Download Geometry Additions Revisions

Geometry Additions/Revisions
Added TEKS Student Expectation GEOM.8F
Use conversions between measurement systems to solve problems in real-world situations.
Learning Focus 3.1 – Similarity
Students solve geometric problems involving similarity, proportionality, and dilation.
HISD Objectives
Add revised HISD
objectives (revisions
indicated by
underlined text) to
page 1 of existing
HAPG
GEOM.8F
Use conversions between
measurement systems to
solve problems in realworld situations.
Time
Allocation
Assessment
Connections
Instructional Considerations
Instructional Strategies
Resources
Existing information of special note:
Add to existing information:
Existing information of special note:
Add to existing information:
Existing resources of
special note:
Connections to Future Objectives/
Assessments
Students will be asked to problem solve
with proportions and similar figures on
the TAKS tests at all levels. College and
Career Readiness Standards include
conversion between measurement
systems. Connections will be made in
measurements for perimeter, area, and
volume in Learning Foci 4.2, 5.1, and
5.2.
To engage and explore conversions,
students working in pairs should measure
the height of an object in inches and in
centimeters. Ask students, “what is the ratio
of centimeters to inches?” Divide the
number of inches into the number of
centimeters. Students should get a quotient
of approximately 2.54 cm/inch. Next, ask
the students to measure their own height in
inches. Use dimensional analysis to convert
their height to centimeters, then to meters.
Clarifying Activities:
These activities provide
short and succinct
review of similar figures
and proportionality.
 TAKS Objective 9;
 What Makes
Geometric Figures
Similar?.
Essential Understandings/
Guiding Questions
Properties and attributes of
proportionality, congruency, dilation, and
similarity assist in solving geometric
problems.
1. How are proportionality and similarity
used to solve problems with geometric
figures?
2. How does the geometric mean affect
the proportionality of geometric
figures?
3. In what way are conversions between
measuring systems verified through
ratios and proportions?
Background Knowledge for Teachers
Critical Content:
 Ratios;
 Proportions;
 Conversion Factors;
 Dimensional Analysis;
 Geometric Means;
 Proportions in Geometry; and
 Similar Polygons.
? cm  63 inches 
2.54 cm
 160.02 cm
1 inch
160.02 cm 
1m
 1.6002 m
100 cm
Homework and Practice
Include other measuring systems by
instructing students to take measurements
made in customary units and convert the
unit of measure either to another customary
or to a metric measure. For instance, in
simplifying ratios on text page 360 #4-10,
connect these problems to other systems
by asking students to change the units in
the ratio to either customary or metric.
Geometry, McDougalLittell, 2007:
 Ratios, Proportions,
and Geometric Mean
pp. 356-361;
 Use Proportions to
Solve Geometry
Problems pp. 364366;
 Similar Polygons
pp.371-375.
Look for similar figures in every-day life at
school or in your classroom. Develop the
concept for conversion factors by having
students convert these every-day
measurements to metric or customary
units.
Vocabulary
Academic
Think-pairshare
Content-Specific
Ratio
Proportion
Geometric mean
Reciprocals
Corresponding
angles
Corresponding
sides
Conversion factor
TAKS Tips
This learning focus reviews proportions
from the eighth-grade curriculum (TAKS
Objective 9) and solving algebraic
equations (TAKS Objective 3). All high
school mathematics TAKS test evaluates
these objectives.
 Use What Makes Geometric Figures
Similar? as an engagement activity (see
Resources column). Look for applications in
the McDougal-Littell textbook page 361,
#46-51. Use these problems to assess
students’ ability with more difficult problems
and to generate discussion among
students. Use chart paper for students to
display their results.
1(F), 2(E), 4(F)
Added TEKS Student Expectation GEOM.8F (continued)
Use conversions between measurement systems to solve problems in real-world situations.
Learning Focus 4.2 – Measuring Lengths and Area
Students concretely and algebraically determine the perimeter and area of various geometric figures or portions of the figures.
HISD Objectives
Add revised HISD
objectives (revisions
indicated by
underlined text) to
page 4 of current
HAPG
GEOM.8F
Use conversions
between measurement
systems to solve
problems in real-world
situations.
Time
Allocation
Assessment
Connections
Instructional Considerations
Instructional Strategies
Resources
Existing information of special note:
Add to existing information:
Existing information of special note:
Add to existing information:
Existing resources of
special note:
Connections to Future Objectives/
Assessments
The study of area and perimeter offers
students an opportunity to see real
applications that may arise in their home
and professions. Converting between
measurement systems may be necessary
when buying products made in countries
that use metric units versus products
made in the United States that use
customary units.
Cues, Questions, and Advance
Organizers
 Teacher’s probing questions give focus
for students in Finding the Rectangle,
Application Problems, Find the Missing
Dimension, and Circle Area. Whether
working as a class or in groups, wait time,
exploration time, and monitoring group
interaction creates the student- lead
environment. In each activity, engage
students to report what they have
discovered.
1(D), 2(D), 3(F)
Integrate these
Clarifying Activities to
give students more
hands-on experiences:
 Find the number of
square units in each
figure (various shapes
are superimposed
with a grid so squares
can be counted to
prove the formula for
area);
 Finding the Rectangle
(using rectangles as a
basis of finding area);
 Application Problems
(short assessment on
area of
quadrilaterals);
 Find the Missing
Dimension (working
backwards to find the
missing dimension);
and
 Circle Area (exploring
circle circumference
and area, especially
making sectors
become a
parallelogram).
Background Knowledge for Teachers
Critical Content:
 Area of Geometric Figures;
 Perimeter and Area of Similar Figures;
 Circles;
 Sectors and Arcs;
 Regular Polygons;
 Changing of scale factors;
 Converting between measurement
systems.
Include applications to area of rectangular
figures, with quadratic expressions that
need to be solved, in engagement and
assessment problems. Take this
opportunity in the discussion of area to
review solving quadratic equations using
graphing or algebraic methods (TAKS
Objective 5).
Include application problems, involving
figures within a figure for calculating the
area of shaded regions and compound
figures (two or three shapes attached), in
lessons and assignments (TAKS Objective
8 problems often appear in this form).
In each of these application problems,
extend the questions by having students
convert units of measures to another
system.
Added TEKS Student Expectation GEOM.8F (continued)
Use conversions between measurement systems to solve problems in real-world situations.
Learning Focus 5.2 – Volume
Students build and draw three-dimensional figures and calculate the area of the base and volume of the figure. Students also analyze what occurs when
one or more dimensions are changed by a scale factor greater than one or less than one.
HISD Objectives
Add revised HISD
objectives (revisions
indicated by
underlined text)
GEOM.8F
Use conversions
between measurement
systems to solve
problems in real-world
situations.
Time
Allocation
Assessment
Connections
Instructional Considerations
Add to existing information:
Background Knowledge for Teachers
Critical Content:
 Building 3-D Figures;
 Volume of Regular Figures;
 Volume of Composite or Irregular
Figures ;
 Finding Volume of Similar Figures;
 Conversions between measurement
systems.
Connect concepts of two dimensions and
area with three dimensions and volume.
Deconstruct composite figures to assist
students visualizing how to find the
volume of solids. Use small boxes and
other shapes that are available at hobby
stores to assist with decomposing solids.
Integrate opportunities to question
students on how conversion factors may
be used to change units of measure.
Centimeter grids copied on card stock can
be used to cut out similar prisms that may
just be expanded by a factor of two or
three (Power Objective GEOM.11D).
Instructional Strategies
No additional strategies required
Resources
No additional
resources required
Added TEKS Student Expectation GEOM.8E
Use area models to connect geometry to probability and statistics.
Learning Focus 5.3 – Circles: Sectors and Arc Length
Students apply properties of circles to find the area of a sector and the length of an arc; then students connect circle area to statistical models.
HISD Objectives
Add revised HISD
objectives (revisions
indicated by
underlined text)
GEOM.8E
Use area models to
connect geometry to
probability and
statistics.
Time
Allocation
Assessment
Connections
Instructional Considerations
Existing information of special note:
Prerequisites/Background Knowledge
for Students
Students have used proportionality to
interpret circle graphs.
Students studied underlying concepts of
area of a sector and arc length in
Geometry Learning Focus 4.2.
Connections to Future Objectives/
Assessments
Students will encounter various concepts
of circles on the Pre-Scholastic Aptitude
Test (PSAT) and the Scholastic Aptitude
Test (SAT).
Students will apply statistical concepts in
their Economics classes.
Essential Understandings/
Guiding Questions
The length of a radius of a circle affects its
area, the area of a sector, and the length
of an arc.
1. How is proportionality used to calculate
the area of a sector?
2. How does area of a sector and length of
an arc help in representations of
statistical information?
Instructional Strategies
Existing information of special note:
Add to existing information (see next
page):
KWL
Engage students by having them list
everything that they know about a circle
from previous lessons. Determine if the
students have seen circles used to
represent any real-world problems.
Students may brainstorm what they want to
know by analyzing objectives GEOM.8A
and GEOM.8B. Highlight with the students
what areas are covered in this Learning
Focus. Retain this portion of the KWL for
the end of each lesson to continue
discussions of what the students want to
learn in future lessons.
Cooperative Learning
 In groups, assign students certain
statistical data to collect from the class. For
example, collect data concerning modes of
transportation to and from school, student
participation in clubs, and students’ favorite
sports.
Use this information to make a circle graph
of the data. Students must prove that all the
calculations verify that the circle graph and
the sectors are correctly divided. Students
may work in pairs to review previously
learned knowledge from Learning Focus
4.2 and of proportionality to assist with this
activity.
In developing the lesson for area of a circle
and area of a sector, ask students where
they have seen circles used where the area
of all or part of the circle is important to
interpreting the information.
Resources
Existing information
of special note:
Textbook References:
Geometry, McDougal
Littell, 2007
 Find Arc Measures
pp. 659-663;
 Areas of Circles and
Sectors pp. 755-761;
 Comparing Measures
for parts of Circles
and the Whole Circle
pp. 779;
 Area of Circles and
Sectors Review pp.
782.
Use this lesson to review how to draw a pie
chart for statistical information. Use
proportionality to calculate the measure of a
sector of a circle, using real-world data
such as a household budget or a poll of
student hair colors. This activity and
discussion lead to the use of the area of the
sector or arc length in order to accurately
represent the data. 1(C), 5(G), 2(E)
For connections to probability, ask, “if
students’ participation in clubs at a school
follows this pie graph, then what is P(C),
the probability that a student chosen at
random is in the chess club?
Drama
30%
Chess
X%
Dance
18%
Board
Game
18%
Another example for a circle graph could
show students’ favorite sports. For
example, if the circumference of the graph
is 20π cm, the arc length for the sector
representing “football” is 5 cm, the arc
length for the sector representing “soccer”
is 10 cm, and the remaining circumference
is designated for “basketball,” what is P(B),
the probability that a student chosen at
random reports that his or her favorite sport
is “basketball?”
Assessment for Learning (suggested from
Region 10 ICAT):
 Find the area of a sector as a fractional
part of the area of the circle.
 Find the length of an arc as a fractional
part of the circumference of the circle.
 Divide a circle with a 12-inch diameter
into three sections. Students determine
the approximate length of the arc of a
given section, given the angles of the
other two sections.
 Given the length of the radius and the
measure of a central angle of a circle,
students find the measure of the length of
the arc cut off by the central angle.