Teaching Resources for 6-7 Statistics, Data Analysis, and Probability
- Movement with Functions: Lesson 1 - his investigation uses a motion detector to help students understand graphs and equations. Students experience constant and variable rates of change and are challenged to consider graphs where no movements are possible to create them. Multiple representations are used throughout the lesson to allow students to build their fluency with in how graphs, tables, equations, and physical modeling are connected. The lesson also allows students to investigate multiple function types, including linear, exponential, quadratic, and piecewise.
- Movement with Functions: Lesson 3- In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.
- Math 6 Spy Guys- Interactive lessons that covers multiple standards from the Mathematics Framework for California Schools. Includes a glossary, strategies, and operations.
- Web Math - Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics. The site also offers specific sections on the conversion of units (applicable to the sciences).
- Understanding Probability - This lesson is useful to introduce the idea of probability to students.
Title Movement with Functions: Lesson 1 URL http://illuminations.nctm.org/LessonDetail.aspx?ID=L768 Materials needed CBR2 or other motion detector,
TI Interactive!,
TI-84 graphing calculator,
TI Nspire handheld, or other compatible graphing calculator or computer with compatible software,
How Should I Move? Overhead,
How Should I Move? Graphs Activity Sheet,
How Should I Move? Questions Activity Sheet (several copies per student),
How Should I Move? Questions Answer Key,
Comparing Graph Pairs Activity Sheet and Answer Key,
Graphing Equations Activity Sheet (optional)Learning Objectives Students will:
- Provide a conjecture on the type of motion that creates a provided graph
- Use a motion detector to re-create provided graphs
- Compare pairs of graphs by describing the similarities and differences between them in physical modeling and symbolic representations
- Create tables and equations for the graphs
- Compare the results of several graph investigations to one another
Grade Levels Grade 5, Grade 6, Grade 7, Algebra 1 CA 97 Standards Grade 5 AF 1.1 Use information taken from a graph or equation to answer questions about a problem situation. Grade 5 SDAP 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph. Grade 6 AF 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results. Grade 7 AF 1.5 Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Grade 7 AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Algebra 1 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6.EE. (Cluster statement) Reason about and solve one-variable equations and inequalities.
8-Cluster domain; Use functions to model relationships between quantities. 8-F.1; Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (algebraically, graphically, numerically in tables, or by verbal descriptions). 8-F.5; Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Functions-IF.1; Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Functions-IF.2; Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. How could you use this resource? This could be used by students as a lab to discover how the graphs of equations relate to the symbolic notation. Teachers can use this activity to introduce students to graphing by having students work in groups to determine the equations of multiple graphs and compare similar graphs with other groups. EL and Special Needs This is a discovery activity that can be used for all students. Students will get to work in groups to determine the equations of graphs. Lesson Plans Teacher Comments Cost Free Copyright © 2000-2010 NCTM
Title Movement with Functions: Lesson 3 URL http://illuminations.nctm.org/LessonDetail.aspx?ID=L770 Materials needed Stop watches
Remote-controlled cars (strongly suggested, but alternatives are described below)
Rulers
Colored Masking Tape
Collision Activity Sheet (optional pre-activity)
Road Rage Activity Sheet
What If? Activity Sheet (optional)
Road Rage Answer KeyLearning Objectives Students will:
- Collect data and graph a scatter plot to determine the speed of a remote-controlled car
- Create a line of best fit using estimation and technology
- Use tables, graphs, and algebraic calculation to determine when their cars will crash with another group's car
- Validate their calculations by crashing the cars into each other
- Analyze why their time and location estimates for the crash may not be the same as a real-life trial
Grade Levels Grade 7, Algebra I, Algebra II, Probability & Statistics, AP Probability & Statistics CA 97 Standards Grade 7: AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. Grade 7: SDAP 1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables. Algebra 1: 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Algebra II: 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices. Probability and Statistics: 8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots. AP Probability and Statistics: 12.0 Students find the line of best fit to a given distribution of data by using least squares regression. AP Probability and Statistics 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningGrade 6.SP.4; Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Grade 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Grade 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Grade 8-EE.8: Analyze and solve pairs of simultaneous linear equations.
a. Understand 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.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
c. Solve real-world and mathematical problems leading to two linear
equations in two variables.Algebra-CED.3; Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Algebra-REI.7; Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Algebra-REI.10; Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Algebra-REI.12; Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. plots on a number line, including dot plots, histograms, and box plots. Statistics & Probability-ID.1; Represent data with plots on the real number line (dot plots, histograms, and box plots). Statistics & Probability-ID.5; Summarize categorical data for two categories in two-way frequency tables. Recognize possible associations and trends in the data. Statistics & Probability-ID.6; Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential model
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.Statistics & Probability-ID.7; Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Statistics & Probability-ID.8; Compute (using technology) and interpret the correlation coefficient of a linear fit. How could you use this resource? This activity can be used by students to create a systems of equations using manipulatives and data. Teachers can use this activity to create real life connections to systems of equations. EL and Special Needs This is a hands-on activity using manipulatives and group work to help students make connections for concepts and real life applications. Lesson Plans Teacher Comments Cost Free Copyright © 2000-2010 NCTM
Title Math 6 Spy Guys URL http://www.learnalberta.ca/content/mesg/html/math6web/math6shell.html#
Materials needed Computer with high speed internet
Adobe Flash Player
Learning Objectives Learning objectives vary by lesson selection and concept.
Grade Levels Grade 6 CA 97 Standards Multiple Grade 6 standards from the Mathematics Framework for California Public School. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands that involve: 1) Area and Perimeter, 2) Equations and Expressions, 3) Prime Factorization, 4) Percents, 5) Graphing, 6) Probability, 7) Integers, 8) Geometric Transformations, 9) Volume and 10) Angles. How could you use this resource? Students can use this resource to clarify concepts not understood in class, interact with virtual manipulatives, and see virtual representations of concepts. This resource could be used by a teacher to show to a class as a demonstration or an introduction. This is a fairly simple site to use EL and Special Needs Visual graphics display concepts and are interactive. Lesson Plans Teacher Comments Cost Free Copyright © 2003 Alberta Education
Title Web Math URL Materials needed Computer
Learning Objectives Learning objectives vary by concept.
Grade Levels K-8, Algebra 1, Geometry, Calculus, Trigonometry CA 97 Standards Multiple standards from the Mathematics Framework for California Public School. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoningCovers multiple CaCCSS across grade levels and strands. How could you use this resource? Students could use this as a resource for review or to clarify topics discussed in class but not fully understood. The resource is easy to use and navigate to either specific concepts or broader topics.
The site also offers specific sections on the conversion of units (applicable to the sciences).
EL and Special Needs Visual graphics display concepts and are interactive. Lesson Plans Teacher Comments Cost Free Copyright (c) 2009 WebMath.com
Title Understanding Probability URL http://www.discoveryeducation.com/teachers/free-lesson-plans/understanding-probability.cfm
Materials needed Computer
Learning Objectives Students will
1. understand what probability is,
2. learn different ways to express probability numerically: as a ratio, a decimal and a percentage, and
3. learn how to solve problems based on probability.
Grade Levels Grade 6 CA 97 Standards Grade 6 NS 1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages.
Grade 6 NS 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b). Grade 6 NS 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross- multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Grade 6 AF 2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. Grade 6 SDAP 3.0 Students determine theoretical and experimental probabilities and use these to make predictions about events: Grade 6 SDAP 3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring. Grade 7 NS 1.3 Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. CA Common Core State Standards Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning6RP: (Cluster Statement) Understand ratio concepts and use ratio reasoning to solve problems.
6 NS: (Cluster Statement) Apply and extend previous understanding of numbers to the system or rational numbers. 6.RP.1: Understand the concept of a ration and use ratio language to describe a ratio relationship between two quantities. 6.RP.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to zero, and use rate language in the context or a ratio relationship. 6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6.RP.3a: 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.
6.RP.3b: Solve unit rate problems including those involving unit pricing and constant speed.
6.RP.3c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.3d: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
7.NS.2: Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers: present addition and subtraction on a horizontal or vertical number line.
7.NS.2a: Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property leading to products such as (-1)(-1)=1 and the rule for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.2b: Understand that integers can be divided, provided the divisor is not zero, and every quotient of integers (with non-zero divisors) is a rational number. If p and q are integers, then –(p/q)=(-p)/q=p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.2c: Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.2d: Convert a rational number to a decimal using long division: know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.EE.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities are related. 7.EE.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation or estimation strategies. 7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measure in like or different units. 7.RP.2: Recognize and represent proportional relationships between quantities.
7.RP.2a: Describe whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.2b: Identify constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions or proportional relationships.
7.RP.2c: Represent proportional relationships by equations.
7.RP.3: Use proportional relationships to solve multi-step ratio and percent problems. 7.G.1: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.SP.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.6: Approximate the probability of a chance even 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. 7.SP.7: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain the possible sources of the discrepancy.
7.SP.7a: Develop uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
7.SP.7b: Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
7.SP.8: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulations.
7.SP.8a: Understand that, just as with simple events, the probability of a compound even is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.8b: 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.
7.SP.8c: Design and use a simulation to generate frequencies for compound events.
How could you use this resource? This lesson could be used as an introduction and/or to help students gain practice.
EL and Special Needs Discussion questions and possible modificiations available to allow for differentiated instruction. Lesson Plans Teacher Comments Cost Free Copyright (c) 2011 Discovery Education