Milwaukee Public Schools
Curriculum Framework
for
High School Mathematics
Foundation Level
(Algebra 1 and Geometry 1)
Draft 08/25/08
1
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Introduction
This MPS Curriculum Framework for High School Mathematics, Foundation Level (Algebra 1 and Geometry 1) is based on the
nineteen (19) MPS Foundation Level Learning Targets. These targets represent the Wisconsin mathematics process standard (A) and
the Wisconsin content standards of number operations and relationships (B), geometry (C), measurement (D), statistics and probability
(E) and algebra (F). These learning targets provide the context in which students can develop the knowledge and skills needed for a
solid foundation in algebra and geometry. The nineteen learning targets (revised in summer 2008: Version 2.2) are aligned to state
descriptors. The Wisconsin Assessment Framework for 10th grade was used for the first thirteen (13) learning targets. State standards
for 12th grade were used for the remaining learning targets.
The first learning target addresses process skills: reasoning, using vocabulary, connecting math to real world applications, and
communicating mathematical ideas. These process skills are to be continuously reinforced throughout the implementation of the
Foundation Level courses. Learning targets two through thirteen are to be covered prior to November 1 of the second year, in
preparation for the state of Wisconsin Knowledge and Concepts Examination. These thirteen targets (*in the following document),
including Learning Target 1, integrate the process and content standards related to the Wisconsin Assessment Framework for 10th
grade. Learning targets fourteen through nineteen inform the content for the remainder of the second year math course.
For each learning target, a set of benchmarks (assessable specifications) is listed in a numerical sequence i.e. benchmark 2.1 represents
the first benchmark for Learning Target 2. Every attempt was made to sequence the benchmarks for each target so that both teacher
and student can determine student progress. Although the benchmarks are numbered, this does not designate a specific order of
teaching. The specific pacing guides (designed in the summer of 2008) for the Foundation Level courses provide the sequencing for
the benchmarks aligned with the textbook adopted in the spring 2008.
Reinforcing mathematical process skills, Learning Target 1, and acknowledging the fact that MPS schools are to implement
instructional strategies to build mathematical vocabulary, Benchmark 1.8 was specifically designed: students will be able to
communicate the MPS Academic Vocabulary for mathematics in oral and written presentations using literacy strategies, e.g. graphic
organizers, vocabulary cloze, and word splash activities and cause-effect and similarities-differences diagrams. Under the MPS DIFI
improvement plan, implemented in the fall of 2008, Marzano’s Six-Step Vocabulary Process is to be implemented.
A key criterion for the benchmarks was to provide a variety of thinking levels for each target from lower level thinking skills
(knowledge and recall) to higher level thinking skills (evaluate and synthesize ideas and create models). MPS mathematical thinking
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
classification levels were created in 2005 to analyze the depth of thinking a student is expected to demonstrate when completing an
assessment. An overall depth of thinking must be associated with each benchmark in order to accurately interpret student achievement.
It is important that analyzing the depth of thinking be part of the student feedback process, an important component of the MPS
mathematical framework.
Specific course codes, listed in the Instructional Sequence column, have been given for the foundation level courses: For blocked
schools, the first year instructional sequence includes RC111: Math Investigations I; MA211: Algebra (semester one); MA221:
Algebra (semester two); and RC121: Math Investigations II. For schools on a regular teaching schedule, the instructional sequence
includes MA211: Algebra (semester one) and MA221: Algebra (semester two). Block and regular instructional sequences for the
second year are the same: MA301: Geometry (semester one) and MA311: Geometry (semester two). In the summer of 2008, pacing
guides for the newly adopted Key Curriculum Press textbooks of Discovering Algebra and Discovering Geometry were drafted. These
pacing guides will be under revision during the 2008-09 academic year as teachers use these guides and the textbooks.
Using this curriculum framework necessitates a strong commitment to the use of technology, from graphing calculators to computers
and the newest in software. However, we are aware that not all schools are similarly equipped with the most recent developments. We
can only encourage that technology be integrated to the extent possible in each classroom. The benchmarks for Learning Target 2,
using a variety of mathematical and technological tools, are integrated throughout the pacing guides for these courses. A
recommended school policy on the purchase and use of calculators at the high school has been drafted for fall 2008.
The MPS initiative that all high school teachers use the ESIS grade book was implemented in the fall of 2008. The learning targets
were used as “categories” in the ESIS framework and the benchmarks are listed under “assignment types” as summative assessments.
Each category was given a one or two word indicator which summarized the key content of that learning target. Similarly, each
benchmark under summative assessments is summarized by a one to two word phrase. ESIS grade book also requires that categories
(learning targets) be aligned to specific course codes. The learning targets listed for each course code represent the time frame when
all, or the majority of benchmarks, will have been taught.
MPS teachers designed this curriculum framework for their colleagues. The framework was reviewed by educational faculty from the
University of Wisconsin – Milwaukee. A framework is a constant work in progress and as such we hope that this framework will be
revisited and revised. Reflections by teachers as they use this framework will be valuable and needed as this framework is
implemented in the MPS high school classrooms.
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 1*: Process
Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways, e.g. using words, numbers,
symbols, pictures, charts, tables, diagrams, graphs, and models.
Topics Covered: Mathematical reasoning and communication; MPS Mathematics Academic Vocabulary and literacy strategies.
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
A.1*Use reasoning and logic to perceive
patterns, formulate questions, identify
relationships, pose problems, make and test
conjectures, and evaluate and justify
strategies.
A.2*Communicate mathematical ideas and
reasoning using the vocabulary of
mathematics in a variety of ways e.g. using
words, numbers, symbols, pictures, charts,
tables, diagrams, graphs and models.
A.3* Connect mathematics to the realworld as well as within mathematics.
A.4*Create and use representations to
organize, record, and communicate
mathematical ideas.
A.5* Solve and analyze routine and nonroutine problems.
A.12.1. Use reason and logic to evaluate
information, perceive patterns, identify
relationships, formulate questions, pose
problems, and make and test conjectures
and pursue ideas that lead to further
understanding and deeper insight.
1.1 Defend, orally or in writing, the reasonableness of mathematical and
real-world solutions.
3
1.2 Determine whether or not the reasoning applied to mathematical and
real- world problems make sense.
4
1.3 Write and solve their own mathematical and real-world application
problems.
3
1.4 Write, using appropriate mathematical vocabulary, a defense of their
own thinking in the solving of constructed response items in a test.
3
1.5. Design a PowerPoint or other computer-based application to
demonstrate the connectedness of mathematics to the real-world.
4
1.6. Respond to feedback on classroom assessments based on standards
which was written by the teacher and which uses a scoring rubric.
3
1.7 Read and interpret, in writing, charts, tables, diagrams, graphs, and
models.
3
1.8 Communicate the MPS Academic Vocabulary for mathematics in oral
and written presentations using literacy strategies e.g. graphic
organizers, vocabulary cloze and word splash activities, and causeeffect and similarities-differences diagrams.
2
TL** Instructional Sequence
Block
RC111
MA211
MA221
RC121
MA301
MA311
Regular
MA211
MA221
MA301
MA311
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
10th Grade Assessment
Framework/12th Grade Standards
Benchmarks: Students will be able to:
TL
Instructional Sequence
A.12.2. Communicate logical arguments
and clearly show why a result does or does
not make sense; why the reasoning is or is
not valid, and an understanding of the
difference between examples that support a
conjecture and a proof of the conjecture.
A.12.3. Analyze non-routine problems and
arrive at solutions by various means,
including models and simulations, often
starting with provisional conjectures and
progressing, directly or indirectly, to a
solution, justification, or counter-example.
A.12.4. Develop effective oral and written
presentations employing correct
mathematical terminology, notation,
symbols, and conventions for mathematical
arguments and display of data.
A.12.5. Organize work and present
mathematical procedures and results
clearly, systematically, succinctly, and
correctly.
A.12.6. Read and understand mathematical
texts and other instructional materials and
writing about mathematics, e.g., articles in
journals, mathematical ideas as they are
used in other contexts.
* 10th Grade Assessment Framework Descriptors
No * State 12th grade Standards
** TL means Thinking Levels as defined by MPS 2005
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 2*: Measurement
Use a variety of mathematical and technological tools to determine measurements directly (ruler, protractor, compass) and to check
reasonableness of answers. When students apply technology, e.g. computers and graphing calculators, they will interpret results
appropriately.
Topics Covered: Measurement, estimation, technology and use of math tools e.g. protractor, ruler.
Vocabulary: estimate, dimension, conversion, length, degree, centimeter, kilometer, metric unit, meter, inch, foot, yard, mile,
compass, straightedge, protractor.
10th Grade Assessment
Framework*/12th Grade Standards
B.6* Determine reasonableness of
answers.
D2.* Select and use tools with
appropriate degree of precision to
determine measurements directly.
Benchmarks: Students will be able to
TL
Instructional Sequence
2.1 Estimate measurements with precision, using appropriate
units.
1
Block
Regular
2.2 Measure lengths using a metric/English ruler (to the
appropriate degree of tenths of cm and sixteenths of an
inch).
2
RC111
MA211
MA221
RC121
MA301
MA311
MA211
MA221
MA301
MA311
2
2.3 Measure angles using a protractor (to appropriate degrees).
4
2.4 Justify reasons to interpret reasonableness of answers to
mathematical and real-world applications.
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 3*: Data Analysis
Use data in a variety of formats to summarize, predict, and analyze real-world applications.
Topics Covered: Use of data in various representations; finding measures of variation and measures of central tendency; use of data to
make predications.
Vocabulary: Statistics: survey, sample, population, sampling techniques, validity, mean, median, mode, graph, range, standard
deviation, data, scatter plot, percentiles, box and whisker plot, line of best fit, outliers, histogram, quartiles, stem and leaf, interquartile range, outlier, bias, measures of central tendency.
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
E.1*Organize, display, compare and
interpret data in a variety of ways in
mathematical and real-world contexts,
e.g. histograms, line graphs, stem-andleaf plots, scatter plots, box and
whiskers, bar charts, Venn diagrams,
tables and circle graphs.
E.2* Interpret, analyze and make
predictions from organized and
displayed data, e.g. measures of central
tendency, measures of variation such
as standard deviation, mean, median,
mode, range, dispersion, outliers, line
of best fit, percentiles.
E.3* Analyze, evaluate and critique
methods and conclusions of statistical
experiments, e.g. randomness,
sampling techniques, surveys.
3.1
3.2
3.3
3.4
Organize information into tables and stem-and-leaf plots.
Create histograms and describe the distribution.
Create and interpret bar graphs and circle graphs.
Determine the mean, median and mode of a set of data.
TL
2
3
3
2
3.5 Calculate the 5 number summary (quartiles) and create a box and whisker plot.
3.6 Use measures of center and measures of variation to interpret, analyze and make
predications from the data.
3.7 Display and interpret data using line graphs and scatter plots.
3.8 Use the line of best fit to make predictions from data on a graph.
1
2
3.9 Display and interpret data using Venn diagrams.
3.10 Compare and contrast two sets of data using measures of central tendency
from a box and whisker plot and a stem and leaf plot.
3.11 Choose a graph given a set of data and justify its appropriateness.
3
3
¾
Instructional Time
Block
RC111
Regular
MA211
4
3
4
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 4*: Probability
Use a variety of counting methods to identify all possible occurrences of events and apply this information to analyze, critique, and
predict the results of statistical experiments.
Topics Covered: Probability: Experiments, counting methods, probability of events.
Vocabulary: probability, permutations, combinations, reasonableness, conditional probability, theoretical probability, experimental
probability, dependent events, independent events, bias, randomness.
10th Grade Assessment
Framework*/12th Grade Standards
E.3* Analyze, evaluate and critique
methods and conclusions of
statistical experiments, e.g.
randomness, sampling techniques,
surveys.
E.4* Determine the likelihood of
occurrence of simple and complex
events, e.g., combinations and
permutations, fundamental counting
principle, experimental vs.
theoretical probability and
independent, dependent and
conditional probability.
Benchmarks: Students will be able to
TL
Instructional Time
4.1
4.2
4.3
4.4
2
2
2
2
Block
RC121
Determine the experimental probability of an event.
Determine the theoretical probability of an event.
Determine the probability of independent events.
Determine the likelihood of complex events using
dependent and/or conditional probability.
4.5 Compare the results of an experiment to the theoretical
probability.
4.6 Apply the fundamental counting principle to real-world
applications.
4.7 Use permutations and combinations.
Regular
MA221
4
2
2
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 5*: Number Operations
Compare, perform, explain and evaluate mathematical operations (+, -, *, /, exponents, roots) and properties on real numbers,
emphasizing negative and positive numbers; ratios, proportions and percents; and exponents and roots in mathematical and realworld applications.
Topics Covered: Number System: Order of operations, properties of real numbers, expressions, proportions, percents, roots.
Vocabulary: integer, ratio, proportion, percent, distributive property, associative properties, commutative properties, identity
properties, inverse properties, expression, exponent, power, root, absolute value real numbers, transitivity.
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
B.1* Compare and order real numbers.
B.2* Analyze and solve problems using
percents.
B.3* Apply proportional reasoning and ratios
in mathematical and real-world contexts.
B.4*Compare, perform and explain
operations on real numbers with and without
context, e.g. transitivity, rate of change,
exponential functions, scientific notation,
roots, powers, reciprocals, absolute value,
ratios, proportions, percents.
B.5* Select and use appropriate properties,
computational procedures, and modes of
representation with and without context, e.g.
simple and compound interest, commission,
percents, proportions.
F.6* Demonstrate understanding of
properties by evaluating and simplifying
expressions.
5.1 Solve problems using integers and absolute values.
5.2 Solve problems with fractions, decimals and percents.
5.3 Solve problems using ratios and proportions.
5.4 Solve real-world application problems using scale factors
and scale drawings.
5.5 Solve problems involving exponents and roots.
5.6 Express numbers using scientific notation and perform
operations, using the laws of exponents.
5.7 Identify the properties of real numbers.
5.8 Use the order of operations to evaluate and simplify
expressions.
5.9 Simplify expressions using the commutative, associative
and distributive properties.
5.10 Translate verbal expressions into algebraic expressions.
5.11 Simplify a radical.
TL
2
2
3
4
Instructional Sequence
Block
RC111
Regular
MA211
3
2
1
2
2
2
2
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 6*: Applied Formulas
Identify, describe and use formulas to solve real-world measurement problems such as interest, perimeter, area and volume.
Topics Covered: Formulas: interest, commissions, distance, area, perimeter, circumference, volume.
Vocabulary: variable, formula, substitution, area, perimeter, circumference, volume, interest, commission, prism, quadrilateral,
variable, metric units, meter, gram, liter, length, volume, weight, capacity, square unit, cubic unit, polygon, triangle, right triangle,
acute triangle, isosceles triangle, scalene triangle, equilateral triangle, obtuse triangles, circle, chord, diameter, radius, tangent, arcs in
circles, simple interest, compound interest.
10th Grade Assessment
Framework*/12th Grade Standards
B.5*Select and use appropriate
properties, computational procedures
and modes of representation with and
without context e.g. simple and
compound interest, commission,
percents, proportions.
D.3*Determine the perimeter/area of
two-dimensional figures.
D.4*Determine the surface
area/volume of three-dimensional
figures.
D.7*Use formulas in applications,
e.g. distance formula, simple and
compound interest.
Benchmarks: Students will be able to
TL
6.1 Use formulas in real-world applications, e.g. simple and
compound interest, commission, distance.
3
6.2 Calculate the area, perimeter and/or circumferences of a
triangle, quadrilateral and a circle.
3
6.3 Calculate the volume of rectangular prisms in real-world
applications.
3
6.4 Solve for any variable in a mathematical formula.
2
Instructional Sequence
Block
RC121
Regular
MA221
Draft 08/25/08
10
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 7*: Representations
Create and use tables, graphs, equations and formulas to describe relationships between variables.
Topics Covered: Linear Functions: using tables, graphs (slopes and intercepts) and equations, patterns of change, direct and inverse
variation.
Vocabulary: graph, scale, equation, table, slope, intercept(s), parallel, perpendicular, coordinates, coordinate system, rate of change,
domain, range, function(s), variable, slope-intercept, point slope, direct variation.
10th Grade Assessment
Framework*/12th Grade
Standards
Benchmarks: Students will be able to
F.1* Describe, recognize, interpret and
translate graphical representations of
mathematical and real-world phenomena
on coordinate grids, e.g., slope,
intercepts, rate of change, linear and
non-linear functions, and quadratic,
exponential and constant functions.
F.2*Analyze, generalize and represent
patterns of change, e.g., direct and
inverse variations, including numerical
sequences, patterns to a given term,
algebraic expressions and equations.
F.5*Translate between different
representations and describe the
relationship among variable quantities in
a problem, e.g. tables, graphs, functional
notations, formulas.
7.1 Create tables, including real-world data, using equations and graphs.
7.2 Create graphs, including real-world data, using equations and tables.
7.3 Identify the domain and range for a relation given a set of ordered pairs, a
table or a graph.
7.4 Determine from a set or ordered pairs, a table or a graph whether or not a
relation is a function.
7.5 Interpret the meaning of tables, graphs and equations using data from realworld applications.
7.6 Find and describe intercepts on a graph.
7.7 Find and describe slopes as constant rates of change.
7.8 Describe the relationship between the slopes of parallel and perpendicular
lines.
7.9 Write an equation from tables and graphs using real-world data.
7.10 Describe patterns of change from a table of ordered pairs.
7.11 Use formulas to calculate slope and intercepts from a set of ordered pairs.
7.12 Write a linear equation to represent a pattern in which there is a constant
rate of change between variables.
7.13 Identify the difference between direct and inverse variations in tables,
graphs, and equations.
TL Instructional Sequence
1
1
2
Block
MA211
Regular
MA211
2
3
2
3
3
2
2
4
2
2
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 8*: Coordinate Plane
Use the two-dimensional coordinate plane to analyze, generalize and represent algebraic properties in mathematical and real-world
applications.
Topics Covered: Coordinate plane: distance and midpoint formulas, horizontal and vertical lines.
Vocabulary: symmetry, midpoint, segment, direct variation, inverse variation, axis of symmetry, bisector, perpendicular, parallel,
altitude, angle bisector. Pythagorean Theorem.
10th Grade Assessment
Benchmarks: Students will be able to
TL Instructional Sequence
Framework*/12th Grade Standards
C.7* Use the two-dimensional rectangular
8.1 Determine if lines and segments drawn on a coordinate
2
Block
Regular
coordinate system to describe and
graph are parallel or perpendicular.
MA221
MA221
characterize properties of geometric
figures. Identify and apply symmetry about
an axis.
C.8* Use the two-dimensional rectangular
coordinate system and algebraic
procedures to describe and characterize
geometric properties and relationships (e.g.
slope, intercepts, parallelism, and
perpendicularity, Pythagorean Theorem,
distance formula).
F.1* Describe, recognize, interpret and
translate graphical representations of
mathematical and real-world phenomena
on coordinate grids, e.g. slope, intercepts,
rate of change, linear and non-linear
functions, and quadratic, exponential and
constant functions.
F2* Analyze, generalize and represent
patterns of change, e.g. direct and inverse
variations, including numerical sequences,
patterns to a given term, algebraic
expressions and equations.
8.2 Draw the line of symmetry of a graph (if it exists).
3
8.3 Use a coordinate plane and the distance and midpoint
formulas to determine lengths and midpoints of segments.
2
8.4 Use a coordinate plane and the distance and midpoint
formulas to determine the lengths and midpoints of the
sides of triangles.
2
8.5 Write and graph an equation of a horizontal and a vertical
line.
1
8.6 Analyze graphs of linear, quadratic and exponential
functions to determine how slopes and intercepts affect
rates of change.
4
Draft 08/25/08
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 9*: Modeling
Model and solve mathematical and real-world applications that can be represented by linear equations and systems of equations,
linear inequalities, and quadratic equations.
Topics Covered: Solving equations: One, two and multi-step equations; inequalities; systems; quadratics, properties of equality.
Vocabulary: equation, inverse, matrices, roots, symmetric, transitive, reflexive, systems of equations, vertex, minimum, maximum,
Substitution, inequality, velocity
10th Grade Assessment
Benchmarks: Students will be able to
TL Instructional Sequence
Framework*/12th Grade Standards
F.3* Solve linear and quadratic
9.1 Solve one-step equations using inverse operations.
2
Block
Regular
equations, linear inequalities and
9.2 Solve two-step equations using inverse operations and
2
MA211
MA211
systems of linear equations and
explain the meaning of the solution in context.
inequalities.
9.3 Solve multi-step equations using inverse operations and
3
F.4* Model and solve a variety of
explain the meaning of the solution in context.
mathematical and real-world
9.4 Solve linear inequalities in one or two variables and graph
3
problems by using algebraic
the results.
expressions, equations and
9.5
Solve and graph absolute value equations and inequalities.
3
inequalities, e.g., linear, exponential,
9.6 Solve a system of linear equations and explain the meaning
4
quadratic.
of the solution in context (methods: graphs, table,
F.7* Demonstrate understanding of
substitution, elimination, or matrices).
properties by solving linear and
quadratic equations, linear
9.7 Solve a quadratic equation by finding the roots and
3
inequalities, and systems of linear
identifying the minimum and maximum of a quadratic
equations and inequalities with one or
equation.
two variables.
9.8 Solve linear equations, inequalities, quadratic equations
3
and systems of linear equations to real-world applications.
9.9 Model real-world situations with linear equations, systems
of equations, and inequalities.
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 10*: Functions
Evaluate information, analyze patterns, identify relationships, and represent them using algebraic expressions, functions, and
equations, e.g., linear, exponential and quadratic, recognizing that a family of functions can model a variety of real-world
applications.
Topics Covered: Algebraic Relations: Patterns, graphing inequalities and systems of equations, writing equations for linear, quadratic
and exponential equations.
Vocabulary: pattern, inequality, model, quadratic, exponential function, analyze, acceleration, density.
10th Grade Assessment
Benchmarks: Students will be able to
Framework*/12th Grade Standards
F.2* Analyze, generalize and represent 10.1 Recognize the characteristics of linear, quadratic and
patterns of change, e.g., direct and
exponential families of functions.
inverse variation, including numerical
10.2 Identify the characteristics of an inequality graph.
sequences, patterns to a given term,
algebraic expressions and equations.
10.3 Analyze graphs that represent a given real world
F.4* Model and solve a variety of
applications.
mathematical and real-world problems 10.4 Write equations for linear, quadratic and exponential
by using algebraic expressions,
patterns.
equations, and inequalities, e.g., linear,
exponential, quadratic.
10.5 Identify and interpret the intersection of a system.
10.6 Interpret the properties of linear and quadratic
functions.
10.7 Identify a pattern in a numerical sequence and
complete the pattern to a given term.
10.8 Create models and justify strategies for a variety of
real-world applications using linear, exponential and
quadratic equations.
TL
3
Instructional Sequence
Block
MA221
Regular
MA221
3
3
2
3
3
2
3/4
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Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 11*: Right Triangles
Explain and use the Pythagorean Theorem and right triangle trigonometry as models to solve a variety of right triangle problems in
real-world applications.
Topics Covered: Right triangles.
Vocabulary: Triangles, Pythagorean Theorem, right, obtuse, acute, sine, cosine, tangent, elevation, depression, trigonometry functions,
clinometer.
10th Grade Assessment
Framework*/12th Grade Standards
C2* Present convincing geometric
arguments by means of informal
proof, counter-examples or other
logical means.
C.3* Model problems using the
Pythagorean Theorem and right
triangle trigonometry.
C.4*Use proportional reasoning to
solve congruence and similarity
problems, e.g., scale drawings and
similar geometric figures.
D.6*Use right triangle
trigonometric functions and the
Pythagorean Theorem to solve
right-triangle problems.
Benchmarks: Students will be able to
TL
11.1 Apply the Pythagorean Theorem to find side lengths of a
right triangle in mathematical and real-world applications.
2
11.2 Determine if a triangle is right, obtuse or acute by using the
Pythagorean Theorem.
3
11.3 Solve right triangle problems using trigonometric ratios.
3
11.4 Use trigonometric ratios to find the angle of elevation or
the angle of depression in real world applications.
4
Instructional Sequence
Block
RC121
Regular
MA221
Draft 08/25/08
15
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 12*: Lines and Angles
Identify and apply the definitions related to lines and angles in order to use them to solve problems and perform basic geometric
constructions.
Topics covered: Lines: angles; parallel and perpendicular; proofs.
Vocabulary: line, ray, line segment, plane, parallel, perpendicular, angle, complementary angle, supplementary angle, linear pair,
vertical angles, transversal alternate exterior, alternate interior, corresponding, consecutive interior, proof, collinear, coplanar,
endpoint, congruent, vertex, counterexample, right angle, acute angle, obtuse angle, circumcenter, incenter, centroid, orthocenter
10th Grade Assessment
Framework*/12th Grade Standards
C.1* Identify, describe and analyze
properties of figures, relationships
among figures and relationships
among their parts.
C.2* Present convincing arguments
by means of informal proof, counterexamples or other logical means.
Benchmarks: Students will be able to
TL
12.1 Identify and name points, lines, rays, segments and
planes.
1
12.2 Name and classify angles.
2
12.3 Identify and use angle-pair relationships in
mathematical and real-world applications.
3
12.4 Classify and communicate the differences between pairs
of lines.
3
12.5 Classify the types of angles formed by two lines and a
transversal.
2
12.6 Prove statements about segments, lines, angles and
angle relationships by using properties, postulates, and
theorems.
4
12.7 Perform basic geometric constructions.
3
Instructional Sequence
Block
MA301
Regular
MA301
Draft 08/25/08
16
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 13*: Classify Polygons
Classify and describe polygons and 3-dimensional figures.
Topics covered: Polygons: 3-dimensional figures.
Vocabulary: polygon, acute, obtuse, scalene, equilateral, isosceles, prism, pyramid, cone, cylinder, sphere, convex, concave,
polyhedron, displacement, density, orthographic drawing.
10th Grade Assessment
Framework*/12th Grade Standards
C.1* Identify, describe and analyze
properties of figures, relationships
among figures and relationships
among their parts.
C.6* Visualize 3-dimensional figures
in problem-solving situations
C.7* Use the two-dimensional
rectangular coordinate system to
describe and characterize properties
of geometric figures. Identify and
apply symmetry about an axis.
C.8* Use the two-dimensional
rectangular coordinate system and
algebraic procedures to describe and
characterize geometric properties and
relationships (e.g. slope, intercepts,
parallelism and perpendicularity,
Pythagorean Theorem, distance
formula).
Benchmarks: Students will be able to
TL
13.1 Classify polygons by sides and angles.
2
13.2 Classify triangles by sides and angles.
2
13.3 Classify quadrilaterals by their parts.
2
13.4 Describe properties of a given polygon.
3
13.5 Describe properties of quadrilaterals.
3
13.6 Describe properties of other polygons.
3
13.7 Classify geometric properties by using the coordinate
system (e.g. distance formula, slope).
3
13.8 Identify 3-dimensional objects (e.g. prisms, pyramids,
cones, cylinders and spheres).
2
Instructional Sequence
Block
MA301
Regular
MA301
Draft 08/25/08
17
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 14: Analyze Polygons
Use and analyze geometric relationships and properties of polygons and 3-dimensional figures to solve mathematical and real-world
applications.
Topics Covered: Polygons: Sides and angles in triangles and polygons
Vocabulary: angles, interior angle, exterior angle, vertical angle, polygon, 3-dimensional, similarity, similar polygons.
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
C.12.1 Identify, describe and analyze
properties of figures, relationships
among figures and relationships among
their parts by constructing physical
models, drawing precisely with paperand-pencil, hand calculators, and
computer software, using appropriate
transformations, and using reason and
logic.
C.12.2 Use geometric models to solve
mathematical and real-world
problems.
C 12.3 Present convincing arguments
by means of demonstration, informal
proof, counter-examples or any other
logical means to show the truth of
statements (e.g. these two triangles are
not congruent) and generalizations
(e.g., the Pythagorean theorem holds
for all right triangles).
D.5* Solve for angles and segments in
similar polygons and arcs in circles.
14.1 Determine if lengths of three segments can form a
triangle.
3
14.2 Determine the range in which the length of the third side
of a triangle must be given the lengths of two sides.
3
14.3 Find the measures of segments in similar polygons.
2
14.4 Find the measures of each interior and exterior angle of a
regular polygon.
2
14.5 Find the sum of the measures of the interior and exterior
angles of a convex polygon.
2
14.6 Solve problems involving real-world applications using
the relationships between pairs of angles.
3
14.7 Make a two-dimensional representation of a 3dimensional object.
3
TL
Instructional Sequence
Block
MA301
Regular
MA301
Draft 08/25/08
18
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 15: Congruence/Similarity
Apply the basic theorems of congruence and similarity to solve mathematical and real-world applications.
Topics covered: Congruence, similarity.
Vocabulary: ASA, SSS, SAS, HL, AA, congruent, similar, similarity.
10th Grade Assessment
Benchmarks: Students will be able to
Framework*/12th Grade Standards
C.12.1 Identify, describe, and analyze
15.1 Identify congruent and similar polygons.
properties of figures, relationships among
figures, and relationships among their parts
by constructing physical models, drawing
precisely with paper-and pencil, hand
calculators, and computer software, using
appropriate transformations, and using
reason and logic.
C 12.3 Present convincing arguments by
means of demonstration, informal proof,
counter-examples or any other logical
means to show the truth of statements (e.g.
these two triangles are not congruent) and
generalizations (e.g., the Pythagorean
theorem holds for all right triangles).
C.12.5 Identify and demonstrate an
understanding of the three ratios used in
right-triangle trigonometry (sine, cosine,
tangent). .
D12.3 Determine measurements indirectly,
using proportional reasoning.
F.12.4 Model and solve a variety of
mathematical and real-world problems by
using algebraic expressions, equations and
inequalities.
TL
2
15.2 Justify congruency or similarity of polygons by using
formal and informal proofs.
3
15.3 Prove two triangles are congruent using coordinate
methods (e.g. distance and slope formulas).
4
15.4 Solve real-world applications using the properties of
congruency and similarity.
4
Instructional Sequence
Block
MA301
Regular
MA301
Draft 08/25/08
19
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 16: Transformations
Apply transformations and their compositions to analyze geometric figures and create designs.
Topics Covered: Transformations and tessellations.
Vocabulary: transformation, tessellation, image, pre-image, symmetry, representations, reflection, rotation, isometry, isometrics,
translation, axis of symmetry
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
C.12.1 Identify, describe, and analyze
properties of figures, relationships
among figures, and relationships among
their parts by constructing physical models,
drawing precisely with paper-andpencil, hand calculators, and computer
software, using appropriate transformations
(e.g., translations, rotations, reflections,
enlargements), and using reason and logic.
C.12.2 Use geometric models to solve
mathematical and real-world problems.
C.12.4 Use the two-dimensional
rectangular coordinate system and
algebraic procedures to describe and
characterize geometric properties and
relationships such as slope, intercepts,
parallelism, and perpendicularity.
C.5* Use transformations and symmetry to
solve problems.
16.1 Plot and label geometric figures using a coordinate plane.
2
16.2 Perform transformations using triangles in a coordinate
plane.
3
16.3 Identify, given an image and pre-image, the
transformation that has taken place as a reflection,
rotation, or translation.
2
16.4 Determine whether a figure has point symmetry,
line symmetry or neither.
2
16.5 Identify tessellations in art, construction and nature.
3
16.6 Design a real-world application of transformations and
communicate the process used in the design.
4
TL
Instructional
Sequence
Block
Regular
MA311
MA311
Draft 08/25/08
20
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 17: Circles
Use properties of circles to solve mathematical and real-world applications.
Topics covered: Circles.
Vocabulary: chord, secant, tangent, arc, sector, inscribed, central angle.
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
C.12.1 Identify, describe, and analyze
properties of figures, relationships
among figures, and relationships
among their parts by constructing
physical models, drawing precisely
with paper-and pencil, hand
calculators, and computer software,
using appropriate transformations (e.g.,
translations, rotations, reflections,
enlargements), and using reason and
logic.
C.12.2 Use geometric models to solve
mathematical and real-world problems.
D.12.3 Determine measurements
indirectly using geometric relationships
and properties of circles and polygons
(e.g. size of central angles, area of a
sector of a circle).
17.1 Determine circumference and area of a circle.
2
17.2 Determine measures of inscribed and central angles.
3
17.3 Determine arc lengths and areas of sectors of circles.
3
17.4 Write and solve equations to find the measures of angles
formed by chords, secants, and tangents.
3
17.5 Calculate lengths of chord, tangent and secant segments.
3
17.6 Write an equation of a circle and graph a circle from its
equation.
3
17.7 Solve real-world applications of circles.
4
TL
Instructional Sequence
Block
MA311
Regular
MA311
Draft 08/25/08
21
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 18: Applied Geometry
Use perimeter, area, volume and surface area to solve real-world applications.
Topics covered: Measurements of 2 and 3 dimensional figures.
Vocabulary: perimeter, area, volume, surface area, degree, cubic units, square units, faces, bases, edge, lateral area, perspective,
concurrency
10th Grade Assessment
Framework*/12th Grade Standards
Benchmarks: Students will be able to
C.12.1 Identify, describe, and analyze
properties of figures, relationships among
figures, and relationships among their parts
by constructing physical models, drawing
precisely with paper-and pencil, hand
calculators, and computer software, using
appropriate transformations (e.g.,
translations, rotations, reflections,
enlargements), and using reason and logic.
C.12.2 Use geometric models to solve
mathematical and real-world problems.
D.12.3Determine measurements indirectly
using geometric formulas to derive lengths,
areas, or volumes or shapes and objects.
18.1 Determine the perimeter/circumference and area of
quadrilaterals, triangles and other polygons, using real-world
applications.
18.2 Determine the volume and lateral area and total surface area of
spheres, prisms, pyramids, cylinders and cones.
18.3 Use volume, lateral area and surface area of 3-dimensional
figures to solve real-world application problems.
18.4 Solve real-world application problems for missing lengths in
special right triangles.
18.5 Compare perimeters and areas of similar 2-dimensional figures
using proportions.
18.6 Describe how a change in one measurement affects other
measurements of the object.
18.7 Solve real-world application problems using scale drawings,
perspective drawings, blueprints, and/or computer drawings as
models.
18.8 Make a model of a 3-dimensional figure from a 2-dimensional
drawing.
TL
2
Instructional Sequence
Block
MA311
Regular
MA311
2
3
3
3
3
3
4
Draft 08/25/08
22
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.
Foundation Level
Learning Target 19:
Use increased rigor in justifying conclusions to mathematical and real-world applications and in verifying the truth of geometric
theorems.
Topics covered: Conjectures, proofs
Vocabulary: conjecture, counterexample, deductive logic, inductive logic, proof, theorem, postulate, syllogism, hypothesis,
conclusion, converse, inverse, contra positive, paragraph proof, flow-chart, coordinate proof., Venn diagram, justify, strategy,
reasonableness,, representations, negation.
10th Grade Assessment
Benchmarks: Students will be able to
TL Instructional Sequence
Framework*/12th Grade Standards
A.12.1 Use reason and logic to
19.1 Identify the converse, inverse and contra positive of a
2
Regular
Block
evaluate information, perceive
conditional statement and their relative truth value.
MA311
MA311
patterns, identify relationships,
formulate questions, pose problems, 19.2 Translate short verbal arguments into symbolic form.
2
and make and test conjectures.
B.12.5 Create and critically
19.3 Write conjectures using characteristics, postulates and
3
evaluate numerical arguments
theorems of geometric figures.
presented in a variety of classroom
3
and real-world situations (e.g.,
19.4 Use and interpret Venn diagrams.
political, economic, scientific,
social).
19.5 Diagram conditional statements to solve logic puzzles.
3
C.12.3 Students will present
convincing arguments by means of 19.6 Use valid forms of inductive and deductive reasoning to
3
demonstration, informal proof,
draw conclusions and make conjectures.
counter examples or any other
logical means to show the truth of
19.7 Select and use various types of reasoning and methods to
4
statements and generalizations.
prove characteristics and properties of algebraic models
and geometric figures.
Draft 08/25/08
23
Developed by the Milwaukee Mathematics Partnership (MMP) with support by the National Science Foundation under Grant No. 0314898.