Zoom Lens Slides Graphics Studio 1.0
THE ZOOM LENS
Introduction to Dilations & Scale Factors
The "Perfect Zoom"
Have you ever resized an image on your phone and noticed it got "crunchy" or pixelated? Or maybe you've used a map and needed to know exactly how much the real world was "shrunk" to fit on your screen?
Scaling Up
Scaling Down
DILATION
Definition
A transformation that produces an image that is the same shape as the original, but a different size .
"Think of your pupils dilating in the dark—they stay circular, but they grow larger to let in more light!"
A
A'
THE SCALE FACTOR (\(k\))
1
Enlargement
\(k > 1\)
The image becomes larger than the pre-image.
Example: \(k = 2\) means the figure doubles in size.
2
Reduction
\(0 < k < 1\)
The image becomes smaller than the pre-image.
Example: \(k = \frac{1}{2}\) means the figure is half the size.
What happens if \(k = 1\)? The size stays exactly the same!
STUDIO CHALLENGE: GROW OR SHRINK?
\(k = 3\)
Does the shape grow or shrink?
Reduction
Enlargement
\(k = 0.5\)
Does the shape grow or shrink?
Reduction
Enlargement
\(k = \frac{3}{4}\)
Does the shape grow or shrink?
Reduction
Enlargement
Scale Factor Teacher Guide Scale Factor Guide
Teacher Facilitation Resource
Lesson 1
The Zoom Lens
Learning Objectives
Define dilation as a non-rigid transformation.
Determine if a dilation is an enlargement or reduction based on the scale factor \(k\).
Understand the multiplicative relationship between pre-image and image side lengths.
Key Vocabulary
Pre-Image
The original figure before any transformation.
Image
The resulting figure after a transformation (denoted by prime notation, e.g., \(A'\)).
Scale Factor (\(k\))
The ratio of the image side length to the corresponding pre-image side length.
Center of Dilation
The fixed point from which all points are enlarged or reduced.
Common Pitfalls
Adding instead of Multiplying
Students may think \(k=2\) means "add 2" to the side length. Emphasize that dilations are proportional and multiplicative.
Fractions = Reduction?
Students often assume any fraction results in a reduction. Remind them that improper fractions like \(\frac{3}{2}\) are \(> 1\) and cause enlargement.
Negative Numbers
Note: In 8th grade, we focus on positive scale factors. Negative scale factors involve a rotation and are usually introduced later.
Discussion Framework
1
"If a photo is 4 inches wide and I use a scale factor of \(k=3\), how wide is the new photo?"
Expected answer: 12 inches. Use this to verify the multiplicative rule.
2
"Can a scale factor be a decimal? If \(k=0.75\), is it an enlargement or reduction?"
Expected answer: Reduction, because \(0.75\) is less than \(1\).
3
"If an image is half the size of the pre-image, what is the scale factor?"
Expected answer: \(k = 0.5\) or \(k = \frac{1}{2}\).
Studio Action Items
The Worksheet Hook
When handing out the Zoom Analysis worksheet, tell students they are "Quality Control Technicians" for a graphics studio. They must verify that resized assets are mathematically consistent.
Extension Idea
Challenge advanced students to calculate the scale factor given two side lengths: \(k = \frac{\text{Image}}{\text{Pre-Image}}\). For example, if pre-image is 5 and image is 15, \(k = 3\).
Zoom Analysis Worksheet Zoom Analysis
Graphics Studio Unit: Lesson 1
Name:
Date:
Mission Brief
You are a Quality Control Technician at Dilation Design Studio. Your job is to analyze scale factors and ensure all graphic assets are being resized correctly.
PART 1: SCALE FACTOR CLASSIFICATION
Determine if the following scale factors (\(k\)) will result in an Enlargement , a Reduction , or No Change .
\(k = 4\)
Enlarge
Reduce
\(k = 0.25\)
Enlarge
Reduce
\(k = \frac{1}{3}\)
Enlarge
Reduce
\(k = \frac{5}{2}\)
Enlarge
Reduce
\(k = 1.05\)
Enlarge
Reduce
\(k = 1\)
Enlarge
Reduce
PART 2: STUDIO MEASUREMENTS
Calculate the missing side length for each dilated figure.
Pre-image Length: 8 units
Scale Factor: \(k = 3\)
Calculation Space:
Image Length:
Pre-image Length: 20 units
Scale Factor: \(k = 0.5\)
Calculation Space:
Image Length:
Pre-image Length: 6 units
Image Length: 12 units
Determine the Ratio (Image/Pre-image):
Scale Factor (\(k\)):
PART 3: DESIGNER INSIGHT
You have a logo that is 10cm tall. You need to fit it onto a business card that is only 2cm tall. What scale factor should you use, and why?
Coordinate Growth Slides COORDINATE GROWTH
Mapping Dilations to the Grid
(x, y)
(kx, ky)
THE DILATION RULE
ALGORITHM
To dilate a figure from the origin (0,0), multiply every coordinate of every vertex by the scale factor \(k\).
(x, y)
(kx, ky)
ENLARGEMENT IN ACTION
Pre-image Vertex \(A\):
(2, 4)
Scale Factor \(k = 3\):
(2 \(\cdot\) 3, 4 \(\cdot\) 3) = (6, 12)
A(2, 4)
A'(6, 12)
REDUCTION IN ACTION
Reducing works the same way! Just multiply by the fraction or decimal.
Vertex \(B\): (10, -6)
Scale \(k = 0.5\): Multiply by 0.5
New Vertex \(B'\): (5, -3)
🔬
"Multiplying by 0.5 is the same as dividing by 2."
THE ANCHOR
In our studio, we always dilate from the ORIGIN (0,0) .
Visualizing the Dilation:
Draw a line from the origin through any pre-image vertex. The new image vertex will always lie on that same line!
Origin (0,0)
Grid Growth Practice Sheet Grid Growth
Mapping Transformations with (kx, ky)
Name:
Period:
PART 1: THE MULTIPLICATION LAB
Apply the scale factor (\(k\)) to each pre-image vertex to find the new image coordinates.
Vertex Pre-image \((x, y)\) Scale Factor \(k\) Image \((kx, ky)\) A (2, 3) \(k = 2\) B (-4, 0) \(k = 3\) C (10, -8) \(k = \frac{1}{2}\) D (6, 9) \(k = \frac{1}{3}\) E (-3, 5) \(k = 4\)
PART 2: THE COORDINATE GRID
Step 1: Pre-image
Graph Triangle \(TUV\) with vertices:
\(T (1, 1)\)
\(U (4, 1)\)
\(V (1, 3)\)
Step 2: Apply \(k = 2\)
Find the new coordinates:
\(T' (\_\_\_, \_\_\_)\)
\(U' (\_\_\_, \_\_\_)\)
\(V' (\_\_\_, \_\_\_)\)
Step 3: Graph Image
Plot the new vertices and connect them to form Triangle \(T'U'V'\).
0
2
4
6
8
10
2
4
6
8
10
y-axis
x-axis
Studio Note: Always verify your multiplication before plotting points.
Grid Growth Answer Key ANSWER KEY
Material: Grid Growth Practice Sheet
Teacher Resource
Lesson 2
PART 1: THE MULTIPLICATION LAB
Vertex Calculation Correct Answer A \((2 \cdot 2, 3 \cdot 2)\) (4, 6) B \((-4 \cdot 3, 0 \cdot 3)\) (-12, 0) C \((10 \cdot 0.5, -8 \cdot 0.5)\) (5, -4) D \((6 \cdot \frac{1}{3}, 9 \cdot \frac{1}{3})\) (2, 3) E \((-3 \cdot 4, 5 \cdot 4)\) (-12, 20)
PART 2: THE COORDINATE GRID
Calculated Vertices for \(T'U'V'\):
T' (2, 2) ...was (1,1) \(\cdot\) 2
U' (8, 2) ...was (4,1) \(\cdot\) 2
V' (2, 6) ...was (1,3) \(\cdot\) 2
Check Point:
The area of the image triangle should be 4 times the area of the pre-image triangle (Scale Factor squared), and the side lengths should be exactly double.
Visual Verification
The resulting triangle \(T'U'V'\) should look exactly like triangle \(TUV\) but shifted further from the origin and twice as large. A straight line drawn from the origin (0,0) through vertex \(T\) should pass directly through \(T'\).
Shape Twins Discovery Lab Shape Twins
Discovery Lab: Similarity Inquiry
Lesson 3
Essential Question
How does a dilation change a shape, and what properties stay exactly the same?
STEP 1: THE DATA COLLECTION
Observe Triangle \(ABC\) and its dilated image Triangle \(A'B'C'\) below. The scale factor used was \(k = 2\).
Pre-image measurements
Side \(AB\): 3 cm
Angle \(A\): 90°
Side \(BC\): 5 cm
Angle \(B\): 37°
Side \(AC\): 4 cm
Angle \(C\): 53°
Image measurements (\(k=2\))
Side \(A'B'\): 6 cm
Angle \(A'\): _______
Side \(B'C'\): 10 cm
Angle \(B'\): _______
Side \(A'C'\): 8 cm
Angle \(C'\): _______
STEP 2: ANALYSIS & PREDICTION
1. Look at the side lengths. How do the Image lengths relate to the Pre-image lengths? (Be specific!)
2. Predict: If the original angle \(A\) is 90°, what do you think the dilated angle \(A'\) will be? Why?
Discovery Moment
"During a dilation, angles stay the same. Only side lengths change proportionally."
STEP 3: THE VERDICT
Congruent
Are these triangles congruent? (Same size and same shape)
YES
NO
Similar
Are these triangles similar? (Same shape, proportional size)
YES
NO
Conclusion:
Dilations create figures that are _________________ but NOT _________________.
Similarity Anchor Chart SIMILAR VS CONGRUENT
≅
CONGRUENT
SAME Shape
SAME Size
SAME Angles
Caused By:
Rigid Motions (Translation, Reflection, Rotation)
~
SIMILAR
SAME Shape
DIFFERENT Size
SAME Angles
Caused By:
Dilations
💡
The Proportionality Rule
In similar figures, side lengths are proportional . If one side doubles, every side doubles!
Logo Lab Project Guide LOGO LAB
Design Project Phase 1
Designer Name
Your Creative Brief
You've been hired as the Lead Graphic Designer for a new startup. Your task is to design a unique, geometric logo and then calculate the math needed to scale it for two different applications:
The Billboard: An enlargement of your logo (Scale factor \(k > 1\)).The Stamp: A reduction of your logo (Scale factor \(0 < k < 1\)).
Requirements
Logo must have at least 5 distinct vertices .
All original coordinates must be within the first quadrant \((0-10)\).
You must use at least one fractional scale factor.
Calculations must be shown for all 3 versions.
PHASE 1: THE PRE-IMAGE
List the coordinates for your original design. Aim for clear, simple geometric shapes.
Vertex Pre-image \((x, y)\) Billboard \((k = \_\_\_)\) Stamp \((k = \_\_\_)\) A B C D E F G
Ready to sketch?
Use the Graphic Design Grid to draw your Pre-image first!
Logo Design Grids Sheet GRAPHIC DESIGN GRIDS
Designer:
1. THE ORIGINAL LOGO (Pre-image) Grid: 1 unit per square
2. THE STAMP (Reduction)
k = _________
3. THE BILLBOARD (Enlargement)
k = _________
All coordinates must be relative to the origin (0,0) located at the bottom-left intersection.
Studio Verification Form STUDIO VERIFICATION
Quality Assurance Peer Workshop
Lesson 5: Master Plan
Mission: Math Check
In a real graphics studio, we check each other's work to prevent expensive printing errors. Exchange your Coordinate List with a partner. DO NOT show them your drawings. They must use your coordinates to recreate your logo!
Partner Information
Whose work are you checking?
The Coordinate Audit
Select 3 random vertices and check the partner's multiplication for the Billboard version.
Pre-image \(k\) Calculation
Visual QA Check
Compare the partner's Billboard drawing to their Pre-image .
Are the angles the same? (Similarity)
Is the shape stretched? (Error)
Did it stay in the lines?
Reviewer's Verdict
APPROVED REVISIONS NEEDED
Final Design Reflection
1. What was the hardest part about calculating the Stamp (reduction) coordinates?
2. How did the "Verify" process change how you feel about your math accuracy?
Logo Lab Rubric DESIGN RUBRIC
Logo Lab Final Assessment
FINAL GRADE
/ 20
Criteria Exceptional (5) Proficient (4) Developing (2-3) Mathematical Accuracy All image coordinates are calculated correctly using (kx, ky). No errors. 1-2 minor calculation errors, but overall process is clear. Multiple coordinate errors; scale factor was not applied correctly. Graphic Similarity Scaled images are perfectly similar to the pre-image. Angles and shapes are preserved. Images are mostly similar; minor distortion in line segments. Images are not similar (stretched/squashed); shapes do not match. Scale Factor Selection Used both enlargement (k > 1) and reduction (k < 1) correctly as requested. Used both types, but scale factor calculations were basic. Only one type of scaling used, or scale factors were not used at all. Technical Precision Design is neat, professional, and uses at least 5 vertices. Plotted precisely on grid. Design meets vertex requirement; mostly neat execution. Incomplete design or fewer than 5 vertices used. Messy execution.
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