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Transcript
Youngstown City Schools
MATH: GEOMETRY
Unit 1B: PROVING THE CONGURENCE OF TRIANGLES (2 WEEKS) 2013-2014
SYNOPSIS: In this unit, students work with the Theorem of congruence as it relates to triangles. Students find real-life examples of
congruence in their lives, and explain why they meet the definition of congruence in terms of angles and sides.
STANDARDS
G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure;
given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
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.
G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid
motions.
MATH PRACTICES:
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
LITERACY STANDARDS
L.1
L.2
L.4
L.5
L.8
Learn to read mathematical text - - problems and explanations
Communicate using correct mathematical terminology
Listen to and critique peer explanations of reasoning
Justify orally and in writing mathematical reasoning
Read appropriate text, providing explanations for mathematics concepts, reasoning or procedures
MOTIVATION
TEACHER NOTES
1. Teacher shows students pictures that will be similar to those required in the Authentic Assessment
(e.g., flukes on whales; butterfly wings; a person with fists on hips, etc.); asks students if they see two
congruent triangles, demonstrates how to identify these on samples.
2. To introduce congruent triangles teachers can engage students in a discussion of congruent and not
congruent triangles or use clip from Flatland or United Streaming on congruent triangles below
(G.CO.6, 7, 8) http://player.discoveryeducation.com/index.cfm?guidAssetId=A38D422E-90C9-4758B1C5-3B6A8FE1835F&blnFromSearch=1&productcode=US
3. Preview expectations for end of Unit.
4.
Have students set both personal and academic goals for this Unit or grade period.
TEACHING-LEARNING
6/30/2013
TEACHER NOTES
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
1
TEACHING-LEARNING
Vocabulary:
Pythagorean theorem
Corresponding sides
SSA
TEACHER NOTES
Congruent
Corresponding angles
SAS
ASA
SSS
postulate
1. Teacher demonstrates and defines congruent triangles by showing students two triangles from the
prior unit (remind students that when they did the 3 transformations, the segments remained the same)
modeling for them to see how they are congruent. Ask students what they notice about the triangles,
and have them describe what does not change. Next, have students draw a trapezoid, and then do
the transformations. Students then make a stencil of the original, and place atop the transformations.
Teacher may use TI-Nspire activity for this, but must be connected to triangles from previous unit.
They can also use sketchpad geogebra or other hands-on activity.
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5024&t=5056&id=13159
(G.CO.6) (L.2; L.5) (MP-2; MP-7)
2. Teacher leads discussion about the definition of two congruent triangles and their corresponding
angles and sides through transformation (vocabulary). Students do examples with teacher including
using the distance formula to show sides congruent and then individually (e.g., whiteboards, math
notebook) with different triangles. Use real-life examples such as the spokes in a bicycle, landscaping,
architecture, etc. for students to work. Have student write angles and sides that are congruent, using
the appropriate notation (). The students write these to show congruence of several pairs of triangles.
Make sure students understand that they have to identify corresponding sides and angles. They also
need to solve algebraic problems dealing with corresponding sides and angles, including one and two
variable equations. (G.CO.6, 7) (L.8) (MP-8)
3. Teacher explains that we need not measure all 6 parts to know if triangles are congruent; the
“Theorem” was devised to determine congruence (ASA, SAS, and SSS). (G.CO.8) (L-2; L-5) (MP-4;
MP-5)
4. Teacher give students additional examples with some that are not congruent and have students
“prove” the congruence (or not). (L-1; L-5) (MP-1; MP-5)
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Paper-pencil test with M-C questions and 2-and 4-point questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. Quizzes
2. In-class participation and practice problems for each concept
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
2
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students evaluate goals they set at beginning of unit or on a weekly basis.
2. Each student will be given a “Student Test” to score. The objective is to find if there are errors,
and provide an explanation of what the errors were. On pages 5-12 of this Unit, you will find 4
different “student tests.” Your Geometry students are to review and grade one of the 4 tests
to see where the “student who took that test” may have made errors. Answers are on pages
13-17. For each of the 4 Tests, the “student taking the test” made 12 errors. As the evaluator
of this test, student will earn 2 points for each error found and one point for each error explained
as to what was done incorrectly in the work on the “test.” (G.CO.7, G.CO.8,L.2; L.4; L.5)
3. Each student will: find real-world pictures that illustrate Unit concepts; students “draw”
the triangles on the picture and then use a theorem to prove the congruence; students
must justify orally and in writing why they chose a theorem and why it works for the
picture chosen (page 18). (G.CO.6, G.CO.7, G.CO.8, L.5; L.8)
.
.
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
3
GEOMETRY UNIT 1B AUTHENTIC ASSESSMENT #1 CONGRUENT TRIANGLES
(Standards: G.CO.7 and G.CO.8)
DIRECTIONS FOR THE ASSESSMENT:
Each student will be given a “Student Test” to score. The objective is to find if there are
errors, and provide an explanation of what the errors were.
On the pages 5-12, you will find 4 different “student tests.” Your Geometry students are to
review and grade one of the 4 tests to see where the “student who took that test” may have
made errors. Answers are on pages 13-17
For each of the 4 Tests, the “student taking the test” made 12 errors.
As the evaluator of this test, student will earn 2 points for each error found and one point for
each error explained as to what was done incorrectly in the work on the “test.”
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
4
AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B
STUDENT TEST #1: Analyze the 6 proofs for correctness, including (a) if the labels are
correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or
definitions, but incorrect labels. Correct the errors and write a statement about the
corrections and errors for each problem. Then choose one problem to orally defend.
1. Prove: ∆DAB
∆BCD
A
1. <ADB
<CBD
<CDB
<ABD
2. BD
BD
3. ∆DAB
∆BCD
B
1. Given
2. Reflexive
3. AAS
D
2. Prove: ∆RVT
C
∆RST
1. RV
SR, ST
2. RT
RT
3. ∆RVT
∆STR
R
VT
1. Given
2. Reflexive
3. SSS
V
S
T
3. Prove: ∆DBW
∆TYW
1. DW
WT, WB
<DWB
<TWY
2. ∆DBW
∆TYW
D
WY,
1. Given
W
2. ASA
B
6/30/2013
T
Y
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
5
(Student Test #1 continued)
4. Prove: ∆BSE
∆USL (note: EL and UB intersect at S)
1. <ESB
<EBS
2. ∆BSE
<USL, BS
<LSU
∆USL
SU
E
U
1. Given
S
SSS
2. ASA
B
5. Prove: ∆JOS
L
∆HPE
1. JO
PH, OS
PE,
1. Given
JO is perpendicular to PO
and HP is perpendicular to PO
2. <O
<P
2. All right angles are
3. ∆JOS
∆HPE
3. SSS
J
E
O
P
S
H
6. Prove: ∆PIK
∆NIK
K
1. <PKI
<NKI,
1.
KI is perpendicular to PN
2. KI
KI
2.
3. <KIP
<NIK
3.
4. ∆KPI
∆KNI
4.
Given
Reflexive
All right angles are congruent
SAS
P
I
N
Scoring per problem: Two points for each correction, 2 points for each written explanation.
Oral defense of one problem: 3 points
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
6
AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B
STUDENT TEST #2: Analyze the 6 proofs for correctness, including (a) if the labels are
correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or
definitions, but incorrect labels. Correct the errors and write a statement about the
corrections and errors for each problem. Then choose one problem to orally defend.
1. Prove: ∆DBA
∆BDC
1. AD
<CB
<CBD
<ADB
2. BD
BD
3. ∆DAB
∆BCD
2. Prove: ∆RVT
A
B
D
C
1. Given
2. Reflexive
3. SSS
∆RST
1. RV
SR, ST
2. RT
RT
3. ∆RVT
∆RST
R
VT
1. Given
2. Transitive
3. SAS
V
S
T
3. Prove: ∆DBW
∆TYW
1. DW
WT, WB
<DWB
<TWY
2. ∆DBW
∆WTY
D
WY,
1. Given
T
W
2. ASA
Y
B
(Student Test #2 continued)
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
7
4. Prove: ∆BSE
1. <ESB
<EBS
2. ∆BSE
∆USL (note: EL and UB intersect at S)
<USL, BS
<LUS
∆USL
US
E
U
1. Given
S
SSS
2. SSA
B
5. Prove: ∆JOS
L
∆HPE
1. JO
HP, OS
SE, SE
JO is perpendicular to OP
HP is perpendicular to OP
2. <O
<P
3. OS
PE
4. ∆JOS
∆HEP
PE
1. Given
J
E
2. All right angles are
3. Reflexive
4. SAS
O
P
S
H
6. Prove: ∆PIK
∆NIK
K
1. <PKI
<NKI,
1.
KI is perpendicular to PN
2. KI
IK
2.
3. <KIP
<KIN
3.
4. ∆KPI
∆KNI
4.
Given
Transitive
All right angles are congruent
ASA
P
I
N
Scoring per problem: Two points for each correction, 2 points for each written explanation.
Oral defense of one problem: 3 points
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
8
AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B
STUDENT TEST #3: Analyze the 6 proofs for correctness, including (a) if the labels are
correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or
definitions, but incorrect labels. Correct the errors and write a statement about the
corrections and errors for each problem. Then choose one problem to orally defend.
E
1. Prove: ∆ABE
∆DCE
1. <A
<D
<ABE
<DCE
AB
BC, BC
DC
2. AB
DC
3. ∆ABE
∆ECD
1. Given
2. Reflexive
3. ASA
A
2. Prove: ∆GAP
∆PSG
P
1. SG
AP, SP
AG
2. GP
GP
3. ∆GAP
∆PSG
B
C
D
S
1. Given
2. Reflexive
3. SAS
G
A
G
3. Prove: ∆DGO
∆ CTA
1. <D
<C, <O
<A
DO
CA
2. ∆DGO
∆ ATC
A
6/30/2013
T
1. Given
2. SAS
D
O C
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
9
(Student Test #3 continued)
4. Prove: ∆SRU
∆ STU
1. <RSU
<UST
RS
TS
2. SU
SU
3. ∆SRU
∆ STU
R
1. Given
2. Reflexive
3. AAA
S
U
T
J
5. Prove: ∆JEF
∆ JIF
1. <EJF
<FJI, EJ
2. JF
FJ
3. ∆JEF
∆ JIF
IJ
1. Given
2. Reflexive
3. SAS
E
F
X
6. Prove: ∆RXS
∆ UTW
1. XS is perpendicular to RT
WT is perpendicular to XU
<R
<U
RS
UW
2. <RSX
<TWU
3. ∆RXS
∆ TWU
I
W
U
1. Given
R
S
2. All right angles are congruent
3. ASA
T
Scoring per problem: Two points for each correction, 2 points for each written explanation.
Oral defense of one problem: 3 points
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
10
AUTHENTIC ASSESSMENT GEOMETRY UNIT 1B
STUDENT TEST #4: Analyze the 6 proofs for correctness, including (a) if the labels are
correct and the definitions and/or theorems are incorrect; (b) correct theorems and/or
definitions, but incorrect labels. Correct the errors and write a statement about the
corrections and errors for each problem. Then choose one problem to orally defend.
E
1. Prove: ∆ABE
∆DCE
1. <A
<D
<ABE
<DCE
AB
BC, BC
DC
2. AB
CD
3. ∆ABE
∆DCE
1. Given
2. Transitive
3. AAS
A
2. Prove: ∆GAP
∆PSG
P
1. SG
AP, SP
AG
2. GP
GP
3. ∆GAP
∆PSG
B
C
D
S
1. Given
2. Reflexive
3. SAS
G
A
G
3. Prove: ∆DGO
∆ CTA
1. <D
<C, <O
<A
DO
AC
2. ∆DGO
∆ ATC
A
6/30/2013
T
1. Given
2. ASA
D
O C
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
11
(Student Test #4 continued)
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
12
4. Prove: ∆SRU
∆ STU
1. <RSU
<TSU
RS
ST
2. SU
SU
3. ∆SRU
∆ STU
R
1. Given
2. Transitive
3. SAS
S
U
T
J
5. Prove: ∆JEF
∆ JIF
1. <EJF
<IJF, EJ
2. JF
FJ
3. ∆JEF
∆ JFI
IJ
1. Given
2. Reflexive
3. SAS
E
F
X
6. Prove: ∆RXS
∆ UTW
1. XS is perpendicular to RT
WT is perpendicular to XU
<R
<U
RS
WU
2. <RSX
<UWT
3. ∆RXS
∆ UTW
I
W
U
1. Given
R
S
2. All right angles are congruent
3. SAS
T
Scoring per problem: Two points for each correction, 2 points for each written explanation.
Oral defense of one problem: 3 points
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
13
ANSWERS TO GEOMETRY UNIT 1B AUTHENTIC ASSESSMENT #1
Student Test #1
1.
Step #2: DB BD (letters are to be in the reverse order)
Step #3: ASA
Explanations:
The triangles are set up in the prove statement such that D→B, A→C, and B→D so DB→BD.
ASA because side is included between the two angles.
2.
Step #1: RV RS
Step #3: ∆RVT
∆RST
Explanations:
The triangles are set up in the prove statement such that R→R, V→S, and T→T so RV→RS
(R goes with R not S).
∆RVT → ∆RST because V→S, not T and T→T, not R
3.
Step #1: DW TW
Step #2: SAS
Explanations:
The triangles are set up in the prove statement such that D→T, W→W, and B→Y so
DW→TW.
SAS because the angle is included between the two sides and there are two pairs of sides and
one pair of angles
4.
Step #1: <ESB <LSU, BS
US
Step #3: SAS
Explanations:
The triangles are set up in the prove statement such that E→L, S→S, and B→U so
<ESB→<LSU and BS .
5.
Step #1: JO HP
Step #3: ∆JOS
∆HPE
Explanations:
The triangles are set up in the prove statement such that J→H, O→P, and S→E so JO→HP.
SAS because there is one pair of angles and two pairs of sides congruent and the angle is
included between the two sides.
6.
6/30/2013
Step #3: <KIP <KIN
Step #4: ASA
Explanations:
The triangles are set up in the prove statement such that P→N, I→I, and K→K so
<KIP→<KIN.
ASA because the side is included between the two angles and there are two pairs of angles
and one pair of sides.
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
14
Student Test #2:
1.
Step #2: BD DB (letters are to be reversed)
Step #3: SAS
Explanations:
The triangles are set up in the prove statement such that D→B, B→D, and A→C so BD→DB.
SAS because the angle is included between the two sides and there are two pairs of sides and
one pair of angles congruent.
2.
Step #1: RV RS
Step #2: Reflexive
Explanations:
The triangles are set up in the prove statement such that R→R, V→S, and T→T so RV→RS
(R goes with R not S. V goes with S).
Transitive states that a first quantity is congruent to a second quantity and a second quantity is
congruent to a third quantity, then the first quantity is congruent to the third quantity. We do
not have that here. Reflexive states a quantity is congruent to itself, which is what this step
states.
3.
Step #1: DW TW
Step #2: SAS
Explanations:
The triangles are set up in the prove statement such that D→T, W→W, and B→Y so
DW→TW.
SAS because the angle is included between the two sides and there are two pairs of sides and
one pair of angles
4.
Step #1: <ESB <LSU
Step #3: ASA
Explanations:
The triangles are set up in the prove statement such that E→L, S→S, and B→U so
<ESB→<LSU.
ASA because there are two pairs of angles and one pair of sides congruent and the side is
included between the angles. Also SSA is not a method of proving triangles congruent.
5.
Step #3: Transitive
Step #4: ∆JOS
∆HPE
Explanations:
Transitive states that a first quantity is congruent to a second quantity and a second quantity is
congruent to a third quantity, then the first quantity is congruent to the third quantity, which is
what we have here. Reflexive states a quantity is congruent to itself, which we do not have
that here.
The triangles are set up in the prove statement such that J→H, O→P, and S→E so ∆JOS→
∆HPE.
6.
Step #2: KI <KI
Step #2: Reflexive
Explanations:
The triangles are set up in the prove statement such that P→N, I→I, and K→K so KI→KI.
Transitive states that a first quantity is congruent to a second quantity and a second quantity is
congruent to a third quantity, then the first quantity is congruent to the third quantity, which is
not what we have here. Reflexive states a quantity is congruent to itself, which we do have
here.
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
15
Student Test #3:
1.
Step #2: Transitive
Step #3: ∆ABE
∆DCE
Explanations:
Transitive states that a first quantity is congruent to a second quantity and a second quantity is
congruent to a third quantity, then the first quantity is congruent to the third quantity. We have
that here. Reflexive states a quantity is congruent to itself, which is not what this step states.
The triangles are set up in the prove statement such that A→D, B→C, and E→E so ∆ABE→
∆DCE.
2.
Step #2: GP PG
Step #3: SSS
Explanations:
The triangles are set up in the prove statement such that G→P, A→S, and P→G so GP→PG
(letters are to be reversed).
There are three pairs of sides congruent not two pairs of sides and a pair of angles.
3.
Step #2: ∆DGO
∆ CTA
Step #2: ASA
Explanations:
The triangles are set up in the prove statement such that D→C, G→T, and O→A so
∆DGO→ ∆ CTA.
ASA because the side is included between the two angles and there are two pairs of angles
and one pair of sides.
4.
Step #1: <RSU <TSU
Step #3: SAS
Explanations:
The triangles are set up in the prove statement such that R→T, S→S, and U→U so
<RSU→<TSU.
SAS because there are two pairs of sides and one pair of angles congruent and the angle is
included between the two sides. Also AAA is not a method of proving triangles congruent.
5.
Step #1: <EJF
< IJF
Step #2: JF
JF
Explanations:
The triangles are set up in the prove statement such that J→J, E→I, and F→F so <EJF
<
IJF in step #1 and also JF→ JF in step #2.
SAS because there are two pairs of sides and one pair of included angles congruent, not two
pairs of angles and a pair of sides.
6.
Step #2: <RSX <UWT
Step #3: ∆RXS
∆ UTW
Explanations:
The triangles are set up in the prove statement such that R→U, X→T, and S→W so
<RSX→ <UWT, also when naming the triangles this correspondence is preserved so
∆RXS
∆ UTW.
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
16
Student Test #4:
1.
Step #2: AB
DC
Step #3: ASA
Explanations:
The triangles are set up in the prove statement such that A→D, B→C, and E→E so AB→ DC.
ASA because there are two pairs of angles and the pair of included sides congruent and AAS
is not a way we learned to prove triangles congruent.
2.
Step #2: GP PG
Step #3: SSS
Explanations:
The triangles are set up in the prove statement such that G→P, A→S, and P→G so GP→PG
(letters are to be reversed).
There are three pairs of sides congruent not two pairs of sides and a pair of angles.
3.
Step #1: DO
CA
Step #2: ∆DGO
∆ CTA
Explanations:
The triangles are set up in the prove statement such that D→C, G→T, and O→A, so DO→ CA
and also ∆DGO
∆ CTA.
4.
Step #1: RS TS
Step #2: Reflexive
Explanations:
The triangles are set up in the prove statement such that R→T, S→S, and U→U so RS→TS.
Transitive states that a first quantity is congruent to a second quantity and a second quantity is
congruent to a third quantity, then the first quantity is congruent to the third quantity, which is
not what we have here. Reflexive states a quantity is congruent to itself, which we do have
here.
5.
Step #2: JF
JF
Step #3: ∆JEF
∆ JIF
Explanations:
The triangles are set up in the prove statement such that J→J, E→I, and F→F so JF→ JF and
also with ∆JEF
∆ JIF.
6.
Step #1: RS UW
Step #3: ASA
Explanations:
The triangles are set up in the prove statement such that R→U, X→T, and S→W so RS→UW.
ASA because there are two pairs of angles and one pair of included sides congruent, not two
pairs of sides and one pair of included angles congruent.
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
17
6/30/2013
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
18
Geometry Unit 1B Authentic Assessment #2
Standards: G.CO.6, G.CO.7, G.CO.8
Find three real-world pictures that illustrate two congruent triangles (e.g., flukes of whale’s
tail; wings of butterfly, etc.); draw the triangles on the pictures and then use a theorem to
prove the congruence. Justify orally and in writing why the particular theorem was chosen
and why it works for the picture.
ELEMENTS
OF PROJECT
Submit pictures
0
No pictures
submitted
Draw congruent Triangles not
triangles on
congruent
picture
Written proof
Use incorrect
theorems on all
three
Oral Proof
Use incorrect
theorems on all
three
6/30/2013
Rubric
1
2
3
1 picture
submitted
Congruent on
one drawing
2 pictures
submitted
Congruent on
two drawings
3 pictures
submitted
Congruent on
three drawings
Correct
theorems on
one proof
Correct
theorems on
one proof
Correct
theorems on
two proofs
Correct
theorems on
two proofs
Correct
theorems on
three proofs
Correct
theorems on
three proofs
YCS Geometry: Unit 1B: Proving The Congruence of Triangles 2013-14
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