Milwaukee Public Schools
Curriculum Framework
for
High School Mathematics
Foundation Level
(Algebra 1 and Geometry 1)
Introduction
This MPS Curriculum FrameworkforHigh 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 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) andMA221: 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.
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 / TL** / Instructional SequenceA.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 real-world 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 non-routine 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.
10th Grade Assessment Framework/12th Grade Standards
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. / 1.1 Defend, orally or in writing, the reasonableness of mathematical and
real-world solutions.
1.2 Determine whether or not the reasoning applied to mathematical and
real- world problems make sense.
1.3 Write and solve their own mathematical and real-world application
problems.
1.4 Write, using appropriate mathematical vocabulary, a defense of their
own thinking in the solving of constructed response items in a test.
1.5. Design a PowerPoint or other computer-based application to
demonstrate the connectedness of mathematics to the real-world.
1.6. Respond to feedback on classroom assessments based on standards
which was written by the teacher and which uses a scoring rubric.
1.7 Read and interpret, in writing, charts, tables, diagrams, graphs, and
models.
1.8Communicate 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.
Benchmarks: Students will be able to: / 3
4
3
3
4
3
3
2
TL / Block Regular
RC111 MA211
MA211 MA221
MA221 MA301
RC121 MA311
MA301
MA311
Instructional Sequence
* 10th Grade Assessment Framework Descriptors
No * State 12th grade Standards
** TL means Thinking Levels as defined by MPS 2005
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 / Benchmarks: Students will be able to / TL / Instructional SequenceB.6* Determine reasonableness of answers.
D2.* Select and use tools with appropriate degree of precision to determine measurements directly. / 2.1 Estimate measurements with precision, using appropriate
units.
2.2 Measure lengths using a metric/English ruler (to the
appropriate degree of tenths of cm and sixteenths of an
inch).
2.3 Measure angles using a protractor (to appropriate degrees).
2.4 Justify reasons to interpret reasonableness of answers to
mathematical and real-world applications. / 1
2
2
4 / Block Regular
RC111 MA211
MA211 MA221
MA221 MA301
RC121 MA311
MA301
MA311
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, inter-quartile range, outlier, bias, measures of central tendency.
10th Grade Assessment Framework*/12th Grade Standards / Benchmarks: Students will be able to / TL / Instructional TimeE.1*Organize, display, compare and interpret data in a variety of ways in mathematical and real-world contexts, e.g. histograms, line graphs, stem-and-leaf 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. / 3.1Organize information into tables and stem-and-leaf plots.
3.2 Create histograms and describe the distribution.
3.3 Create and interpret bar graphs and circle graphs.
3.4 Determine the mean, median and mode of a set of data.
3.5Calculate the 5 number summary (quartiles) and create a box and whisker plot.
3.6Use measures of center and measures of variation to interpret, analyze and make predications from the data.
3.7Display and interpret data using line graphs and scatter plots.
3.8Use the line of best fit to make predictions from data on a graph.
3.9 Display and interpret data using Venn diagrams.
3.10Compare and contrast two sets of data using measures of central tendency
from a box and whisker plot and a stem and leaf plot.
3.11Choose a graph given a set of data and justify its appropriateness. / 2
3
3
2
1
2
4
3
3
3
¾
4 / Block Regular
RC111 MA211
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 / Benchmarks: Students will be able to / TL / Instructional TimeE.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. / 4.1 Determine the experimental probability of an event.
4.2 Determine the theoretical probability of an event.
4.3 Determine the probability of independent events.
4.4 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. / 2
2
2
2
4
2
2 / Block Regular
RC121 MA221
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 real-world 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 / TL / Instructional SequenceB.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. / 2
2
3
4
3
2
1
2
2
2
2 / Block Regular
RC111 MA211
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 / Benchmarks: Students will be able to / TL / Instructional SequenceB.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. / 6.1 Use formulas in real-world applications, e.g. simple and
compound interest, commission, distance.
6.2 Calculate the area, perimeter and/or circumferences of a
triangle, quadrilateral and a circle.
6.3 Calculate the volume of rectangular prisms in real-world
applications.
6.4 Solve for any variable in a mathematical formula. / 3
3
3
2 / Block Regular
RC121 MA221
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 / TL / Instructional SequenceF.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.
7.3Identify the domain and range for a relation given a set of ordered pairs, a table or a graph.
7.4Determine from a set or ordered pairs, a table or a graph whether or not a relation is a function.
7.5Interpret the meaning of tables, graphs and equations using data from real-world applications.
7.6Find and describe intercepts on a graph.
7.7 Find and describe slopes as constant rates of change.
7.8Describe 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
rateof change between variables.
7.13 Identify the difference between direct and inverse variations in tables,
graphs, and equations. / 1
1
2
2
3
2
3
3
2
2
4
2
2 / Block Regular
MA211 MA211
Foundation Level