Ratio Rig Facilitation Guide Intervention Facilitation Guide
Lesson: Ratio Rig (Standard HS.G-SRT.C.6)
Tier 2 Small Group
Learning Objective
Students will explain why ratios of side lengths in right triangles are consistent for a given acute angle across similar triangles, and use these ratios to define sine, cosine, and tangent.
Key Vocabulary
Similarity Ratio Hypotenuse Opposite Adjacent Trig Ratios
Materials Needed
Ratio Rig Slide Deck
Side Sleuths Worksheet
Trig Talk Vocab Map
Calculators
Colored highlighters (3 colors)
Instructional Routine
1
Engage
5 min
The Similarity Hook
Goal: Activate prior knowledge of AA similarity in right triangles.
Do: Project Slide 2 showing two different-sized right triangles with a shared \(30^{\circ}\) angle.
Say: "Look at these two triangles. They are different sizes, but they both have a \(90^{\circ}\) and a \(30^{\circ}\) angle. How do we know for sure they are similar?"
Ask: "If they are similar, what do we know about the relationships between their sides?"
2
Explore
15 min
The Ratio Reveal
Goal: Discover that side ratios are constant for a specific angle.
Do: Hand out "Side Sleuths Worksheet". Have students measure or use provided lengths to calculate \(\frac{\text{height}}{\text{hypotenuse}}\) for both triangles.
Say: "We are going to find a 'hidden constant'. Even though the sides are longer in the big triangle, let's see what happens to the ratio."
Look For: Students struggling with decimal division; provide calculators and encourage rounding to two decimal places.
3
Explain
10 min
Formalizing the Ratios
Goal: Introduce Sine, Cosine, and Tangent using the SOH CAH TOA mnemonic.
Do: Use the "Trig Talk Vocab Map". Model how to identify the Reference Angle first. Use highlighters to color-code sides: Hypotenuse (Red), Opposite (Blue), Adjacent (Green).
Say: "Mathematicians gave these ratios special names because they are so reliable. Sine is just the name for the ratio of the Opposite side to the Hypotenuse."
Scaffold: For students struggling with 'Opposite', use the 'Flashlight Method'—the angle shines a light on the opposite side.
4
Apply
10 min
Guided Practice
Goal: Calculate ratios for different angles.
Do: Complete Part 2 of the "Side Sleuths Worksheet" together. Focus on identifying how 'Opposite' and 'Adjacent' switch depending on which acute angle we choose.
Common Misconceptions
The "Hypotenuse" Switch
Students may try to label the hypotenuse as 'Opposite' or 'Adjacent'.
Fix: Always label the Hypotenuse first (across from the right angle).
Angle vs. Side
Students may try to put the degree measure into the ratio instead of the side length.
Fix: Emphasize "Ratio = Side / Side". The angle just tells us which sides to pick.
Progress Monitoring
Level Student Can... Next Step Beginning Identify the hypotenuse but confuses opposite/adjacent sides. Use the 'Flashlight' visual for the reference angle. Developing Set up the ratios correctly using SOH CAH TOA but makes calculation errors. Provide a calculator check and verify numerator vs denominator. Proficient Correctly calculates ratios and explains why similarity makes them constant. Introduce solving for missing sides using the ratios.
Ratio Rig Slides Ratio Rig
Bridging Similarity & Trig
The Similarity Connection
When two right triangles share an acute angle, they are similar by AA similarity.
Big Idea:
"If they have the same shape, their side ratios must be the same!"
30° 10 cm 5 cm Bigger version?
Identifying Our Tools
H
Hypotenuse
The longest side. Always across from the right angle.
O
Opposite
The side "across the way" from our chosen angle.
A
Adjacent
The side "next to" our angle (not the hypotenuse!).
The "Trig" Trio
Sine
=
Opposite
Hypotenuse
Cosine
=
Adjacent
Hypotenuse
Tangent
=
Opposite
Adjacent
The Secret Code
SOH
Sine | Opp | Hyp
CAH
Cos | Adj | Hyp
TOA
Tan | Opp | Adj
"Some Old Hippie Caught Another Hippie Tripping On Acid"
(Or your favorite school-appropriate version!)
Side Sleuths Worksheet Side Sleuths
Ratio Investigation Worksheet
Name:
Date:
Part 1: The Ratio Reveal
Below are two similar right triangles. They share a \(37^{\circ}\) angle. Use the given side lengths to calculate the ratios.
37° 4 cm (Adj) 3 cm (Opp) 5 cm (Hyp) A
\(\frac{\text{Opposite}}{\text{Hypotenuse}} =\)
\(\frac{\text{Adjacent}}{\text{Hypotenuse}} =\)
37° 8 cm (Adj) 6 cm (Opp) 10 cm (Hyp) B
\(\frac{\text{Opposite}}{\text{Hypotenuse}} =\)
\(\frac{\text{Adjacent}}{\text{Hypotenuse}} =\)
🔍 What do you notice?
Part 2: The Flashlight Test
Your Reference Angle determines which side is 'Opposite'.
θ 12 5 13
Focus on angle θ (theta)
Opposite:
Adjacent:
Hypotenuse:
Sine θ = ____ / ____
β 12 5 13
Focus on angle β (beta)
Opposite:
Adjacent:
Hypotenuse:
Sine β = ____ / ____
Part 3: The Big Trio
Calculate the decimal value (to 2 decimal places) for the ratios below.
28° 15 8 17
SIN (28°)
Opp / Hyp
COS (28°)
Adj / Hyp
TAN (28°)
Opp / Adj
Trig Talk Vocab Map Trig Talk
Visual Vocabulary Organizer
The Reference Frame
θ Reference Angle Hypotenuse Opposite Adjacent
1
Pick a Reference Angle (not the 90° angle!).
2
Label the Hypotenuse (the longest side).
3
Label the Opposite side (across from the angle).
4
Label the Adjacent side (next to the angle).
SOH CAH TOA
The Universal Mnemonic
SOH
CAH
TOA
Sine
SOH
The ratio of the side opposite to the hypotenuse .
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Cosine
CAH
The ratio of the side adjacent to the hypotenuse .
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Tangent
TOA
The ratio of the side opposite to the adjacent .
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Why does this work?
Because of similarity , these ratios are the same for every right triangle with the same angle \(\theta\).
Trig Check Exit Ticket Trig Check: Exit Ticket
Ratio Rig
Name:
1. For the angle marked with X, label the sides as Opposite , Adjacent , or Hypotenuse .
X
Long Side:
Across Side:
Next-to Side:
2. Write the Sine ratio for angle X in the triangle above using these lengths: Opp=12, Hyp=13.
Sine X =
3. TRUE or FALSE?
"If I double the size of a right triangle but keep the angles the same, the ratio of its sides will also double."
TRUE
FALSE
Trig Check: Exit Ticket
--- DUPLICATE COPY ---
