Size Me Up Lesson Plan Field Guide
Size Me Up: Similar Figures
GRADE 9
Duration: 65 Minutes
LEARNING OBJECTIVE
Students will be able to use proportions to find missing side lengths in similar figures and apply this skill to solve real-world problems involving indirect measurement (shadow reckoning).
KEY CONCEPTS
Similar Figures Proportions Scale Factor Indirect Measurement Shadow Reckoning
EQUIPMENT
Meter sticks (1 per pair)
Tape measures
Scientific calculators
Slide Deck & Projector
Printed Activity Sheets
Instructional Timeline
00-10m
Introduction & Hook
Challenge students to estimate the height of the school's flagpole or tallest tree without touching it. Introduce the concept of "Shadow Reckoning" used by ancient mathematicians like Thales.
10-25m
Direct Instruction
Define similarity (same shape, different size). Review the properties: corresponding angles are congruent, corresponding sides are proportional. Model finding scale factor and solving for unknown sides using \( \frac{a}{b} = \frac{c}{x} \).
25-35m
Guided Practice
Walk through nested triangles (triangles within triangles) and mirror-on-the-ground problems. Ensure students identify corresponding parts correctly before setting up cross-multiplication.
TEACHER NOTES & STRATEGIES
Common Misconceptions
Mismatched ratios (e.g., small side over large side = large side over small side).
Confusing similarity with congruence.
Incorrectly identifying corresponding sides in rotated figures.
Failing to include units in final answers.
Check for Understanding
Questions to ask during instruction:
"If I double all the side lengths, does the shape change? Does the scale factor change?"
"Why do we need the sun's rays to be parallel for shadow reckoning to work?"
"How can we verify that these two triangles are actually similar?"
Differentiation
Support
Provide highlighters to color-code corresponding sides. Use a simplified worksheet with the proportion structure \( \frac{\text{__}}{\text{__}} = \frac{\text{__}}{\text{__}} \) pre-drawn.
Extension
Challenge students to calculate the height of an object when shadows aren't available (e.g., using a mirror on the ground and eye-level height).
Success Criteria
Set up a proportion correctly for two similar triangles.
Solve for a missing variable using cross-multiplication.
Translate a word problem into a geometric diagram.
Vocabulary Review
Similar Same shape, diff size
Congruent Identical size/shape
Ratio Comparison of 2 nums
Proportion Two equal ratios
Size Me Up Slides GEOMETRY UNIT 4
SIZE ME UP
Similar Figures & Indirect Measurement
How Tall is That?
Imagine you need to know the height of the school's flagpole or a giant redwood tree...
"How can you measure something you can't reach?"
No ladders allowed!
No drones allowed!
Just geometry.
Similar Figures
THE CRITERIA
Two figures are similar if they have the same shape but not necessarily the same size.
1
Corresponding Angles
Are congruent (the same!)
2
Corresponding Sides
Are proportional
A
B
The Proportion Formula
Side 1 (Small)
Side 1 (Large)
=
Side 2 (Small)
Side 2 (Large)
RULE OF THUMB
Keep your "categories" consistent! If the small triangle is on top in the first ratio, it MUST be on top in the second ratio.
WORKED EXAMPLE
Level 1
6 cm 4 cm 15 cm x
1. Set up the proportion
4 / x = 6 / 15
2. Cross-multiply
6x = 60
x = 10 cm
Shadow Reckoning
Over 2,500 years ago, a Greek philosopher named Thales used a simple stick to measure the height of the Great Pyramid of Giza.
THE SECRET
"The rays of the sun hit the earth at the same angle for both objects, creating two similar right triangles."
Height (H) Shadow (L) h s
FIELD LAB
01
MEASURE the height and shadow of your meter stick.
02
MEASURE the shadow of your "tall object."
03
CALCULATE the actual height using proportions.
"Precision counts! Measure to the nearest millimeter."
Similar Figures Practice Worksheet SURVEYOR'S FIELD NOTES
Similar Figures & Proportions
Surveyor:
Date:
The Rule of Ratios
Two figures are similar if their corresponding angles are equal and their sides are proportional: \( \frac{\text{Side A}_1}{\text{Side B}_1} = \frac{\text{Side A}_2}{\text{Side B}_2} \)
1 IDENTIFYING SIMILARITY & SCALE FACTOR
1. Figure A and Figure B are similar. Find the Scale Factor from A to B.
4 cm Fig A 8 cm Fig B
Scale Factor:
2. Determine if these two triangles are similar. Compare the ratios of corresponding sides.
5 4 10 8
Ratios & Conclusion:
2 SOLVING FOR MISSING SIDES
14 10 28 x
3. Find the value of \( x \). Figures are similar.
Show Work (Proportion):
4 NESTED TRIANGLES (TRIANGLE WITHIN A TRIANGLE)
12 m 6 m 4 m y
4. The two triangles share an angle and are similar. Solve for \( y \).
Hint: Identify the base and height for both the small triangle and the large overall triangle.
Work Space:
INDIRECT MEASUREMENT CHALLENGE
A surveyor wants to find the height of a radio tower. At the same time of day, a 5-foot fence post casts a 2-foot shadow. The radio tower casts a 36-foot shadow.
Step 1: Sketch & Label
Sketch triangles here
Step 2: Calculate
Final Height = __________ feet
GEO-UNIT-04-SIMILARITY PAGE 2 OF 2
Measure a Tall Object Lab FIELD OPERATION
Operation: Shadow Reckoning
SURVEYOR LAB 4.2
Location: ________________
Surveyor:
Partner:
The Mission
Determine the height of a Tall Object (Flagpole, Tree, or Wall) using only a meter stick and the power of the Sun. You cannot reach the object directly.
Step 1: Raw Measurements (Units: _______ )
Target Object Height Shadow Length Meter Stick (Reference) Tall Object (Target) ??? Find Me ???
Step 2: Geometric Sketch
Sketch the two similar triangles formed by the Sun's rays. Label all known measurements and use 'H' for the unknown height.
[Drawing Space]
Step 3: Proportion & Calculation
A. Set up your ratio equation:
=
B. Show the cross-multiplication steps:
Final Determination
Based on our field measurements and geometric calculations, the estimated height of the ________________ is:
____________ UNITS
Debriefing
1. Why is it important that you measure the meter stick's shadow and the object's shadow at the exact same time of day?
2. Name one factor in the environment that might have made your measurement slightly inaccurate (error analysis).
Size Me Up Exit Ticket FINAL CHECK-IN
Size Me Up: Similar Figures
Surveyor:
1. A person who is 6 feet tall casts a 4-foot shadow. At the same time, a nearby tree casts a 10-foot shadow. How tall is the tree?
6' 4' x 10'
Show Work:
Height = __________ ft
2. List the two requirements for two triangles to be considered similar:
CUT HERE
FINAL CHECK-IN
Size Me Up: Similar Figures
Surveyor:
1. A person who is 6 feet tall casts a 4-foot shadow. At the same time, a nearby tree casts a 10-foot shadow. How tall is the tree?
6' 4' x 10'
Show Work:
Height = __________ ft
2. List the two requirements for two triangles to be considered similar:
Size Me Up Answer Key ANSWER KEY
Surveyor's Field Notes & Assessments
Teacher Resource
1 Practice Worksheet Solutions
1. Scale Factor (A to B)
Answer: k = 2
\( \text{Scale Factor} = \frac{\text{New}}{\text{Original}} = \frac{8}{4} = 2 \)
2. Similarity Check
Answer: YES, Similar
Compare ratios: \( \frac{8}{4} = 2 \) and \( \frac{10}{5} = 2 \). Since ratios are equal, triangles are similar.
3. Solving for X
PROPORTION
\( \frac{10}{x} = \frac{14}{28} \)
\( 14x = 280 \)
\( x = 20 \)
x = 20
4. Nested Triangles (y)
ADVANCED
\( \frac{4}{y} = \frac{6}{12} \)
\( 6y = 48 \)
\( y = 8 \)
y = 8 m
Tower Challenge Key
Ratio: \( \frac{\text{Fence}}{\text{Tower}} = \frac{\text{Shadow 1}}{\text{Shadow 2}} \)
\( \frac{5}{H} = \frac{2}{36} \)
\( 2H = 180 \)
Height = 90 ft
Exit Ticket Key
Q1: Tree Calculation
Answer: 15 feet
\( \frac{6}{x} = \frac{4}{10} \rightarrow 4x = 60 \)
Q2: Similarity Requirements
Corresponding angles are congruent.
Corresponding sides are proportional.
