Download HSCE: G1 - Math Companion Documents

HSCE: G1.1.1 Solve multi-step problems and construct proofs involving
vertical angles, linear pairs of angles supplementary angles,
complementary angles, and right angles.
HSCE: G1.1.2 Solve multi-step problems and construct proofs involving
corresponding angles, alternate interior angles, alternate
exterior angles, and same-side (consecutive) interior angles.
Clarification statements:
Students have had experience with these geometric figures and their
properties in middle school. The 6th grade GLCE (G.GS.06.01) addresses
basic knowledge about geometric figures and their properties, but this HSCE
asks students to solve multi-step problems. (Insert Link to GCLES here)
Students also have an introduction to the concept of justification in middle
school, where they have to justify simple constructions and (in 8th grade) a
proof of the Pythagorean theorem (G.SR.06.05, G.SR.07.02, G.GS.08.01).
This HSCE does not require a specific type of proof:
Paragraph proofs, algebraic proofs, two-column proofs are all appropriate.
What is central to the idea of proof in this HSCE is that students provide
some type of mathematical argument (justifying conclusions, providing
support for steps in the proof).
“Proof has three functions in mathematics (Bell, 1976): (1) verification –
concerned with establishing the truth of a proposition; (2) illumination –
concerned with conveying insight into why a proposition is true; and (3)
systematization – concerned with organizing propositions into a deductive
system. Too many U.S. students do not appreciate or experience these
functions.” (From A Research Companion to PSSM, NCTM, 2003, p. 167168)
Also note that this HSCE asks for proofs involving the various figures and
their properties. That is, students will be held responsible for proving more
than just the basic fact about vertical angles (that they are equal), linear
pairs (that they are supplementary), etc. For example, proving that an
exterior angle is equal to the sum of the two remote angles involves linear
pairs and is multi-step.
Sample Activity:
Good problems are available in most geometry textbooks to illustrate both
“multi-step” problem solving and different types of proofs that involve the
figures and their properties. We strongly encourage teachers to require
students to think their way through problems and proofs rather than follow a
simple example in rote, thoughtless way. The use of manipulatives is helpful
for discovering approaches to solving the problems or constructing the
proofs.
Suggestions for differentiation1
Content can be differentiated by the complexity of the multi-step problems or
the proofs they are required to do, as long as all students are required to do
multi-step problems, not single step problems. Teachers can give additional
scaffolding support or more highly structured directions for those students
who need it, while working on the same problem.
Not all students need to start with two-column proofs. Some may feel more
comfortable with paragraph proofs when they begin to learn how to justify
their conclusions. Some may even need to explain verbally to teachers, oneon-one, their thinking about a relationship they have been asked to prove.
HSCE: G1.1.3 Perform and justify constructions, including midpoint of a line
segment and bisector of an angle, using straightedge and
compass.
Clarification statements:
Students will perform geometric investigations and make conjectures.
Remind students of the following: Don’t erase your construction marks when
you complete your construction. These marks are an important part of your
reasoning and conjecturing.
Justification of constructions can be done by measuring or paper folding. It
does not need to be a formal proof at this point.
Example 1:
Draw a line segment and find it’s midpoint by construction using a
straightedge and compass. Explain how you know that this is the midpoint of
the line segment. (Conjectures should include: Why is the midpoint on the
perpendicular bisector of the segment? What is true of all points on the
perpendicular bisector?)
Example 2:
Draw and angle and construct its angle bisector. Explain how you know that
this is the line that bisects the angle. (Conjectures should include: What are
the measures of the small angles created by the bisecting line?)
http://en.wikipedia.org/wiki/Compass_and_straightedge The_basic_constructions
HSCE: G1.1.4 Given a line and a point, construct a line through the point
that is parallel to the original line using straightedge and
compass; given a line and a point, construct a line through the
point that is perpendicular to the original line; justify the steps
of the constructions.
Clarification statements:
Justification does not have to be in proof form.
Remind students that they should not erase marks make to complete
constructions. These marks are part of their work.
Example 1:
Construct a line. Select a point not on the line. Construct a second line
parallel to the first line that passes through the point using a compass and a
straight edge. (Justification should include: Equidistant measurement
between both lines.)
Example 2:
Construct a line through the given point that is perpendicular to the give line.
Explain how you know that the line is perpendicular. (Justification should
include: The line forms a 90-degree angle with the given line. The
perpendicular line bisects the given line.)
HSCE: G1.1.5 Given a line segment in terms of its endpoints in the
coordinate plane, determine its length and midpoint.
Clarification statements:
Use the distance formula to find the length of a line segment between Point 1
(x1, y1) and Point 2 (x2, y2), and the midpoint formula to find the
midpoint of the segment between the two coordinates:
Distance formula: d 
Midpoint formula:
( x 2  x1 ) 2  ( y 2  y1 ) 2
x2  x1 y 2 y1
,
2
2
Sample Activities:
www.purplemath.com/modules/distform.htm
The Distance Formula is a variant of the Pythagorean Theorem. This page
shows how we get from the one to the other.
www.purplemath.com/modules/midpoint.htm
Sometimes you need to find the point that is exactly between two other
points. For instance, you might need to find a line that bisects (divides into
equal halves) a given line segment. This middle point is called the
"midpoint". The concept doesn't come up often, but the Formula is quite
simple and obvious, so you should remember it for later. This website
develops the formula in a way that makes sense, including examples.
HSCE: G1.1.6 Recognize Euclidean Geometry as an axiom system; know
the key axioms and understand the meaning of and distinguish
between undefined terms (e.g., point, line, plane), axioms,
definitions, and theorems.