Introduction to patterns

Introduction to patterns
Teaching & Learning Plans
Introduction to Patterns
Junior Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on teacher input
and gives details of the key student tasks and teacher questions which move the
lesson forward.
ii.
Student Activities – Possible and Expected Responses: Gives details of
possible student reactions and responses and possible misconceptions students
may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan:
Introduction to Patterns
Aims
• Recognise a repeating pattern
• Represent patterns with tables, diagrams and graphs
• Generate arithmetic expressions from repeating patterns
Prior Knowledge
Students should have some prior knowledge of drawing basic linear graphs on the
Cartesian plane.
Learning Outcomes
As a result of studying this topic, students will be able to:
• use tables, graphs, diagrams and manipulatives to represent a repeatingpattern situation
• generalise and explain patterns and relationships in words and numbers
• write arithmetic expressions for particular terms in a sequence
• use tables, diagrams and graphs as tools for representing and analysing
patterns and relations
• develop and use their own generalising strategies and ideas and consider
those of others
• present and interpret solutions, explaining and justifying methods, inferences
and reasoning
Catering for Learner Diversity
In class, the needs of all students whatever their level of ability are equally important. In
daily classroom teaching, teachers can cater for different abilities by providing students
with different activities and assignments graded according to levels of difficulty so that
students can work on exercises that match their progress in learning. For less able students,
activities may only engage them in a relatively straightforward way and more able students
can engage in more open–ended and challenging activities. This will cultivate and sustain
their interest in learning. In this T & L Plan for example teachers can provide students with
the same activities but with variations on the theme e.g. draw a picture, put it in words,
write a multiplication sentence, apply the algorithm. Teachers can give students various
amounts and different styles of support during the class for example, providing more clues.
In interacting with the whole class, teachers can make adjustments to suit the needs of
students. For example, they can ask less able students simple questions (e.g. Student
Activity 1A and 1B) and reserve more challenging questions for the more able students
(e.g. Student Activity 4A and 4B).
© Project Maths Development Team 2011
www.projectmaths.ie
1
Teaching & Learning Plan: Introduction to Patterns
Besides whole-class teaching, teachers can consider different grouping strategies to
cater for the needs of students and encourage peer interaction. Students are also
encouraged in this T & L Plan to verbalise their mathematics openly and share their
work in groups to build self-confidence and mathematical knowledge.
Relationship to Junior Certificate Syllabus
Topic Number
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 and so
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.ms
involving fractional amounts.
• use tables to represent a
repeating-pattern situation
4.2 Representing
situations
with tables,
diagrams and
graphs.
• generalise and explain
patterns and relationships
in words and numbers
• write arithmetic
expressions for particular
terms in a sequence
Relations derived from some
• use tables, diagrams
and graphs as tools for
kind of context – familiar,
representing and analysing
everyday situations, imaginary
contexts or arrangements of
patterns and relations
tiles or blocks. They look at
• develop and use their own
various patterns and make
generalising strategies and
predictions about what comes
ideas and consider those
next.
of others
• present and interpret
solutions, explaining
and justifying methods,
inferences and reasoning
Resources Required
Unifix cubes (optional), coloured paper, graph paper
Introducing the Topic
Ask students to give you examples of patterns e.g.
• 5, 10, 15, 20
• 50, 100, 150, 200
• Apple, orange, apple, orange
• Sue, Bob, Tom, Sue, Bob, Tom, Sue
© Project Maths Development Team 2011
www.projectmaths.ie
2
Teaching & Learning Plan: Introduction to Patterns
Real Life Context
The following examples could be used to explore real life contexts.
• The pattern that traffic lights follow
• The Fibonacci Sequence in nature, e.g. leaves on trees
• The pattern of the tides in the oceans
• The pattern an average school day follows
© Project Maths Development Team 2011
www.projectmaths.ie
3
Teaching & Learning Plan: Introduction to Patterns
Lesson Interaction
Student Learning Tasks: Teacher
Input
Student Activities: Possible and
Expected Responses
Teacher’s Support
and Actions
Assessing the Learning
• 5,10, 15»
»» Write student
examples of
patterns on the
board.
»» Are students able to
provide examples of
patterns?
Teacher Reflections
»» Today we are going to discuss
patterns in mathematics; e.g. 2,
4, 6, 8 10, 20, 30, 40
»» Can anyone give me some
examples of number patterns?
• 3, 6, 9, 12,»
• 2, 4, 6, 8, 10
»» Could the numbers 1, 5, 12,
13, 61 be a pattern? Give me a
reason for your answer.
• No, because you can’t predict
the next number.»
»» Can students give a reason
why this sequence of
numbers could not be a
pattern?
• There isn’t any logical sequence
in the list of numbers.
»» What do you think the
properties or characteristics of a
pattern could be?
• You have to be able to predict
what number comes next.»
• If it’s a pattern then it will have
a logical sequence.»
»» Begin to make
a list of the
“properties”
patterns usually
have on the
board.
»» Are students able to list the
properties which define a
pattern?
• Most patterns have some kind
of rule that helps you figure
out what the next number
could be.
»» Do you think patterns can only
occur in numbers? What about
colours? e.g. Traffic lights: red,
amber, green, or letters e.g. A,
B, C, A, B, C, A., could these be
called patterns?
© Project Maths Development Team 2011
• Once there is a logical sequence
and you can predict what
comes next, then yes, colours or
letters or anything could be a
pattern.
www.projectmaths.ie
»» Can students explain why
patterns are not limited to
“lists” of numbers?
»» Can students explain why
almost anything could be
a pattern if it has a logical
predictable sequence?
KEY: » next step
• student answer/response
4
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks: Student Activities: Possible Teacher’s Support and Actions
Teacher Input
and Expected Responses
Assessing the Learning
»» Can you give me an
example of a pattern
that doesn’t contain
numbers?
»» Are students able to give
examples of patterns
which don’t contain
numbers?
• Yellow, Green, Blue,
Yellow, Green, Blue.»
• Toffee, Chocolate, Toffee,
Chocolate.
»» Working in pairs,
complete Student
Activity 1A
• Students report back to
the rest of the class on
Student Activity 1A.
© Project Maths Development Team 2011
Teacher Reflections
»» Can students represent
the given pattern, using
diagrams, or unifix cubes,
»» Distribute resources which may be
required, e.g. unifix cubes, coloured
or a table?»
markers.»
»» Are students interacting
and cooperating with
»» Circulate around the room to
each other to find the
see what students are doing and
solutions on Student
provide help where necessary.»
Activity 1A?
»» Get a selection of students to
report back on the various solutions »» Can students verbalise
they found when completing
their thinking when
Student Activity 1A.
reporting back to the
class?
»» Distribute Student Activity 1A.
»» We are now going
to look at an activity
about patterns.»
»» What do you think
the key to the pattern
in this activity was?
»» Write these new examples of
patterns without numbers on the
board.
• The red blocks were
always odd numbers, and
the black blocks were
always even numbers.
www.projectmaths.ie
»» Were students able to
identify and verbalise
what the key to the
pattern was?
KEY: » next step
• student answer/response
5
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
»» We are now going to
look at a slightly more
complicated pattern.»
»» Distribute Student Activity 1B
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.»
»» Working in pairs, complete
Student Activity 1B.
• Students report back to
the rest of the class on
Student Activity 1B.
»» How does this represent a
pattern?»
»» Can students represent
the pattern?»
Teacher Reflections
»» Are students interacting
and cooperating with
each other?»
»» Get a selection of students
»» Have students identified
to report back on the various
the key to this pattern
solutions they found when
and can they verbalise
completing Student Activity 1B.
their thinking?
»» Write the following on the
board: “Starting from today, I
am going to save 5 euro every
day, for the next 7 days”
»» We are now going to
look at a sentence which
might describe a number
pattern. Let me give you an
example.
»» Do you think that this
sentence could represent a
pattern?»
Assessing the Learning
• Yes»
• Because you can predict
the next number every
time.»
»» Can students work out
the answer?»
»» Do students know if
the word sentence
represents a pattern?»
• It has a logical sequence»
»» So, looking at this, how
much will I have saved in 7
days?
© Project Maths Development Team 2011
• 35 euro
www.projectmaths.ie
»» Can students give a
reason why the word
sentence is a pattern?
KEY: » next step
• student answer/response
6
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Support and
and Expected Responses
Actions
»» We are now going to look
at another type of pattern
in Student Activity 2A.
• Students report back to
rest of class on Student
Activity 2A.
• Yes»
»» Sometimes, we can
understand a pattern
more easily if we can draw • No
a graph of it.»
Assessing the Learning
»» Distribute Student Activity
2A to students.
»» Can students represent the
pattern?»
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.»
»» Are students interacting and
cooperating with each other?»
»» Get a selection of students
to report back on the
various solutions they found
when completing Student
Activity 2A.
»» If students are confused
on how to plot points on
coordinate plane, a quick
recap will be required.
Teacher Reflections
»» Have students identified the
key to this pattern and can
they verbalise their thinking?»
»» Did students realise that the
start amount of money could
be included in the answer to
question 7?
»» Are all students familiar with
how to plot points on the
coordinate plane?
»» Does everyone remember
how to plot points on the
coordinate plane?
»» We are now going to look
at Student Activity 2B.
© Project Maths Development Team 2011
»» Distribute Student Activity
2B.
www.projectmaths.ie
KEY: » next step
• student answer/response
7
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Support and
and Expected Responses
Actions
Assessing the Learning
• Students report back to
rest of class on Student
Activity 2B.
»» Can students represent the
pattern?»
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.»
»» Get a selection of students
to report back on the
various solutions they found
when completing Student
Activity 2B.
Teacher Reflections
»» Are students interacting and
cooperating with each other?»
»» Have students identified the
key to this pattern and can they
verbalise their thinking?»
»» Were students able to
correctly plot the points on the
coordinate plane?
»» Do you think that
drawing a graph to
show the relationship
between the number
of days gone by and
the amount of money
in the money box has
any advantages?
• Yes, because you can see
how it increases all the
time. »
»» What things did you
notice about this
graph?
• It goes up»
»» Can students explain the
advantages of drawing a graph
to represent a pattern?
»» Write comments about
the graph on the board.
Introduce the phrase
“linear” for a straight line
graph.
»» Could students describe the
properties of the graph?
• Yes, because you can
extend the graph and
find out the value for any
day, without having to
build a table
• It’s a straight line»
• It doesn’t start at zero
© Project Maths Development Team 2011
»» Write the benefits of
drawing a graph on the
board.
www.projectmaths.ie
KEY: » next step
• student answer/response
8
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
»» To ensure that everyone
is clear on how to draw
a graph of a pattern, can
you all complete Student
Activity 2C.
»» Distribute Student Activity 2C
• Students report back to
rest of class on Student
Activity 2C.
Assessing the Learning
»» Can the students complete
the activity?»
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.»
»» Are students able to choose
a relevant scale for drawing
a coordinate graph?»
»» Get a selection of students
to report back on the various
solutions they found when
completing Student Activity 2C.
»» Have students identified
the key to this pattern and
can they verbalise their
thinking?
Teacher Reflections
»» Can students represent
both patterns on the same
coordinate plane?»
»» Were students able to
correctly plot the points on
the coordinate plane?
»» Graphs can be a very
useful way of comparing
two things.»
»» Distribute Student Activity 3A.
»» For example, let’s just
look at the question on
Student Activity 3A
• Probably Bill, because he
»» Before we begin the
starts with 30 texts and
question, can you predict
gets an extra 3 free every
who might have the
day.
better deal for the most
texts in a month?
© Project Maths Development Team 2011
www.projectmaths.ie
KEY: » next step
• student answer/response
9
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
»» The next activity we
• Students report back to rest
are going to look at
of class on Student Activity
requires you to draw
3A.
a graph of 2 different
patterns on the same
graph paper, and
compare them. When
we have done this, we
will know for sure which
person has the better
deal from the mobile
phone company
»» What do you think
was the benefit of
representing the
information graphically
in Student Activity 3A?
© Project Maths Development Team 2011
• It was much easier to figure
out who had the better deal»
Teacher’s Support and
Actions
Assessing the Learning
»» Walk around the
room and listen to
what students are
saying. Offer help,
guidance where it is
required.»
»» Were students able to correctly
complete the table?»
»» Get a selection of
students to report
back on the various
solutions they found
when completing
Student Activity 3A.
»» Could students correctly answer
the questions in the activity?»
Teacher Reflections
»» Were students able to represent
both sets of data on the same
graph?»
»» Were students able to give a
reason why Amy had the better
deal for free texts each month?
»» Were students able to give the
benefits of drawing a graph for
this question?
• You can just extend your
graph and see who gets the
most texts, you don’t have
to keep filling in the table.
www.projectmaths.ie
KEY: » next step
• student answer/response
10
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
»» Now we are going to
• Students report back
try Student Activity 3B.
to the rest of class on
This time you will have
Student Activity 3B.
to figure out how you
are going to represent
the information and
find the answers to the
questions asked.
Teacher’s Support and
Actions
Assessing the Learning
»» Distribute Student Activity
3B
»» Were students able to construct
a table to represent the data?»
»» Walk around the room and
listen to what students
are saying. Offer help,
guidance where it is
required.»
»» Were students able to choose an
appropriate scale for the graph?»
»» Get a selection of students
to report back on the
various solutions they
found when completing
Student Activity 3B.
• They have to have a
»» So, what have we
logical sequence»
learned about patterns
so far?
• They don’t always have to
be numbers»
»» Write student answers on
the board.
Teacher Reflections
»» Could students draw both sets of
data on the same axes?»
Were students able to answer
the questions based on the
graph correctly?»
»» Could students verbalise their
mathematical thinking?
»» Are students able to state what
they have discovered about
patterns to date?
• You can represent them
in lots of ways, e.g. tables,
graphs, diagrams»
• Sometimes a graph is a
good way to represent a
pattern, because you can
see what will happen over
a period of time.
© Project Maths Development Team 2011
www.projectmaths.ie
KEY: » next step
• student answer/response
11
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
Assessing the Learning
HIGHER LEVEL MATERIAL: Student Activity 4 and Student Activity 5 contain optional activities for students taking higher level.
»» Now we are going
to look at a pattern
containing some
blocks.
• Selection of students
report back to rest of
class on Student Activity
4A.
»» Distribute Student Activity 4A
»» Distribute materials which
may be required for task e.g.
coloured blocks, or coloured
paper etc »
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.»
Teacher Reflections
»» Were students able to
model the pattern using
coloured blocks or paper?»
»» Did students successfully
create a table / diagram to
represent the pattern?»
»» Were students able to
verbalise their mathematical
thinking to you and their
colleagues?
»» Get a selection of students
to report back on the various
solutions they found when
completing Student Activity 4A.
»» May have to explain what
the term “algebraic formula”
means.
»» The final step we
are going to do on
patterns is to take a
look at how we might
represent a pattern
using an algebraic
formula.»
»» Does everyone
understand what
the term “algebraic
formula” means?
© Project Maths Development Team 2011
• Is it when we write the
formula using x’s and y’s?»
• Is it when we try to
explain a pattern using
algebra?
www.projectmaths.ie
KEY: » next step
• student answer/response
12
Teaching & Learning Plan: Introduction to Patterns
Student Learning Tasks: Student Activities: Possible Teacher’s Support and Actions
Teacher Input
and Expected Responses
Assessing the Learning
»» We are now going
to look at Student
Activity 4B
»» Are students interacting
and cooperating with each
other to find the solutions on
Student Activity 4B?
»» Distribute Student Activity 4B
»» Walk around the room and
listen to what students are
saying. Offer help, guidance
where it is required.
Teacher Reflections
»» Can students accurately
model the problem using the
resources supplied
»» Get a selection of students
to report back on the various
solutions they found when
completing Student Activity 4B.
»» Were students able to
develop a formula which
would describe the pattern?»
»» Did students verify their
equation for the pattern?»
»» Are students able to verbalise
their mathematical thinking
and defend their answers?
© Project Maths Development Team 2011
www.projectmaths.ie
KEY: » next step
• student answer/response
13
Teaching & Learning Plan: Introduction to Patterns
Student Activity 1A
1.Represent this repeating pattern - red, black, red, black,
red, black, – by building it with blocks or colouring it in
on the number strip below:
1
2
3
4
5
6
7
8
9
10
2.Complete the following table based on your diagram
above:
Block position number on strip
1
2
3
4
5
6
7
8
9
10
Colour
Red
Black
3.Answer the following questions:
1. List the position numbers of the first 5 red blocks:_______________________________
2. What do you notice about these numbers?______________________________________
3. List the position numbers of the first 5 black blocks:_ ____________________________
4. What do you notice about these numbers?______________________________________
5. What colour is the 100th block?_ _______________________________________________
6. What colour is the 101st block?_________________________________________________
7. What position number on the strip has the 100th black block?____________________
8. What position number on the strip has the 100th red block?______________________
9. What colour will the 1000th block be?_ _________________________________________
Explain how you found your answer for question 8:_________________________________
______________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
14
Teaching & Learning Plan: Introduction to Patterns
Student Activity 1B
1.Represent this repeating pattern – yellow, black, green,
yellow, black, green – by building it with blocks or
colouring it in on a number strip or drawing a table or
in any other suitable way.
1
2
3
4
5
6
7
8
9
10
1. List the numbers of the first 3 yellow blocks. Is there a pattern in these numbers?
______________________________________________________________________________
2. List the numbers of the first 3 black blocks. Is there a pattern in these numbers?
______________________________________________________________________________
3. List the numbers of the first 3 green blocks. Is there a pattern in these numbers?
______________________________________________________________________________
4. What colour is the 6th block?___________________________________________________
5. What colour is the 18th block?_ ________________________________________________
6. What colour is the 25th block?_ ________________________________________________
7. What colour is the 13th block?_ ________________________________________________
8. What colour will the 100th block be in the sequence?____________________________
9. What colour will the 500th block be in this sequence?____________________________
10. Explain how you found your answers to questions 8 and 9,
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
11. What rule could you use to work out the position number of any of the (i) yellow
blocks, (ii) black blocks, (iii) green blocks?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
15
Teaching & Learning Plan: Introduction to Patterns
Student Activity 2A
Money box problem
John receives a gift of a money box with €4 in it for his
birthday. This is his starting amount. John decides he will
save €2 a day. Represent this pattern by drawing a table,
or a diagram, or by building it with blocks.
1. How much money will John have in his money box on the following days?
Day 5: ___________ Day: 10: ___________ Day 14: ___________ Day 25: ___________
2. By looking at the pattern in this question, can you explain why the amount of
money John has on day 10 is not twice the amount he has on day 5?______________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
3. How much money does John have in his money box on day 100?_________________
4. Explain how you found your answer for the amount for day 100:_________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5. How much money has John actually put in his money box after 10 days? Explain
how you arrived at this amount.________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6. John wants to buy a new computer game. The game costs €39.99. What is the
minimum number of days John will have to save so that he has enough money to
buy the computer game?_ _____________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
16
Teaching & Learning Plan: Introduction to Patterns
Student Activity 2B
Annie has a money box; she starts with €2 and adds €4 each day.
Create a table showing the amount of money Annie has each
day over a period of 10 days.
1. Using the axes below, draw a graph to show how much
money Annie has saved over 6 days
32
30
28
26
24
Number of days
22
20
18
16
14
12
10
8
•
6
4
2
•
0
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Start Amount (Euros)
2. List two things that you notice about this graph
a. ______________________________________________________________________________
b. ______________________________________________________________________________
3. Could you extend the line on this graph to find out how much money Annie has in
her money box on day 10?
a. Amount on day 10 =___________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
17
Teaching & Learning Plan: Introduction to Patterns
Student Activity 2C
Owen has a money box; he starts with €1 and adds €3 each day.
1. Draw a table showing the amount of money Owen has each day.
2. Draw a graph to show the amount of money Owen has saved over 10 days. Hint: Think carefully about the following before you draw your graph:
a._ Where will you put “Number of days” and “Amount of money” on your graph?
b._What scale will you use for the amount of money? (Will you use 1, 2, 3,... or will
you decide to use 5, 10, 15, 20..... or perhaps a different scale?)
© Project Maths Development Team 2011
www.projectmaths.ie
18
Teaching & Learning Plan: Introduction to Patterns
Student Activity 3A
Using Graphs to represent information
Amy and Bill are discussing phone network offers. Bill says that on his network he begins each
month with 30 free texts and receives 3 additional free texts each night. Amy says that she
gets no free texts at the beginning of the month but that she receives 5 free texts each night.
To see how many texts each person has over a period of time, complete the tables below.
Amy’s texts
Time
Start amount
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Bill’s texts
Time
Start amount
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Number of free texts
0
5
10
Number of free texts
30
33
36
1. Who has the most free texts after 10 days?______________________________________
Number of Texts
2. Using the graph paper provided, draw a graph showing the number of texts Amy
has, and using the same axes draw a graph of the texts Bill has, for 10 days.
60
55
50
45
40
35
30
25
■■
■
Graph Key:
Amy =
Bill = ■
20
15
10
5
••
0
1
•
•
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Time (Days)
3. What do you notice about each graph?_ ________________________________________
______________________________________________________________________________
4. Extend the lines of each graph to “Day 20”. In your opinion, who do you think has
the better deal on free texts Bill or Amy? Why?__________________________________
______________________________________________________________________________
5. Will Amy and Bill ever have the same number of texts on a particular day? If so,
which day? If not, why?_ ______________________________________________________
______________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
19
Teaching & Learning Plan: Introduction to Patterns
Student Activity 3B
Examine the situation below
Liam begins the month with 20 free texts and receives 2 additional free texts each
night.
Jessie does not have any free texts at the beginning of the month but receives 3 free
texts each night.
1. Draw a table showing the following information.
a. The number of free texts Liam has after 10 days
b. The number of free texts Jessie has after 10 days
2. Represent this information on a graph, (note: show Liam’s and Jessie’s number of
free texts on the same graph)
3. Will Liam and Jessie ever have the same number of free texts on a certain day? If so, which day? If not, why not?_______________________________________________
______________________________________________________________________________
4. Who in your opinion has the better deal for free texts each month? Give a reason
for your answer._ _____________________________________________________________
______________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
20
Teaching & Learning Plan: Introduction to Patterns
Student Activity 4A (Higher Level Material)
Finding Formulae
The figures below are made up of white and red square tiles. The white squares are in
the middle row and have a border of red tiles around them. For 1 white tile, 8 red tiles
are needed; for 2 white tiles, 10 red tiles are needed, and so on.
Using the information above, create a table or diagram which will show how the
number of red tiles increases as the number of white tiles increases.
(Hint: Look at the way the number of red tiles change each time a white tile is added,
can you see a pattern?)
© Project Maths Development Team 2011
www.projectmaths.ie
21
Teaching & Learning Plan: Introduction to Patterns
Student Activity 4B (Higher Level Material)
Breaking down the pattern and developing your own
formula
Let’s look at how each shape is built.
Complete the blank spaces below
To make the next shape in the
sequence, I have to add ____ white
tile(s) to the middle and _____ extra
red tiles to the outside
So each time I make a new figure the number of white tiles increases by ______ and the
number of red tiles increases by _______; complete the pattern in the table below.
+1
Number of white tiles
1
Number of red tiles
8
2
10
© Project Maths Development Team 2011
www.projectmaths.ie
+2
22
Teaching & Learning Plan: Introduction to Patterns
Student Activity 4B (continued)
Let the number of white tiles = n. We know from our first shape that the white tile is
surrounded by 8 red tiles, and each time we add a white tile we must add two red tiles.
+1
White
If n is the number of white tiles and there are the same number of red tiles above and
below it (i.e. a total of 2n) and two lots of 3 at either side (i.e. +6). Then the general
formula (or expression) for the number of red tiles must be 2n + 6.
(Remember we are calculating the total number of RED tiles, not the total number of tiles in the
shape)
Using the logic above, can you develop another formula or expression based on the
information below; (let the number of white tiles = n)
Hint: The number of red tiles above and below is 2 more than the number of white tiles and 2
extra at the sides.
1 How many red tiles will there be if there are 100 white tiles? (Check your answer
using both equations, i.e. 2n + 6 AND the one you have found above.)____________
_ ____________________________________________________________________________
_ ____________________________________________________________________________
© Project Maths Development Team 2011
www.projectmaths.ie
23
Teaching & Learning Plan: Introduction to Patterns
Student Activity 5 (Higher Level Material)
Group Work: Graphing Functions
Set 1.
1. I have €36 to buy a gift. The amount of money I have left depends on the price of
the purchase. (x-axis, price of purchase; y-axis, amount of money I have left).
2. A straight line, 36cm long, is divided into 2 parts A and B. The length of part B
depends on the length of part A. (x-axis, length in cm of part A; y-axis, length in cm
of part B ).
For each of the sets assigned to your “expert” group:
• Find some data points that fit the given problem, e.g. (1, 35) or (10, 26) etc
• Organise the data into a table. Use the quantity identified as the x-axis for the left
column and the quantity identified as the y-axis for the right column.
• Organise the data into a graph. Use the quantity identified as the x-axis for the
horizontal axis and the quantity identified as the y-axis for the vertical axis.
IMPORTANT: Each member of the group will need a copy of each graph to share with
the next group.
• Share and compare your graphs with other groups
• Identify similarities (things that are the same) and differences among the various
graphs.
© Project Maths Development Team 2011
www.projectmaths.ie
24
Teaching & Learning Plan: Introduction to Patterns
Student Activity 5 (continued)
Set 2.
3. The amount of money collected for a particular show is dependent on the number of
tickets sold. Tickets cost €8 each. (x-axis, number of tickets sold; y-axis, amount of
money collected).
4. The number of empty seats in a cinema that seats 350 people depends on the number of
seats that are sold (x-axis, number of seats that are sold; y-axis, number of seats that are
empty).
For each of the sets assigned to your “expert” group:
• Find some data points that fit the given ticket problem, e.g. (1,8) or (2,16) etc
• Find some data points that fit the cinema probelm, e.g. (0,350) or (20,330) etc
• Organise the data into a table. Use the quantity identified as the x-axis for the left
column and the quantity identified as the y-axis for the right column.
• Organise the data into a graph. Use the quantity identified as the x-axis for the
horizontal axis and the quantity identified as the y-axis for the vertical axis.
IMPORTANT: Each member of the group will need a copy of each graph to share with the
next group.
• Share and compare your graphs with other groups
• Identify similarities (things that are the same) and differences among the various graphs.
Set 3.
5. The perimeter of a square depends on the length of a side (x-axis, length of side; y-axis,
perimeter.)
6. If I fold a 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? (x-axis, number of folds; y-axis, number of
sections )
7. The area of a square depends on the length of a side. (x-axis, length of side; y-axis, area
of the square).
For each of the sets assigned to your “expert” group:
• Find some data points that fit the given problem.
• Organise the data into a table. Use the quantity identified as the x-axis for the left
column and the quantity identified as the y-axis for the right column.
• Organise the data into a graph. Use the quantity identified as the x-axis for the
horizontal axis and the quantity identified as the y-axis for the vertical axis.
IMPORTANT: Each member of the group will need a copy of each graph to share with the
next group.
• Share and compare your graphs with other groups
• Identify similarities (things that are the same) and differences among the various graphs.
© Project Maths Development Team 2011
www.projectmaths.ie
25
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement