TASKS FOR THE MATHEMATICAL PROBLEM SOLVING

ON THE SITUATIONS OF THE FLOOD

(for secondary students: Grade 8 & 10)

Tran VuiHue University of Education, Vietnam

APEC-Tsukuba International Conference VIIInnovation of Mathematics Education through Lesson Study

Challenges to Emerency Preparedness for Mathematics Focusing on Flood and Typhoon

14-17 February 2013Tokyo, Japan

1. Introduction: Typhoons & Floods in Vietnam

People and Flood in Vietnam

50 years Typhoons in Vietnam1954-2006

380 Typhoons Affected to Vietnam8 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

• 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?

2. A point of View to Design Mathematical Problems

• 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.

MATHEMATISATIONThe process of “converting”

from Real World Model into Mathematical Model

Translate

MATHEMATIZATION: Simplified Mathematical ModelingSince 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

Concrete

Abstract

2. Mathematical ProblemsKnowledge PrerequisitesGrade 8: 2nd Semester & Grade 10: 1st Semester, Vietnamese Curriculum.

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 MeasurementSemester 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 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.

The Flowing of Rivers and Streams

Bankfull cross section

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 1How to estimate the area of the bankfull cross sectional area? Show your mathematical reasoning.

Problem 2The 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 3After 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?

Velocity with and against the waterSemester 2, Grade 8, 13-14 years. 45 minutes

LESSON 2

LINEAR EQUATIONS

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.

( 65 - x) 2 = (65 + x) 1.25

Mathematical Model

Real World Model

Visual Representation

Algebraic Representation

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 hoursDistance = 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 2Sketch 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 xx increases 1 unit then the difference d(x) decreases 3.25 units.

The difference d(x) = 048.75 = 3.25 xx = 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.

605.2

150 yx

405.2

100 yx

5010024060

xxyxyx

a) With the water:

b) Against the water:

c) We have a system of 2 equations:

km/h (*)

km/h (**)

HINT:

LESSON 3

THE SUM OF TWO VECTORSSITUATION: CROSS A RIVER BY rescue boat IN A FLOODSemester 1, Grade 10, 15-16 years. Time: 45 minutes

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.

Problem 1

Mathematical Model

Real World Model

a) Where does the rescue boat reach to another bank?b) What happens if v1 = v2?c) What happens if v1 < 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?