Author:
Connecticut Department of Education
Subject:
Mathematics
Material Type:
Unit of Study
Level:
Middle School
Grade:
7
Provider:
CT State Department of Education
Provider Set:
CSDE - Public
Tags:
Language:
English
Media Formats:
Text/HTML

Education Standards

Unit 6 Overview: Probability

Unit 6 Overview: Probability

Overview

Unit Overview/Summary - FOCUS: 

This unit focuses on Statistics and Probability. Learning this unit will enable students to: 

  • Investigate chance processes and develop, use, and evaluate probability models. 

Relevant Standards:

Major work of the grade is in bold.   

The standards that are to be addressed during this unit of study include: 

7.SP.C.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 

7.SP.C.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. 

7.SP.C.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. 

a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? 

7.SP.C.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 

a. 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. 

b. 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 events. 

c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40 percent of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood 

Examples and Explanations:

7.SP.C.5  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.  

Marble Mania http://www.sciencenetlinks.com/interactives/marble/marblemania.html  

Random Drawing Tool - http://illuminations.nctm.org/activitydetail.aspx?id=67  

 

Example:  

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. 

 

 

7.SP.C.6  Students can collect data using physical objects or graphing calculator or web-based simulations. Students can perform experiments multiple times, pool data with other groups, or increase the number of trials in a simulation to look at the long-run relative frequencies.  

Example:  

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?). 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.) Students try the experiment and compare their predictions to the experimental outcomes to continue to explore and refine conjectures about theoretical probability. 

 

7.SP.C.7  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).  

Example:  

If you choose a point in the square, what is the probability that it is not in the circle?  

 

 

7.SP.C.8  Examples: 

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.  

 

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  

                            

Transfer Goal: Aligned to district portrait or vision of the learner

  • Make sense of problems and persevere in solving them. 

  • Model with mathematics. 

  • Look for and make use of structure. 

Grade level content is in bold 

Coherence: 

How does this unit build on and connect to prior knowledge and learning?   

Students build on their previous understanding of: 

  • Analyzing proportional relationships and use them to solve real-work mathematical problems; 

How does this unit prepare students for future learning?   

The learning of this unit serves as a foundation for content that will be addressed in future years.  Specifically, students will utilize this learning to: 

  • Use random sampling to draw inferences about a population; 

  • Understand And Evaluate Random Processes Underlying Statistical Experiments;  

  • Understand Independence And Conditional Probability And Use Them To Interpret Data; and 

Essential Questions:

Essential Questions can be approached in multiple ways. There should be no more than 2-3 essential questions and they should align with your topics. Questions can be repeated throughout a course or over years, with different enduring understandings.  

  • How is probability approximated? 

  • How are probabilities of compound events determined? 

  • What is the difference between predicted outcomes and observed outcomes? 

Enduring Understanding:

The major ideas you want students to internalize and understand deeply. These understandings should be thematic in nature. They are not the end all, be all of the question. They are focused to align to the focus (unit overview). 

Students understand: 

  • The probability of a chance event is approximated by collecting data on the chance process that produces it, observing its long-run relative frequency, and predicting the approximate relative frequency given the probability. 

  • Various tools are used to find probabilities of compound events. 

  • Statistics and probabilities are used in everyday life including sports, forecasting weather, predicting diseases, political campaigns, etc. 

What Students Will Know:

  • Probabilities are between 0 and 1, including 0 and 1 

  • The formulas for relative frequency, experimental probability, and theoretical probability  

  • Outcomes of two or more events can be found using fundamental counting principle 

  • Probability of a compound even can be found using tree diagrams and fundamental counting principle 

  • Probability of independent events can be found by multiplying the probabilities of individual events together 

  • probability of two dependent events A and B is the probability of A times the probability of B after A occurs 

What Students Will Do:

  • Represent the probability of a chance event as a number between 0 and 1 

  • Use the terms “likely”, “unlikely,” to describe the probability represented by the fractions used 

  • Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency 

  • Predict the approximate relative frequency of a chance event given the probability 

  • Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events 

  • Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process 

  • Compare probabilities from a model to observed frequencies 

  • If the agreement between a model and observed frequencies is not good, explain possible sources of the discrepancy 

  • Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation 

  • Represent the probability of a compound event as the fraction of outcomes in the sample space for which the compound event occurs 

  • 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 

  • Design and use a simulation to generate frequencies for compound events

Unit Specific Vocabulary and Terminology:

The purpose of vocabulary work should be to allow all students to access mathematics. Vocabulary is a way to provide opportunities for students to use mathematical language to communicate about how they solved a problem, describe their reasoning, and demonstrate understanding of mathematical content. Vocabulary is inclusive of key words and phrases.

Often multilingual learners/English learners (MLs/ELs) are perceived as lacking academic language and needing remediation. Research shows that MLs/ELs bring standards-aligned background knowledge and experiences to the task of learning, and they need opportunities to extend their language for academic purposes. When considering the language demand of a lesson (at the word level), you can check for cognates and polysemous words. Pointing out these words to students can help them activate and build background knowledge assumed in lessons. TESOL professionals can assist with the identification of cognates and polysemous words, and they can provide guidance about the background knowledge MLs/ELs bring or may need.

The words with an * are words that appear in the Smarter Balanced Construct Relevant Vocabulary for Mathematics intended to ensure that teachers remember to embed these terms in their instruction.

Academic Vocabulary 

  • Chance   

  • Data   

  • Events  

  • Fair 

  • Frequencies 

  • Likelihood* 

  • Outcome

  • Percent 

  • Probability 

  • Probability model*  

  • Simulation 

  • Unfair 

Content Vocabulary 

  • Combination 

  • Complimentary Event 

  • Compound Event

  • Counting Principle 

  • Dependent Event 

  • Empirical probability 

  • Equally likely 

  • Experimental probability 

  • Independent Event* 

  • Organized list  

  • Permutation 

  • Probability of a chance event   

  • Random sample 

  • Relative Frequency

  • Sample space 

  • Simple event

  • Tables  

  • Theoretical probability 

  • Tree diagram* 

  • Uniform probability model

Vocabulary resources: 

English Math Vocabulary Cards

Spanish Math Vocabulary Cards

Chinese Math Vocabulary Cards

French Math Vocabulary Cards

Aligned Unit Materials, Resources and Technology:

High-quality instructional resources are critical for improving student outcomes. The alignment guidance is intended to clarify content and support understanding for clear implementation and coherence. 

Materials selection is a local control decision and these documents have been provided from participating publishers to assist districts in implementation. Use of the materials from these publishers is not required. These aligned core programs meet expectations as reported by edReports. Strong alignment of curricula and instructional materials have the potential to support student engagement and teacher growth. 

Opportunities for Interdisciplinary Connections:

Science: 

  • MS-LS1-4 Use argument based on empirical evidence and scientific reasoning to support an explanation for how characteristic animal behaviors and specialized plant structures affect the probability of successful reproduction of animals and plants respectively. 

  • MS-LS1-5 Construct a scientific explanation based on evidence for how environmental and genetic factors influence the growth of organisms. 

  • MS-LS1-8 Gather and synthesize information that sensory receptors respond to stimuli by sending messages to the brain for immediate behavior or storage as memories. 

  • MS-LS2-1 Analyze and interpret data to provide evidence for the effects of resource availability on organisms and populations of organisms in an ecosystem. 

  • MS-LS2-2 Construct an explanation that predicts patterns of interactions among organisms across multiple ecosystems. 

  • MS-LS2-4 Construct an argument supported by empirical evidence that changes to physical or biological components of an ecosystem affect populations. 

  • MS-LS2-5 Evaluate competing design solutions for maintaining biodiversity and ecosystem services. 

  • MS-LS4-1 Analyze and interpret data for patterns in the fossil record that document the existence, diversity, extinction, and change of life forms throughout the history of life on Earth under the assumption that natural laws operate today as in the past. 

  • MS-LS4-2 Apply scientific ideas to construct an explanation for the anatomical similarities and differences among modern organisms and between modern and fossil organisms to infer evolutionary relationships. 

  • MS-LS4-4 Construct an explanation based on evidence that describes how genetic variations of traits in a population affects individuals’ probability of surviving and reproducing in a specific environment. 

  • MS-LS4-6 Use mathematical representations to support explanations of how natural selection may lead to increases and decreases of specific traits in populations over time. 

  • MS-ESS2-3 Analyze and interpret data on the distribution of fossils and rocks, continental shapes, and seafloor structures to provide evidence of the past plate motions. 

  • MS-ESS2-5 Collect data to provide evidence for how the motions and complex interactions of air masses results in changes in weather conditions. 

  • MS-ESS3-2 Analyze and interpret data on natural hazards to forecast future catastrophic events and inform the development of technologies to mitigate their effects. 

  • MS-ESS3-3 Apply scientific principles to design a method for monitoring, evaluating, and managing a human impact on the environment. 

ELA 

  • CCSS.ELA-LITERACY.RI.7.1 Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text. 

  • CCSS.ELA-LITERACY.W.7.7 Conduct short research projects to answer a question, drawing on several sources and generating additional related, focused questions for further research and investigation. 

  • CCSS.ELA-LITERACY.W.7.8 Gather relevant information from multiple print and digital sources, using search terms effectively; assess the credibility and accuracy of each source; and quote or paraphrase the data and conclusions of others while avoiding plagiarism and following a standard format for citation. 

ISTE 

  • 1c Empowered Learner 

  • 5b Computational Thinker 

Computer Science 

  • 2-DA-07 Represent data using multiple encoding schemes.  

  • 2-DA-08 Collect data using computational tools and transform the data to make it more useful and reliable. 

  • 2-DA-09 Refine computational models based on the data they have generated. 

Opportunities for Application of Learning:

Defined Learning provides an open access online library of standards-aligned project-based lessons to help students meet the expectations of the Standards. Each project is based on a situation in a relevant career to help students connect classroom content to career pathways. This supplemental resource is available at no cost to teachers and districts. Create an account and log in to access this free resource to support your curriculum.

The tasks below provide additional opportunities to apply the content of this unit.

Critical Consciousness for Diversity and Equity:

Culturally relevant mathematics engages and empowers students. Opportunities for teachers to orchestrate discussions where students share not only connections to prior mathematics learned but also to their lived experiences must be provided. It is important to dig deep to find ways to link students’ home cultures and the mathematics classroom. Build authentic relationships with families through two-way, reciprocal conversations that acknowledge families’ cultures as assets for teaching and learning. As you plan to implement this unit, focus on designing experiences that have students at the center. In addition to keeping students engaged, ensure the learning experiences have a context that reflects lived experiences (mirror) or provide opportunities to view and learn about the broader world (window). 

One crucial link to students’ home cultures is through their language. Students’ language repertoires –all the languages and language varieties they use everyday– are a valuable resource to be engaged in the mathematics classroom. This approach is referred to as a translanguaging stance. It is based on a dynamic view of bilingualism that understands individuals as having one linguistic repertoire composed of various named languages (such as English and Spanish) and/or language varieties on which they draw to make meaning.

The following questions are intended to assist in promoting diverse voices and perspectives while avoiding bias and stereotyping:

  • How will students share their experiences with others while attending to the mathematics in the unit?
  • What opportunities are there for students to make connections from their life to the mathematics?
  • What do I know or need to learn about my students to create lessons free from bias and stereotypes?
  • In what ways can the mathematical thinking already taking place in the classroom and community be honored?
  • How is relevant background knowledge developed so that all students can access the mathematics of the unit?
  • What opportunities are there for students to use their full language repertoires during mathematical discussions and practice? Where can I create these opportunities?
  • What do I know or need to learn about students’ languages and how they use them? How can I learn this?

* Resources to support diversity and equity in the classroom please visit the DEI Collection

Supporting Multilingual Learners/English Learners (ML/EL):

Mathematical symbols, expressions, and methods are not universal; ways of doing math differ across cultures. When working with diverse students, especially with those from different countries, it is important to be aware that these differences exist.

It is also important to remember that communicating about mathematical content and practices requires complex language. Since conceptual learning and language learning are interconnected and acquired through participation in meaningful activities, the research-based strategies listed below focus on making content comprehensible (accessible) and creating opportunities for student voice, both verbal and written.

Additional resources for ML/EL

CELP Standards--Linguistic Supports

ML/EL Support Collection for Math  

  • Explicit instruction with regard to understanding the contexts for probability models.  

  • Explicit vocabulary instruction with regard to probability language. 

This unit presents opportunity to address the following CELP Standards: 

CELP Standard 1: Construct meaning from oral presentations and literary and informational text through grade-appropriate listening, reading, and viewing 

CELP Standard 2: Participate in grade-appropriate oral and written exchanges of information, ideas, and analyses, responding to peer, audience, or reader comments and questions 

CELP Standard 3: Speak and write about grade-appropriate complex literary and informational texts and topics 

CELP Standard 5: Conduct research and evaluate and communicate findings to answer questions or solve problems 

CELP Standard 6: Analyze and critique the arguments of others orally and in writing 

CELP Standard 7: Adapt language choices to purpose, task, and audience when speaking and writing 

CELP Standard 8: Determine the meaning of words and phrases in oral presentations and literary and informational text 

CELP Standard 9: Create clear and coherent grade-appropriate speech and text 

The document below provides guidance on these standards that is grade appropriate and broken down by language level descriptors which will assist the teacher in making the content of the unit accessible to all students.

Grade 7 Language Level Descriptors