The document discusses two approaches to calculating the distance between two points: the traditional distance formula approach and an alternative approach using a drawing and the Pythagorean theorem. It provides an example of finding the distance between points (-3,4) and (5,-3) and step-by-step works through the alternative approach of drawing a right triangle between the points and applying the Pythagorean theorem to calculate the distance as the hypotenuse of 10.6 units. The document suggests asking students to estimate distances, draw diagrams, calculate, and reflect on their work.

1. Topic: the distance between two points.
TRADITIONAL APPROACH:
Distance-formula:
y
Q
(x2,y2)
P (x ,y )
1 1
0 x
The distance between P and Q ( PQ ) is
(x 2 −1 )2 +y 2 − 1 )2
x ( y
Example: Find the distance between the points with
coordinates (-3,4) and (5,-3).
Solution: ( (−3) − 5) 2 + ( 4 − (−3) ) 2 = 64 + 49 = 113 ≈ 10.6
ALTERNATIVE APPROACH:
Start:
Find the distance between the points with coordinates (-3,4)
and (5,-3).
STEP 1: Draw the situation! (The situation will get more
meaning)

2. y
(-3,4)
1
0 1 x
(5,-3)
STEP 2: So, what do you think the distance between the two
points will be? 3? Or 5? Or greater ? Why?
(ESTIMATE!!)
STEP 3: We are going to find out. We will use a right
triangle, and Pythagorean theorem.
y
C
(-3,4)
1
0 1 x
B
A (5,-3)
Recall:Pythagorean Theorem for the right triangle above:
AB2+AC2=BC2
• What is the distance between A and B? (You can just
count: 8) (USE OF COMMON SENSE, OF WHAT YOU
ALREADY KNOW)
(If you want to express it already in terms of the x-coordinates: |
5-(-3)|=8; but maybe better to first show that you don’t need any
‘complex’ notations…)

3. • What is the distance between A and C? (You can just
count: 7) (USE OF COMMON SENSE, OF WHAT YOU
ALREADY KNOW)
(If you want to express it already in terms of the y-coordinates: |
(-3)-4|=7; but maybe better to first show that you don’t need any
‘complex’ notations…)
• So what is the distance between B and C? With use of
Pythagorean theorem:
(BC)2= 82+72 = 64+49 = 113, so the distance between B and C
is 113 = 10.6
(CALCULATE!!)
STEP 4: From your first estimation and the drawing, you
think 10.6 could be the correct answer? So you did not make
any mistakes?
(REFLECTION!!)
After this, you can give one other example, or you can
already let them work. You could even ask the students to
give a general formula of the distance between two points, if
they have practised enough exercises.
You can also ask a CONTEXT-problem in which distance is
concerned, for instance:

4. Mark lives 3 km from school and Rona lives 5 km from
school.
a) What is the maximum possible distance between their
two houses and what is the minimum possible distance
between their two houses?
b) If the situation is as illustrated below, how far do Mark
and Rona live apart?
Rona
school Mark
c) If you know that Rona and Mark live 4 km apart, draw in
the graph below the possible locations for Mark’s
house:
Rona
school
How did you find the locations?