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```									Patterns and Algebra
The case of the disappearing M & Ms and other
interesting stuff

MANSW Annual Conference

21st Sept 2008
George Anderberg
Mathematics Consultant
The Mathematics Connection
Ph: 0421151043
Fax: 02 49538320
Web: www.mathcon.com.au
EXPONENTIAL DECAY

Syllabus Ref: PAS3.1a/4.2/Stage 6 Ext 1 Log and Exp functions

1. Students take a 250 gm pack of M & Ms and spread them over a sheet of butcher’s
paper.

2. Students count all the M & Ms and plot this on the graph at “eating No.” zero. They
should also complete the table on the next page.

3. Students take note of which of the M & Ms have the M facing up and they eat or remove
these.

4. Students count all the remaining M & Ms and plot this on the graph at “eating No 1”
and they should also complete the table on the next page.

5. The remaining M & Ms are put into the paper cup and shaken before again being
spread on the butcher’s paper. Students take note of which of the M & Ms have the M
facing up and they eat or remove these.

6. Students count all the remaining M & Ms and plot this on the graph at “eating No.” 2 and
they should also complete the table on the next page.

7. This process is repeated until no M & Ms remain.

8. When no M & Ms remain and all the points are plotted students attempt to draw a
smooth curve joining, as far as possible, all the points.

9. If time permits students can calculate the theoretical values (to the nearest whole M &
M.).

Tables of Values

Record of M&Ms after each eating.

# of eatings
0       1      2      3      4     5      6      7      8      9      10
# of M&Ms

Theoretical Record of M&Ms after each eating (what you think should have happened)

# of eatings
0       1      2      3      4     5      6      7      8      9      10
# of M&Ms

2
Graph of the decay in M & Ms after each eating

310

290

270

250

230

210

190
Number of M & Ms

170

150

130

110

90

70

50

30

10

0
0   1       2       3       4      5    6   7   8   9   10

Number of Eatings

3
SPAGHETTI MATHS
Syllabus Ref: PAS 3.1a/4.2

This unit incorporates:
 Measurement
 Algebra/equations
 Graphs/number plane
 Problem solving

AIM: 1. To investigate the relationship between the number of strands of
spaghetti and the load they can withstand without breaking.
2. To estimate the number of strands of spaghetti required to suspend a
50 kg student.

EQUIPMENT:
 paper clip
 plastic cup
 nails
 graphical calculator
 dry strands of spaghetti

METHOD: A plastic cup is suspended from a strand of spaghetti by a paper clip.
Nails gently placed into the cup until the spaghetti breaks..
The number of nails the strand of spaghetti could hold before
breaking is recorded.

Strand(s) of spaghetti

desk                                                            desk
paper clip
gap betw. desks approx. 10cm

nails                           plastic cup

The gap between the desks is held constant and the experiment
Repeated for 2, 3, 4 or 5 strands of spaghetti.

The number of strands of spaghetti and the number of nails are
recorded and graphed on a graphical calculator.

The constants of the resulting equation are optimized by trial
and error to obtain the “line of best fit”.

4
Sample set of Results:

Strands of           Number of nails
spaghetti                                                       The mass of one nail was
1                 12                               calculated by averaging the mass of
2                 18                               10 nails.
3                 30
4                 34                               mass of 10 nails = 64 g
5                 42
mass of 1 nail = 64 ÷ 10
= 6·4 g
Number of nails
(dependent variable)   50     y
y = 8x + 2
40                                   X
X
30                      X

20
X
X
10

x
1     2      3        4   5
Number of strands of spaghetti
(independent variable)

student mass = 50 kg = 50 000 g and if 1 nail = 6·4 g then
student mass ≡ 50 000 = 7813 nails
6·4
therefore y = 7813 and substituting this in y = 8x +2, we get x = 7811 ÷ 8 = 976
i.e. the number of sticks (x) to support a 50 kg student is approx. 1000

CONCLUSION:
1. The graph was a straight line i.e. the relationship between the number of strands
of spaghetti and the mass they can support is a linear relationship.
2. Based upon this experiment, approximately 1000 strands of spaghetti would be
needed to support a 50 kg student.

During the optimizing of the equation it was noted that the number multiplying the
x (8) changed the slope of the line and that the number at the end of the equation
(+2) was where the graph line crossed the y axis.

5
Tables of Values

Record of the number       Theoretical Record of
of nails supported by      the number of nails
the spaghetti              supported by the
spaghetti

# of      # of nails      # of      # of nails
strands of                strands of
spaghetti                 spaghetti

1                          1
2                          2
3                          3
4                          4
5                          5
6                          6

6
Graph of the number of nails that are supported by the strands of
spaghetti

150

140

130

110

100

90
Number of Nails

80

70

60

50

40

30

20

10

0
0   1    2     3     4     5     6   7    8    9    10

Number of Strands of spaghetti

7
LESSON PLAN FOR PATTERNS AND ALGEBRA, “FINDING A RULE”

Date:

The Following NSW Mathematics 7-10 Syllabus outcomes will be addressed.

Patterns and Algebra Strand 3.1a/4.2 (PAS 3.1a/PAS 4.2) Students learn to “create, record,
analyse and generalise number patterns using words and algebraic symbols in a variety of
ways.

Goals: Students use algebraic methods to explore, model, and describe patterns.

Students learn how to develop a table, a graph and an algebraic rule from a word description of

a function.

Students develop the problem solving skills, of understanding the problem, questioning and
investigating, reasoning and reflection. They also learn some techniques such as trial and error,
working backwards and systematic lists.

Standard    Strategy                              Questions you might ask.            Duration
#2          Select 6 students to be the           Who can tell me;                    15mins
This        narrator and actors.                  How many bread rolls did Kath start
section     Give the students the scripts at
out with?
uses        least 5 minutes before the
the play    lesson.
Kath’s      Sit back with the rest of the class   How could we find out?
hot         and watch them act out the            Is there another way we could act this
rolls       At the end of the play pose the       How many bread rolls did Kath bring
question in bold opposite and         home.
The         allow a discussion to develop.        How many did her friend Adriana get?
script is   If students want to do a trial and    How did you work that out?
attache     error process give them some          How many did Matt the surfer get?
d           counters or blocks to represent       Is there a way to work how many she
bagels and let them get on with       started out with?
QT 1.6,     it.
2.2, 3.1.
Give the class a copy of the          How would Kath explain to her          20 mins
QT 1.4     script and ask them to rewrite it     daughter Kim how the rolls
(or rethink it) in such a way that    disappeared?
WMS        they will get the answer.             If Kim ran next door to Adriana’s
4.2, 4.4.   Ideally you want them to do the       house and invited her to breakfast and
play backwards starting with          asked her to bring the rolls with her
Kylie discovering that she only       how many would they have?
had 2 rolls left. They could do a     If Kath only gave each person half of
flashback.                            what she had left how many would
If students are having difficulty     she have started out with?
suggest that they try the
scenario without Kath giving out
8
the two extra rolls each time.

Only do      At this point the students have      How about we rewind the tape a bit?      10 mins
this if no   probably worked out that Kath        If Kath got home with two rolls how
student      started out with 44 rolls. If not    many did she have before she gave
has          give the whole class some            Adriana the extra two?
come up      concrete materials (blocks,          How many did she have before she
with the     markers, pennies etc) and            gave Adriana half?
answer.      encourage them to model the          Can you keep working backwards
process backwards.                   from here?
QT 2.3       Students need to recognize that
the answer to the problem (the
PAS          dependant variable) is the
4.2.1        number of rolls that Kath starts
with.
In this section students             If Kath got home with a different        25mins
QT 1.4,      investigate other possible           number of rolls could you work out
1.6         outcomes. Such as Kath ending        how many she started out with?
up with zero, 1, 3, 4, 5, _, _, _,   Kath and Kim like to have 2 rolls each
PAS         rolls.                               for breakfast, how many would Kath
4.2.1,      Make sure students record their      need to start out with for this to
4.2.2       results in a systematic fashion.     happen?
An in – out or function table        Try some other cases, for example
WMS         would be best.                       Kath gets home with none, 3, 6 and
4.5                      In Out                 so on.
(ending rolls) (starting rolls)   How should we keep track of our
0    ___             findings?
1    ___
2     44
3    ___
4    ___
5    ___

At this point students should be     Kath is a very generous person and       25 mins
encouraged to go from the            always gives to the needy, but she
QT 1.2      specific to the general.             wants to be sure she has enough rolls
They should be able to describe      when she gets home for her needs.
WMS         a way of answering the problem       Can you describe a way Kath can
4.2, 4.4,    for any number of rolls.             easily calculate how many rolls she
4.5        Students should also have            needs to buy so she can have the
emphasized that a function can       exact amount she will need for that
be described in four ways:           day? Assume that she will always
 As a story,                    meet 3 needy people on her way
 as a table,                    home.
 as a graph,
 and as an algebraic rule.
This could be a good time to
introduce the graphics
calculator.
Students should discover that
the general rule is:
#Of starting rolls = 8 times the #
of rolls left + 28.
Or R = 8r + 28

9
Extension:                       Can any one work out a general rule if
Students could try seeing what   the number of needy people Kath
happens when they alter the      meets is four instead of three?          15 mins
number of people given rolls.
They could add or delete         What would be the outcome if Matt
characters from the original     was still surfing and did not meet
Total time
script.                          Kath?
85 to 110 mi

Quality teaching elements that are promoted:
1.1? 1.2, 1.3? 1.4, 1.5? 1.6.
2.2, 2.3, 2.6?
3.1, 3.3?, 3.4, 3.6?

10
ACT IT OUT

Cast:
Narrator
Kath, a lady who goes shopping for bread rolls for her and her daughter.
Dorothy, a social worker who runs a homeless shelter,
Matt, a fanatical surfer who rides his surfboard in all weather.
Kim, Kath’s daughter.

Scene: The road from the bakery to Kath’s house. Kath is walking along with a LARGE
shopping bag.

Narrator: Kath is walking home from the hot bread shop. She has bought a LOT of hot
rolls for her and her daughter’s breakfast. Now who is this, why it is Dorothy, the social
worker who runs the homeless shelter.

Dorothy waves to Kath and stops her to chat.

Dorothy: My, Kath, those bread rolls sure do smell good. Can you spare some for the
people in my homeless shelter

Kath: Why sure I can Dorothy, here have half of them and two more for good measure

Narrator: So Kath gives Dorothy half of her rolls plus two more

Kath waves goodbye to Dorothy and walks on still carrying her shopping bag, Dorothy
walks off stage with her rolls.

Narrator: As Kath walks on she walks past a beach and sees her friend Matt who has
been out for an early morning surf in the cold, cold ocean

Matt comes running up to Kath and gives her a hug.

Matt: My, Kath, those bread rolls sure do smell good. Can you give me some; it’s so cold
that I am very hungry.

Kath: Why sure I can Matt, here have half of them and two more for good measure.

Narrator: So Kath gives Matt half of her rolls plus two more.

Kath waves goodbye to Matt and walks on still carrying her shopping bag, Matt walks off
stage with his rolls.

11
Narrator: Kath is nearly home, as she walks past the house next door she sees her elderly
neighbor, Adriana Appleby sitting on her porch, she waves and goes up to her and gives
her a hug.

Kath: How are you this morning Adriana.

Adriana: (In a quivery voice) Not so good today darl. My knees are acting up again; you
young people don’t know what it’s like to be old. (She smells the bread rolls)

My, Kath, those bread rolls sure do smell good. Can you spare me some I don’t think I
will be able to go shopping today and they smell so good.

Kath: Why sure I can Adriana, here have half of them and two more for good measure.

Narrator: So Kath gives Adriana half of her rolls plus two more.

Kath waves goodbye to Adriana and walks next door to her house still carrying her
shopping bag, Adriana hobbles inside/offstage with her rolls.

Narrator: Kath walks up her front path and her daughter Kim runs out to meet her.

Kim runs towards Kath

Kim: Mum, Mum, your home at last, I am soooo hungry. Have you got the rolls?

Kath hands the shopping bag to Kim who eagerly peers inside her expression turning to
one of dismay.

Kim: Mum !!! There are only two rolls here; you said that you were going to buy heaps!

Kath looks surprised.

Narrator: Poor Kath, only two bread rolls left, I wonder how many rolls she
started with.

12

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