeter of the base in terms of the edge
length.
9. Plot a graph to show the above relationship.
10. Check if values for perimeter predicted by the formula agree with
values predicted by the graph.
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Patterns a Relations Approach to Algebra
Relationship between surface area and volume for a cube.
Do you think the volume and surface area of a cube will ever be equal
numerically?
Do you think the volume will be ever be numerically greater than the
surface area?
Check by completing the following table.
Edge
1
2
3
4
5
6
7
8
9
n
Area of base
Total surface
area
Volume
Surface Area
Volume
At what point is the ratio of surface area to volume equal to 1?
When is volume less than surface area?
When is volume greater than surface area?
How can you explain the rapid growth of volume and the slower
growth of surface area?
Example 2 (LC HL) Designing the largest box – cuboid shaped – from a
rectangular sheet of fixed dimensions
You are making boxes from
cardboard for personalised
Christmas presents. You wish
to put as many sweets as
9cm
possible into the boxes using
rectangular based boxes but
you are limited by the size
of the sheets of cardboard
available. (Alternative: design
a suitcase to hold maximum volume)
12cm
Start with a rectangular sheet of cardboard 9cm x 12cm. From each
corner cut squares of equal size. Fold up the four flaps and tape them
together forming an open box. Depending on the size of the squares
cut from the rectangular sheet, the volume of the box will vary.
Investigate how the volume of the box will vary.
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Patterns a Relations Approach to Algebra
(The student will engage in problem solving using their experiences
from previous activities. One of the aims of this approach is to empower
students to use the multiple representations to help solve problems
and some representations may be more useful than others. Below are
suggested prompt questions where students are having difficulty.)
1. When the box is formed which dimension of the box will be
represented by x?
2. If we cut out squares of side x from the cardboard, predict what
value of x will give you the largest volume box.
3. Write a formula for the volume V of the box in terms of the length
of the box l, its width w, and height x.
4. Use a table to test out values of x to see which value will give the
box of largest volume. Check them using the formula.
x/cm
l/cm
w/cm
V/cm
0.0
0.5
1.0
1.5
5. Between what two x values do you expect the largest volume to
be?
6. Do you expect a graph of V against x to be linear, quadratic or
cubic? Explain your answer. Plot the graph of V against x.
7. From the graph approximately what value of x will yield the
maximum volume?
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Patterns a Relations Approach to Algebra
Example 3 The painted cube (LC HL)
You have a cube like the one shown on the right, but it
has been thrown into a bucket of paint. The outside is covered with
paint but the inside is not. You break it apart into unit cubes to see how
many of the unit cubes have 0 faces painted, 1 face painted, 2 faces
painted, 3 faces painted , 4 faces, 5 faces, or 6 faces painted.
You are looking for a pattern between the number of painted faces
and the original size of the cube. Explore this for a 2x2x2 cube up to
a 6x6x6 cube. For each cube, count the number of blocks in each of
the categories 0 faces painted, 1 face painted, 2 faces painted, 3 faces
painted, 4 faces, 5 faces, or 6 faces painted.(Students can come up
with the table themselves as they should be used to doing this as a
problem solving strategy.)
Dimensions
2x2x2
0 faces
painted
1 face
painted
2 faces
painted
3 faces
painted
4 faces
painted
5 faces
painted
6 faces
painted
Total
number
of unit
blocks
0
0
0
8
0
0
0
8
3x3x3
4x4x4
24
0
64
5x5x5
6x6x6
96
nxnxn
n3
(n>=2)
Verify the equivalent expression for n3 by adding up all the expressions
in columns 2 – 8 of the last row.
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Patterns a Relations Approach to Algebra
Example 4 (extension for HL)
Find a relationship between total surface area (TSA) of a cube and the
perimeter (P) of one side.
Edge length/cm
P of one side
TSA
1
4 (1) = 4
6 (1)2 = 6
2
4 (2) = 8
6 (2)2 = 24
3
4
5
6
n
1. Predict what you expect a graph of total surface area plotted
against perimeter of a cube to look like. Explain your prediction.
2. Using the formulae for perimeter and total surface area, derive a
formula for total surface area in terms of perimeter.
3. Using the formula derived check that it correctly predicts values of
total surface area given the perimeter values in the table.
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Patterns a Relations Approach to Algebra
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64
Chapter 8: Appendix
containing some notes and
suggested solutions
65
Patterns a Relations Approach to Algebra
Graphs for Student Activity Page 18
Sketched line in black
Situations a and b
25
20
15
10
a: y = 2x + 3
b: y = 2x + 6
c: y = 2x + 5
5
0
1
2
3
4
5
6
7
8
9
Situations b and c
25
20
15
10
a: y = 4x + 6
b: y = 2x + 6
c: y = 3x + 6
5
0
1
2
3
4
5
6
7
8
9
Situations c and d
25
20
15
10
d: y = 2x + 8
c: y = 3x + 6
b: y = 2x + 6
5
0
© Project Maths Development Team 2011
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2
3
4
5
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6
7
8
9
66
Patterns a Relations Approach to Algebra
Graphs for growing rectangles, staircase towers, and Walkasaurus
Graphs for Student Activity Page 16
11
10
Growing Rectangles
9
8
7
6
5
4
3
Staircase Towers
2
1
0
Graphs for Student Activity
Page 31
18
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Graphs for Student Activity Page 31
Walkasarus
16
14
12
10
8
6
Staircase Towers
4
2
0
2
4
6
8
10
Graphs for Student Activity Page 34
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Patterns a Relations Approach to Algebra
Area in m2
Graphs for Student Activity Page 38
Graphs for Student Activity Page 38
Graphs for Student Activity Page 47
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Patterns a Relations Approach to Algebra
Graphs for Student Activity Page 46
Spreadsheet for Example 6 Compound Interest
1
676
2
703.04
3
731.16
4
760.41
5
790.82
6
822.4528
7
855.3509
8
889.5649
9
925.1475
10
962.1534
11
1000.64
12
1040.665
13
1082.292
14
1125.583
15
1170.607
16
1217.431
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Patterns a Relations Approach to Algebra
Graphs for Student Activity Page 48
Graphs for Student Activity Page 51
Graphs for Student Activity Page 54
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Patterns a Relations Approach to Algebra
Graphs for Student Activity Page 59
Sketched line in black
Showing growth of surface area and growth of
volume for a cube of side x
9
7
6
4
3
1
0
1
2
3
4
5
6
7
Implications for faster growth rate of volume compared to surface area
Surface area is important as it effects how fast an object cools off
(greater area equals quicker cooling. Machines that need to get rid of
extra heat have little metal fins stuck to them to increase their surface
area).
Babies have a large surface area compared to their volume and lose
heat quickly – hence they must be wrapped up very well to prevent
heat loss. As they grow the ratio of surface area to volume decreases
(volume grows faster than surface area) and so they do not lose heat as
quickly.
Surface area affects how quickly a droplet evaporates – greater area
gives faster evaporation.
Surface area affects how quickly a chemical reaction proceeds – greater
area gives faster reaction.
Insects breathe through their surface area (x2) but their need for oxygen
is in proportion to their volume (x3) – hence they can’t become the size
of humans as their surface area would not supply enough oxygen for
such a large volume.
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Patterns a Relations Approach to Algebra
Why can’t giants exist?
Consider an adult woman 160cm tall and a giant woman 10 times as
tall. How tall is the giant? 16m
Assume the giant is similar to the woman in composition of bones,
flesh etc. If the woman weighs 50kg, what does the giant weigh?
(Weight, like volume, increases as height3)
Giant weighs 50 x 10 x 10 x 10 = 50,000kg
The woman’s weight is supported by her two leg bones. If the cross
sectional area of each of her leg bones is 10cm2, how much weight is
supported by each cm2 of leg bone?
50/20 kg/cm2 = 2.5 kg/cm2
What is the cross sectional area of each of the giant’s bones?
10 x 102 = 1,000cm2
How much weight is supported by each cm2 of the giant woman’s leg
bones?
50,000/2000 = 25 kg/cm2
If the giant’s bones are the same density as those of the woman of
average height, what will happen to the giant’s leg bones when she
stands up?
They will break because they have 10 times the weight to carry for the
same surface area.
How could this problem be sorted? Denser bones for the giant, or walk
on all fours to distribute the weight.
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Patterns a Relations Approach to Algebra
Graphs for Student Activity Page 60
Making a box with the largest volume, given a
rectangle of fixed dimensions
x
width
length
Volume
0
9
12
0
0.5
8
11
44
1
7
10
70
1.5
6
9
81
2
5
8
80
2.5
4
7
70
3
3
6
54
3.5
2
5
35
4
1
4
16
4.5
0
3
0
5
-1
2
-10
5.5
-2
1
-11
6
-3
0
0
6.5
-4
-1
26
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Patterns a Relations Approach to Algebra
The Painted Cube, page 62
Dimensions
0 faces
painted
1 face
painted
2 faces
painted
3 faces
painted
4 faces
painted
5 faces
painted
6 faces
painted
Total
number
of unit
blocks
2x2x2
0
0
0
8
0
0
0
8
3x3x3
1
6
12
8
0
0
0
27
4x4x4
8
24
24
8
0
0
0
64
5x5x5
27
54
36
8
0
0
0
125
6x6x6
64
96
48
8
0
0
0
216
(n-2) 3
6(n-2)2
12(n-2)
8
0
0
0
n3
nxnxn (n>=2)
(n-2)3 =
n3-6n2 +12n-8
6 (n-2)2
=
6n2-24n+24
12 (n-2)
=
12n-24
+8
=
8
}
Total n3
Explanation:
For the 2 x 2 x 2, no cubes are painted on all faces
The cubes with 3 painted faces are always on the corners and there are
8 of those.
The cubes with 2 painted faces occur on the edges between the
corners. There are 12 of those if it is a 3 x 3 cube and 12 (n-2) if an n
by n cube (take away the 2 on the corners).
The cubes with 1 painted face occur as squares on each of the 6 faces
and there will be 6 (n-2)2 of those.
Those with no painted faces are on the inside and are given by (n-2)3
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Patterns a Relations Approach to Algebra
Suggested Solution, Student Activity page 68
Relationship between TSA and Perimeter of a
face for a cube
P of one side
TSA
Couples
1cm
4(1)
6(1)2
(4, 6)
2cm
4(2)
6(2)2
(8, 24)
3cm
4(3)
6(3)2
(12, 54)
4cm
4(4)
6(4)2
(16, 96)
ncm
4(n)
6(n)2
Relationship between TSA and perimeter of a cube.
Graph going through above set of couples
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Patterns a Relations Approach to Algebra
The equation of this graph is y = 0.375x2.
I then asked the question “Is it possible for the students to derive the
equation of the quadratic from the table before the graph is drawn
with the couples?” The following approach works.
1. We are trying to find an equation which links Perimeter (P) and
Total Surface Area (TSA).
2. From the table (the Tn)
P = 4n and TSA = 6n2
This is the equation of the quadratic graph y = 0.375x2
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