Courses of Study : Mathematics

Students:

• Compute a unit rate for ratios that compare quantities with different units.
• Determine the unit rate for a given ratio, including unit rates expressed as a complex fraction.

Example: if a runner runs 1/2 mile every 3/4 hour, a student should be able to write the ratio as a complex fraction.)

Students know:

• What a unit rate is and how to calculate it given a relationship between quantities.
• Quantities compared in ratios are not always whole numbers but can be represented by fractions or decimals.
• A fraction can be used to represent division.

Students are able to:
Compute unit rates associated with ratios of fractional

• lengths.
• Areas.
• quantities measured in like or different units.

Students understand that:

• Two measurements that create a unit rate are always different (miles per gallon, dollars per hour)

Learning Objectives:
M.7.1.1: Define unit rate, proportions, area, length, and ratio.
M.7.1.2: Recall how to find unit rates using ratios.
M.7.1.3: Recall the steps used to solve division of fraction problems.

Students:

• Decide whether a relationship between two quantities is proportional.
• Recognize that not all relationships are proportional.
• Use equivalent ratios in a table or a coordinate graph to verify a proportional relationship.
• Identify the constant of proportionality when a proportional relationship exists between two quantities.
• Use a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to demonstrate the constant of proportionality.
• Explain the meaning of a point (x, y) in the context of a real-world problem.
Example, if a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1,6) means that after 1 hour of working the boy makes $6, which shows the unit rate of $6 per hour.

Students know:

• (2a) how to explain whether a relationship is proportional.
• (2b) that the constant of proportionality is the same as a unit rate. Students know:
  • where the constant of proportionality can be found in a table, graph, equation or diagram.
  • (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.

Students are able to:

• (2a) determine if a proportional relationship exists when given a table of equivalent ratios or a graph of the relationship in the coordinate plane.
• (2b) identify the constant of proportionality and express the proportional relationship using a variety of representations including tables, graphs, equations, diagrams, and verbal descriptions.
• (2c) model a proportional relationship using coordinate graphing.
• Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.

Students understand that:

• (2a) A proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.
• (2b) The constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context. (2c) The context of a problem can help them interpret a point on a graph of a proportional relationship.

Learning Objectives:
M.7.2.1: Define proportions and proportional relationships.
M.7.2.2: Demonstrate how to write ratios as a fraction.
M.7.2.3: Define equivalent ratios and origin.
M.7.2.4: Locate the origin on a coordinate plane.
M.7.2.5 Show how to graph on Cartesian plane.
M.7.2.6: Determine if the graph is a straight line through the origin.
M.7.2.7: Use a table or graph to determine whether two quantities are proportional.
M.7.2.8: Define a constant and equations.
M.7.2.9: Create a table from a verbal description, diagram, or a graph.
M.7.2.10: Identify numeric patterns and finding the rule for that pattern.
M.7.2.11: Recall how to find unit rate.
M.7.2.12: Recall how to write equations to represent a proportional relationship.
M.7.2.13: Discuss the use of variables.
M.7.2.14: Define ordered pairs.
M.7.2.15: Show how to plot points on a Cartesian plane.
M.7.2.16: Locate the origin on the coordinate plane.

Students:

• Use proportional reasoning strategies including setting up and solving proportions to solve problems involving simple interest, tax, gratuities, commissions, fees, markups, percent increase, markdowns or percent decrease.

Example: Students might be asked to "order" from a menu for lunch then calculate the tax and gratuities to determine the total cost.

Students know:

• how to interpret a real-world problem to determine what is being asked.
• Techniques for calculating and using percents to solve problems in context.
• how to interpret the solution in the context of the problem.

Students are able to:

• Write and solve proportions to help them solve real-world problems involving percent.
• Solve problems that require them to calculate: simple interest, tax, gratuities, commission, fees, mark ups, markdowns, percent increase and percent decrease.

Students understand that:

• percents relate to real-world contexts, and how to determine the reasonableness of their answers based on that context.

Learning Objectives:
M.7.3.1: Define interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and percent error.
M.7.3.2: Apply definitions to context in real-world problems.
M.7.3.3: Solve proportional problems.
M.7.3.4: Recall how to find percent and ratios.
M.7.3.5: Recall steps for solving multi-step problems.

Students:

• Explain situations where opposite quantities combine to make zero, known as additive inverses.
• Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
• Add and subtract rational numbers using number lines to show connection to distance
• Explain the connection between subtraction and addition of additive inverses.
• Model multiplication and division of rational numbers (number horizontal and vertical number lines, integer chips, bar models).
• Use properties of operations to multiply signed numbers.
• Convert rational numbers to a decimal using long division and determine if the result is terminating or repeating.

Students know:

• a number and its opposite have a sum of 0.
• A number and its opposite are called additive inverses.
• Strategies for adding and subtracting two or more numbers.
• Absolute value represents distance on a number line, therefore it is always non-negative.
• Strategies for multiplying signed numbers.
• Every quotient of integers (with non-zero divisor) is a rational number.
• If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
• The decimal form of a rational number terminates or eventually repeats.

Students are able to:

• add rational numbers.
• Subtract rational numbers.
• Represent addition and subtraction on a number line diagram.
• Describe situations in which opposite quantities combine to make 0.
• Find the opposite of a number.
• Interpret sums of rational numbers by describing real-world contexts.
• Show that the distance between two rational numbers on the number line is the absolute value of their difference.
• Use absolute value in real-world contexts involving distances.
• Multiply and divide rational numbers.
• Convert a rational number to a decimal using long division.

Students understand that:

• finding sums and differences of rational numbers (negative and positive) involves determining direction and distance on the number line.
• Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
• If a factor is multiplied by a number greater than one, the answer is larger than that factor.
• If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
• Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
• Integers can be divided, provided that the divisor is not zero.

Learning Objectives:
M.7.4.1: Define rational numbers, horizontal, and vertical.
M.7.4.2: Recall how to extend a horizontal number line.
M.7.4.3: Recall how to extend a vertical number line.
M.7.4.4: Demonstrate addition and subtraction of whole numbers using a horizontal or vertical number line.
M.7.4.5: Give examples of rational numbers.
M.7.4.6: Define absolute value and additive inverse.
M.7.4.7: Explain that the sum of a number and its opposite is zero.
M.7.4.8: Locate positive, negative, and zero numbers on a number line.
M.7.4.9: Recall properties of addition and subtraction.
M.7.4.10: Model addition and subtraction using manipulatives.
M.7.4.11: Show addition and subtraction of 2 or more rational numbers using a number line within real-world context.
M.7.4.12: Define absolute value and additive inverse.
M.7.4.13: Show subtraction as the additive inverse.
M.7.4.14: Give examples of the opposite of a given number.
M.7.4.15: Show addition and subtraction using a number line.
M.7.4.16: Discuss various strategies for solving real-world and mathematical problems.
M.7.4.17: Identify properties of operations for addition and subtraction.
M.7.4.18: Recall the steps for solving addition and subtraction of rational numbers.
M.7.4.19: Identify the difference between two rational numbers on a number line.
M.7.4.20: Recall the steps of solving multiplication of rational numbers.
M.7.4.21: Identify the pattern for multiplying signed numbers.
M.7.4.22: Recall the steps of solving division of rational numbers.
M.7.4.23: Explain that dividing a rational number zero is undefined.
M.7.4.24: Recall that a fraction can be written as a division problem.
M.7.4.25: Recall the steps to divide two rational numbers.
M.7.4.26: Identify whether a decimal is terminating or repeating.

Students:

• Apply their knowledge of addition, subtraction, multiplication and division of rational numbers to describe real-world contexts.
• Solve multi-step problems using numerical expressions that involve addition, subtraction, multiplication, and/or division of rational numbers, including problems that involve complex fractions

Students know:

• how to model real-world problems to include situations involving elevation, temperature changes, debits and credits, and proportional relationships with negative rates of change.
• how to evaluate numerical expressions with greater fluency, using the properties of operations when necessary.

Students are able to:

• Solve real-world and mathematical problems involving the four operations with rational numbers.

Students understand that:

• rational numbers can represent values in real-world situations.
• properties of operations learned with whole numbers in elementary apply to rational numbers.

Learning Objectives:
M.7.5.1: Discuss various strategies for solving real-world and mathematical problems.
M.7.5.2: Recall steps for solving fractional problems.
M.7.5.3: Identify properties of operations for addition and multiplication.
M.7.5.4: Recall the rules for multiplication and division of rational numbers.
M.7.5.5: Recall the rules for addition and subtraction of rational numbers.

Students:

• Use properties of operations to produce combined and re-written forms of the expressions that are useful in resolving mathematical and contextual problems.

Students know:

• how to add, subtract, multiply, and divide rational numbers.
• A(b + c) = ab + ac.
• how to find the greatest common factor of two or more terms.

Students are able to:

• apply properties of operations as strategies to add and subtract linear expressions with rational coefficients.
• Apply properties of operations as strategies to factor linear expressions with rational coefficients.
• Apply properties of operations as strategies to expand linear expressions with rational coefficients.

Students understand that:

• only like terms can be combined, e.g., x + y = x + y but x + x = 2x.
• To factor an expression, one must factor out the greatest common factor.
• There are many different ways to write the same expression.

Learning Objectives:
M.7.6.1: Define linear expression, rational, coefficient, and rational coefficient.
M.7.6.2: Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y).
M.7.6.3: Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
M.7.6.4: Recognize the property demonstrated in a given expression.
M.7.6.5: Combine like terms of a given expression.
M.7.6.6: Recall how to find the greatest common factor.
M.7.6.7: Give examples of the properties of operations including distributive, commutative, and associative.

Students:

• Write an expression for a situation and determine equivalent expressions for the same situation.

Students know:

• properties of operations can be used to identify or create equivalent linear expressions.
• Equivalent expressions can reveal real-world and mathematical relationships, and some forms of equivalent expressions can provide more insight than others.

Students are able to:

• determine whether two expressions are equivalent.
• Rewrite expressions into equivalent forms by combining like terms, using the distributive property, and factoring.

Students understand that:

• rewriting expressions in multiple equivalent forms allows for thinking about problems in different ways and highlights different aspects/relationships of quantities in problems.

Learning Objectives:
M.7.7.1: Define expression, equivalent, and equivalent expressions.
M.7.7.2: Recall mathematical terms such as sum, difference, etc.
M.7.7.3: Recognize that a variable without a written coefficient is understood to have a coefficient of one.
M.7.7.4: Recall how to convert mathematical terms to mathematical symbols and numbers and vice versa.
M.7.7.5: Restate numerical expressions with words.

Students:

• Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form.
• Interpret solutions of problems with rational numbers in the context of the problem.
• Assess the reasonableness of answers using mental computation and estimation strategies.
• Use variables to represent quantities in a real-world or mathematical problem.

Students know:

• techniques for converting between fractions, decimals, and percents.
• Techniques for estimation, mental computations, and how to assess the reasonableness of their answers.

Students are able to:

• convert between different forms of a rational number.
• Add, subtract, multiply and divide rational numbers. -translate verbal forms of problems into algebraic symbols, expressions, and equations.
• Use estimation and mental computation techniques to assess the reasonableness of their answers.

Students understand that:

• One form of a number may be more advantageous than another form, based on the problem context.
• Using estimation strategies helps to determine the reasonableness of answers.

Learning Objectives:
M.7.8.1: Define estimation, rational numbers, and reasonable.
M.7.8.2: Recall mental calculation strategies.
M.7.8.3: Recall estimation strategies.
M.7.8.4: Analyze the given word problem to set up a mathematical problem.
M.7.8.5: Recognize the mathematical operations of rational numbers in any form, including converting between forms. (Ex. 0.25=1/4 =25%).
M.7.8.6: Recognize the rules of operations of positive and negative numbers.
M.7.8.7: Recognize properties of numbers (Distributive, Associative, Commutative).
M.7.8.8: Recall problem solving methods.

Students:

• Write and solve mathematical equations (or inequalities) to model real-world problems.
• Interpret the solution to an equation in the context of a problem
• Interpret the solution set of an inequality in the context of a problem.
• Graph the solution to an inequality on a number line.

Students know:

• p(x + q) = px + pq, where p and q are specific rational numbers.
• When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.
• In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).

Students are able to:

• use variables to represent quantities in a real-world or mathematical problem.
• Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.
• Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.
• Graph the solution set of an inequality.

Students understand that:

• Real-world problems can be represented through algebraic expressions, equations, and inequalities.
• Why the inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number.

Learning Objectives:
M.7.9.1: Define equation, inequality, and variable.
M.7.9.2: Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
M.7.9.3: Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.4: Test the found number or number set for accuracy by substitution.
M.7.9.5: Recall solving one step equations and inequalities.
M.7.9.6: Recognize properties of numbers (Distributive, Associative, Commutative).
M.7.9.7: Define equation and variable.
M.7.9.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.
M.7.9.9: Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.10: Test the found number for accuracy by substitution.
Example: Is 5 an accurate solution of 2(x + 5)=12?.
M.7.9.11: Identify the unknown, in a given situation, as the variable.
M.7.9.12: List given information from the problem.
M.7.9.13: Recalling one-step equations.
M.7.9.14: Explain the distributive property.
M.7.9.15: Define inequality and variable.
M.7.9.16: Set up an inequality to represent the given situation, using correct mathematical operations and variables.
M.7.9.17: Calculate a solution set to an inequality by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.18: Test the solution set for accuracy.
M.7.9.19: Identify the unknown, of a given situation, as the variable.
M.7.9.20: List information from the problem.
M.7.9.21: Recall how to graph inequalities on a number line.
M.7.9.22: Recall how to solve one step inequalities.

Students:

• distinguish between a population and a sample population, and identify both for statistical questions.
• Understand that a population characteristic is determined using data from the entire population, whereas a sample statistic is determined using data from a sample of the population.
• Describe different ways that data can be collected to answer a statistical question.
• Understand why a sample of a population may be useful or necessary to answer a statistical question.

Students know:

• a random sample can be found by various methods, including simulations or a random number generator.
• Samples should be the same size in order to compare the variation in estimates or predictions.

Students are able to:

• determine whether a sample is random or not and justify their reasoning.
• Use the center and variability of data collected from multiple same-size samples to estimate parameters of a population.
• Make inferences about a population from random sampling of that population.
• Informally assess the difference between two data sets by examining the overlap and separation between the graphical representations of two data sets.

Students understand that:

• statistics can be used to gain information about a population by examining a sample of the populations.
• Generalizations about a population from a sample are valid only if the sample is representative of that population.
• Random sampling tends to produce representative samples and support valid inferences
• The way that data is collected, organized and displayed influences interpretation.

Learning Objectives:
M.7.10.1: Recall how to calculate range, outlier, ratio, and proportion.
M.7.10.2: Define sample, data, variation, prediction, estimation, validity, population, inference, random sampling, statistic, and generalization.
M.7.10.3: Explain the validity of random sampling.
M.7.10.4: Differentiate the appropriate sampling method.
M.7.10.5: Analyze attributes of sample size.
M.7.10.6: Compare and contrast the random sampling data to the population.
M.7.10.7: Compare sample size with population to check for validity.
M.7.10.8: Analyze conclusions of the sample to determine its appropriateness for the population.
M.7.10.9: Predict an outcome of the entire population based on random samplings.
M.7.10.10: Discuss real-world examples of valid sampling and generalizations.
M.7.10.11: Recall the nature of the attribute, how it was measured, and its unit of measure.
M.7.10.12: Collect data from population randomly, choosing same size samples. (Ex. If population is your school, different random samplings should be same number of students).
M.7.10.13: Define and discuss bias.
M.7.10.14: Compare and contrast statistical situations to determine if statistical bias exists.

Students:

• Determine which measure of center best represents the typical value in the data set.
• Calculate measures of variability (range, interquartile range, and mean absolute deviation), noting how larger values indicate that values are more spread out from the center of the distribution.

Students know:

• populations can be compared using measures of center and measures of variability

Students are able to:

• informally assess the degree of visual overlap of two numerical data distributions with similar variabilities.
• Measure the difference between the centers by expressing it as a multiple of a measure of variability.

Students understand that:

• outliers skew data, which in turn affects the display.
• Measures of center give information about the location of mean, median, and mode, whereas measures of variability give information about how spread out the data is.

Learning Objectives:
M.7.11.1: Define measure of variability, distribution, and measure of center.
M.7.11.2: Analyze the skew of the distributions and recognize the difference in measure of center and variability.
M.7.11.3: Compare the measure of center and measure of variability of two distributions.
M.7.11.4: Relate the measure of variation with the concept of range.
M.7.11.5: Relate the measure of the center with the concept of mean.
M.7.11.6: Recall how to calculate measure of center and measure of variability.
M.7.11.7: Discuss how to read and interpret a graph.

Students:

• Use measures of center for numerical data from random samples to draw informal comparative inferences about two populations.
• Use measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

Students know:

• measures of center are insufficient to compare populations. measures of variability are necessary to assess if data sets are significantly different or not.
• Mean is the sum of the numerical values divided by the number of values.
• Median is the number that is the midpoint of an ordered set of numerical data.
• Mode is the data value or category occurring with the greatest frequency (there can be no mode, one mode, or several modes).
• Mean absolute deviation of a data set is found by the following steps: 1) calculate the mean 2) determine the deviation of each variable from the mean 3) divide the sum of the absolute value of each deviation by the number of data points.
• Range is a number found by subtracting the minimum value from the maximum. value.

Students are able to:

• find the measures of center of a data set.
• Find the interquartile range of a data set and use to compare variability between data sets.

Students understand that:

• outliers skew data, which in turn affects the display.
• Measures of center give information about the location of mean, median and mode, whereas measures of variability give information about how spread out the data is.
• The mean absolute deviation of a data set describes the average distance that points within a data set are from the mean of the data set.

Learning Objectives:
M.7.12.1: Define measure of variability, measure of center, inference and mean absolute deviation.
M.7.12.2: Recall how to calculate measure of center and measure of variability.
M.7.12.3: Recall that center is related to measure of center and measure of variability is related to variation.
M.7.12.4: Compare and contrast the measure of center and measure of variability of two numerical data sets.
M.7.12.5: Calculate the mean absolute deviation of a data set in context.

Students:

• Accurately describe the likelihood of an event occurring.
• Describe the probability of an event occurring on a scale of 0 to 1 and using appropriate vocabulary based on the scale.
• Categorize and order the probabilities of events by their likelihood.
• Use words like impossible, very unlikely, unlikely, equally likely/unlikely, likely, very likely, and certain to describe the probabilities of events.

Students know:

• probability is equal to the ratio of favorable number of outcomes to total possible number of outcomes.
• As a number for probability increases, so does the likelihood of the event occurring.
• A probability near 0 indicates an unlikely event.
• A probability around 1/2 indicates an event that is neither unlikely nor likely.
• A probability near 1 indicates a likely event.

Students are able to:

• approximate the probability of a chance event.
• Use words like impossible, very unlikely, unlikely, equally likely/unlikely, likely, very likely, and certain to describe the probabilities of events.

Students understand that:

• the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.
• An event that is equally likely or equally unlikely has a probability of about 0.5 or ½.
• The sum of the probabilities of an event and its complement must be 1.

Learning Objectives:
M.7.13.1: Define probability and event.
M.7.13.2: Recall the order of fractions on a number line.
M.7.13.3: Recall how to compare fractions with like denominators.
M.7.13.4: Demonstrate how to compare fractions with different denominators.
M.7.13.5: Determine the likelihood of an event occurring.

Students:

• Develop uniform (all outcomes have the same probability) and non-uniform (outcomes with different probabilities) probability models and use them to find probabilities of simple events.
• Explain possible sources of discrepancy if the agreement between the probability model and observed frequencies is not good.
• Estimate the probability of an event happening in an experiment.
• Compare the accuracy of estimated probabilities from different experiments to the actual probability.

Students know:

• the probability of any single event can be expressed using terminology like impossible, unlikely, likely, or certain or as a number between 0 and 1, inclusive, with numbers closer to 1 indicating greater likelihood.
• A probability model is a visual display of the sample space and each corresponding probability
• probability models can be used to find the probability of events.
• A uniform probability model has equally likely probabilities.
• Sample space and related probabilities should be used to determine an appropriate probability model for a random circumstance.

Students are able to:

• make predictions before conducting probability experiments, run trials of the experiment, and refine their conjectures as they run additional trials.
• Collect data on the chance process that produces an event.
• Use a developed probability model to find probabilities of events.
• Compare probabilities from a model to observed frequencies
• Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Students understand that:

• long-run frequencies tend to approximate theoretical probability.
• predictions are reasonable estimates and not exact measures.

Learning Objectives:
M.7.14.1: Define probability of chance, probability of events, outcome, and probability of observed frequency.
M.7.14.2: Compare and contrast probability of chance and probability of observed frequency.
M.7.14.3: Display all outcomes in a graphic representation (probability model-tree diagram, organized list, table, etc.).
M.7.14.4: Demonstrate how to write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.
M.7.14.5: Recall how to simplify fractions to lowest terms.
M.7.14.6: Recognize equivalent fractions.
M.7.14.7: Recall how to create a table or graphic display of data.
M.7.14.8: Define probability of chance, outcome, and event.
M.7.14.9: List all possible outcomes using a graphic representation (probability model-tree diagram, organized list, table, etc.).
M.7.14.10: Using the model, count the frequency of the desired outcome.
M.7.14.11: Demonstrate how to write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.
M.7.14.12: Recall how to simplify fractions to lowest terms.
M.7.14.13: Recognize equivalent fractions.
M.7.14.14: Recall how to create a table or graphic display of data.
M.7.14.15: Analyze collected data to predict probability of events.
M.7.14.16: Define probability of observed frequency, outcome, discrepancy and event.
M.7.14.17: List all actual outcomes using a graphic representation (probability model-tree diagram, organized list, table, etc.).
M.7.14.18: Using the model, count the frequency of the actual outcome.
M.7.14.19: Demonstrate how to write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.
M.7.14.20: Recall how to simplify fractions in lowest terms.
M.7.14.21: Recognize equivalent fractions.
M.7.14.22: Recall how to create a table or graphic display of data.

Students:

• Predict the approximate relative frequency of an event given the probability.
• Compare the accuracy of estimated probabilities from different experiments to the actual probability.
• Describe how a single event can be simulated using an experiment.

Students know:

• relative frequencies for experimental probabilities become closer to the theoretical probabilities over a large number of trials.
• Theoretical probability is the likelihood of an event happening based on all possible outcomes.
• long-run relative frequencies allow one to approximate the probability of a chance event and vice versa.

Students are able to:

• approximate the probability of a chance event.
• observe an event's long-run relative frequency.

Students understand that:

• real-world outcomes can be simulated using probability models and tools.

Learning Objectives:
M.7.15.1: Define probability of chance, outcome, theoretical probability, experimental probability and event.
M.7.15.2: Recognize the difference between possible outcomes and likely outcomes.
M.7.15.3: Write the probability as a fraction, with likely outcomes as the numerator and possible outcomes as the denominator.
M.7.15.4: Recall how to simplify fraction to lowest terms.
M.7.15.5: Recognize equivalent fractions.
M.7.15.6: Define relative frequency.

Students:

• Conduct probability experiments to quantify and interpret likeliness of an event occurring.
• Design and use a simulation to generate frequencies for compound events
• Analyze the results from a simulation of a compound event to estimate the probability of the compound event.
• Represent probabilities as percents, decimals, and fractions.

Students know:

• how the sample space is used to find the probability of compound events.
• A compound event consists of two or more simple events.
• A sample space is a list of all possible outcomes of an experiment.
• how to make an organized list.
• how to create a tree diagram.

Students are able to:

• find probabilities of compound events using organized lists, tables, tree diagrams and simulations
• Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.
• For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
• Design a simulation to generate frequencies for compound events.
• Use a designed simulation to generate frequencies for compound events.

Students understand that:

• the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
• A compound event can be simulated using an experiment.

Learning Objectives:
M.7.16.1: Define simple events and compound events.
M.7.16.2: Discover when to add or multiply events to find probability of compound events.
M.7.16.3: Recall how to find the probability of simple events.
M.7.16.4: Demonstrate adding and multiplying fractions.
M.7.16.5: Recognize how to obtain a common denominator when adding fractions.
M.7.16.6: Recall how to add fractions with like denominators.
M.7.16.7: Define simulation, frequency, and compound events.
M.7.16.8: Recall how to find the probability of compound events.
M.7.16.9: Create a tree diagram including all possible outcomes.
M.7.16.10: Choose appropriate model to display outcomes (tree diagram, organized list, or table).
M.7.16.11: Identify the desired outcomes in model. M 7.16.12: Create and use a simulation to illustrate compound events.
M.7.16.13: Recall when to add or multiply events to find probability of compound events.
M.7.16.14: Recall how to find the probability of simple events.
M.7.16.15: Demonstrate adding and multiplying fractions.
M.7.16.16: Recognize how to obtain a common denominator when adding fractions.
M.7.16.17: Recall how to add fractions with like denominators.
M.7.16.18: Recall how to construct a table.

Students:

• Solve problems involving scale drawings.
• Use a scale factor to reproduce a scale drawing at a different scale.
• Determine the scale factor for a scale drawing.

Students know:

• how to calculate actual measures such as area and perimeter from a scale drawing.
• Scale factor impacts the length of line segments, but it does not change the angle measurements.
• There is a proportional relationship between the corresponding sides of similar figures.
• A proportion can be set up using the appropriate corresponding side lengths of two similar figures.
• If a side length is unknown, a proportion can be solved to determine the measure of it.

Students are able to:

• find missing lengths on a scale drawing.
• Use scale factors to compute actual lengths, perimeters, and areas in scale drawings.
• Use a scale factor to reproduce a scale drawing at a different scale.

Students understand that:

• scale factor can enlarge or reduce the size of a figure.
• Scale drawings are proportional relationships.
• Applying a scale factor less than one will shrink a figure.
• Applying a scale factors greater than one will enlarge a figure.

Learning Objectives:
M.7.17.1: Define scale, scale drawings, length, area, and geometric figures.
M.7.17.2: Locate/use scale on a map.
M.7.17.3: Identify proportional relationships.
M.7.17.4: Recognize numeric patterns.
M.7.17.5: Recall how to solve proportions.

Students:

• Determine if a unique triangle can be made when given three specific conditions of a triangle.
• Explain why three given conditions about a triangle may result in more than one or no triangles.

Students know:

• if three side lengths will create a unique triangle or no triangle.

Students are able to:

• freehand, draw geometric shapes with given conditions.
• Using a ruler and protractor, draw geometric shapes with given conditions.
• Using technology, draw geometric shapes with given conditions.
• Construct triangles from three measures of angles or sides.
• Identify the conditions that determine a unique triangle, more than one triangle, or no triangle.

Students understand that:

• from their experiences with constructions, what conditions are necessary to construct a triangle.
• only certain combinations of angle and side measures will create triangles.
• Constructing a triangle requires a specific relationship between the legs of the triangle and a specific sum between the angles of the triangle.

Learning Objectives:
M.7.18.1: Demonstrate how to use a protractor to draw an angle.
M.7.18.2: Construct segments of a given length using a ruler.
M.7.18.3: Recognize attributes of geometric shapes.

Students:

• Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Students know:

• the difference between a two-dimensional and three-dimensional figure.
• The names and properties of two-dimensional shapes.
• The names and properties of three-dimensional solids.

Students are able to:

• Discover two-dimensional shapes from slicing three-dimensional figures. For example, students might slice a clay rectangular prism from different perspectives to see what two-dimensional shapes occur from each slice.

Students understand that:

• slicing he prism from different planes may provide a different two-dimensional shape.
• There are specific two-dimensional shapes resulting from slicing a three-dimensional figure.

Learning Objectives:
M.7.19.1: Define two-dimensional figure, three-dimensional figure, and plane section.
M.7.19.2: List attributes of three-dimensional figures.
M.7.19.3: List attributes of two-dimensional figures.
M.7.19.4: Describe the relationship between two- and three-dimensional figures.
M.7.19.5: Recognize symmetry.

Students:

• Solve problems with the circumference and area of a circle.

Students know:

• that the ratio of the circumference of a circle and its diameter is always π.
• The formulas for area and circumference of a circle.

Students are able to:

• use the formula for area of a circle to solve problems.
• Use the formula(s) for circumference of a circle to solve problems.
• Give an informal derivation of the relationship between the circumference and area of a circle.

Students understand that:

• area is the number of square units needed to cover a two-dimensional figure.
• Circumference is the number of linear units needed to surround a circle.
• The circumference of a circle is related to its diameter (and also its radius).

Learning Objectives:
M.7.20.1: Define diameter, radius, circumference, area of a circle, and formula.
M.7.20.2: Identify and label parts of a circle.
M.7.20.3: Recognize the attributes of a circle.
M.7.20.4: Apply the formula of area and circumference to real-world mathematical situations.

Students:

• Find the values of angles using complementary and supplementary angle relationships and equations.
• Identify angle relationships in angle diagrams involving vertical, supplementary, and complementary angles.
• Write equations to represent relationships between known and unknown angle measurements.
• Determine the measures of unknown angles and judge the reasonableness of the measures.

Students know:

• supplementary angles are angles whose measures add to 180 degrees.
• Complementary angles are angles whose measures add to 90 degrees.
• vertical angles are opposite angles formed when two lines intersect.
• Adjacent angles are non-overlapping angles which share a common vertex and side.

Students are able to:

• write a simple equation to find an unknown angle.
• Identify and determine values of angles in complementary and supplementary relationships.
• Identify pairs of vertical angles in angle diagrams.
• Identify pairs of complementary and supplementary angles in angle diagrams.
• Use vertical, complementary, and supplementary angle relationships to find missing angles.

Students understand that:

• vertical angles are the pair of angles formed across from one another when two lines intersect, and that the measurements of vertical angles are congruent.
• Complementary angles are angles whose measures add up to 90^o, and supplementary angles are angles whose measures add up to 180^o.
• Relationships between angles depends on where the angles are located.

Learning Objectives:
M.7.21.1: Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
M.7.21.2: Discuss strategies for solving multi-step problems and equations.
M.7.21.3: Identify all types of angles.
M.7.21.4: Identify right angles and straight angles.

Students:

• Find efficient ways to determine surface area of right prisms and right pyramids by analyzing the structure of the shapes and their nets.
• Use the formulas for volume of prisms and pyramids to solve multi-step real-world problems.
• Use the formula for volume to find missing measurements of a prism.

Students know:

• that volume of any right prism is the product of the height and area of the base.
• The volume relationship between pyramids and prisms with the same base and height.
• The surface area of prisms and pyramids can be found using the areas of triangular and rectangular faces.

Students are able to:

• find the area and perimeter of two-dimensional objects composed of triangles, quadrilaterals, and polygons.
• Use a net of a three-dimensional figure to determine the surface area.
• Find the volume and surface area of pyramids, prisms, or three-dimensional objects composed of cubes, pyramids, and right prisms.

Students understand that:

• two-dimensional and three-dimensional figures can be decomposed into smaller shapes to find the area, surface area, and volume of those figures.
• the area of the base of a prism multiplied by the height of the prism gives the volume of the prism.
• the volume of a pyramid is 1/3 the volume of a prism with the same base.

Learning Objectives:
M.7.22.1: Define volume, surface area, triangles, quadrilaterals, polygons, cubes, and right prisms.
M.7.22.2: Discuss strategies for solving real-world mathematical problems.
M.7.22.3: Recall formulas for calculating volume and surface area.
M.7.22.4: Identify the attributes of triangles, quadrilaterals, polygons, cubes, and right prisms.