1 Learning Targets for Mathematics Milwaukee Public Schools
2 Grades K 4 Mathematics Learning Targets Milwaukee Public Schools Number Operations & Relationships Geometry Kindergarten (1) Use strategies for counting and keeping track of quantities to 20 (e.g. counting all in a set, counting all in two sets, counting on). (2) Connect number words and numerals to the quantities they represent and compare the quantities. (3) Use strategies to find combinations of numbers with sums to 6 in realworld situations or with objects (e.g. dots on dominoes). (4) Compare and sort threedimensional shapes and real-world objects. Grade 1 (1) Knows combinations of 10 (e.g , 8 + 2). (2) Use strategies fluently (e.g. doubles plus one, use ten) to determine addition facts (sums through 10) and the corresponding subtraction facts. (3) Record strategies for adding and subtracting quantities up to 20 in realworld contexts including situations with money (e.g. pennies, nickels, and dimes). (4) Sort, classify, and describe the attributes of two- and three-dimensional shapes. Grade 2 (1) Use strategies fluently (e.g. doubles plus one, use ten) to determine addition facts (sums through 20) and the corresponding subtraction facts. (2) Use and explain strategies for fluently solving real-world addition and subtraction situations (e.g. money and menus) with whole numbers up to 100. (3) Describe the relationships and predict the results of combining and subdividing twodimensional shapes (e.g. two trapezoids can form a hexagon). Grade 3 (1) Communicate use of place value to add and subtract whole numbers with strategies that are fluent and flexible. (2) Pose, estimate answers, represent, and solve a variety of real-world multiplication and division situations. (3) Create visual and concrete models (e.g. repeated groups and arrays) of multiplication facts, and use strategies fluently to determine multiplication facts that have one factor of 1, 2, 3, 4, or 5. (4) Compare and describe attributes of shapes and structures in their environment. (5) Predict and describe transformations of shapes (e.g. sliding, flipping, and turning). Grade 4 (1) Use strategies fluently in problem-solving situations to determine multiplication and division facts that have factors 1 through 9. (2) Represent simple fractions (e.g. halves, fourths, tenths, unit fractions) and commonly used decimals and use informal reasoning to compare, add, and subtract them. (3) Describe, compare, and classify two- and threedimensional figures according to their properties including symmetry. (4) Use coordinate systems to specify locations and to represent simple figures.
3 Measurement Statistics & Probability Kindergarten (5) Compare and order objects according to attributes of length, capacity, weight, and mass (e.g. balance scale) using informal strategies. (6) Collect data to answer questions that have two responses (e.g. yes or no) and report the results. Grade 1 (5) Use nonstandard units to measure and quantify attributes of objects. (6) Collect data to answer questions about themselves and their world, collect, organize, represent, and describe the data. Grade 2 (4) Select and use standard units and tools to measure and quantify attributes of objects and use common references to make comparisons and estimates. (5) Compare durations of time (e.g. how long is a minute or an hour) and read time to the nearest hour and half-hour using analog and digital clocks. (6) Pose questions, construct representations for categorical and numerical data, and describe parts of the data in relation to the set of data as a whole. Grade 3 (6) Use, read, and interpret measuring instruments in both customary and metric systems, and make simple unit conversions within a system of measurement. (7) Tell time to the nearest minute using analog and digital clocks and determine elapsed time in contextualized situations. (8) Describe the shape and important features of a data set with an emphasis on its distribution (e.g. range, clusters, gaps) in relation to the context of the data. (9) Describe the degree of likelihood of events (e.g. certain, equally likely, impossible) in everyday contexts. Grade 4 (5) Determine measurements, including area and perimeter, to specified degrees of precision, and use benchmarks to approximate measurements. (6) Design investigations to answer questions, represent data, use the median to summarize and compare data sets, and interpret results. (7) Make and compare predictions with outcomes of simple experiments and describe likelihood of future outcomes. Algebraic Relationships (7) Recognize patterns made with colors and shapes and found in their world. (7) Identify the basic unit in repeating patterns, and describe, create, and extend patterns. (8) Use invented and conventional symbolic notations (e.g. concrete, pictorial, and verbal representations) to represent solution strategies. (7) Describe how repeating and growing patterns change and are generated. (8) Communicate with and interpret written symbols (e.g. the equal sign denotes the relation between two equal quantities). (10) Describe, extend, and make generalizations about geometric and numeric patterns and properties of operations. (8) Use words, symbols, and notation accurately to represent problem situations and to record solution strategies.
4 Number Operations & Relationships Geometry Measurement Grade 5 students know (1) In problem-solving situations, use strategies fluently to multiply and divide whole numbers, evaluate strategies for efficiency, and determine reasonableness of results. (2) Generate equivalent forms of commonly used fractions, decimals, and percents, and use and evaluate informal strategies to compare and add commonly used fractions and decimals. (3) Describe properties of a class of two- and three-dimensional shapes (e.g. parallel sides, measures of angles, congruent faces, regularity, line and rotational symmetry). (4) Move between threedimensional objects and twodimensional representations of shapes by building and drawing and evaluate the results. (5) Explain reasoning for estimates of measures, and determine measurements (i.e. weight, size of angle) with tools to specified degrees of accuracy. Grades 5 8 Mathematics Learning Targets Milwaukee Public Schools Grade 6 students know (1) Use factors, multiples, and prime factorization of whole numbers to solve and explain problems. (2) Use explain, and evaluate, strategies to estimate results and use varied tools (e.g., mental computation, technology, paper and pencil) for comparing and computing with fractions to solve problems. (3) Identify and contrast properties of two-dimensional shapes, and use specifications to draw, construct, and label shapes. (4) Construct and explain informal strategies (e.g. transparent grids, string, tiles, cutting and rearranging) for finding the area and perimeter of shapes and for solving applied problems. (5) Describe relationships between area and perimeter and how areas of shapes are related to each other (e.g. triangles and rectangles). Grade 7 students know (1) Represent (e.g. number line or counters), explain, and evaluate strategies for computing with integers to solve problems. (2) Interpret fractions as percents, ratios, rates, or as parts to whole comparisons (e.g. scale factors and proportional relationships). (3) Explain reasoning and justification for classification of shapes by their defining properties. (4) Enlarge and shrink plane figures, identify scale factors between figures, and apply properties of similar figures to solve real-world problems. (5) Construct and describe procedures to determine the area and volume of shapes, including complex shapes, and justify measurement formulas for area and circumference. Grade 8 students know [Embedded in other targets, e.g. see measurement.] (1) Perform transformations of figures, including reflections, rotations, and translations; analyze effects of transformations on the figures; and use appropriate mathematical vocabulary, symbols, and notation. (2) Use the Pythagorean theorem, square roots, and irrational numbers to determine lengths of sides of a triangle in problemsolving situations and explain procedures.
5 Statistics & Probability Algebraic Relationships Grade 5 students know (6) Make context-based conclusions and predictions about a given data set, evaluate a sample for biases, and design further investigations. (7) Describe relationships between verbal descriptions (e.g. equally likely, certain) and numeric measures of the likelihood of an event, and explain why the measures are represented by numbers from 0 to 1. (8) Represent and analyze patterns and functions, using words, tables, and graphs. (9) Use generalized relationships and properties to compute with whole numbers (e.g. 4 x 27 = (4 x 25) + (4 x 2) or zero times any number is 0) and justify mathematical relationships using words and equations. Grade 6 students know (6) Formulate questions, collect data and analyze these data using measures of central tendency (mean, median, and mode) and graphs (line plot, stem and leaf, scatter plot, and bar graph). (7) Gather data from experiments, and justify possible and favorable outcomes as expressed by probabilities. (8) Describe how two sets of data are related to each other by using coordinate graphs to identify relationships among variables. (9) Identify, describe, and justify generalized properties and relations (e.g. commutative, associative, distributive, inverses, identities). Grade 7 students know (6) Design and conduct a simulations using a variety of strategies to represent and interpret a range of data sets and make predictions, judge fairness of events, and determine expected values. (7) Describe relationships among written descriptions, tables, graphs, and symbolic rules, and reason with the different representations to interpret linear relationships. Grade 8 students know (3) Formulate questions that lead to data collection and analysis, design and conduct a statistical investigation, represent data in appropriate plots (e.g. line, box, scatter), and communicate the results. (4) Design experiments, use strategies to identify the likeliness of possible outcomes (e.g. tree diagrams, lists) of simple events, and justify the selection of the chosen strategy. (5) Represent and describe functional relationships in tables, graphical representations, and symbolic forms, and identify whether they translate into linear or exponential relationships. (6) Use reasoning abilities to perceive patterns and identify relationships in order to generalize a rule that characterizes the rate of change among variables in functional relationships. (7) Represent and solve equations and inequalities using different methods (e.g. informally, graphically, using generalized properties, and with technology) and communicate why a results makes sense.
6 Grades 9 12 Mathematics Learning Targets Milwaukee Public Schools Foundation Level 1 (1) Explain and analyze applications involving linear models using graphs, charts, scatterplots, calculators, computers, and appropriate tools (2) Solve linear equations, linear inequalities, exponential equations, systems of linear equations, and quadratic equations numerically, graphically and symbolically. Apply appropriate technology (e.g., computers and graphing calculators) to interpret solutions. (3) Organize and display 2-variable data sets from statistical investigations using scatterplots and linear regression models incorporating the use of technology to organize data sets. Interpret a line s slope and intercepts using appropriate units based on the applications represented by the data set. (4) Evaluate information, analyze patterns, identify relationships, and represent them using algebraic expressions and equations (e.g., linear, exponential, and quadratic), recognizing that a family of functions can model a variety of real-world situations. (5) Represent and interpret linear, exponential, and quadratic functions using tables, graphs, algebraic equations and verbal descriptions that incorporate correct mathematical vocabulary, symbols an notation. (6) Perform and explain operations of real numbers (add, subtract, multiply, divide, and raise to a power). (7) Determine measurements indirectly using formulas in applications (e.g., compound interest) including the selection and use of appropriate computational procedures, properties (commutativity, associativity, and inverses), and modes of representation (e.g., rationals as repeating decimals). (8) Determine the likelihood of occurrence of complex events by conducting an experiment, designing and conducting simulations, and applying theoretical probabilities. (9) Determine equivalent forms of expressions and equations by selecting and applying appropriate computational procedures and properties (e.g., simplifying polynomials). Foundation Level 2 (1) Explain and analyze solutions to problems involving geometric representations and relationships. (2) Use a two-dimensional rectangular coordinate system and algebraic procedures to describe and interpret geometric properties and relationships including slope, intercepts, parallelism, perpendicularity, distance, and midpoint. (3) Identify patterns, and create and test conjectures involving transformations (e.g., translations, rotations, reflections, and enlargements). (4) Derive measurements indirectly using geometric formulas of lengths, areas, and volumes together with proportional reasoning involving squaring and cubing (e.g., reasoning that areas of circles are proportional to the squares of their radii).
7 (5) Apply deductive logic to describe, analyze and classify plane figures (e.g., quadrilaterals and triangles) by examining sides, angles, proportions based on similarity, and the properties of shapes. (6) Use the three ratios in right-triangle trigonometry (sine, cosine, and tangent) and the Pythagorean Theorem to solve symbolic and real-world problems. (7) Communicate logical, convincing and clear arguments by means of demonstration, informal proof or counter examples in applications involving similarity and congruence. (8) Select and use appropriate tools to determine measurements directly (e.g., ruler and protractor). (9) Represent probability through geometric models and/or representations of area and proportions. Intermediate Level (1) Evaluate and formulate questions and problems involving rate of change as evident in physics, economics, and other applications. (2) Derive, apply, and interpret the intersection of graphs involving linear, power, and exponential functions as related to real-world applications. (3) Compute, interpret, and analyze the mean, median, and mode graphically and numerically as collected from simulations or actual sampling. (4) Derive and interpret properties of the quadratic functions related to minimums, maximums, intercepts, and symmetry representing real-world applications. (5) Interpret expected values as an estimation of an outcome from a specific problem or simulation. (6) Translate different forms of linear and nonlinear functions to tables, graphs, functional notation, and formulas. (7) Apply symbolic algebraic operations and properties to solve problems. (8) Define, compute and interpret sine, cosine, and tangent functions as related to applications involving the laws of sines and cosines and other trig formulas. (9) Represent data in matrix form and apply appropriate matrix computations to solve or simplify system of equations representing authentic applications. Advanced Level Category A: Statistics and Probability (1) Analyze statistical case studies regarding experimental design and inferential conclusions. (2) Interpret graphical displays of univariate data involving shape, center (mean and median), clusters and gaps. (3) Analyze patterns and relationships in scatter plots, boxplots, cumulative frequency plots, and histograms. (4) Interpret marginal and joint relative frequencies from two-way tables as tests for independence and dependence. (5) Develop simulations to identify the variability of sample statistics from a known population. Construct sampling distributions from the simulations.
9 Advanced Level Category B: Pre-Calculus (1) Design and interpret properties of, and representations for, the addition and multiplication of vectors and matrices. (2) Interpret asymptotes in terms of graphical behavior. (3) Derive and interpret range and domain for continuous functions over the set of real numbers. (4) Analyze the effects of transformations and their compositions to applications. (5) Analyze algebraic, exponential, logarithmic, and trigonometric functions graphically, symbolically, and numerically. Advanced Level Category C: Calculus (1) Classify and interpret graphs and rate of change involving exponential, polynomial, and logarithmic functions. (2) Model problems involving rate of change, including velocity, acceleration, and other physics applications. (3) Represent translations, reflections, rotations, and dilations of objects in the plane through various representations. (4) Identify, analyze, and compare situations with constant or varying rates of change using differentiation. (5) Interpret continuous functions, (e.g., relative minimums and maximums) based on the slope of tangent lines, differentiation, and areas underneath their curves by integrals.