What do you need for a **translation**?

An object

How far right/left, how far up/down

In our high school geometry class we can use a directed segment or vector to indicate how far right/left and how far up/down.

How does vector XY tell us how far right/left and up/down?

Students had the opportunity to **look for and make use of structure**.

We talked about a right triangle with segment XY as its hypotenuse. The horizontal and vertical legs tell us how far right/left and how far up/down.

When we translate a triangle using a given segment (or vector), what is congruent?

Students made a list of congruent objects. They shared their list with their teams, and then we discussed with the whole class.

The triangles are congruent. Because one is a translation of the other.

The corresponding segments are congruent. Because the triangles are congruent.

The corresponding angles are congruent.

What else is congruent?

The distance from C to C’ is the same as the distance from B to B’.

What else is congruent?

CC’=BB’=AA’

(Yes. I know that I am interchanging equality and congruence. I actually used to spend time specifically teaching notation in geometry. Now students learn notation by observation.)

What else is congruent?

CC’=XY

CX=C’Y

What is CXYC’?

We ended the lesson with a triangle that had been translated. How can you show that one triangle is a translation of the other?

One student noted aloud that we could show that the triangles are congruent.

Is showing the triangles are congruent **necessary** for proving that one triangle is a translation of the other?

Is showing the triangles are congruent **sufficient** for proving that one triangle is a translation of the other?

What information is **sufficient** for proving that one triangle is a translation of the other?

Is it enough to connect A to A’, B to B’, and C to C’?

What must be true about those segments?

And so the journey continues …

howardat58

August 29, 2014 at 6:29 pm

I have a real problem with the CCSS definition of congruent figures. I did a post on it in July. You may find it interesting. Here it is

http://howardat58.wordpress.com/2014/07/07/congruence-transformations-definitions-unnecessary/

jwilson828

September 2, 2014 at 5:53 am

My students and I have really enjoyed thinking about congruence in terms of rigid motions. Our whole geometry plays out differently because of starting with transformations; what students notice and conjecture is enhanced by their understanding of congruence and similarity in terms of transformations.