STANDARD 4.4: Mathematics

STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS, AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

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.

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

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.

· Factorial notation

· Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)

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 (9^th, 10^th, 11^th, and 12^th grade) gets when the classes have unequal sizes (apportionment)