• # Middle School Mathematics Grade 7 Unit 7

Subject: Mathematics
Timeline: 22 days
Unit 7 Title: What Do You Expect?

Unit Overview:

This unit is a combination of parts of two books: How Likely Is It? Investigation 4 and What Do You Expect? Investigations 1-3.  These books are the two probability units found in the Connected Mathematics curriculum.  Students will gain an understanding of experimental and theoretical probabilities and the relationship between them.  Students will also learn about equally likely events and fair/unfair games.  These units explore different types of probability questions using tables to find theoretical probabilities of outcomes in contexts that are interesting to students, such as games and genetics.  Area models are used to analyze theoretical probabilities of two-stage events.  Students will also determine long-range average or expected value in a variety of different probability settings (How Likely Is It?, pg. 3 and What Do You Expect?, pg. 3).

Unit Objectives:

At the end of this unit, all students must:
• Understand that there are two ways to build probability models: by gathering data from experiments (experimental probability) and by analyzing the possible equally likely outcomes (theoretical probability)
• Interpret experimental and theoretical probabilities and the relationship between them
• Distinguish between equally likely and non-equally likely outcomes
• Review strategies for identifying possible outcomes and analyzing probabilities
• Determine if a game is fair or unfair
• Analyze situations that involve two stages (or two actions)
• Use area models to analyze situations that involve two stages
• Determine the expected value and predict what will happen over the long run in a probability situation
• Use probability and expected value to make decisions

Focus Standards:

PA.CCSS.Math.Content.CC.2.4.7.B.3 Investigate chance processes and develop, use and evaluate probability models. (7.SP.5, 7.SP.6, 7.SP.7.a.-b., 7.SP.8.a.-c.)

Mathematical Practice Standards:

#1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

#3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

#4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

#7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.  Structure can be found in tables, lists, area models and other diagrams.

Concepts - Students will know:
• Experimental probabilities for certain genetic traits
• The power of probability for making predictions
• Strategies for finding theoretical probabilities involving genetics
• The power of probability in determining strategies for winning a game
• Strategies for finding both experimental and theoretical probabilities of winning a game
• Basic probability concepts, such as fair game and fraction notation for expressing probabilities
• Payoff in consideration of the fairness of a game
• The use of probability and payoff to calculate the long-term average result of a game of chance
• Area models to analyze the theoretical probabilities for two-stage outcomes
• Equally likely and non-equally likely outcomes by collecting data and analyzing experimental probabilities
• Expected value in a probability situation
Competencies -Students will be able to:
• Students will be able to:
• Identify genetic traits of classmates
• Find the probability of a person chosen at random for genetic traits
• Play the Roller Derby game
• Find all of the possible outcomes of rolling two number cubes
• Identify equally likely events
• Play the Match/No Match game
• Find the experimental probabilities of outcomes
• List all possible outcomes of a game
• Use outcomes to determine theoretical probabilities
• Determine if outcomes are equally likely
• Compare experimental and theoretical probabilities
• Decide if games are fair or unfair
• Make predictions about outcomes of a game
• Make tree-diagrams of outcomes
• Play the Multiplication game
• Play the Making Purple game
• Use and area model to determine theoretical probability
• Use spinners to simulate outcomes
• Shade area models to determine expected value or long-term average
• Complete a table showing outcome percentages and expected values or long-term averages
• Graph (percentages, averages)

Assessments:

Formative Assessments:
• Informal assessments on learning targets
• Check-up Quiz 1
• Partner Quiz
• Check-up Quiz 2
Summative Assessment:
• Common Core Unit Assessment- What Do You Expect?

Elements of Instruction:

In Grade 6, instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

What Do You Expect? expands on the Grade 6 foundations of multiplication of fractions (percents).  This unit also builds upon how to organize data collected from experiments and games.

Questions about how likely an event is are asked every day.  Such questions ask about the probability of an event happening, and the answers are important to many people.  Spinners, choosing marbles from two buckets and rolling two number cubes provide the settings.  These situations also introduce two-stage events.  A two-stage event such as a one-and-one free-throw situation is used to determine experimental probability and theoretical probability is derived from area models of outcomes.  Students then determine long-term averages.

Major misconceptions by and struggles for students in this unit include:
1. expressing probability as a part-to-whole fraction
2. generalizing data collected from small sample to a large population
3. making Punnett squares to solve probability problems
4. listing all possible outcomes of a situation and not missing any
5. creating theoretical probabilities from experimental probabilities
6. contextualizing theoretical probabilities into situations
7. determining greatest chances of outcomes
8. creating and double shading area models
9. graphing (percentages, averages)
10. comparing probabilities from two different situations with different numbers of outcomes

Differentiation:

Each lesson has differentiation options for each portion of the lesson. Additional differentiation options are listed with directions and student masters in the Teacher’s Guide.
• Special Needs Handbook
• Unit Projects
• Spanish Additional Practice and Skills guide
• Strategies for English Language Learners Guide
• “Extension” homework questions
• District-created notebooks

Interdisciplinary Connections:
• Mathematical Reflections
• “Did You Know?” sections
• phschool.com and web codes
• “Connections” homework questions
• The real-world context embedded in lesson problems

CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success.
• Additional Practice worksheets per investigation
• Skills Review worksheets to target key components of each investigation
• Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and expectations of the coming unit.
• Assessment Resources workbook with extra test items (multiple choice, essay, open ended, question bank, etc)
• Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all other items students may need to complete the investigation and all its parts.  Students are provided with one per unit.
• Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.
• phschool.com and web codes
• http://www.keypress.com/
• www.math.msu.edu/cmp
• http://ti.com/
• Roller Derby Game
• Match/No-Match Game
• Multiplication Game
• Making Purple Game
• One-and-one Free-throw spinner simulation
• Transparencies of models, graphs, etc used within lesson(s)