HANDS-ON MATHS: EXPLORING SIMILARITY WITH SHADOWS AND MIRRORS

Compartido por Francisca Arpa:


One of the most effective ways to shape knowledge and cognitive skills is conducting manipulative activities or projects to engage students in the hands-on learning of History.

Option 1: Hands-on Maths

SIMILAR TRIANGLES : EXPLORING SIMILARITY WITH SHADOWS AND MIRRORS

The aim of this Project is  to identify similar triangles, corresponding sides and angles and to apply Thales’s theorem to calcule measurements of distances to inaccesible points .

Thales of Miletus wondered about the height of the  Great Pyramid in  Egypt. Thales notices that the sun’s  shadows  fell from every object in the desert at the same angle, creating similar triangles from every object. Thales’s research allowed him to use similar triangles to measure the height of the pyramids of Egypt and the distance to a ship at sea

MEASUREMENTS OF DISTANCES TO INACCESSIBLE POINTS

1.- Calculation of the height of the cypress in the schoolyard

We just need measuring tape, paper, pen and a sunny day

The students will be divided into groups of four, one of them will be the benchmark for the measurement, the other two will measure the height of the student and the respective measurements of the shadows of the tree and the student, the fourth member of the group will write down measures.

Then using Thales they will calculate the height of the cypress

The groups will present the measurement obtained and check if they have reached a similar result

2.- Working with a mirror

A mirror placed on the floor can also be used to determinate measures indirectly. When teh mirror is placed at a particular distance from the wall, the distance that and observer stands from the mirror determines the reflection that the observer sees in the mirror.

In groups of four students, they will perform the following steps

     Find a spot on the floor 8 m away from one of the walls of your classroom.

     Place a mirror on the floor, 2m from that wall

     Each gropu member should take a turn standing on the spot 10m from the wall and look into the mirror. Other group member should help the observer  locate the point on the Wall that the observer sees in the mirror and the measure the height of this point above the floor.

     Before moving the mirror, each group member should take a turn as the observer.

     Repeat the same process by moving the mirror to locations that are 3 m and 4m away from the Wall

I.-The students can use this table to record results:

Distance from the Wall to the mirror (in m)	Height of the Point on the Wall reflected in the mirror ( in m)
	Person A	Person B	Person C	Person D
2
3
4

b) Measure the eye-level height for each member of the group and record it in the table :

eye-level height for each group member
Person A	Person B	Person C	Person D

c) - Consider the data collected when the mirror was 2m from the wall

     On the diagram below, label the height of each  group member and the height of the point on th e wall determined by the group member

d)- For each person in the group, determine the ratio of the height of the point on the wall to the eye-level height of the observer

Ratio  of height of the point on the wall  tp eye-level of observer		Person A	Person B	Person C	Person D
	Ratio as a fraction
	Ratio as a decimal

e).- Repeat when the mirror was 3 from the wall and  when the mirror was 4m from the wall

 f).- Express regularity in repeated reasoning. What appears to be true about the ratios you found?

     Finally you can propose to the whole class that they discuss how they would use the mirror method to calculate the height of their classroom