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Rigid Motions: Translations

29 Aug

What do you need for a translation?

An object

How far right/left, how far up/down

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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.

Screen Shot 2014-08-14 at 9.23.34 AM

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.

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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’?

Screen Shot 2014-08-14 at 9.39.35 AM

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

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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?

Screen Shot 2014-08-14 at 9.42.38 AM

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 …

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2 Comments

Posted by on August 29, 2014 in Geometry, Rigid Motions

 

Tags: , , , ,

2 responses to “Rigid Motions: Translations

  1. 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.

       

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