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Instructional Unit Plan School: CMS, GMS, and STMS Teacher: Grade Level: GISD 7th Grade Mathematics Teachers 7th Grade Content Area(s) Implementation Dates: Mathematics: Comparing and Scaling Unit Focus: Ratio, Proportion and Percent. Investigation 1 To explore several ways to make comparisons To begin to understand how to determine when comparisons can be made using multiplication or division versus addition or subtraction To begin to develop ways to use ratios, fractions, rates, and unit rates to answer questions involving proportional reasoning Investigation 2 To further develop the ability to make sensible comparisons of data using ratios, fractions, and decimal rates, with a focus on percents To develop the ability to make judgments about rounding data to estimate ratio comparisons To observe what is common about situations that call for a certain type of ratio comparison Investigation 3 To recognize situations in which ratios are a useful form of comparison To form, label, and interpret ratios from numbers given or implied in a situation To explore several informal strategies for solving scaling problems involving ratios (which is equivalent to solving proportions) Investigation 4 To find unit rates To represent data in tables and graphs To look for patterns in tables in order to make predictions beyond the tables To connect unit rates with the rule describing a situation To begin to recognize that constant growth in a table will give a straight line graph To find the missing value in a proportion Investigation 5 To use geometric scaling to estimate population counts To apply proportional reasoning to situations in which capture-tag-recapture methods are appropriate for estimating population counts To use ratios and scaling up or down (finding equivalent ratios) to find the missing value in a proportion To use rates to describe population and traffic density (space per person or car) Investigation 6 To select and apply appropriate strategies to make comparisons To review when ratio and difference strategies are useful in solving problems To use proportional reasoning to fairly apportion available space so that the group is representative of the larger community 1 New Mexico Content Standards, Benchmarks and Performances Addressed: Mathematics Seventh Grade Strand 1: Number And Operations Standard: Students will understand numerical concepts and mathematical operations. 5-8 Benchmark 1: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Illustrate the relationships among natural (i.e., counting) numbers, whole numbers, integers, rational and irrational numbers. Use properties of the real-number system to explain reasoning and to formulate and solve real-world problems. Simplify numerical expressions using order of operations. 5-8 Benchmark 2: Understands the meaning of operations and how they relate to one another. Add, subtract, multiply, and divide rational numbers (e.g., integers, fractions, terminating decimals) and take positive rational numbers to whole-number powers. Convert terminating decimals into reduced fractions. Calculate given percentages of quantities and use them to solve problems (e.g., discounts of sales, interest earned, tips, markups, commission, profit, simple interest). Add and subtract fractions with unlike denominators. 5-8 Benchmark 3: Compute fluently and make reasonable estimates. Use estimation to check reasonableness of results, and use this information to make predictions in situations involving rational numbers, pi, and simple algebraic equations. Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. Calculate the percentage of increases and decreases of a quantity. Add and subtract fractions with unlike denominators. Strand 2: Algebra Standard: Students will understand algebraic concepts and applications. 5-8 Benchmark 1: Understand patterns, relations, and functions Identify and continue patterns presented in a variety of formats. Represent a variety of relationships using tables, graphs, verbal rules, and possible symbolic notation, and recognize the same general pattern presented in different representations. Simplify numerical expressions by applying properties of rational numbers, and justify the process used . Solve problems involving rate, average speed, distance, and time. Strand 4: Measurement Standard: Students will understand measurement systems and applications. 5-8 Benchmark 1: Understand measurable attributes of objects and the units, systems, and process of measurement. Choose appropriate units of measure and ratios to recognize new equivalences (e.g., 1 square yard equals 9 square feet) to solve problems. Select and use the appropriate size and type of unit for a given measurement situation. Use measures expressed as rates and measures expressed as products to solve problems, check the units of the solutions, and analyze the reasonableness of the answer. 5-8 Benchmark 2: Apply appropriate techniques, tools, and formulas to determine measurements. Solve problems involving scale factors, ratios, and proportions. Strand 5: Data Analysis And Probability Standard: Students will understand how to formulate questions, analyze data, and determine probabilities. 5-8 Benchmark 1: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Describe how data representations influences interpretation. Select and use appropriate representation for presenting collected data and justify the selection. Identify and explain the misleading representations of data. Collect, organize, and represent data sets that have one or more variables and identify relationships among variables within a data set. 2 Compute the minimum, lower quartile, median, upper quartile, and maximum of a data set. Use and explain sampling techniques (e.g., observations, surveys, and random sampling) for gathering data. Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, and selecting, collecting, and displaying appropriate data to address the problem. 5-8 Benchmark 2: Select and use appropriate statistical methods to analyze data. Use the analysis of data to make convincing arguments. Use appropriate technology to gather and display data sets and identify the relationships that exist among variables within the data set. Use data samples of a population and describe the characteristics and limitations of the sample. Identify data that represent sampling errors and explain why the sample and the display might be biased. Identify claims based on statistical data and evaluate the validity of the claims. 5-8 Benchmark 4: Understand and apply basic concepts of probability. Describe the probability of events using fractions, decimals, and percents. Use probability to generate convincing arguments, draw conclusions, and make decisions in a variety of situations. Determine the probability of a simple event or a compound event composed of a simple, independent events. Paraphrase knowledge and skills required by the Standards: Use informal language to ask comparison questions, such as: o “What fraction of the class is going to the picnic?” o “What percent of the girls play basketball?” o “Which model of car has the best fuel economy?” Decide when the most informative comparison is the difference between two quantities and when it is ratios between pairs of quantities. Develop the ability to make judgments about rounding data to estimate ratio comparisons. Find equivalent ratios to make more accurate and insightful comparisons. Scale a ratio or fraction up or down so that a larger or smaller object or population has the same relative characteristics as the original. Represent data in tables and graphs. Apply proportional reasoning to situations in which capture-tag-recapture methods are appropriate for estimating population counts. Set up and solve proportions that arise in applications. Look for patterns in tables that will allow predictions. Connect unit rates with a rule describing the situation. Begin to recognize that constant growth in a table will give a straight-line graph. Use rates to describe population and traffic density. Vocabulary: Comparison Concentrate Context Decimal Denominator Difference Equation Equivalent, Equivalent Ratio Fraction Numerator Part to Part Ratio Part to Whole Ratio Percent Population density Proportion 3 Rate Ratio Scale, Scaling Unit rate Description of the Assessments that will be used to provide evidence of student learning that targets the standards: Within the unit, ACE questions, Checkups, and Quizzes may be used as ongoing student assessment. Investigation I ACE questions: #1-5 and #14 Mathematical Reflections on page 15 Investigation 2 Check up I, pages 86-87 Mathematical Reflections on page 15 Investigation 3 Mathematical Reflections on page 36 Investigation 4 Check up 2, page 88 Quiz, page 89-90 Mathematical Reflections on page 51 Investigation 5 Mathematical Reflections on page 64 Investigation 6 Mathematical Reflections on page 81 Unit Test, pages 96-97 Paper Pool Project, pages 98-99 Description of the rubrics or criteria that will be used to assess student performance: Students will be assessed according to their Understanding Procedures, Strategies, Reasoning Communications Unit-Test Rubric—Teacher formatted Mathematical Reflection: (19 pts total) o A point value will be assigned to each of the 5 questions: #1: 3pts total; 1 for each correct rate. #2-5: 4 pts. each; 2 for appropriate answer and 2 for appropriate explanation. 4 Check-Up 2: Key for the test is provided. Learning Activities – Description of the learning activities that will develop the knowledge and skills required by the performance standard: Investigation 1: Making Comparisons Objective: SWBAT to explore several ways to make comparisons using ratios, fractions and percents. Activities: 1.1 Writing Ads Problem 1.1 and follow –up page 6. 1.2 Targeting an Audience Problem 1.2 page 7 Problem 1.2 follow-up page 8. Can be done as a whole group activity. 1.3 Getting the Message Across Problem 1.3 and follow-up page 9 Homework/Practice Problems: ACE questions: #1-5 and #14. Quiz: #6-11 and the Mathematical Reflections on page 15. Investigation 1 Resources/Materials: Pencils, paper Textbook Overhead projector Transparencies 1.1 – 1.3 Large sheets of paper Calculator (optional) 5 Key Links Practice for Investigation 1, Comparing and Scaling 1.Which one of the following is not equivalent? 7 a) 8 b) 0.875 c) 88% 875 d) 1000 1a. Write or illustrate three different ways of expressing the same number. (Fraction, decimal and percent) 6 Key Links Practice for Investigation 1, Comparing and Scaling 2a. If you combine the total number of students in Ms. Ochoa’s and Mr. Moreno’s classrooms there are 45 students. One-third of the students are boys. How many are girls? a) 30 girls b) 15 girls c) 8 girls d) 5 girls 2b. The ratio of boys to girls in Ms. Ochoa’s class is 7 to 12. What is the ratio of boys to girls in Mr. Moreno’s class? Explain your reasoning. 2c. Write three possible combinations that will produce a group of students of boys to girls, with a ratio of 1 to 5. 7 Investigation 2: Comparing and Finding Percents Objectives: To further develop the ability to make sensible comparisons of data using ratios, fractions, and decimals rates with a focus on percents. To develop the ability to make judgments about rounding data to estimate ratio comparisons. To observe what is common about situations that call for a certain type of ratio comparison. Activities: 2.1 Comparing Leisure Activities 2.1 Follow-up A.C.E. questions 1-8, 17-22 2.2: Comparing Your Class to the Nation 2.2 Follow-up A.C.E. questions 9-16, 23-26 Homework/Practice Problems: Mathematical Reflections, page 25 Assessment Check up 1, pages 86-87 Investigation 2 Resources/Materials: Graphing calculators Centimeter grid paper. 8 Key Links Practice for Investigation 2, Comparing and Scaling 1. A homeroom class of 32 eighth graders at Santa Teresa Middle School completed a survey about their participation in team sports. Each student was asked to list any sport he or she liked to play. The results for four of the most popular sports are given in the following table. Sport Female Male Basketball 14 13 Track and Field 7 13 Softball 10 8 Football 4 11 Total Surveyed 17 15 1a. Which of the following best represents the fraction of the class that is female? 15 17 32 a. 32 b. 32 c. 17 d. none of these. 1b. In which sport does the greatest percent of the class participate? a. Basketball b. Track and Field c. Softball d. Football 1c. Explain the strategy you used to determine your answer to question 1b, give details. 9 Key Links Practice for Investigation 2, Comparing and Scaling 2. 12 ft 9 ft 16 ft 12 ft Above are the floor plans for two college dorm rooms. One is for two students, and the other is for one student. 1. Are the floor plans similar rectangles? Explain why or why not. 2. Select the ratio that best represents a comparison of the two dorm rooms: 2 9 3 3 a. 3 b. 16 c. 4 d. 2 10 Investigation 3: Comparing by Using Ratios Objectives To recognize situations in which ratios are a useful form of comparison To form, label, and interpret ratios from numbers given or implied in a situation To explore several informal strategies for solving scaling problems involving ratios (which is equivalent to solving proportions) Pacing Table: 2-3 Days Activities 3.1 Mixing Juice pg. 28 A-C Follow-Up, pg. 28 1-2 ACE Questions: pg. 31-35: 1,2,12 3.2 Helping the Cook pg. 29 A-B Follow-Up pg. 29 1-2 ACE Questions: pg. 31-35: 3,6,13-23 3.3 Sharing Pizza pg. 30 A-B Follow-Up pg. 30 1-2 ACE Questions: pg. 31-35: 4, 8, 25 Reflections: Problems: pg. 36 3-4 Big Math Idea (Summary) Investigation 3 Resources/Materials: Centimeter grid paper Transparencies 3.1-3.3 Orange and White chips or Orange and white square of paper (about 25 orange and 100 white per group) Orange juice concentrate Pitcher 11 Key Links Practice for Investigation 3, Comparing and Scaling 1. You are a guest at a Pizza Party. There are 3 tables, set up for guests. You many sit anywhere you choose. The pizzas at each table must be shared equally. Since you are especially hungry and pizza is your favorite food, which of the following tables would you sit at in order to get the most pizza possible? a) Table1: 5 seats and 2 pizzas b) Table 2: 7seats and 3 pizzas c) Table 3: 12 seats and 5 pizzas d) All tables will allow for the same amount of pizza Show proof of how your choice allows you to receive more pizza than the other options. You may use pictures, numbers and words to demonstrate your understanding. Extension: Another table was added at the pizza party, with 10 seats and 4 pizzas. Will this option allow you to have more pizza than any of the other three tables? Explain why or why not. 12 Key Links Practice for Investigation 3, Comparing and Scaling 2. Alex and Dora are making punch for a class party. The directions on the liquid punch mix say to use 3 cups of mix for every 7 cups of water. They must make a total of 50 cups of punch for the party. How many cups of mix and how many cups of water will they need in order to serve 50 cups of punch? a) 20 cups of mix and 30 cups of water b) 25 cups of mix and 25 cups of water c) 15 cups of mix and 35 cups of water d) 12 cups of mix and 38 cups of water Show how you arrived at your answer. If 20 unexpected guests arrived at the party and each drank 2 cups of punch, how much mix and water would be needed? 13 Investigation 4: Comparing by Finding Rates Objective: Activities: 4.1 Comparing Fuel Economy, page 38-39 4.1 Follow-Up ACE questions 4 & 5 (page 46) 4.2 Using Unit Rates, page 40-41 4.2 Follow-Up [optional for practice in graphing] ACE questions 1, 6, 10, & 11 4.3 Solving Problems with Rates, page 42 4.3 Follow-Up as a journal entry ACE question 8, 9, 17 & 18 ACE question #2 if you need to retouch skills required for #1. 4.4 Buying Beads, page 43 4.4 Follow-Up ACE questions 7, 12, 13, 14, 15 & 16 Investigation 4 Resources/Materials: Comparing and Scaling textbooks Centimeter graph paper Centimeter graph transparency (for students) Markers Transparencies 4.1-4.4 (for teacher) Vocabulary: rates scaling up and down (higher and lower terms) unit rates 14 Key Links Practice for Investigation 4, Comparing and Scaling Tony can type at a constant rate of 55 words per minute. 1a. Write an equation for the number of words, W, Tony can type in T minutes. 1b. How many words can Tony type in 20 minutes? a) 75 words b) 1100 words c) 75 words per minute d) 1100 words per minute 1c. If Tony has a half hour to type a 1600 word essay, will he have time to type the entire essay? Explain your reasoning. 15 Key Links Practice for Investigation 4, Comparing and Scaling A veterinarian clinic has a patient load of 150 cats and dogs. The ratio of cats to dogs is 4 to 8. 2a. How many patients are cats? a) 100 b) 80 c) 50 d) 40 2b. How many patients are dogs? a) 100 b) 80 c) 50 d) 40 2c. If the ratio of cats to dogs would have been 1 to 2, would answers for questions a & b change? Explain why or why not. 16 Investigation 4: Comparing by Finding Rates Objective: Activities: 5.1 Estimating Populations and Population Densities Ace questions: 3 and 10 5.2 Estimating a Deer Population using beans Ace questions: 1,4,11,13 5.3 Finding Population Densities Ace questions: 6,12 5.4 Comparing the Dakotas (if time allows) Ace questions: 12 5.5 Predicating Traffic Jams (if time allows) Ace questions: 12 Mathematical Reflections pg. 64 Investigation 5 Resources/Materials: Transparencies centimeter & inch grids containers white beans (700-800) markers Comparing and Scaling Unit Materials: Graphing calculators Containers (large enough so students can mix contents) of 300–800 white beans, with lid Scoops for sampling (optional) Marker Large sheets of paper Centimeter and inch grid paper (provided as BLM) Advertisements containing comparisons (optional) Can of orange juice concentrate and pitcher (optional) News article that reports an estimate of crowd size (optional) 17 Key Links Practice for Investigation 5, Comparing and Scaling 1. Michael plays a lot of basketball, and he keeps a record of his free throw attempts in practices and games. Michael has made 48 out of 100 free- throw attempts in one week. What fraction below is closest to his free-throw attempts? a. ¾ b. ½ c. 9/10 d. 5/8 How did you arrive at your answer? Explain. How would his success rate change if he made the next 10 free throw attempts? 18 Key Link Practice For Investigation 5, Comparing And Scaling 2. At Gadsden Middle School, Mr. Hamel’s students conducted a class survey about their favorite football teams. Of the 27 students in the class, 18 picked the Dallas Cowboys, 6 picked the New England Patriots, and 3 picked the Arizona Cardinals. If you randomly selected a person in the hallway and asked them to choose between the Cowboys, Patriots, or Cardinals what are the chances that the person will pick the Patriots as the favorite team? a. 2/3 b. 1/9 c. 2/9 d. ½ If there were1000 students in Gadsden Middle School what percent of the students would choose the Arizona Cardinals as their favorite team? Explain your reasoning behind your answer. 19