Grade 8 Mathematics, Quarter 1, Unit 1.1 Create and Display Data Overview Number of instructional days: 13 (1 day = 45 minutes) Content to be learned Create scatter plots to answer questions related to data, analyze data, formulate or justify conclusions, make predications, and solve problems. Identify which representation would best display a given set of data or situation (line graph, scatter plot, histogram, box-and-whisker plot). Create box-and-whisker plots to answer questions related to data, analyze data, formulate or justify conclusions. Essential questions How can you use a scatter plot to make a prediction and draw a conclusion? What do you do when you analyze data? Mathematical practices to be integrated Attend to precision. Specify units/labels (i.e., graphs, axes, titles all representations). Model with mathematics. Map the relationships using such tools as diagrams and graphs. Use appropriate tools strategically. Consider/use appropriate/available tools (i.e., graphing calculators, Excel, statistical software, protractors). How can you use a box-and-whisker plot to answer questions related to data and draw conclusions? How can you determine which data display is best for a given situation? Cumberland, Lincoln, and Woonsocket Public Schools C-1
Grade 8 Mathematics, Quarter 1, Unit 1.1 Create and Display Data (13 days) Written Curriculum Grade-Level Expectations M(DSP) 8 3 Organizes and displays data using scatter plots to answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems; or identifies representations or elements of representations that best display a given set of data or situation, consistent with the representations required in M(DSP) 8 1. (Local) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 8 2.) Clarifying the Standards Prior Learning In kindergarten through grade 2, this GLE was not addressed. Beginning in third grade, students identified or described representations that best display a given set of data or situation. This process continued through fifth grade. In sixth grade, students organized and displayed data using tables, line graphs, or stem-and-leaf plots to answer questions related to data, analyze data to formulate or justify conclusions, make predictions, or solve problems. In grade 7, students organized and displayed data through scatter plots to answer questions, analyze data, formulate or justify conclusions, make predications, or solve problems. Current Learning Students organize and display data through scatter plots, histograms, box-and-whisker plots, and line graphs. They use these displays to answer questions, analyze data, formulate or justify conclusions, make predictions, or solve problems. They also continue to identify representations that best display a given set of data or situation. In 1.3, students will use this knowledge to display and analyze data from their survey, experiment, or observation. Future Learning In high school, students will organize and display one- and two-variable data using a variety of representations to analyze the data to formulate or justify conclusions, make predictions, or solve problems, with or without using technology. Additional Research Findings According to Curriculum Focal Points, building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatter plots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures (p. 40). C-2 Cumberland, Lincoln, and Woonsocket Public Schools
Create and Display Data (13 days) Grade 8 Mathematics, Quarter 1, Unit 1.1 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-3
Grade 8 Mathematics, Quarter 1, Unit 1.1 Create and Display Data (13 days) C-4 Cumberland, Lincoln, and Woonsocket Public Schools
Grade 8 Mathematics, Quarter 1, Unit 1.2 Interpret and Analyze Data Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Interpret line graphs, histograms, box-andwhisker plots, and scatter plots. (Box-andwhisker plots and scatter plots are introduced.) Analyze data to formulate or justify conclusions, make predictions, or solve problems. Calculate and interpret measures of central tendency (mean, median, and mode). Calculate measures of variation (range, outliers, and introduce quartile values). Determine an estimated line of best fit to analyze situations or solve problems. Determine whether a sample is biased or unbiased (random or non-random). Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze relationships. Explain relationships between tables and graphs. Construct viable arguments and critique the reasoning of others. Use inductive reasoning. Justify conclusions. Make plausible arguments. Model with mathematics. Analyze relationships mathematically to draw conclusions. Interpret their mathematical results Use appropriate tools strategically. Use tools to solve, explore, compare and visualize problems and to deepen their knowledge/understanding. Essential questions How do you analyze data from a line graph, scatter plot, histogram, and box and whisker plot? What effects do outliers have on measures of central tendency? How is an estimated line of best fit helpful in analyzing data? How do you determine which measure of central tendency best describes the data? How can measures of variation be used to describe the data? How does a sample affect the validity of the data? How do you calculate and interpret measures of central tendency? Cumberland, Lincoln, and Woonsocket Public Schools C-5
Grade 8 Mathematics, Quarter 1, Unit 1.2 Interpret and Analyze Data (8 days) Written Curriculum Grade-Level Expectations M(DSP) 8 1 Interprets a given representation (line graphs, scatter plots, histograms, or box-andwhisker plots) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems. (Local) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 8 2.) M(DSP) 8 2 Analyzes patterns, trends, or distributions in data in a variety of contexts by determining or using measures of central tendency (mean, median, or mode), dispersion (range or variation), outliers, quartile values, or estimated line of best fit to analyze situations, or to solve problems; and evaluates the sample from which the statistics were developed (bias, random, or non-random). (Local) Clarifying the Standards Prior Learning In kindergarten, students interpreted a given representation using models and tally charts to answer questions. Students also analyzed patterns, trends, or distributions of data in a variety of contexts. In first grade, students interpreted pictographs (one-to-one correspondence) and tables. Second-graders were introduced to line plots. Third-graders were introduced to bar graphs and made predications based on their graphs. They analyzed trends in data using mode. Fourth-graders interpreted pictographs and circle graphs and justified their conclusions. They determined and used median and mode to analyze data. In grade 5, students interpreted line graphs; and mean was used to analyze data. Stem-and-leaf plots were introduced in grade 6 as well as dispersion (range) to analyze situations or solve problems. In grade 7, students interpreted histograms or scatter plots that represented discrete linear relationships. Outliers were used to analyze situations to determine their effect on the measures of central tendencies. Students also evaluated a sample from which the statistics were developed (bias). Current Learning In grade 8, students learn to interpret line graphs, scatter plots, histograms, or box-and-whisker plots. They analyze data to formulate or justify conclusions, make predictions, or solve problems. Students calculate and use measures of central tendency (mean, median, or mode) and measures of variation (range, outliers, quartile values). Students estimate a line of best fit to analyze situations or solve problems and determine whether a sample is biased or unbiased (random or non-random). Box-andwhisker plots, quartiles, and line of best fit are introduced at this grade level. Students are developing and applying the concepts learned in unit 1.1 throughout this unit and unit 1.3. Future Learning In high school, students will interpret a variety of representations to make observations, answer questions, and analyze data to formulate and justify conclusions, as well as make predictions. They will analyze patterns, trends, or distributions in data using measures of central tendency, dispersions, and variations. Students will also continue to learn, and eventually master, line of best fit and quartile values. Cumberland, Lincoln, and Woonsocket Public Schools C-6
Interpret and Analyze Data (8 days) Grade 8 Mathematics, Quarter 1, Unit 1.2 Additional Research Findings According to Principles and Standards for School Mathematics, students should interpret a given representation and analyze patterns, trends, or distributions of data in a variety of contexts by using measures of central tendencies, dispersion, and outliers (pp. 248 254). Curriculum Focal Points states, students use descriptive statistics, including mean, median, and range to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose (p. 40). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-7
Grade 8 Mathematics, Quarter 1, Unit 1.2 Interpret and Analyze Data (8 days) Cumberland, Lincoln, and Woonsocket Public Schools C-8
Grade 8 Mathematics, Quarter 1, Unit 1.3 Developing a Survey Overview Number of instructional days: 6 (1 day = 45 minutes) Content to be learned Generate a question or hypothesis and decide the most effective method to collect that data (e.g., survey, observation, experimentation). Collect, organize, analyze, and appropriately display the data. Draw conclusions about the question or hypothesis being tested. Make predictions, ask new questions, and make connections to real-world situations when appropriate. Introduce the concept that limitations, which could affect interpretations of the results, need to be considered. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Make conjectures and build a logical progression of statements to explore the truth of conjectures. Justify conclusions. Make plausible arguments. Analyze and interpret arguments. Take into account the context from which the data arose. Use appropriate tools strategically. Consider/use appropriate/available tools i.e. graphing calculators, protractors, statistical software, Excel. Attend to precision. Specify units/labels (i.e., graphs, axes, titles, all representations). Model with mathematics. Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Apply what they know to simplify a complicated situation and realize that revisions may be needed later. Interpret mathematical results in the context of the situation and reflect on whether the results make sense (possibly improving the model if it has not served its purpose). Cumberland, Lincoln, and Woonsocket Public Schools C-9
Grade 8 Mathematics, Quarter 1, Unit 1.3 Developing a Survey (6 days) Essential questions How can you determine if your sample was biased? How could your survey be relevant in a realworld situation? Why did you choose your method for collecting and displaying the data? What conclusions can you make about your question or hypothesis? How can you modify your question or hypothesis to further your study of the topic? What are some limitations to your questions or hypothesis that could cause your results to be biased? Written Curriculum Grade-Level Expectations M(DSP) 8 6 In response to a teacher or student generated question or hypothesis decides the most effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical) necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the data to draw conclusions about the question or hypothesis being tested while considering the limitations that could affect interpretations; and when appropriate makes predictions; and asks new questions and makes connections to real world situations. (Local) (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP) 8 2.) Clarifying the Standards Prior Learning In grades 2 3, students started to respond to teacher- or student-generated questions and decided the most effective method to collect the data. They displayed and analyzed the data to draw conclusions about the question or hypothesis being tested and, when appropriate, made predictions. In grades 4 6, students asked new questions and made connections to real-world situations. In grade 7, students were asked to consider the limitations that could affect interpretations. Current Learning In eighth grade, students collect, organize, analyze, and appropriately display the data generated by a survey. They continue to draw conclusions about the question or hypothesis being tested and, when appropriate, make predictions, ask new questions, and make connections to real-world situations. Students must also consider the limitations that could affect the interpretation of the results. These topics are previously introduced in units of study 1.1 and 1.2, and students apply their knowledge from these units to this current unit. C-10 Cumberland, Lincoln, and Woonsocket Public Schools
Developing a Survey (6 days) Grade 8 Mathematics, Quarter 1, Unit 1.3 Future Learning In high school, students will continue to work on all aspects of the current GLE and use them in more complex situations. Additional Research Findings According to Principles and Standards for School Mathematics, In collecting and representing data, students should be driven by a desire to answer questions on the basis of the data. In the process, they should make observations, inferences, and conjectures and develop new questions. The book further states, Middle-grades students should formulate questions and design experiments or surveys to collect relevant data so that they can compare characteristics within a population or between populations. Students would plan experiments, collect data or find relevant data in other resources. Mathematics teachers may find it useful to collaborate with science teachers so that they are consistent in their design of experiments (pp. 248 249). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-11
Grade 8 Mathematics, Quarter 1, Unit 1.3 Developing a Survey (6 days) C-12 Cumberland, Lincoln, and Woonsocket Public Schools
Grade 8 Mathematics, Quarter 1, Unit 1.4 Probability Overview Number of instructional days: 13 (1 day = 45 minutes) Content to be learned Solve problems involving combinations or permutations using organized lists, tables, tree diagrams, models, and the Fundamental Counting Principle. Determine the experimental or theoretical probability of an event in a problem-solving situation (the sample space may or may not contain equally-likely outcomes). Predict the theoretical probability of an event and test the prediction through experiments and simulations. Compare and contrast theoretical and experimental probabilities. Mathematical practices to be integrated Attend to precision. Use appropriate math language ; precise definitions. Clarify the meaning of symbols. Look for and express regularity in repeated reasoning. Notice if calculations are repeated, and look for general methods and shortcuts. Use appropriate tools strategically. Use tools to solve, explore, compare and visualize problems and to deepen their knowledge/understanding. Consider/use appropriate/available tools and recognize limitations (i.e., calculators). Model with mathematics. Analyze relationships mathematically to draw conclusions. Interpret their mathematical results. Essential questions What strategies can be used to determine the number of permutations and combinations in a given situation? How can you determine whether an event is equally likely to occur? How do you determine the total number of outcomes? How do you determine the experimental probability of an event? How do you determine the theoretical probability of an event? How can an experiment validate a prediction of the theoretical probability of an event? How do theoretical and experimental probabilities compare? Cumberland, Lincoln, and Woonsocket Public Schools C-13
Grade 8 Mathematics, Quarter 1, Unit 1.4 Probability (13 days) Written Curriculum Grade-Level Expectations M(DSP) 8 4 Uses counting techniques to solve problems in context involving combinations or permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or sc others). (Local) M(DSP) 8 5 For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation; and predicts the theoretical probability of an event and tests the prediction through experiments and simulations; and compares and contrasts theoretical and experimental probabilities. (Local) Clarifying the Standards Prior Learning This GLE was not addressed in kindergarten. In first grade, probability of events was introduced with a sample space that may or may not contain equally-likely outcomes. In second grade, students used counting techniques to solve problems using a variety of strategies. The ideas of certain and probable were introduced in the context of determining the probability of an event. In grades 3 4, counting techniques and simple permutations were used. In grade 4, part-to-whole relationships were discussed; fractions were used in grade 5. In sixth grade, students used counting techniques to solve problems involving simple permutations using a variety of strategies. The Fundamental Counting Principle was introduced as well as the concept of fair games. In seventh grade, permutations moved beyond simple to more complex situations. Theoretical and experimental probabilities were also introduced. Current Learning In grade 8, students solve problems involving combinations or permutations using organized lists, tables, tree diagrams, models, and the Fundamental Counting Principle. They determine the experimental or theoretical probability of an event in a problem-solving situation. (The sample space may or may not contain equally-likely outcomes.) They predict the theoretical probability of an event and test the prediction through experiments and simulations. Lastly, students compare and contrast theoretical and experimental probabilities. These topics are introduced and reinforced at this level. Future Learning In high school, students will continue to solve problems involving combinations or permutations, using counting techniques. These concepts will continue to be used in more complex situations. Additional Research Findings According to Principles and Standards for School Mathematics, Students will compute probabilities for simple compound events using such methods as organized lists, tree diagrams, and area models. Students also need to develop their probablistic thinking by frequent experience with actual experiments. Many can be quite simple. For example, students could be asked to predict the probability for various outcomes of flipping two coins sixty times. Then they could discuss whether the results of the experiment are consistent with their predications. If students are accustomed to reasoning form and about data, they will C-14 Cumberland, Lincoln, and Woonsocket Public Schools
Probability (13 days) Grade 8 Mathematics, Quarter 1, Unit 1.4 understand that discrepancies between predictions and outcomes from a large and representative sample must be taken seriously (pp. 248, 254). Curriculum Focal Points states, One of the expectations of the content standards for data analysis and probability for grades 6 8 states that students will compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models (p. 39). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-15
Grade 8 Mathematics, Quarter 1, Unit 1.4 Probability (13 days) C-16 Cumberland, Lincoln, and Woonsocket Public Schools