School Curriculum: Data Analysis, Probability, & Discrete Mathematics
This page is designed to enable parents to understand what their child should be learning, when they should be learning it, and what degree of mastery the child should have attained (at a median level) by a certain grade level. For Homeschoolers, we hope that this page will serve as a valuable asset in establishing a baseline curriculum. For parents whose children attend public or private schools (or for the inquisitive student) this page should give some guidance as to whether or not the school curriculum and methods are providing students with an adequate standard of education.
What is meant by "Data Analysis, Probability, & Discrete Mathematics," why is it important, and how is it approached ? Below is a description of the core discipline and its components, and the answers to why-how-when these components are taught. Data Analysis, Probability, & Discrete Mathematics components have median level goals to be attained by the end of Grade 2, by the end of Grade 3, by the end of Grade 4, by the end of Grade 5, by the end of Grade 6, by the end of Grade 7, by the end of Grade 8, and by the end of Grade 12.
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Data Analysis, Probability, & Discrete Mathematics: A. Data Analysis (Statistics), B. Probability, C. DM--Systematic Listing and Counting, D. DM--Vertex-Edge Graphs and Algorithms
Descriptive Statement: Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics. Each provides students with powerful mathematical perspectives on everyday phenomena and with important examples of how mathematics is used in the modern world. Two important areas of discrete mathematics are addressed in this standard; a third area, iteration and recursion, is addressed in Standard 4.3 (Patterns and Algebra).
Data Analysis (or Statistics). In today’s information-based world, students need to be able to read, understand, and interpret data in order to make informed decisions. In the early grades, students should be involved in collecting and organizing data, and in presenting it using tables, charts, and graphs. As they progress, they should gather data using sampling, and should increasingly be expected to analyze and make inferences from data, as well as to analyze data and inferences made by others.
Probability. Students need to understand the fundamental concepts of probability so that they can interpret weather forecasts, avoid unfair games of chance, and make informed decisions about medical treatments whose success rate is provided in terms of percentages. They should regularly be engaged in predicting and determining probabilities, often based on experiments (like flipping a coin 100 times), but eventually based on theoretical discussions of probability that make use of systematic counting strategies. High school students should use probability models and solve problems involving compound events and sampling.
Discrete Mathematics—Systematic Listing and Counting. Development of strategies for listing and counting can progress through all grade levels, with middle and high school students using the strategies to solve problems in probability. Primary students, for example, might find all outfits that can be worn using two coats and three hats; middle school students might systematically list and count the number of routes from one site on a map to another; and high school students might determine the number of three-person delegations that can be selected from their class to visit the mayor.
Discrete Mathematics—Vertex-Edge Graphs and Algorithms. Vertex-edge graphs, consisting of dots (vertices) and lines joining them (edges), can be used to represent and solve problems based on real-world situations. Students should learn to follow and devise lists of instructions, called "algorithms," and use algorithmic thinking to find the best solution to problems like those involving vertex-edge graphs, but also to solve other problems.
These topics provide students with insight into how mathematics is used by decision-makers in our society, and with important tools for modeling a variety of real-world situations. Students will better understand and interpret the vast amounts of quantitative data that they are exposed to daily, and they will be able to judge the validity of data-supported arguments.
Fostering respect for the power of mathematics. All students should learn that mathematics is integral to the development of all cultures and civilizations, and in particular to the advances in our own society. They should be aware that the adults in their world (parents, relatives, mentors, community members, role models) use mathematics on a daily basis. And they should know that success in mathematics may be a critical gateway to success in their careers, citizenship, and lives.
Setting high expectations. All students should have high expectations of themselves. These high expectations should be fostered by their teachers, administrators, and parents all of whom should themselves believe that all students can and will succeed in mathematics. This belief in his or her abilities often makes it possible for a child to succeed.
Providing opportunities for success. High expectations can only be achieved if students are provided with the appropriate opportunities. At all grade levels, students should receive instruction by teachers who have had the training and professional development appropriate for their grade level. Students should receive prompt and appropriate services essential to ensure that they can learn the mathematical skills and concepts included in the core curriculum, and to ensure that their weaknesses do not result in trapping them in a cycle of failure. Students should receive equitable treatment without regard to gender or ethnicity, and should not be conditioned to fail by predetermined low expectations.
Encouraging all students to go beyond the standards. Teachers should help students develop a positive attitude about mathematics by engaging them in exploring and solving interesting mathematical problems, by using mathematics in meaningful ways, by focusing on concepts and understanding as well as on rules and procedures, and by consistently expecting them to go beyond repetition and memorization to problem solving and understanding. Every effort should be made to ensure that all students are continuously encouraged, nurtured, and challenged to maximize their potential at all grade levels and to become prepared for college-level mathematics. Students who have achieved the standards should be encouraged to go beyond the standards. If schools challenge all students at lower grade levels, they will attain the goal of having advanced mathematics classrooms whose students reflect the diversity of the school’s total population.
Strands and Cumulative Progress Indicators
By the end of Grade 2, students will:
A. Data Analysis
1. Collect, generate, record, and organize data in response to questions, claims, or curiosity.
Data collected from students’ everyday experiences
Data generated from chance devices, such as spinners and dice
2. Read, interpret, construct, and analyze displays of data.
Pictures, tally chart, pictograph, bar graph, Venn diagram
Smallest to largest, most frequent (mode)
B. Probability
1. Use chance devices like spinners and dice to explore concepts of probability.
Certain, impossible
More likely, less likely, equally likely
2. Provide probability of specific outcomes.
Probability of getting specific outcome when coin is tossed, when die is rolled, when spinner is spun (e.g., if spinner has five equal sectors, then probability of getting a particular sector is one out of five)
When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven
C. Discrete Mathematics—Systematic Listing and Counting
1. Sort and classify objects according to attributes.
Venn diagrams
2. Generate all possibilities in simple counting situations (e.g., all outfits involving two shirts and three pants).
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Follow simple sets of directions (e.g., from one location to another, or from a recipe).
2. Color simple maps with a small number of colors.
3. Play simple two-person games (e.g., tic-tac-toe) and informally explore the idea of what the outcome should be.
4. Explore concrete models of vertex-edge graphs (e.g. vertices as "islands" and edges as "bridges").
Paths from one vertex to another
Building upon knowledge and skills gained in preceding grades, by the end of Grade 3, students will:
A. Data Analysis
1. Collect, generate, organize, and display data in response to questions, claims, or curiosity.
Data collected from the classroom environment
2. Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data.
Pictograph, bar graph, table
B. Probability
1. Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.
Likely, unlikely, certain, impossible
More likely, less likely, equally likely
2. Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).
What students think will happen (intuitive)
Collect data and use that data to predict the probability (experimental)
C. Discrete Mathematics—Systematic Listing and Counting
1. Represent and classify data according to attributes, such as shape or color, and relationships.
Venn diagrams
Numerical and alphabetical order
2. Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.
Organized lists, charts
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).
2. Explore vertex-edge graphs
Vertex, edge
Path
3. Find the smallest number of colors needed to color a map.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 4, students will:
A. Data Analysis
1. Collect, generate, organize, and display data in response to questions, claims, or curiosity.
Data collected from the school environment
2. Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data.
Pictograph, bar graph, line plot, line graph, table
Average (mean), most frequent (mode), middle term (median)
B. Probability
1. Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.
Likely, unlikely, certain, impossible, improbable, fair, unfair
More likely, less likely, equally likely
Probability of tossing "heads" does not depend on outcomes of previous tosses
2. Determine probabilities of simple events based on equally likely outcomes and express them as fractions.
3. Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).
What students think will happen (intuitive)
Collect data and use that data to predict the probability (experimental)
Analyze all possible outcomes to find the probability (theoretical)
C. Discrete Mathematics—Systematic Listing and Counting
1. Represent and classify data according to attributes, such as shape or color, and relationships.
Venn diagrams
Numerical and alphabetical order
2. Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.
Organized lists, charts, tree diagrams
Dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results)
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).
2. Play two-person games and devise strategies for winning the games (e.g., "make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).
3. Explore vertex-edge graphs and tree diagrams.
Vertex, edge, neighboring/adjacent, number of neighbors
Path, circuit (i.e., path that ends at its starting point)
4. Find the smallest number of colors needed to color a map or a graph.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:
A. Data Analysis
1. Collect, generate, organize, and display data.
Data generated from surveys
2. Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.
Bar graph, line graph, circle graph, table
Range, median, and mean
3. Respond to questions about data and generate their own questions and hypotheses.
B. Probability
1. Determine probabilities of events.
Event, probability of an event
Probability of certain event is 1 and of impossible event is 0
2. Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).
Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked
Given data obtained experimentally, what is the likely distribution of items in the bag
3. Model situations involving probability using simulations (with spinners, dice) and theoretical models.
C. Discrete Mathematics—Systematic Listing and Counting
1. Solve counting problems and justify that all possibilities have been enumerated without duplication.
Organized lists, charts, tree diagrams, tables
2. Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions.
Building upon knowledge and skills gained in preceding grades, by the end of Grade 6, students will:
A. Data Analysis
1. Collect, generate, organize, and display data.
Data generated from surveys
2. Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.
Bar graph, line graph, circle graph, table, histogram
Range, median, and mean
Calculators and computers used to record and process information
3. Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.
B. Probability
1. Determine probabilities of events.
Event, complementary event, probability of an event
Multiplication rule for probabilities
Probability of certain event is 1 and of impossible event is 0
Probabilities of event and complementary event add up to 1
2. Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).
Given numbers of various types of items in a bag, what is the probability that an item of one type will be picked
Given data obtained experimentally, what is the likely distribution of items in the bag
3. Explore compound events.
4. Model situations involving probability using simulations (with spinners, dice) and theoretical models.
5. Recognize and understand the connections among the concepts of independent outcomes, picking at random, and fairness.
C. Discrete Mathematics—Systematic Listing and Counting
1. Solve counting problems and justify that all possibilities have been enumerated without duplication.
Organized lists, charts, tree diagrams, tables
Venn diagrams
2. Apply the multiplication principle of counting.
Simple situations (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).
Number of ways a specified number of items can be arranged in order (concept of permutation)
Number of ways of selecting a slate of officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 x 22 x 21)
3. List the possible combinations of two elements chosen from a given set (e.g., forming a committee of two from a group of 12 students, finding how many handshakes there will be among ten people if everyone shakes each other person’s hand once).
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions.
2. Analyze vertex-edge graphs and tree diagrams.
Can a picture or a vertex-edge graph be drawn with a single line? (degree of vertex)
Can you get from any vertex to any other vertex? (connectedness)
3. Use vertex-edge graphs to find solutions to practical problems.
Delivery route that stops at specified sites but involves least travel
Shortest route from one site on a map to another
Building upon knowledge and skills gained in preceding grades, by the end of Grade 7, students will:
A. Data Analysis
1. Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).
Type of display most appropriate for given data
Box-and-whisker plot, upper quartile, lower quartile
Scatter plot
Calculators and computer used to record and process information
2. Make inferences and formulate and evaluate arguments based on displays and analysis of data.
B. Probability
1. Interpret probabilities as ratios, percents, and decimals.
2. Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.
Frequency, relative frequency
3. Estimate probabilities and make predictions based on experimental and theoretical probabilities.
4. Play and analyze probability-based games, and discuss the concepts of fairness and expected value.
C. Discrete Mathematics—Systematic Listing and Counting
1. Apply the multiplication principle of counting.
Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)
2. Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).
3. Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Use vertex-edge graphs to represent and find solutions to practical problems.
Finding the shortest network connecting specified sites
Finding the shortest route on a map from one site to another
Finding the shortest circuit on a map that makes a tour of specified sites
Building upon knowledge and skills gained in preceding grades, by the end of Grade 8, students will:
A. Data Analysis
1. Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).
Type of display most appropriate for given data
Box-and-whisker plot, upper quartile, lower quartile
Scatter plot
Calculators and computer used to record and process information
Finding the median and mean (weighted average) using frequency data.
Effect of additional data on measures of central tendency
2. Make inferences and formulate and evaluate arguments based on displays and analysis of data.
3. Estimate lines of best fit and use them to interpolate within the range of the data.
4. Use surveys and sampling techniques to generate data and draw conclusions about large groups.
B. Probability
1. Interpret probabilities as ratios, percents, and decimals.
2. Determine probabilities of compound events.
3. Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red).
4. Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.
Frequency, relative frequency
5. Estimate probabilities and make predictions based on experimental and theoretical probabilities.
6. Play and analyze probability-based games, and discuss the concepts of fairness and expected value.
C. Discrete Mathematics—Systematic Listing and Counting
1. Apply the multiplication principle of counting.
Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)
Factorial notation
Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)
2. Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).
3. Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems.
Finding the shortest network connecting specified sites
Finding a minimal route that includes every street (e.g., for trash pick-up)
Finding the shortest route on a map from one site to another
Finding the shortest circuit on a map that makes a tour of specified sites
Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100)
Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:
A. Data Analysis
1. Use surveys and sampling techniques to generate data and draw conclusions about large groups.
Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)
2. Evaluate the use of data in real-world contexts.
Accuracy and reasonableness of conclusions drawn
Bias in conclusions drawn (e.g., influence of how data is displayed)
Statistical claims based on sampling
3. Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.
4. Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.
5. Analyze data using technology, and use statistical terminology to describe conclusions.
Measures of dispersion: variance, standard deviation, outliers
Correlation coefficient
Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)
B. Probability
1. Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.
2. Use concepts and formulas of area to calculate geometric probabilities.
3. Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.
4. Determine probabilities in complex situations.
Conditional events
Complementary events
Dependent and independent events
5. Estimate probabilities and make predictions based on experimental and theoretical probabilities.
6. Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).
C. Discrete Mathematics—Systematic Listing and Counting
1. Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).
2. Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.
3. Justify solutions to counting problems.
4. Recognize and explain relationships involving combinations and Pascal’s Triangle, and apply those methods to situations involving probability.
D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms
1. Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems.
Circuits that include every edge in a graph
Circuits that include every vertex in a graph
Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloring
Applications to science (e.g., who-eats-whom graphs, genetic trees, molecular structures)
2. Explore strategies for making fair decisions.
Combining individual preferences into a group decision (e.g., determining winner of an election or selection process)
Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)