Vertical Alignment Colorado Academic Standards 6 th - 7 th - 8 th

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1 Vertical Alignment Colorado Academic Standards 6 th - 7 th - 8 th Standard 3: Data Analysis, Statistics, and Probability 6 th Prepared Graduates: 1. Solve problems and make decisions that depend on un 2. Understanding, explaining, and quantifying the variability in data 7 th Prepared Graduates: 1. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions 2. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts 8 th Prepared Graduates: 1. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data Concepts and Skills students master: 1. Visual displays and summary statistics of one-variable data condense the information in data sets into usable knowledge Concepts and Skills students master: 1. Statistics can be used to gain information about populations by examining samples 2. Mathematical models are used to determine probability Concepts and Skills students master: 1. Visual displays and summary statistics of two-variable data condense the information in data sets into usable knowledge Evidence Outcomes Evidence Outcomes Evidence Outcomes Variability Variability Variability 1 a. Identify a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. (CCSS: 6.SP.1) 1c. Explain that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. (CCSS:6.SP.3) 1d. 3. Give quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking a population.(ccss: 7. SP) iv. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. (CCSS: 7. SP.2) two populations (CCSS:7.SP) i. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple

2 deviations from the overall pattern with reference to the context in which the data were gathered. (CCSS: 6.SP.5c) 1d.4. Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. (CCSS: 6.SP.5d) of a measure of variability. (CCSS: 7. SP.3) Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. (CCSS: 7.SP.4) Distributions Distributions Distributions 1b. Demonstrate that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.(ccss:6.sp.2) 1d. Summarize and describe distributions. (CCSS:6.SP) two populations. (CCSS:7.SP.2) Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. (CCSS: 7.SP.4) Measures of Center Measures of Center Measures of Center 1c. Explain that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. (CCSS:6.SP.3) 1d. 3. Give quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. (CCSS: 6.SP.5c) 1d.4. Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. (CCSS: 6.SP.5d) two populations (CCSS:7.SP) i. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability. (CCSS: 7. SP.3) Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. (CCSS: 7.SP.4) Graphical Displays Graphical Displays Graphical Displays 1d. Summarize and describe distributions. (CCSS:6.SP) i. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.(ccss: 6.SP.4) simulation. (CCSS: 7. SP.8) Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. (CCSS: 7.SP.8b) 1a. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. (CCSS: 8.SP.1) 1c. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the

3 closeness of the data points to the line. (CCSS:8.SP.2) 1e. Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. (CCSS: 8.SP.4) i. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (CCSS: 8.SP.4) Interpreting Data Interpreting Data Interpreting Data 1d. Summarize and describe distributions. (CCSS:6.SP) Summarize numerical data sets in to their context. (CCSS: 6.SP.5 1. Report the number of observations. (CCSS:6.SP.5a) 2. Describe the nature of the attribute under investigation, including how it was measured and its units of measurement. (CCSS: 6.SP.5b) a population.(ccss: 7. SP) iv. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. 1 (CCSS: 7.SP.2) two populations (CCSS:7.SP) Use measures of center and measures of random samples to draw informal comparative inferences about two populations. (CCSS:7.SP.4) 1a. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. (CCSS: 8.SP.1) 1b. Describe patterns such as clustering, outliers, positive or negative association, linear association, and non-linear association. (CCSS: 8.SP.1) 1c. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (CCSS:8.SP.2) 1d. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (CCSS:8.SP.3) 1e. Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. (CCSS: 8.SP.4) i. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (CCSS: 8.SP.4) Use relative frequencies calculated for rows or columns to describe possible association between the two variables. (CCSS:8.SP.4) Continued in Standard 2 (Algebra):

4 Data Sampling a population (CCSS: 7. SP) i. Explain that generalizations about a population from a sample are valid only if the sample is representative of iv. that population. (CCSS: 7 SP.2) Explain that random sampling tends to produce representative samples and support valid inferences. (CCSS: 7 SP.1) Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. (CCSS: 7. SP.2) Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates of 2b. Analyze and solve pairs of simultaneous linear equations. (CCSS: 8.EE.8) i. Explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (CCSS: 8. EE.8a) 3a. Define, evaluate, and compare functions (CCSS: 8.F) iv. Interpret the equation y= mx +b as defining a linear function, whose graph is a straight line. (CCSS: 8.F.3) 3b. Use functions to model relationships between quantities. (CCSS: 8.F) Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.4) iv. Describe qualitatively the functional relationship between two quantities by analyzing a graph. 8 (CCSS:8.F.5)

5 Probability Because an understanding of ratio is required for working with probability, this standard has been aligned with probability in order to emphasize an area where ratio will be applied. 1a. Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. 1 (CCSS: 6.RP.1) 1b. Apply the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. 2 (CCSS:6.RP2) predictions. (CCSS: 7.SP) Probability 2a. Explain that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. (CCSS: 7. SP.5) 2b. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. (CCSS: 7. SP.6) 2c. Develop a probability model and use it to find probabilities of events (CCSS: 7.SP.7) i. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of discrepancy. (CCSS; 7.SP.7) Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. (CCSS: 7.SP.7a) Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process (CCSS: 7.SP. 7b) organized lists, tales, tree diagrams, and simulation. (CCSS7.SP.8) i. Explain that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. (CCSS:7.SP.8a) 2b. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers including integers. v. Convert a rational number to a 1b. Demonstrate informally that every number has a decimal expansion. (CCSS: 8.NS.1)

6 1c. Use ratio and rate reasoning to solve real-world and mathematical problems. 3 v Express the comparison of two whole number quantities using differences, part-to-part ratios, and part-to-whole ratios in real contexts. 1c. Use ratio and rate reasoning to solve real-world and mathematical problems. 3 i. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. (CCSS:6.RP.3a) vi. decimal using long division. (CCSS:7.NS.2d) Show that the decimal form of a rational number terminates in 0 s or eventually repeats. (CCSS: 7.NS.2d) Tables/Lists/Diagrams Tables/Lists/Diagrams Tables/Lists/Diagrams simulation. (CCSS: 7.SP.8) i. Represent sample spaces for compound events using methods such as organized lists, tables, tree diagrams. (CCSS: 7.SP.8b) 1c. Identify and represent proportional relationships between quantities. (CCSS: 7.RP.2) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (CCSS: 7.RP.2b) Sample Space 2d.Find probabilities of compound events using simulation. For an event 8 described in everyday language identify the outcomes in the sample space which compose the event. (CCSS:7.SP.8b) Compound Events 1e. Explain patterns of association seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. (CCSS: 8.SP.4) i. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. (CCSS: 8.SP.4) Continued in Standard 2 (Algebra): 3b.Use functions to model relationships between quantities. (CCSS: 8.F) Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or graph. (CCSS: 8.F.4) Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (CCSS: 8.F.4)

7 simulation. iv. Design and use a simulation to generate frequencies for compound events. 9 (CCSS:7.SP.8c) Simulations a population. (CCSS: 7.SP) iv. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. (CCSS: 7. SP.2) simulation. (CCSS: 7.SP.i) iv. Design and use a simulation to generate frequencies for compound events. 9 (CCSS:7.SP.8c) Author: B. Orona February 2012

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