Dilation and Scale Factor

This lesson removes the physical apparatus and asks students to reason from a diagram. The task uses the same structure as Lesson 6.1 — a shadow created by a point light source, with an object and image on parallel planes — but now students work from a diagram with some measurements given and others to be found. The opening task is accessible while offering a high ceiling: groups that finish quickly can explore how perimeter and area scale, planting seeds for Lesson 6.5.

 

The critical consolidation moment in this lesson is making both methods explicit. Students who used scale factor in Lesson 6.1 should see it formalized here. Students who used proportions should see that formalized too. Both are correct, both are useful, and both must be understood. Show the same missing-length problem solved both ways. Ask: which method felt more natural? Can you do it the other way? When might one method be more efficient than the other?

 

The group note-making format works well here: ask groups to construct notes at the boards addressing dilation, scale factor, what stays the same, and one worked example. Groups share and the class builds shared notes from the best of what they produced. The premade note sheet is distributed after groups have written their own version, not before.

 

One item that requires explicit attention: correspondence order in a similarity statement. Just as in Unit 3 with congruence statements, the order of vertices in a similarity statement encodes which parts correspond. Triangle ABC ∼ Triangle DEF means A corresponds to D, B to E, C to F. A student who writes the statement with the wrong order has made a different claim. Push for precision here — it is a direct carry-over of the Unit 3 lesson on correspondence.

 

The extension questions about perimeter and area scaling are worth giving to groups that are ready, but not resolving formally. Lesson 6.5 returns to these ideas with full treatment. Students who conjecture that area scales by the square of the scale factor are doing exactly the right kind of thinking — celebrate the conjecture, record it, and tell them the unit will return to it.

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Before the note-making, address the two methods directly. Write a missing-length problem on the board and solve it using scale factor, then solve the same problem using a proportion. Name both. Ask students which method they used in the task. Ask if they can do it the other way.

 

Scale factor method: find k, multiply each known length by k to find corresponding unknown lengths. Efficient when k is known or easily computed.

 

Proportion method: set up a ratio of corresponding lengths and solve for the unknown. More flexible when the scale factor is not immediately obvious, and more naturally expresses the relationship between multiple pairs of sides simultaneously.

 

Both methods rest on the same mathematical fact: corresponding sides of similar figures are proportional. The scale factor is the constant of proportionality. A student who understands this connection understands why both methods always give the same answer.

 

Then move into group note-making on dilation and similarity. Cover: dilation as a transformation from a fixed center, what stays the same (angles, ratios of corresponding lengths), what changes (side lengths, distances from center), the scale factor as the ratio of corresponding lengths or corresponding distances from the center, and the similarity statement with correct correspondence order.