MATH 130 · Unit 2

Polygon Area, Triangles, and Similarity

Angle sums, area strategies, similarity, and navigation angles

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    compute polygon angle sums and areas

  2. 2

    select and apply triangle area and Pythagorean formulas

  3. 3

    solve similarity proportions and convert navigation angles

Activate · 0.5 minSource: Authored

Chapter transition

Session 2 · Polygons, Triangles, and Applications

Choose formulas strategically and justify every geometric relationship.

Activate · 0 minSource: Authored

Predict: compute polygon angle sums and areas

Predict first

Why is the exterior-angle sum always 360° for a convex polygon?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

compute polygon angle sums and areas

1

Interior-angle sum: $(n-2)180^\circ$.

2

Exterior-angle sum: $360^\circ$.

3

Compound-shape area comes from decomposing into familiar pieces.

Explain · 2.5 minSource: 20, 21

Model the Skill: compute polygon angle sums and areas

1

$(8-2)180^\circ=1080^\circ$.

2

$A=\tfrac12(12+20)(7)=112$ square units.

3

Angle measure uses degrees; area uses square units.

Model · 3.0 minSource: 22, 23

You Try: compute polygon angle sums and areas

Student practice

A regular polygon has exterior angle 24°. Find the number of sides and each interior angle.

Practice · 3.0 minSource: 24, 25

Check and Diagnose: compute polygon angle sums and areas

Use n for the number of sides, not the number of triangles drawn.

Subtract holes only after computing the outer area.

Feedback · 2.0 minSource: 26, 27

Pacing checkpoint

Make It Stick: compute polygon angle sums and areas

Describe a decision process for selecting a polygon-area formula.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 28, 29

Predict: select and apply triangle area and Pythagorean formulas

Predict first

What information determines whether to use $A=bh/2$, Heron’s formula, or the equilateral formula?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

select and apply triangle area and Pythagorean formulas

1

Use $A=bh/2$ when a base and perpendicular height are known.

2

Use Heron’s formula with three sides.

3

For an equilateral triangle, $A=s^2\sqrt3/4$.

Explain · 2.5 minSource: 31, 32

Model the Skill: select and apply triangle area and Pythagorean formulas

1

$A=s^2\sqrt3/4=1600^2\sqrt3/4$.

2

$A=640000\sqrt3\approx1108512.52$ square kilometers.

3

The exact radical form is preferred unless an approximation is requested.

Model · 3.0 minSource: 33, 34

You Try: select and apply triangle area and Pythagorean formulas

Student practice

Find the area of a triangle with sides 7, 8, and 9 using Heron’s formula.

Practice · 3.0 minSource: 35, 36

Check and Diagnose: select and apply triangle area and Pythagorean formulas

Height must be perpendicular to the chosen base.

The Pythagorean theorem applies only to right triangles.

Feedback · 2.0 minSource: 37, 38

Pacing checkpoint

Make It Stick: select and apply triangle area and Pythagorean formulas

Name the given information that triggers each triangle-area method.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 39, 40

Predict: solve similarity proportions and convert navigation angles

Predict first

How does vertex order prevent a similarity proportion from using mismatched sides?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

solve similarity proportions and convert navigation angles

Concept
  • Corresponding vertices must remain in the same order.
  • DMS conversion uses 60 minutes per degree and 60 seconds per minute.
  • One nautical mile corresponds to one minute of latitude.

Explain · 2.5 minSource: 42, 43

Model the Skill: solve similarity proportions and convert navigation angles

Solve $6/9=10/x$, then convert $45^\circ22'18''$ to decimal degrees.

  1. $6x=90$, so $x=15$.
  2. $45+22/60+18/3600=45.371667^\circ$.
  3. Corresponding sides stay in the same order and DMS uses factors of 60.

Model · 3.0 minSource: 44, 45

You Try: solve similarity proportions and convert navigation angles

Student practice

Convert $73.625^\circ$ to DMS. A larger similar triangle has scale factor $3/2$ from a smaller triangle whose corresponding side is 14; find the larger side.

Practice · 3.0 minSource: 46, 47

Check and Diagnose: solve similarity proportions and convert navigation angles

Error analysis
  • Do not pair noncorresponding sides.
  • Carry minutes before seconds when converting DMS.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 49, 50

Pacing checkpoint

Make It Stick: solve similarity proportions and convert navigation angles

Explain how proportional reasoning connects similar triangles and map/navigation scale.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 51, 52

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 53, 54, 55, 57