This Math Talk focuses on congruence and similarity. It encourages students to think about the similarities and differences between the two terms and to rely on what they know about dilations and rigid transformations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students determine if two figures are similar or not.
In explaining whether a statement is always true, sometimes true, or never true, students need to be precise in their word choice and use of language (MP6).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Decide mentally whether each statement is always true, sometimes true, or never true.
If two figures are congruent, then they are similar.
If two figures are similar, then they are congruent.
If a triangle is dilated with the center of dilation at one of its vertices, the side lengths of the new triangle will change.
If a triangle is dilated with the center of dilation at one of its vertices, the angle measures of the triangle will change.
Always true. Sample reasoning: Congruent figures can be taken from one to the other by using translations, rotations, and reflections. Similar figures use these same transformations, but also use dilations. If we don’t do a dilation, the figures are still similar.
Sometimes true. Similar figures can be taken from one to the other using translations, rotations, reflections, and dilations. Congruent figures do not allow dilations. Sample reasoning: Two figures can be similar but be different sizes, like a square with side length 1 and a square with side length 2. These figures are not congruent, but 2 squares with the same side length would be similar and congruent.
Sometimes true. Sample reasoning: The side lengths of the new triangle will change based on the scale factor. The side lengths will not change when the scale factor is 1.
Never true. Sample reasoning: While a dilation will change the side lengths of the triangle (unless the scale factor is 1), the angle measures will always stay the same.
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
All skills for this lesson
No KCs tagged for this lesson
This Math Talk focuses on congruence and similarity. It encourages students to think about the similarities and differences between the two terms and to rely on what they know about dilations and rigid transformations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students determine if two figures are similar or not.
In explaining whether a statement is always true, sometimes true, or never true, students need to be precise in their word choice and use of language (MP6).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Decide mentally whether each statement is always true, sometimes true, or never true.
If two figures are congruent, then they are similar.
If two figures are similar, then they are congruent.
If a triangle is dilated with the center of dilation at one of its vertices, the side lengths of the new triangle will change.
If a triangle is dilated with the center of dilation at one of its vertices, the angle measures of the triangle will change.
Always true. Sample reasoning: Congruent figures can be taken from one to the other by using translations, rotations, and reflections. Similar figures use these same transformations, but also use dilations. If we don’t do a dilation, the figures are still similar.
Sometimes true. Similar figures can be taken from one to the other using translations, rotations, reflections, and dilations. Congruent figures do not allow dilations. Sample reasoning: Two figures can be similar but be different sizes, like a square with side length 1 and a square with side length 2. These figures are not congruent, but 2 squares with the same side length would be similar and congruent.
Sometimes true. Sample reasoning: The side lengths of the new triangle will change based on the scale factor. The side lengths will not change when the scale factor is 1.
Never true. Sample reasoning: While a dilation will change the side lengths of the triangle (unless the scale factor is 1), the angle measures will always stay the same.
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”