Download Activity Review - Math Interventions Matrix

Congruence
At a Glance
Student Probe
Is ∆ABC congruent to ∆DEF?
How do you know?
A
F
B
C
D
E
Answer: Yes, because the corresponding parts
are congruent.
This may be shown by measuring the sides and
angles, or by using patty paper to trace and fit
one triangle on the other.
Lesson Description
In this lesson students verify that two polygons
are congruent using transformations
(translations, reflections, or rotations). Students
use patty paper to fit one geometric shape onto
another to determine that the corresponding
parts are congruent.
Rationale
What: Congruence in geometric shapes.
Common Core State Standard: CC.9-12.G.CO.7.
Use the definition of congruence in terms of
rigid motions to show that two triangles are
congruent if and only if corresponding pairs of
sides and corresponding pairs of angles are
congruent.
Matched Arkansas Standard: AR.9-12.T.G.2.1
(T.2.G.1) Apply congruence (SSS ...) and
similarity (AA ...) correspondences and
properties of figures to find missing parts of
geometric figures and provide logical
justification
Mathematical Practices:
Reason abstractly and quantitatively.
Model with mathematics.
Attend to precision.
Who: Students who do not understand
congruence.
Grade Level: Geometry
Prerequisite Vocabulary: congruent,
corresponding parts
Prerequisite Skills: None
Delivery Format: Individual, pairs or small
groups
Lesson Length: 15 min.
Materials, Resources, Technology: patty paper,
pencil, straight edge, geometry software
(optional)
Student Worksheets: Polygons (.pdf)
To solve many mathematical problems, it is
advantageous to recognize the relationships
among abstract objects, such as numbers or
functions. The study of congruence allows students to develop this manner of thinking by
investigating the relationships among concrete objects. Additionally, the study of congruence
can promote students’ development of visualization and spatial reasoning.
Preparation
Prepare copies of Polygons for each student. Provide patty paper and straight edges (rulers).
Lesson
The teacher says or does…
Expect students to say or do…
1. Trace polygon ABCD on
patty paper.
Compare polygon ABCD
to polygon A’B’C’D’.
Trace polygon ABCD and place
on top of polygon A’B’C’D’.
2. How do the polygons
compare?
Answers may vary, but listen
for, “They are equal”, or “they
are the same”, or “they are
congruent”.
3. Let’s make a list of the
AB and A’B’
parts of the two polygons BC and B’C’
that are the “same”.
CD and C’D’
DA and D’A’
A and A '
If students do not, then the
teacher says or does…
Model for students.
Are corresponding parts
matched?
Model for students, if
necessary.
Which parts “fit on top” of
each other?
Which parts “match”?
Prompt students, if necessary.
B and B '
C and C '
D and D '
4. These “matching” parts
are called corresponding
parts.
How can we tell that the They are the same measure.
corresponding sides and
angles are the same?
5. Polygons whose
corresponding parts all
have the same measures
are called congruent.
Mathematicians use this
symbol for congruent: .
6. How do the areas of both Answers may vary, but listen
polygons compare?
for, “The areas are the same”.
7. Trace triangle ABC on
patty paper and compare
to triangle A’B’C’.
Trace triangle ABC and place
on top of triangle A’B’C’.
Corresponding sides and
angles are the same. Be
certain that correct vertices
and sides are matched.
Congruent polygons are exact
copies of each other.
What is area?
Is the same amount of space
covered?
Remind student to be precise
with tracing and correctly label
the original triangle.
The teacher says or does…
Expect students to say or do…
8. How do the triangles
compare?
How do you know?
They are congruent.
9. Suppose you are told
that DEF XYZ .
What do you know about
the triangles?
(See Teacher Notes.)
The corresponding sides and
angles have the same
measure.
DE and XY
EF andYZ
DF and XZ
D and X
If students do not, then the
teacher says or does…
Are corresponding parts
matched?
Let’s list the corresponding
parts.
Let’s list the corresponding
parts.
What do we know about the
corresponding parts of
congruent polygons?
E and Y
F and Z
have the same measure.
10. Repeat with additional
polygons as needed.
Teacher Notes
1. Congruence is a special case of similarity in that corresponding angles are congruent and
corresponding sides are in a 1:1 ratio.
2. When two figures are congruent, one can be moved so that it fits exactly on the other.
Those moves may be a translation, a reflection, or a rotation.
3. Congruent polygons have congruent corresponding parts—their matching sides and angles
are congruent.
4. When naming congruent polygons, the corresponding vertices are always listed in the same
order.
Variations
1. Students may not recognize that when working with triangles it is not necessary to show
that all six pairs of corresponding parts of the triangles are congruent in order to conclude
that the triangles are congruent. This lesson might be extended by investigating which pairs
of corresponding parts in triangles will determine congruence.
2. Use interactive geometry software such as Geometer’s Sketchpad, Cabri, or Geogebra for
this lesson.
Formative Assessment
Is ∆ABC congruent to ∆DEF? How do you know?
D
E
F
A
B
C
References
Driscoll, M. (2007). Fostering Geometric Thinking. Portsmouth NH: Heinemann.
Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in
Mathematics. Retrieved 2 25, 2011, from rti4sucess.