wi10
Mathematics for Teaching Portfolio
EDSE 437
Reflective Response to Mathematical Tasks
TITLE: Measuring Distance and Height with Similar Triangles
Curricular Connection: Alberta Mathematics & ICT Program of
Studies
Grade Level 9
(& Stream) Shape and Space (3-D Objects and 2-D Shapes)
Topic
3. Demonstrate an understanding of similarity of polygons.
Specific
outcomes
Solve a given problem that involves the properties of similar triangles.
Mathematics and Teaching
This activity applies the knowledge of similar triangles to angular
measurements so that the heights or distances of objects can be
found. Perfectly suitable to outdoor measurements to find the
heights of trees or buildings, or in the classroom if necessary.
Meter sticks and other linear measuring devices are the necessary
materials, along with recording material.
Students will work in small groups of about three each, where
Description of some students can measure, while others record data and calculate
Activity;
the resulting height or distance of the object.
materials and
special
preparation
required;
Teaching
considerations
Firstly, properties of similar triangles should be reviewed with the
wi10
Mathematics for Teaching Portfolio
EDSE 437
class, namely that the proportions of the sides are constant in
similar figures. Define the characteristics of similar figures and
why the proportions would not be constant in non-similar figures.
Next, present the class with the above figure and have them
identify the two similar triangles and justify how they know they
are similar.
-all angles are the same
-side lengths are different
-Thus the ratio of side lengths will remain constant
Optional
If students are not too familiar or need more work with similar
triangles, selected questions from the attached handout can be
completed to aid in understanding the concepts.
Indirect Measurement Using Similar Triangles
In groups, students will choose objects in the classroom or outside
(such as trees) to indirectly measure their height.
The students will measure the apparent height of an object at a set
distance using meter sticks or other measuring devices, and using
this information in combination with the total distance to the
object, they can find the actual height.
Object
Apparent Height
Distance at
Apparent Height
Total Distance to
Object
Tree
2m
3m
30m
Building
The students will then find the common factor between the
apparent height and apparent distance. This factor should be the
same between the total height and total distance.
wi10
Mathematics for Teaching Portfolio
EDSE 437
MEASUREMENT: Some attributes of objects are measurable and can be
quantified using unit amounts.
The Big Ideas ORIENTATION & LOCATION: Objects in space can be oriented in an infinite
(Overarching number of ways, and an object’s location in space can be described
mathematical quantitatively.
themes);
ESTIMATION: Numerical calculations can be approximated by replacing
numbers with other numbers that are close and easy to compute with mentally.
Measurements can be approximated using known referents as the unit in the
measurement process.
Program of Studies: Mathematical Processes
CN Connections
Through this activity, students will see how the proportions of
similar triangles remain constant, and can connect this with the
general polygon case.
R Reasoning
Students must provide arguments as to why two triangles in a
Mathematical
diagram are similar. They will see that two angles are necessary
Processes
the same, and will use reasoning to know that the third angle must
then be congruent too in a triangle. The reasoning also extends to
justifying the ratio of sides in similar triangles.
V Visualization
By performing the measurements, students can see how the
triangle created by measuring the apparent sizes of the objects is
similar in properties to the triangle made with the object itself.
Differentiating Instruction
Extensions, Extensions of similar triangles can go into more abstract
diversity and definitions and proofs of properties (such as introducing Angleremediation; Angle congruency, Side-Side-Side congruency, etc.) for advanced
learners. The properties of angles can also be investigated, and
some trigonometry can be introduced.
Remediation in this activity can be done through co-operative
group work. Students of varying ability can be placed together to
assist each other with the learning of the concepts and
measurements.
Assessment
wi10
Mathematics for Teaching Portfolio
EDSE 437
Have students work through the attached sheet labeled 'Space
Math.' This is a good formative assessment for the students.
A summative assessment would fit better for an end-of unit exam
including similar polygons and scale factor.
Reflection
This activity is a very hands-on approach to applying similar triangles. The
distinctive feature of this lesson is showing the utility of mathematics in the real
world. By using similar triangles in a real application, the students will get a
good 'feeling' for how the mathematics behind it works. Rather than being able
to quickly compute what an unknown side length of a polygon is, this lesson is
aimed at forming a relational understanding of similar triangle (Skemp, 1986).
This lesson is also very open to modifications and varying applications. For
example, if the height of a distant building is known, the distance to it can be
measured. Similar triangles can be applied in many other situations, as shown
on the handout. Students should feel free to experiment and apply this
knowledge to different cases, or to calculate the missing side in various ways, as
outlined in Bass's paper on algorithms (Bass, 2003).