# tasks for the mathematical problem solving on the situations of the

```APEC-Tsukuba International Conference VII
Innovation of Mathematics Education through Lesson Study
Challenges to Emerency Preparedness for Mathematics
Focusing on Flood and Typhoon
14-17 February 2013
Tokyo, Japan
THE MATHEMATICAL PROBLEM SOLVING
ON THE SITUATIONS OF THE FLOOD
(for secondary students: Grade 8 &amp; 10)
Tran Vui
Hue University of Education, Vietnam
1. Introduction: Typhoons &amp; Floods in Vietnam
People and Flood in Vietnam
50 years Typhoons in Vietnam
1954-2006
380 Typhoons Affected to Vietnam
8 typhoons/year
North
31 %
118
Middle
36 %
137
South
33 %
125
Vietnam at high risk of natural disasters
in Asia-Pacific
Friday, 04 January 2013
• The ADB’s respond to natural disasters and disaster
risks report stated that Vietnam, Bangladesh and the
Philippines are the countries at high risk of natural
disasters in the Asia-Pacific region.
• Every year, the Asia-Pacific region suffered more
than \$ 40 billion in losses because of natural
disasters.
Vietnam is one the most natural disaster-prone countries
2. A point of View
to Design Mathematical Problems
• Some mathematics that can be used to
understand water speed in a flood.
• Learning mathematics in a real life context.
• How can we bring real life situations into
class to teach mathematics?
• Use simple mathematics which we teach in our
schools to explain some “Everyday Knowledge of
the situations” related to water speed in a flood.
• Developing teaching materials for school use
against the flood in mathematics education.
• Well instructed lesson plans with mathematisation.
MATHEMATISATION
The process of “converting”
from Real World Model into Mathematical Model
Translate
MATHEMATIZATION: Simplified Mathematical Modeling
Since within a classroom activity the results are never put into operation
there is no real problem of validation.
Mathematisation vs. Demathematisation
Abstract Mathematics vs. Applied Mathematics
Abstract
Concrete
2. Mathematical Problems
Knowledge Prerequisites
Algebra
- Create linear equations
- Solving linear equations.
Geometry
- Similar triangles
- Diagonal of a parrallelogram
- Area and Volume
- Sum of 2 vectors.
Trigonometry
- Trigonometric ratio in a right triangle.
LESSON 1
DEDUCE FLOOD HEIGHT
Measurement
Semester 2, Grade 8, 13-14 years.
45 minutes
A Longitudinal profile of a stream beginning in mountains and
flowing across a plain into the sea.
Why the plain gets flood?
The Flowing of Rivers and Streams
Bankfull cross section
The water in rivers and streams is in constant motion. It moves
faster with a steep gradient, a narrow, curving or a high volume of
water.
The average speed of moving water is about 5 km/h,
The speed can range up to 30 km/h during floods.
After 100 years of development, people lived in a city realized that
The flood height increases. With your mathematical knowledge, can you have
any suggestion to deduce the flood height.
A. Before
Development
C. After
Development
B. Increases
in flood height
Problem 1
How to estimate the area of the bankfull cross sectional area?
Problem 2
The figure shows the bank full cross section of a river before the
floodplain development.
The bankfull width of the river is 20 meters. In a field work
we collected the data for the depth in every 2 meters cross
the river.
0
2
4
6
8
10 12 14 16 18 20
0
1
2
4
4
4
7
8
4
3
0
From the data collected find the bankfull cross sectional area?
Can you generalize a pattern to estimate the cross sectional
area?
Plot points from the data collected. Find the area of the cross
section.
Without counting, how can you find a way to estimate the area of
the blue region?
In general,
you can plot the width and depth readings on graph paper, then count the area of
the stream. Mathematics types may wish to use the trapezium rule to calculate the
area. Can you derive to the formula stated in the above figure?
Problem 3
After 100 years of floodplain development, some fill remains on
the river bed. If the average speed of moving water is ranging
from 5 to 30 km/h. From the figure, can you estimate the flood
height?
LESSON 2
LINEAR EQUATIONS
Velocity with and against the water
Semester 2, Grade 8, 13-14 years.
45 minutes
Problem 1
A rescue boat can average 65 km/h in still
water. In a flood, if a trip takes 2 hours one
way and the return takes 1 hour and 15
minutes. Find the speed of water, assuming
it is constant.
Real World
Model
Visual
Representation
Mathematical
Model
Algebraic
Representation
( 65 - x) 2
= (65 + x) 1.25
Let speed of water be x km/h.
From the model of distance showing in above figure.
a) Find the distance of the boat in x when it runs against the
water in 2 hours.
b) Find the distance of the boat in x when it runs against the
water in 1 hour and 15 minutes.
c) Observe the area model, explain when the two distances are
equal?
d) Which value of x the two distances are the same?
Hint
a) Against the water
Speed = 65 - x
Time = 2 hours
Distance = Speed x Time = ( 65 - x) 2 = 130 - 2x (*)
b) With the water
Speed = 65 = x; Time = 1.25 hours
Distance = Speed x Time = ( 65 + x) 1.25 = 81.25 + 1.25x (**)
Hint
c)
Green + Blue
=
Green + Yellow
Blue
=
Yellow
1.25(65 – x) + 2.5x
=
1.25 (65 – x) + 0.75 (65 – x)
2.5 x
=
48.75 – 0.75x
d) Now distance traveled is same: (*) = (**).
130 - 2x
= 81.25 + 1.25x
48.75
= 3.25 x
x
= 15 km/h
Problem 2
Sketch the graphs of two straight lines y = f(x) = 2(65 – x) and y
= g(x) = 1.25(65 + x) on a rectangular grid coordinate system
such that you can see their intersection.
a) Find the difference between f(x) and g(x) when x = 0.
b) What is the change of the difference between f(x) and g(x)
when x increases 1 units?
c) At which value of x the change is equal to 0?
The difference d(x) = f(x) – g(x) = 48.75 – 3.25 x
x increases 1 unit then the difference d(x) decreases 3.25 units.
The difference d(x) = 0
48.75 = 3.25 x
x = 15 km/h
Problem 3 (Further Exploration)
With the water, a rescue boat can run 150 km in 2 hours and
30 minutes. Against the water, it can run only 100 km in the
same time.
a) Find the speed of the rescue boat with the water.
b) Find the speed of the rescue boat against the water.
c) Find the speed of the rescue boat in still water.
HINT:
a) With the water:
150
x y
 60 km/h (*)
2 .5
b) Against the water:
100
x y
 40 km/h (**)
2.5
c) We have a system of 2 equations:
 x  y  60
 2 x 100 x  50

 x  y  40
LESSON 3
THE SUM OF TWO VECTORS
SITUATION: CROSS A RIVER BY rescue boat IN A FLOOD
Semester 1, Grade 10, 15-16 years. Time: 45 minutes
Problem 1
In a flood, a rescue boat needs to cross a
river with the width of 400 meters. The
rescue boat speed v1 is 10 m/s, and the
water speed v2 is 5 m/s. The boat starts
from port A and needs to reach to port B,
assuming that AB is perpendicular to the
river bank.
Real World
Model
Mathematical
Model
a) Where does the rescue boat reach to another bank?
b) What happens if v1 = v2?
c) What happens if v1 &lt; v2?
The diagonal of the rectangle.
Materialized Mathematical Model
with Dynamic Software
Problem 2
In a flood, a rescue boat crosses a river with speed v1
of 4 m/s , starting from port A and needs to reach
to port B, assuming that AB is perpendicular to the
river bank.
If the direction of the rescue boat is always
perpendicular to the river bank, but the water
speed is big, and after 100 seconds the rescue boat
reach to position C in another bank away 200
meters from B.
a) Find water speed v2?
b) Find the width of the river?
Problem 3
A rescue boat crosses a river with the width of 600 m. The boat speed v1 is
6 m/s and the water speed v2 is 3 m/s.
The boat starts from A, and needs to reach B, assuming that AB is always
perpendicular to the river bank.
a) At which angle that the boat direction should make with the river bank
such that the boat will reach B.
b) How long does the boat reach to port B?
```