Mathematics Learning Goals

• Understand relationships among factors, multiples, divisors, and products

• Recognize and use properties of prime and composite numbers, even and odd numbers, and square numbers

• Use rectangles to represent the factor pairs of numbers

• Develop strategies for finding factors and multiples, least common multiples, and greatest common factors

• Recognize and use the fact that every whole number greater than one can be written in exactly one way as a product of prime numbers

• Use factors and multiples to solve problems and to explain some numerical facts of everyday life

• Develop a variety of strategies for solving problems-building models, making lists and tables, drawing diagrams, and solving simpler problems

• Build an understanding of fractions, decimals, and percents and the relationships between and among these concepts and their representations

• Develop ways to model situations involving fractions, decimals, and percents

• Understand and use equivalent fractions to reason about situations

• Compare and order fractions

• Move flexibly among fraction, decimal, and percent representations

• Use benchmarks such as 0, 1/2, and 1 to help estimate the size of a number or sum

• Develop and use benchmarks that relate different forms of representations of rational numbers (for example, 50% can be represented as 0.5)

• Use physical models and drawings to help reason about a situation

• Look for patterns and describe how to continue the pattern

• Use context to help reason about a situation

• Use estimation to understand a situation

• Understand area and relate area to covering a figure

• Understand perimeter and relate perimeter to surrounding a figure

• Develop strategies for finding areas and perimeters of rectangular shapes and non- rectangular shapes

• Discover relationships between perimeter and area, including that each can vary while the other stays fixed

• Understand how the areas of simple geometric figures relate to each other (e.g. the area of a parallelogram is twice the area of a triangle with the same base and height)

• Develop formulas and procedures-stated in words and/or symbols-for finding areas and perimeters of rectangles, parallelograms, triangles, and circles

• Develop techniques for estimating the area and perimeter of an irregular figure

• Recognize situations in which measuring perimeter or area will help answer practical questions

• Build on knowledge of operations on fractions and whole numbers

• Develop and use benchmarks and other strategies to estimate the answers to computations with decimals

• Develop meaning of and algorithms for operations with decimals

• Use the relationship between decimals and fractions to develop and understand why decimal algorithms work

• Use the place value interpretation of decimals to make sense of short-cut algorithms for operations

• Generalize number patterns to help make sense of decimal operations

• Choose between addition, subtraction, multiplication or division as an appropriate operation to use to solve a problem

• Understand that decimals are often associated with measurements in real world situations

• Solve problems using operations on decimals

• Use understanding of operations and the meaning of percents to solve percent problems of the form a% of b equals c for any one of the variables a, b, or c.

• Create and interpret circle graphs

• Understand that probabilities are useful for predicting what will happen over the long run

• Understand the concepts of equally likely and not-equally likely

• Understand that fairness implies equally likely outcomes.

• Understand that there are two ways to build probability models: by gathering data from experiments (experimental probability) and by analyzing the possible equally likely outcomes (theoretical probability)

• Understand that experimental probabilities are better estimates of theoretical probabilities when they are based on larger numbers of trials

• Develop strategies for finding both experimental and theoretical probabilities

• Critically interpret statements of probability to make decisions or answer questions

• Understand and use the process of data investigation by posing questions, collecting data, analyzing data distributions, and making interpretations to answer questions

• Represent data distributions using line plots, bar graphs, stem-and-leaf plots, and coordinate graphs

• Compute the mean, median, or mode and the range of the data

• Distinguish between categorical data and numerical data and identify which graphs and statistics may be used to represent each kind of data

• Make informed decisions about which graph or graphs and which of the measures of center (mean, median, or mode) and range may be used to describe a data distribution

• Develop strategies for comparing data distributions

• Read and make two-dimensional representations of three-dimensional cube buildings

• Observe that the back view of a cube building is the mirror image of the front view and that the left view is the mirror image of the right view

• Explain how drawings of the base outline, front view, and right view describe a building

• Construct cube buildings that fit two- dimensional building plans

• Develop a way to describe all buildings that can be made from a set of plans

• Understand that a set of plans can have more than one minimal building but only one maximal building

• Explain how a cube can be represented on isometric dot paper, how the angles on the cube are represented with angles on the dot paper, and how the representations fit what the eye sees when viewing the corner of a cube building

• Make isometric drawings of cube buildings

• Visualize transformations of cube buildings and make isometric drawings of the transformed buildings

• Reason about spatial relationships

• Use models and representations to solve problems

Grade 7 Goals

• Recognize problem situations in which two or more quantitative variables are related to each other

• Identify quantitative variables in situations

• Describe patterns of change between two variables that are shown in words, tables and graphs of data

• Construct tables and graphs to display relations among variables

• Observe relationships between two variables as shown in a table, graph, or equation and describe how the relationship can be seen in each of the other forms of representation

• Use algebraic symbols to write equations relating variables

• Use tables, graphs, and equations to solve problems

• Use graphing calculators to construct tables and graphs of relations between variables and to answer questions about these relations

• Identify similar figures by comparing corresponding parts

• Use scale factors and ratios to describe relationships among the side lengths of similar figures

• Construct similar polygons

• Draw shapes on coordinate grids and then use coordinate rules to stretch and shrink those shapes

• Predict the ways that stretching or shrinking a figure affect lengths, angle measures, perimeters, and areas

• Use the properties of similarity to calculate distances and heights that can't be directly measured

• Analyze comparison statements made about quantitative data

• Use ratios, fractions, differences, and percents to form comparison statements in a given situation, such as:

  • "What is the ratio of boys to girls in our class?"

  • "What fraction of the class is going to the spring picnic?"

  • "What percent of the girls play basketball?"

  • "Which model of car has the best fuel economy?"

  • "Which long-distance telephone company is more popular?"

• Judge whether comparison statements make sense and are useful

• See how forms of comparison statements are related, for example, a percent and a fraction comparison

• Make judgments about which statements are most informative or best reflect a particular point of view

• Decide when the most informative comparison is to find the difference between two quantities and when it is to form ratios between pairs of quantities

• Scale a ratio, rate, or fraction up or down to make a larger or smaller object or population with the same relative characteristics as the original

• Represent related data in tables

• Look for patterns in tables that will allow predictions to be made beyond the tables

• Write an equation to represent the pattern in a table of related variables

• Apply proportional reasoning to solve for the unknown part when one part of two equal ratios is unknown

• Set up and solve proportions that arise in applications

• Recognize that constant growth in a table is related to proportional situations

• Connect unit rates with the equation describing a situation

• Use appropriate notation to indicate positive and negative numbers

• Locate rational numbers (positive and negative fractions and decimals and zero) on a number line

• Compare and order rational numbers

• Understand the relationship between a positive or negative number and its opposite (additive inverse)

• Develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers and write mathematics sentences to show relationships

• Write and use related fact families for addition/subtraction and multiplication/division to solve simple equations with missing facts

• Use parentheses and order of operations to make computational sequences clear

• Understand and use the Commutative Property for addition and multiplication of negative and positive numbers

• Apply the Distributive Property with positive and negative numbers to simplify expressions and solve problems

• Use positive and negative numbers to graph in four quadrants, model and answer questions about applied settings

• Recognize problem situations in which two or more variables have a linear relationship to each other

• Construct tables, graphs, and symbolic equations that express linear relationships

• Translate information about linear relations given in a table, a graph, or an equation to one of the other forms

• Understand the connections between linear equations and patterns in the tables and graphs of those relations-rate of change, slope, and y-intercept

• Solve linear equations

• Solve problems and make decisions about linear relationships using information given in tables, graphs, and symbolic expressions

• Use tables, graphs, and equations of linear relations to answer interesting questions

• Understand volume as a measure of filling an object and surface area as a measure of wrapping or covering an object

• Use flat patterns to visualize and calculate surface areas of prisms and cylinders

• Develop formulas for the volumes of prisms, cylinders, cones, pyramids, and spheres either directly or by comparison with known volumes

• Understand that three-dimensional figures may have the same volume but quite different surface areas or they may have the same surface areas but different shapes and volumes

• Use surface area and volume to solve a variety of real-world problems

• Understand how changes in one or more dimensions of a rectangular prism or cylinder affects the prism's volume

• Extend students' understanding of similarity and scale factors to three-dimensional figures

• Understand the effect on surface area and volume of applying a scale factor to a rectangular prism

• Apply the process of statistical investigation to pose questions, identify ways data are collected, determine strategies for analyzing data and interpreting the analysis in order to answer the questions posed

• Compare the distributions of data using their related centers, variability, and shapes

• Use the shape of a distribution to estimate the mean and median

• Recognize that variability occurs whenever data are collected and use properties of distributions to describe the variability in a given data set

• Identify sources of variability, including natural variability and variability that results from errors in measurement

• Decide if a difference among data values and/or summary measures matters

• Understand and decide when to use the mean and median to describe a distribution

• Make effective use of a variety of representations to display distributions, including tables, value bar graphs, dot or line plots, and bar graphs

• Understand and use counts or percents to report frequencies of occurrence of data

• Develop and use strategies for comparing equal- size and unequal-size data sets to solve problems

• Interpret experimental and theoretical probabilities and the relationship between them

• Distinguish between equally likely and non- equally likely outcomes

• Review strategies for identifying possible outcomes and analyzing probabilities, such as using lists or counting trees

• Understand that fairness implies equally likely outcomes

• Analyze situations that involve two-stages (or two actions)

• Use area models to analyze situations that involve two stages

• Determine the expected value of a probability situation

• Analyze binomial situations

• Use probability and expected value to make decisions

• Choose sensible units for measuring

• Understand that a measurement has two components, a unit of measure and a count

• Build a repertoire of benchmarks to relate the measures of unfamiliar objects or events to the measures of objects or events that are personally meaningful

• Review the concept of place value as it relates to reading, writing, and using large numbers

• Read, write, and interpret the large numbers that occur in real-life measurements using standard, scientific, and calculator notation

• Review and extend the use of exponents

• Use estimates and rounded values for describing and comparing objects and events

• Develop strategies for operating with large numbers

• Choose sensible ways of comparing counts and measurements, including using differences, rates, and ratios

• Draw sensible conclusions from given information.

Grade 8 Goals

• Recognize linear and non-linear patterns in contexts, tables and graphs and describe those patterns using words and symbolic expressions

• Write equations to express linear patterns appearing in tables, graphs, and verbal contexts

• Write linear equations when specific information, such as two points or a point and a slope, is given for a line

• Approximate linear data patterns with graph and equation models

• Solve linear equations

• Interpret inequalities

• Write equations describing inverse variation

• Use linear and inverse variation equations to solve problems and to make predictions and decisions

• Relate the area of a square to the length of a side of the square

• Estimate square roots

• Develop strategies for finding the distance between two points on a coordinate grid

• Understand and apply the Pythagorean Theorem

• Use the Pythagorean Theorem to solve a variety of problems

• Recognize situations where one variable is an exponential function of another variable

• Recognize the connections between exponential equations and growth patterns in tables and graphs of those relations

• Construct equations to express exponential patterns that appear in data tables, graphs, and problem conditions

• Understand and apply the rules for operating on numerical expressions with exponents

• Solve problems about exponential growth and decay in a variety of situations such as science or business

• Compare exponential and linear relationships

• Recognize the patterns of change for quadratic relationships in a table, graph, equation, and problem situation

• Construct equations to express quadratic relationships that appear in tables, graphs and problem situations

• Recognize the connections between quadratic equations and patterns in tables and graphs of those relationships

• Use tables, graphs, and equations of quadratic relationships to locate maximum and minimum values of a dependent variable and the x- and y-intercepts and other important features of parabolas.

• Recognize equivalent symbolic expressions for the dependent variable in quadratic relationships

• Use the distributive property to write equivalent quadratic expressions in factored form or expanded form

• Use tables, graphs, and equations of quadratic relations to solve problems in a variety of situations from geometry, science, and business

• Compare properties of quadratic, linear, and exponential relationships

• Understand important properties of symmetry

• Recognize and describe symmetries of figures

• Use tools to examine symmetries and transformations

• Make figures with specified symmetries

• Identify basic design elements that can be used to replicate a given design

• Perform symmetry transformations of figures, including reflections, translations, and rotations

• Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation

• Give precise mathematical directions for performing reflections, rotations, and translations

• Draw conclusions about a figure, such as measures of sides and angles, lengths of diagonals, or intersection points of diagonals, based on symmetries of the figure

• Understand that figures with the same shape and size are congruent

• Use symmetry transformations to explore whether two figures are congruent

• Give examples of minimum sets of measures of angles and sides that will guarantee that two triangles are congruent

• Use congruence of triangles to explore congruence of two quadrilaterals

• Use symmetry and congruence to deduce properties of figures

• Write coordinate rules for specifying the image of a general point (x, y) under particular transformations

• Use transformational geometry to describe motions, patterns, designs, and properties of shapes in the real world

• Model situations with symbolic statements

• Write equivalent expressions

• Determine if different symbolic expressions are mathematically equivalent

• Interpret the information equivalent expressions represent in a given context

• Determine which equivalent expression to use to answer particular questions;

• Solve linear equations involving parentheses

• Solve quadratic equations by factoring

• Use equations to make predictions and decisions

• Analyze equations to determine the patterns of change in the tables and graphs that the equation represents

• Understand how and when symbols should be used to display relationships, generalizations, and proofs

• Write and use equations of circles

• Determine lines are parallel or perpendicular by looking at patterns in their graphs, coordinates, and equations

• Find coordinates of points that divide line segments in various ratios

• Write inequalities that satisfy given situations

• Find solutions to inequalities represented by a graph or an equation

• Solve systems of linear equations by graphing, combining equations, and by substitution

• Write linear inequalities in two variables to match constraints in problem conditions

• Graph linear inequalities and systems of inequalities and use the results to solve problems

• Revisit and use the process of statistical investigation to explore problems

• Distinguish between samples and populations and use information drawn from samples to draw conclusions about populations

• Explore the influence of sample size and of random or nonrandom sample selection

• Apply concepts from probability to select random samples from populations

• Compare sample distributions using measures of center (mean or median), measures of dispersion (range or percentiles), and data displays that group data (histograms and box-and-whisker plots)

• Explore relationships between paired values of numerical attributes

• Recognize situations in which counting techniques apply

• Construct organized lists of outcomes for complex processes and uncover patterns that help in counting the outcomes of those processes

• Use diagrams, tables, and symbolic expressions to organize examples in listing and counting tasks

• Analyze the usefulness of counting trees and use counting trees

• Use mental arithmetic to make estimates in multiplication and division calculations

• Invent strategies for solving problems that involve counting

• Analyze counting problems involving choices in various contexts

• Differentiate among situations in which order does and does not matter and in which repeats are and are not allowed

• Analyze the number of paths through a network

• Compare the structures of networks with problems involving combinations

• Create networks that satisfy given constraints

• Apply thinking and reasoning skills to an open-ended situation in which assumptions must be made and create a persuasive argument to support a conjecture