Download Math 7 - Unit 6 Blueprint

Math 7: Unit 6 – Probability
(4 weeks)
Unit Overview:
Grade 7 is the introduction to the formal study of probability. (This is students’ first exposure to probability in the Common Core.)
Through multiple experiences, students begin to understand the probability of chance (simple and compound), develop and use
sample spaces, compare experimental and theoretical probabilities, develop and use graphical organizers, and use information from
simulations for predictions.
Content Standards:
Investigate chance processes and develop, use, and evaluate probability models.
MCC.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 1/2
indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. (ITBS)
MCC.7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing
its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a
number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. (ITBS)
MCC.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 possible sources of the discrepancy. (ITBS)
MCC.7.SP.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine
probabilities of events. (ITBS)
MCC.7.SP.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance
process. (ITBS)
MCC.7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (ITBS)
MCC.7.SP.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the
sample space for which the compound event occurs. (ITBS)
MCC.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. (ITBS)
MCC.7.SP.8c Design and use a simulation to generate frequencies for compound events. (ITBS)
Standards for Mathematical Practice:
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Diagnostic: Prerequisite Assessment
Standards for Mathematical Practice (4)
EQ:
Learning Targets:
I can …
 Make comparisons and formulate predictions.
 Use experiments or simulations to generate data sets and create probability models.
Concept Overview:
MP7 Look for and make use of structure.
Students routinely seek patterns or structures to model and solve problems. Students examine tree diagrams or systematic lists to
determine the sample space for compound events and verify that they have listed all possibilities.
MP8 Look for and express regularity in repeated reasoning.
In grade 7, students use repeated reasoning to understand algorithms and make generalizations about patterns. During multiple
opportunities to solve and model problems, they create, explain, evaluate, and modify probability models to describe simple and
compound events.
Resources:
MP7 Inside Mathematics Website
MP8 Inside Mathematics Website
Introduction to Probability
Investigate chance processes and develop, use, and evaluate probability models.
MCC.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 1/2
indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
MCC.7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing
its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a
number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
EQ: How is probability used to make informed decisions about uncertain events?
Learning Targets:
I can …
• represent the probability of an event as a fraction or decimal from 0 to 1 or percent from 0% to 100 (SP5)
• explain why a probability of 0 is impossible. (SP5)
• explain why probabilities near 0 are unlikely to occur. (SP5)
• explain why probabilities of .5 are equally likely and unlikely. (SP5)
• explain why probabilities near 1 are more likely to occur. (SP5)
• explain why a probability of 1 is certain. (SP5)
• perform an experiment and collect data on a chance event. (SP6)
• relate the results of an experiment to the theoretical relative frequency of an event. (SP6)
• use the results of an experiment to estimate the probability of an event. (SP6)
• estimate the long-run relative frequency of an event given the probability of the event. (SP6)
Concept Overview: In Grade 7, students build their understanding of probability on a relative frequency view of the subject,
examining the proportion of “successes” in a chance process—one involving repeated observations of random outcomes of a given
event, such as a series of coin tosses. “What is my chance of getting the correct answer to the next multiple choice question?” is not
a probability question in the relative frequency sense. “What is my chance of getting the correct answer to the next multiple choice
question if I make a random guess among the four choices?” is a probability question because the student could set up an
experiment of multiple trials to approximate the relative frequency of the outcome. And two students doing the same experiment
will get nearly the same approximation. These important points are often overlooked in discussions of probability.
Students begin by relating probability to the long-run (more than five or ten trials) relative frequency of a chance event, using coins,
number cubes, cards, spinners, bead bags, and so on. Hands-on activities with students collecting the data on probability
experiments are critically important, but once the connection between observed relative frequency and theoretical probability is
clear, they can move to simulating probability experiments via technology (graphing calculators or computers).
It must be understood that the connection between relative frequency and probability goes two ways. If you know the structure of
the generating mechanism (e.g., a bag with known numbers of red and white chips), you can anticipate the relative frequencies of a
series of random selections (with replacement) from the bag. If you do not know the structure (e.g., the bag has unknown numbers
of red and white chips), you can approximate it by making a series of random selections and recording the relative frequencies. This
simple idea, obvious to the experienced, is essential and not obvious at all to the novice. The first type of situation, in which the
structure is known, leads to “probability”; the second, in which the structure is unknown, leads to “statistics.”
Help students understand the probability of chance is using the benchmarks of probability: 0, 1 and ½. Provide students with
situations that have clearly defined probability of never happening as zero, always happening as 1 or equally likely to happen as to
not happen as 1/2. Then advance to situations in which the probability is somewhere between any two of these benchmark values.
This builds to the concept of expressing the probability as a number between 0 and 1. Use this concept to build the understanding
that the closer the probability is to 0, the more likely it will not happen, and the closer to 1, the more likely it will happen.
Probability can be expressed in terms such as impossible, unlikely, likely, or certain or as a number between 0 and 1 as illustrated on
the number line. Students can use simulations such as Marble Mania on AAAS or the Random Drawing Tool on NCTM’s Illuminations
to generate data and examine patterns.
Students learn to make predictions about the relative frequency of an event by using simulations to collect, record, organize and
analyze data. Students also develop the understanding that the more the simulation for an event is repeated, the closer the
experimental probability approaches the theoretical probability. The focus of this standard is relative frequency.
Vocabulary:
probability: A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a
coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).
likely: An event that is likely to happen has a probability ratio closer 1.
unlikely: An event that unlikely has a probability ratio closer to 0.
theoretical probability: The mathematical calculation that an event will happen in theory.
experimental probability: The ratio of the number of times an outcome occurs to the total amount of trials performed.
Experimental Probability refers the chance that something happens based on repeating experiments and observing the outcomes.
frequency: The number of times an item, number, or event occurs in a set of data.
relative frequency: Relative frequency is another term for proportion; The value calculated by dividing the number of times an
event occurs by the total number of times an experiment is carried out. The probability of an event can be thought of as its long-run
relative frequency when the experiment is carried out many times. If an experiment is repeated n times, and event E occurs r times,
then the relative frequency of the event E is defined to be rfn (E) = r/n.
Sample Problem(s): Solutions to Sample Problems
Sample Problem 1:
The container below contains 2 gray, 1 white, and 4 black marbles. Without looking, if you choose a marble from the container, will
the probability be closer to 0 or to 1 that you will select a white marble? A gray marble? A black marble? Justify each of your
predictions. (SP5)
Sample Problem 2:
There are three choices of jellybeans – grape, cherry and orange. If the probability of getting a grape is 3/10 and the probability of
getting a cherry is 1/5, what is the probability of getting an orange? (SP5)
Sample Problem 3:
Using a six-sided number cube, have students create events that are impossible, unlikely, as likely as unlikely, likely, and certain.
(SP5)
Sample Problem 4:
Each group receives a bag that contains 4 green marbles, 6 red marbles, and 10 blue marbles. Each group performs 50 pulls,
recording the color of marble drawn and replacing the marble into the bag before the next draw. Students compile their data as a
group and then as a class. They summarize their data as experimental probabilities and make conjectures about theoretical
probabilities (How many green draws would you expect if you were to conduct 1000 pulls? 10,000 pulls?). (SP6)
Sample Problem 5:
Students create another scenario with a different ratio of marbles in the bag and make a conjecture about the outcome of 50 marble
pulls with replacement. (An example would be 3 green marbles, 6 blue marbles, 3 blue marbles.) (SP6)
Sample Problem 6:
If Portia were to flip a coin one hundred times could the outcomes be 80 heads and 20 tails? Explain your reasoning. (SP6)
Standard
MCC.7.SP.5
Topic
Understanding
probability
Resources
Holt 3
Probability
Section 10 – 1 pgs. 522 – 524
Pearson 2
Probability
Section 12 – 1 pgs. 580 – 581
 Marble Mania (U)
 NCTM Illuminations
Random Drawing Tool (U)
 Rock, Paper, Scissors (S, K)
 What Are My Chances? (K,
U)
Teacher Notes
Model Lesson: Probability
Student Misconceptions:
Students think the odds of an event occurring is the same as
probability.
Students mistakenly believe probability can be a number
larger than 1.
Math 7 Unit 6- Good Questions
Math 7 Unit 6- Differentiation Strategies
Cooperative Learning Strategy: Probability Activity Centers
Literacy Strategy: Vocabulary Strategy
MCC.7.SP.6
Relative
frequency
Holt 3
Experimental Probability
Section 10 – 2 pgs. 527 – 528
Pearson 2
Experimental Probability
Student Misconceptions:
Students sometimes do not make the connection that
histograms of experimental data should be proportional in
size to another histogram from the same experiment with a
different sample size.
Section 12 – 2 pgs. 586 – 587
Math 7 Unit 6- Good Questions
 Gumball Machine (U)
 Video Simulations of
Relative Frequency (S)
 Relative Frequency & TwoWay Tables (S, K)
Math 7 Unit 6- Differentiation Strategies
Literacy Strategy: Word Web
**See Is It Fair task below.
Developing Probability Models
Investigate chance processes and develop, use, and evaluate probability models.
MCC.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 possible sources of the discrepancy.
MCC.7.SP.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine
probabilities of events.
MCC.7.SP.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance
process.
EQ: How is probability used to make informed decisions about uncertain events?
Learning Targets:
I can …
 use theoretical probabilities to create a probability model (e.g. table showing the potential outcomes of an experiment or
random process with their corresponding probabilities) in which all outcomes are equally likely (uniform). (SP7a)
 use observed frequencies to create a probability model for the data generated from a chance process. (SP7)
 use probability models to find probabilities of events. (SP7b)
 compare theoretical and experimental probability. (SP7)
Concept Overview: A probability model provides a probability for each possible non-overlapping outcome for a chance process so
that the total probability over all such outcomes is unity (i.e. the sum of all probabilities in the sample space is 1). A probability
model is defined by its sample space, events within the sample space, and probabilities associated with each event.





The collection of all possible individual outcomes is known as the sample space for the model. For example, the sample
space for the toss of two coins (fair or not) is often written as {TT, HT, TH, HH}.
The probabilities of the model can be either theoretical (based on the structure of the process and its outcomes) or
empirical (based on observed data generated by the process).
Example 1: In the toss of two balanced coins, the four outcomes of the sample space are given equal theoretical
probabilities of 1/4 because of the symmetry of the process—because the coins are balanced, an outcome of heads is just
as likely as an outcome of tails.
Example 2: Randomly selecting a name from a list of ten students also leads to equally likely outcomes with probability
0.10 that a given student’s name will be selected. If there are exactly four seventh graders on the list, the chance of
selecting a seventh grader’s name is 0.40.
Example 3: On the other hand, the probability of a tossed thumbtack landing point up is not necessarily 1/2 just because
there are two possible outcomes; these outcomes may not be equally likely and an empirical answer must be found by
tossing the tack and collecting data.
Students need multiple opportunities to perform probability experiments and compare these results to theoretical probabilities.
Critical components of the experiment process are making predictions about the outcomes by applying the principles of theoretical
probability, comparing the predictions to the outcomes of the experiments, and replicating the experiment to compare results.
Experiments can be replicated by the same group or by compiling class data. Experiments can be conducted using various random
generation devices including, but not limited to, bag pulls, spinners, number cubes, coin toss, and colored chips. Students can collect
data using physical objects or graphing calculator or web-based simulations. Students can also develop models for geometric
probability (i.e. a target).
Vocabulary:
probability model: A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of
the process. The set of all outcomes is called the sample space, and their probabilities sum to 1
uniform probability model: A probability model which assigns equal probability to all outcomes. (ex: a spinner that has 4 equal
sections)
sample space: The set of all possible outcomes from an experiment.
event: Any possible outcome of an experiment in probability.
Sample Problem(s): Solutions to Sample Problems
Sample Problem 7:
If you choose a point in the square, what is the probability that it is not in the circle? (SP7)
Sample Problem 8:
Juan rolled 15 fours when rolling a fair die 60 times. Would you expect this result? Justify your answer. (SP7)
Sample Problem 9:
The results of a spinner experiment are 50% red, 10% blue, 20% yellow, and 20% green. Draw the spinner. (SP7)
Standard
MCC.7.SP.7
MCC.7.SP.7a
MCC.7.SP.7b
Topic
Probability
models
Resources
Holt 3
Theoretical Probability
Section 10 – 4 pgs. 540 – 541
Counting Principles
Section 10 – 8 pgs. 558 – 560
Pearson 2
Sample Space
Section 12 – 3 pg. 591 - 593
 Interactive Probability
Models (S, K, U)
 Learning Task: What are the
Chances? (U)
 Learning Task: Designing
Simulations (U)
Teacher Notes
Student Misconceptions:
Students might get confused when a probability problem
involves a fraction of area.
Math 7 Unit 6- Good Questions
Math 7 Unit 6- Differentiation Strategies
Cooperative Learning Strategy:
SKUNK Game
Sticks and Stones Game
Literacy Strategy: Math Journal
Probability of Compound Events
Investigate chance processes and develop, use, and evaluate probability models.
MCC.7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
MCC.7.SP.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the
sample space for which the compound event occurs.
MCC.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.
MCC.7.SP.8c Design and use a simulation to generate frequencies for compound events.
EQ: How is probability used to make informed decisions about uncertain events?
Learning Targets:
I can …
 represent probabilities of simple and compound events as a fraction, decimal, or percent. (SP8a)
• find the sample space of a compound event. (SP8b)
• create organized lists, tables, tree diagrams, and simulations to determine the probability of compound events. (SP8)
• generate frequencies for compound events using random number generators (e.g. tables, calculators, manipulatives).
(SP8c)
Concept Overview: The product rule for counting outcomes for chance events should be used in finite situations like tossing two or
three coins or rolling two number cubes. There is no need to go to more formal rules for permutations and combinations at this
level. Students should gain experience in the use of diagrams, especially trees and tables, as the basis for organized counting of
possible outcomes from chance processes.7.SP.8 For example, the 36 equally likely (theoretical probability) outcomes from the toss
of a pair of number cubes are most easily listed on a two-way table. An archived table of census data can be used to approximate the
(empirical) probability that a randomly selected Florida resident will be Hispanic.
After the basics of probability are understood, students should experience setting up a model and using simulation (by hand or with
technology) to collect data and estimate probabilities for a real situation that is sufficiently complex that the theoretical probabilities
are not obvious. For example, suppose, over many years of records, a river generates a spring flood about 40% of the time. Based on
these records, what is the chance that it will flood for at least three years in a row sometime during the next five years?
Students should begin to expand the knowledge and understanding of the probability of simple events, to find the probabilities of
compound events by creating organized lists, tables and tree diagrams. This helps students create a visual representation of the
data; i.e., a sample space of the compound event. From each sample space, students determine the probability or fraction of each
possible outcome. Students continue to build on the use of simulations for simple probabilities and now expand the simulation of
compound probability. Providing opportunities for students to match situations and sample spaces assists students in visualizing the
sample spaces for situations.
Students often struggle making organized lists or trees for a situation in order to determine the theoretical probability. Having
students start with simpler situations that have fewer elements enables them to have successful experiences with organizing lists
and trees diagrams. Ask guiding questions to help students create methods for creating organized lists and trees for situations with
more elements.
Students often see skills of creating organized lists, tree diagrams, etc. as the end product. Provide students with experiences that
require the use of these graphic organizers to determine the theoretical probabilities. Have them practice making the connections
between the process of creating lists, tree diagrams, etc. and the interpretation of those models. Additionally, students often
struggle when converting forms of probability from fractions to percents and vice versa. To help students with the discussion of
probability, don’t allow the symbol manipulation/conversions to detract from the conversations. By having students use technology
such as a graphing calculator or computer software to simulate a situation and graph the results, the focus is on the interpretation of
the data. Students then make predictions about the general population based on these probabilities.
Vocabulary:
compound event: An event that consists of two or more events that are not mutually exclusive.
Independent events: Events whose outcomes do not influence each other.
dependent events: Two or more events in which the outcome of one event does affect the outcome of the other event or events.
mutually exclusive events: Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in
common).
tree diagram: A diagram that shows all the possible outcomes of an event.
sample space: The set of all possible outcomes from an experiment.
outcomes: A possible result of an experiment.
favorable outcomes: The number of outcomes of a desired event in an experiment.
Sample Problem(s): Solutions to Sample Problems
Sample Problem 10:
What is the probability of a family with five children having exactly two boys? (SP8)
Sample Problem 11:
Students conduct a bag pull experiment. A bag contains 5 marbles. There is one red marble, two blue marbles and two purple
marbles. Students will draw one marble without replacement and then draw another. What is the sample space for this situation?
Explain how you determined the sample space and how you will use it to find the probability of drawing one blue marble followed by
another blue marble. (SP8)
Sample Problem 12:
Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn at
random, what is the probability that you will draw the letters F-R-E-D in that order? What is the probability that your “word” will
have an F as the first letter? (SP8)
Sample Problem 13:
Create a tree diagram for illustrating the outcomes for a car that has two or four doors and is red, black, or silver. Create questions
that can be answered based on the diagram. (SP8)
Standard
MCC.7.SP.8
MCC.7.SP.8a
MCC.7.SP.8b
MCC.7.SP.8c
Topic
Probability of
compound
events
Resources
Teacher Notes
Holt 3
Independent and Dependent
Events
Section 10 – 5 pgs. 545 – 547
Model Lesson: Compound Probability
Pearson 2
Compound Events
Section 12 – 4 pgs. 598 – 600
Math 7 Unit 6- Good Questions
 Two Dice (U)
 Two Hospitals (U)
 Performance Task: Is It
Fair? (U)
Cooperative Learning Strategy:
Is the game Rock, Paper, Scissors fair?
Student Misconceptions: List common places of confusion
for students and hints for addressing those misconceptions.
Math 7 Unit 6- Differentiation Strategies
Literacy Strategy: Reword and Regroup
**Is it Fair Task also covers
MCC 7.SP.6.
Math 7 Unit 6- Summative Assessment (ONL)
Math 7 Unit 6- Summative Assessment (ADV)