Grade 7
The Human Knowledge Project
Unit 5 — Geometry, Transformations, Probability, and Statistics
Welcome back!
Mathematics helps us describe shapes, patterns, movement, and uncertainty.
Geometry allows us to measure and design the world around us.
Probability helps us predict how likely events are to occur.
Statistics helps us understand information and make informed decisions.
Today you will strengthen your understanding of geometry, transformations, probability, and data analysis.
Geometric Transformations
A transformation changes the position, orientation, or size of a figure.
The four basic transformations are:
- translation
- reflection
- rotation
- dilation
Transformations help mathematicians describe how figures change while preserving certain properties.
Translation
A translation slides a figure without changing its size or shape.
Every point moves:
- the same distance
- in the same direction
Translations preserve:
- length
- angle measure
- area
- orientation
Reflection
A reflection flips a figure across a line called the line of reflection.
The reflected figure is a mirror image of the original.
Reflections preserve:
- size
- shape
- angle measure
- distance
Orientation changes after a reflection.
Rotation
A rotation turns a figure around a fixed point.
Rotations may be:
- clockwise
- counterclockwise
Common rotations include:
- 90°
- 180°
- 270°
Rotations preserve both size and shape.
Dilation
A dilation changes the size of a figure using a scale factor.
If the scale factor is:
- greater than 1, the figure enlarges.
- between 0 and 1, the figure reduces.
Dilations preserve shape but do not preserve size.
Circles
A circle is the set of all points that are the same distance from a center.
Important vocabulary includes:
- center
- radius
- diameter
- circumference
- chord
- arc
These measurements help describe circular objects found throughout nature and engineering.
Probability
Probability measures how likely an event is to occur.
Probability ranges from:
- 0 (impossible)
- 1 (certain)
Examples include:
- drawing a card
- rolling dice
- weather forecasting
- scientific experiments
Understanding probability helps us evaluate uncertainty.
Statistics
Statistics helps us collect, organize, analyze, and interpret data.
Common displays include:
- bar graphs
- line graphs
- circle graphs
- histograms
- box plots
- scatter plots
Different graphs reveal different kinds of information.
Interpreting Data
When examining data, ask:
- What patterns appear?
- Are there unusual values?
- What conclusions are supported?
- Are there possible sources of error?
- What additional information would help?
Good statisticians rely on evidence rather than assumptions.
Activity
Complete the following:
- perform five translations
- perform five reflections
- perform five rotations
- perform five dilations
- solve ten probability problems
- analyze three sets of statistical data
- explain your reasoning for every solution
Practice
Answer these questions.
What is a geometric transformation?
How does a translation differ from a reflection?
What changes during a dilation?
What does probability measure?
Why are statistics useful?
How can graphs help us understand data?
Review
Today you learned:
- transformations describe changes in geometric figures
- translations, reflections, and rotations preserve size and shape
- dilations change size while preserving shape
- circles contain several important measurements
- probability measures likelihood
- statistics help organize and interpret information
- data analysis supports informed decision-making
Geometry, probability, and statistics allow us to understand patterns, movement, measurement, and uncertainty in both mathematics and everyday life.
Looking Ahead
Next we will begin Grade 7 Science by studying the structure of matter, atoms, elements, compounds, and the physical principles that govern the material world.