Unit Three Organizer: RATIONAL REASONING (5 weeks)

Transcription

1 The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org. OVERVIEW: Unit Three Organizer: RATIONAL REASONING (5 weeks) In this unit students will explore positive and negative numbers, extending many previously learned concepts to include the use of all rational numbers. Algorithms for computing with rational numbers should be investigated, discovered and formalized by students using models, diagrams, manipulatives, and patterns. Students will: investigate the use of positive and negative numbers in real contexts; plot positive and negative numbers in the coordinate plane; order positive and negative rational numbers and plot them on a number line; use absolute value to explore the relationship between a number and its additive inverse; develop algorithms for computing with positive and negative rational numbers; simplify and evaluate algebraic expressions involving positive and negative rational numbers; extend properties of real numbers to include all rational numbers; solve one- and two-step linear equations in one variable; and, solve problems by defining a variable, writing and solving an equation, and interpreting the solution of the equation in the context of the original problem. To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the task listed under Evidence of Learning be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources. ENDURING UNDERSTANDINGS: Negative numbers are used to represent quantities that are less than zero such as temperatures, scores in games or sports, Unit 3 Organizer RATIONAL REASONING September 20, 2006 Page 1 of 35

2 and loss of income in business. Absolute value is useful in ordering and graphing positive and negative numbers. Computation with positive and negative numbers is often necessary to determine relationships between quantities. Models, diagrams, manipulatives and patterns are useful in developing and remembering algorithms for computing with positive and negative numbers. Properties of real numbers hold for all rational numbers. Positive and negative numbers are often used to solve problems in everyday life. ESSENTIAL QUESTIONS: When are negative numbers used and why are they important? Why is it useful for me to know the absolute value of a number? What strategies are most useful in helping me develop algorithms for adding, subtracting, multiplying, and dividing positive and negative numbers? What properties and conventions do I need to understand in order to simplify and evaluate algebraic expressions? STANDARDS ADDRESSED IN THIS UNIT Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics. KEY STANDARDS: M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation. a. Find the absolute value of a number and understand it as a distance from zero on a number line. b. Compare and order rational numbers, including repeating decimals. c. Add, subtract, multiply, and divide positive and negative rational numbers. d. Solve problems using rational numbers. M7A1. Students will represent and evaluate quantities using algebraic expressions. a. Translate verbal phrases to algebraic expressions. September 20, 2006 Page 2 of 35

3 b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate. c. Add and subtract linear expressions. M7A2. Students will understand and apply linear equations in one variable. a. Given a problem, define a variable, write an equation, solve the equation, and interpret the solution. b. Use the addition and multiplication properties of equality to solve one- and two-step linear equations M7A3. Students will understand relationships between two variables. a. Plot points on a coordinate plane. b. Represent, describe, and analyze relations from tables, graphs, and formulas. c. Describe how change in one variable affects the other variable. M7P2. Students will reason and evaluate mathematical arguments a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. RELATED STANDARDS: M7P1. Students will solve problems (using appropriate technology) a. Build new mathematical knowledge through problem solving, b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M7P3. Students will communicate mathematically a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. M7P4. Students will make connections among mathematical ideas and to other disciplines a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. September 20, 2006 Page 3 of 35

4 M7P5. Students will represent mathematics in multiple ways a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. CONCEPTS/SKILLS TO MAINTAIN: It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas. Inverse operations Basic mathematical operations with decimals, fractions, and whole numbers Modeling multiplication with rectangular arrays Graphing data SELECTED TERMS AND SYMBOLS: The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. Absolute value: The distance between a number and zero on the number line. The symbol for absolute value is shown in this equation 8 = 8. Associative property: In addition or multiplication, the result of the expression will remain the same regardless of grouping. Examples: a + (b+c) = (a+b) + c; a(bc) = (ab)c Commutative property: The sum or product of numbers is the same no matter how the numbers are arranged. Examples: a + b = b + a; ab = ba September 20, 2006 Page 4 of 35

5 Distributive property: The sum of two addends multiplied by a number will be the sum of the product of each addend and the number. Example: a(b + c) = ab + ac Integers: The set of whole numbers and their opposites { -3, -2, -1, 0, 1, 2, 3, } Natural numbers: The set of numbers {1, 2, 3, 4, }. Natural numbers can also be called counting numbers. Negative Numbers: The set of numbers less than zero. Opposite Numbers: Two different numbers that have the same absolute value. Example: 4 and 4 are opposite numbers because both have an absolute value of 4. Positive Numbers: The set of numbers greater than zero. Rational Numbers: The set of numbers that can be written in the form a/b where a and b are integers and b 0. Sign: a symbol that indicates whether a number is positive or negative. Example: in 4, the ( ) sign shows this number is read negative four. Whole numbers: The set of all natural numbers and the number zero. You may visit and click on dictionary to see definitions and specific examples of terms and symbols used in this seventh grade GPS unit. EVIDENCE OF LEARNING: By the conclusion of this unit, students should be able to demonstrate the following competencies: Distinguish between natural numbers, whole numbers, integers and other rational numbers. Find the absolute value of a number. Compare two positive and/or negative rational numbers using <, >, or =. Order a set of positive and/or negative rational numbers from least to greatest or greatest to least and plot the numbers on a number line. Illustrate computation of positive and negative numbers using models, diagrams, manipulatives or patterns, including fact families. Compute with positive and/or negative rational numbers. Use the properties of real numbers (commutative, associative, distributive, inverse and identity) and the order of operations to simplify and evaluate simple numeric and algebraic expressions. Solve problems involving applications of positive and/or negative rational numbers. The following task represents the level of depth, rigor, and complexity expected of all 7 th grade students. This task or a task of similar depth and rigor should be used to demonstrate evidence of learning. September 20, 2006 Page 5 of 35

6 Culminating Activity: A Poster This activity requires that students apply the skills and knowledge they have learned in this unit. Students will demonstrate proficiency in finding absolute value and comparing rational numbers. They will create a variety of word problems with solutions as well as number sentences that involve the four basic mathematical operations with positive and negative rational numbers. By stating rules in their own words, giving useful tips, and identifying common misconceptions, students will use content vocabulary to communicate their thinking. STRATEGIES FOR TEACHING AND LEARNING: The tasks provided are representative of the types of tasks and problems that may be used in the classroom; however, they should not be construed as a complete curriculum. Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using Graphs, Tables, Pictures, Symbols and Words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection. September 20, 2006 Page 6 of 35

7 TASKS: The collection of the following tasks represents the level of depth, rigor and complexity expected of all seventh grade students to demonstrate evidence of learning. Connect the Dots Helicopters and Submarines Using the Number Line Sums and Products Always, Sometimes, Never Making a Test Working with Integers Designing a Poster TASK: The collection of the following tasks represents the level of depth, rigor and complexity expected of all sixth grade students to demonstrate evidence of learning. Connect the Dots September 20, 2006 Page 7 of 35

8 Connect the Dots Do you remember the activity books from elementary school? There were dots on a page numbered one to a larger number and you connected the dots to form a picture. Your task is to create a series of points on the coordinate plane that, when connected, will form a picture. You must then create a script that tells someone how to connect the dots. Your picture must have a minimum of eight points and there must be at least one point in each of the four quadrants. The example below is a simple version of what you could create. Script: Start at (-1,1) and draw a line segment to (-4, 1). Connect (-4, 1) to (-3,-1). Connect this dot to (3, -1) and continue with a line segment to (4,1). Connect this dot to (-1, 1). Continuing from (-1, 1), connect to (-1, 6). This will then be connected to (-4, 2). Connect this to (0,2). Finish the drawing by drawing a line segment from here to (-1,6). ********************************************************************************* Discussions, Suggestions, Possible solutions This task helps students apply their knowledge of points on a coordinate plane. It also can be used as a partner task. Each student could create a script and then have the partner create the image from this script. Teachers may find it beneficial to have students draw their pictures first and then identify the points to use in the script For students who need extra practice, there are many commercially prepared activity books that have series of September 20, 2006 Page 8 of 35

9 points listed for graphing that when completed, create a picture. Another activity would have teacher-created scripts for the vertices of basic geometric plane figures for students to plot and identify. A common game used in classrooms is a version of Hangman. Twenty-six points are located on the coordinate plane and are labeled with the letters of the alphabet. Students then ask their partners to identify a word clue by its coordinates. Teachers can also make up practice problems by posing silly riddles with the letters to the answers spelled out in point coordinates. This task can also be used in conjunction with most graphing calculators. Many calculatorscan be programmed to create the students pictures once the coordinate points have been listed into the program. Texas Instruments has published several activities using this skill. Going to the website may yield other activities suited to extension of this concept. By going to the site students can demonstrate their knowledge of coordinate graphing by navigating a robot through a minefield. Also allows students to identify points by their coordinates interactively. The site also has many other games that students can use to increase their skill level in identifying and placing points on the coordinate plane. Helicopters and Submarines September 20, 2006 Page 9 of 35

10 Helicopters and Submarines You are an engineer in charge of testing new equipment that can detect underwater submarines from the air. Part 1: The first three hours During this part of the test, you are in a helicopter 250 feet above the surface of the ocean. The helicopter moves horizontally to remain directly above a submarine. The submarine begins the test positioned at 275 feet below sea level. After one hour, the submarine is feet below sea level. After two hours, the submarine dives another 23 feet. After three hours, the submarine dives again, descending by an amount equal to the average of the first two dives. Make a table/chart with five columns (Time, Position of Submarine, Position of Helicopter, Distance between Helicopter and Submarine, and a Mathematical Sentence showing how to determine this distance) and four rows (start, one hour, two hours, three hours). Make a graphical display which shows the positions of the submarine and helicopter using the information in your table/graph. Part 2: The next three hours The equipment in the helicopter is able to detect the submarine within a total distance of 750 feet. At the end of the fourth hour, the helicopter remains at 250 feet. At the end of the fifth hour, the submarine returns to the same depth that it was at the end of the third hour. At the end of the sixth hour, the submarine descends to three times its second hour position. September 20, 2006 Page 10 of 35

11 For each scenario, determine the maximum or minimum location for the other vehicle in order for the helicopter to detect the submarine; and write a mathematical sentence to show your thinking. Determine the ordered pairs for these additional hours and include them on your graph. ********************************************************************************* Discussion, Suggestions, Possible Solutions Part 1: The first three hours The Mathematical Sentence column shows two different approaches that students might use. Should a student receive a negative answer in this column, they should recognize the distance between the vehicles as absolute value. For students who still have difficulty with absolute value, has a variety of explanations. Time Position of Position of Distance between Mathematical Submarine Helicopter Submarine and Sentence Helicopter Start = hour (-325.8) = hours = hours (-385.7) = For the graph, the following information could be useful. The helicopter would be plotted at +250 and the submarine at 275. Points above sea level should be September 20, 2006 Page 11 of 35

12 noted as a positive number while positions below sea level would be negative numbers. In line with the first hour, the submarine would be plotted at and the helicopter would remain at 250. The new position of the submarine is feet. Helicopter remains at 250. The position of the submarine will be feet. Helicopter remains at 250. Part 2: The next three hours The mathematical sentence could be = 500 feet. This would be a good place for teachers to point out that depth refers to distance rather than position. At the end of the third hour, the submarine was at feet. Therefore, a mathematical sentence could be (-385.7) = feet. This one is impossible because the submarine would be at a depth of 3(348.8) = feet which is out of the range of the equipment. A sample mathematical sentence is 750 3(348.8) = which shows that the helicopter would need to be under water. Using the Number Line Using the Number Line Graph these numbers on the number line and then answer questions , , 16, 0, 1.8, , -.2, How did you scale your number line? Explain why you chose this increment. September 20, 2006 Page 12 of 35

13 2. Which number has the larger absolute value 1.8 or 1 6 8? How do you know? 3. Look at the fractions and mixed numbers in this list. Which of these numbers, when written as decimals, are repeating decimals? Which form terminating decimals? Can you tell, without dividing, which fractions will repeat and which will terminate? How do you know? 4. Compare your number line with a partner. a. Did you both use the same increment? Is one choice better than the other? Why or why not? b. Explain how you placed your numbers. Are your numbers in the same order? If not, decide who is correct and why. ********************************************************************************* Discussions, Suggestions, Possible solutions To best complete this task, students should be able to: convert among different forms of rational numbers; compare sizes of numbers in the same form; and, use the idea that the absolute value of a number is the distance between the number and 0 to order rational numbers on a number line. 1 = , 1 2 = , 1 6 = 1 3 = -1.75, and 1 2 = From least to greatest the numbers 1are: 6,, -1.2, 1 -.2, 1 2, 0,,, and Students may scale their number lines using different increments. Tenths work well when plotting these numbers but scaling in tenths is not the only correct possibility. September 20, 2006 Page 13 of 35

14 The best way to compare the two numbers is to write them both in the same form 8 is in decimal form. 1.8 has the larger absolute value. Students should notice that 1.8 is farther away from 0 on the number line than is 1 6. They should also be able to compare the decimal forms 1.8 (1.80) and 1.75 realizing that is larger /7 and 3 form repeating decimals. 8, and 5 form terminating decimals. Students need to see that if the denominator of a fraction, in lowest terms, is a factor of (or divides evenly into) 10, 100, 1000, etc., (powers of 10) then the fraction will terminate. If not, it will repeat. Numbers whose only prime factors are 2 and/or 5 will divide evenly into powers of Give students time to compare their work. Encourage discussion about the different increments students used and why they chose those increments. Ask which they think work best and why. Sums and Products Sums and Products Each of the puzzles in this task has a space for two different rational numbers, their sum, and their product. You are given the position of each value and a sample puzzle. Your job is to find the missing numbers. For each space you fill, write a number sentence that results in the missing value. Show how you found your sentence using any of the methods you have learned-facts families, number lines, positive/negative counters, area models, equations, etc. September 20, 2006 Page 14 of 35

15 For each puzzle, values should be placed in the following positions. a product sum b -2 Suppose you are given that a is -2 and b is The problem you use to find the sum might read: = 1. Here is a set of positive and negative counters. Use it to illustrate the number sentence September 20, 2006 Page 15 of 35

16 The problem you use to find the product, might read: 3-2 = -6. Here is a number line. Use it to represent the number sentence September 20, 2006 Page 16 of 35

17 / /8 September 20, 2006 Page 17 of 35

18 7. -1/5 8. 1/4 10/9-2/ ********************************************************************************* Discussions, Suggestions, Possible solutions Teacher notes and key for task. You may want to let students work on this task alone and then compare their methods and their answers with a partner. Allowing pairs of students to post and share their illustrations and explanations at the end of this task is also useful. September 20, 2006 Page 18 of 35

19 1. In this problem students should add -3 and = -8 A number line or counters may be used. To find the product, multiply -3 and x -5 = 15. Students should use any method they have learned to illustrate why the product of two negative numbers is positive. 2. In Problem 2, a is 2 and the sum is -6. Again students may use any method to find b. 2 +? = -6. Using fact families, we know that -6-2 = b so b is -8. The product is In Problem 3, we are given b and the product. a -4 = 28. a is = In Problem 4, we are given the product and the sum. This problem requires some trial and error on the part of students. We need two numbers that will multiply to give -18 and add to give 3. Students should realize immediately that the numbers must have different signs. The larger of the numbers must be positive to add to give a positive value. They are 6 and a + 11 = 4. a = -7. The product is ½ + 3/8 = 7/8. ½ 3/8 = 3/16. A number line would be a good representation for the sum. An area model would be excellent for the product /5 +? = ¼. Fact families are again useful here. ¼ - -1/5 = 5/20 + 4/20 = 9/20. The product is -9/ /3? = 10/9. 10/9-2/3 = 10/9-3/2 = -5/3. The product is -5/3 + -2/3 = -7/3. 9. Sum = -.8. Product = b =.5. The sum is = 0. September 20, 2006 Page 19 of 35

20 Always, Sometimes, Never Always, Sometimes, Never In the table below, you are given ten statements. You are to choose whether each statement is always true (A), sometimes true (S), or never true (N). If the statement is always true, you should be able to give a rule or a property that justifies your claim. If the statement is sometimes true, you should be able to give an example showing that it can be true and a counterexample showing that it can be false. If the statement is never true, you should again be able to give a rule or a property that is contradicted by the statement. Statement A, S, or N Justification 1. 3a > 3 2. a + 2 = 2 + a 3. -2b < 0 4. a + -a = 0 5. a 6 > a 6. 5a + 5b = 5(a + b) 7. a - 2 = 2 - a 8. a + 10 > If two different numbers have the same absolute value, their sum is zero. 10. If the sum of two numbers is negative, their product is negative. September 20, 2006 Page 20 of 35

21 ********************************************************************************* Discussions, Suggestions, Possible solutions The mathematics addressed in this task includes: reasoning and proof example and counterexample properties of real numbers, specifically the commutative property of addition, the distributive property, and the property of additive inverses absolute value computation with positive and negative numbers Student responses to this task need not be formal. A counterexample may be explained as simply an example that shows that a given statement is false. This task works nicely when students work in pairs and discuss their thinking problem by problem. 1. Sometimes true. If a is negative or a fractional value less than 1, 3 times a will be less than 3. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group. 2. Always true. This is the commutative property for addition. 3. Sometimes true. -2b < 0 if b is a positive number and -2b > 0 if b is a negative number. If b = 0, the statement is also false because -2 x 0 = 0 which is not less than zero. Some students may come up with this example. If not, it is worth asking the students to investigate what happens when b = Always true. This is the property of additive inverses. 5. Never true. If we take 6 away from a no matter what number a might be, the result will be smaller. If a is -2, -2-6 = is smaller than -2. If a is 0, 0 6 = is less than -2, etc. 6. Always true. The distributive property. 7. Sometimes true but only when a = 2. Then we would get 0 = 0. For all other values of a, this statement is false. It is important to bring out here that subtraction is not commutative. September 20, 2006 Page 21 of 35

22 8. Sometimes true. If a is any positive number, a + 10 >10. If a is 0 or a negative number a + 10 < 10. Any example showing that the statement is sometimes true and a counterexample showing that the statement is sometimes false is a sufficient student response. However, you may want to discuss exactly when the statement is true and when it is false as a group. 9. Always true. The key word here is different. If two different numbers have the same absolute value, the numbers will always be opposites and therefore by the property of additive inverses their sum will be Sometimes true. If the numbers have opposite signs, the product will be negative = x 3 = -18. However, if both numbers are negative, the sum will be positive. If one of the numbers is 0, the sum may be negative but the product will be 0 which is neither positive nor negative = x -5 = 0. Making a Test Making a Test In this task, your job is to create a multiple choice test. You are given ten problems. For each problem, you are to write four possible answer choices. Your choices should include exactly one correct answer and at least one choice that contains common errors or misconceptions. On a separate sheet of paper, for each problem, show how you got the correct answer; and explain the choices you made based on common errors or misconceptions. So, here is the test! Evaluating and Simplifying Algebraic Expressions 1. Which of the following is the value of 3a 2b when a = 2 and b = -6? 2. Which of the following is the value of 3a 2 4 when a = -5? September 20, 2006 Page 22 of 35

23 3. Which of the following is the value of 2(4 b) 2 b when b = 5? 4. Which of the following is the value of ( 10a 6) when a = 2/5? 4 5. Which of the following is the value of (a + 7) -3b when a = -3 and b = 2? 6. Which of the following shows the expression -3m 2 + 5m 8 in simplest possible form with no parentheses and no like terms? 7. Which of the following shows the expression 4(3a 5) + 2(-3a + 7) in simplest possible form with no parentheses and no like terms? 8. Which of the following shows the expression 2/3(1/2x + 9) + 3 in simplest possible form with no parentheses and no like terms? 9. Which of the following shows the expression x/4 1/8 x/6 + 1/6 in simplest possible form with no parentheses and no like terms? 10. Which of the following shows the expression 2ab + a ab - 7 in simplest possible form with no parentheses and no like terms? ********************************************************************************* Discussions, Suggestions, Possible solutions Please note that most of these expressions are fairly simple. Students will continue to increase their skill in simplifying and evaluating expressions throughout their mathematical career but particularly in Grade 8 and Math 1. A class discussion of how students found the correct answers and why they chose specific misconceptions is particularly useful at the end of this task. 1. The correct answer is = = 18. Students may select any other choices they choose. One of the most common mistakes is to misunderstand that you are subtracting -12. Missing the sign, a student might get 6 12 or -6 rather than The correct answer is 3 (-5) 2-4 = = 75 4 = 71. One mistake student often make is to square the product of 3 and -5. This would give an answer of 221. September 20, 2006 Page 23 of 35

24 3. The correct answer is 2(4 5) 2-5 = 2(-1) 2 5 = = 2 5 = T he correct answer is (10 2/5 6)/4 = (20/5 6)/4 = (4 6) /4 = -1/2. A common mistake is to divide part of the numerator by the denominator and not the whole numerator. For example, if a student calculated (4-6)/4 and divided only the first term in the numerator (4) by 4, the result would be 4/4 6 = 1 6 = -5. Of course, there are also lots of mistakes that students might make in using the fractional value 2/5 for a. 5. The correct answer is (-3 + 7) = = 4 6 = -2. One common mistake at the step is to subtract 3 from 4 before multiplying 3 times 2. This would yield an answer of The correct answer is: -3m 2 + 5m 8 = 2m 10. The commutative and associative properties of addition are used to simplify this expression. The signs when collecting the constants in a problem of this nature are often missed. 7. The correct answer is 4(3a 5) + 2(-3a + 7) = 12a a + 14 = 6a 6. The distributive property and the commutative and associative properties of addition are used to simplify this expression. 8. The correct answer is 2/3(1/2x + 9) + 3 = x / = x/ Common mistakes include using the distributive property incorrectly and multiplying fractions incorrectly. 9. The correct answer is x/4 1/8 x/6 + 1/6 = 3x/12 are common. 3/24 2x/12 + 4/24 = x/12 + 1/24. Mistakes with fractions 10. The correct answer is 2ab + a ab 7 = ab + a 7. Common mistakes include combining the unlike term a with the terms 2ab and ab and not recognizing ab as having the same result as adding -1ab. September 20, 2006 Page 24 of 35

25 Working with Integers Working with Integers Adapted from Achieve Even and Odd Numbers 1. The algebraic expression 2n is often used to represent an even number. Why do you think this is true? Illustrate your explanation with pictures and/or models. If 2n represents an even number, could n represent any number? Why or why not. 2. Write an algebraic expression that could represent an odd number. Explain your thinking and illustrate with a picture or a model. 3. What kind of number do you get when you add two even numbers? Justify your answer two different ways, by using models and by using algebraic expressions. 4. What kind of number do you get when you add two odd numbers? Justify your answer two different ways, by using models and by using algebraic expressions. Consecutive Integers I The lengths of the sides of the triangle below are consecutive integers. September 20, 2006 Page 25 of 35

26 1. What are consecutive integers? Give examples. 2. How can you represent the lengths of the sides of the triangle in terms of one variable? Explain your thinking. 3. If the perimeter of the triangle is 81 inches, what are the lengths of the sides? Justify your answer by writing and solving an equation. Consecutive Integers II Suppose three consecutive integers have a sum of If x denotes the middle integer, how can you represent the other two in terms of x? 2. Write an equation in x that can be used to find the integers. 3. Show that the sum of any three consecutive integers is always a multiple of three. ********************************************************************************* Discussions, Suggestions, Possible solutions Even and Odd Numbers 1. The algebraic expression 2n is often used to represent an even number. Why do you think this is true? Illustrate your explanation with pictures and/or models. September 20, 2006 Page 26 of 35

27 If 2n represents an even number, could n represent any number? Why or why not. Even numbers are numbers that are divisible by 2. Students may see this algebraically (2 is a factor of 2n and so is evenly divisible by 2) or geometrically (2n means n groups of two). Pictures should show objects that can be paired with no single object left over. Ask students what kind of number n must be for 2n to be even. Ask if n can be negative. Can n be zero? Make sure they understand that n must be an integer and can be any integer (positive, negative or 0). 2. Write an algebraic expression that could represent an odd number. Explain your thinking and illustrate with a picture or a model. If 2n, where n is an integer, is always an even number, 2n + 1 would always represent an odd number. Models should show groups of 2 with one object remaining. 3. What kind of number do you get when you add two even numbers? Justify your answer two different ways, by using models and by using algebraic expressions. This problem asks students to prove that the sum of two even numbers is even. Informal proofs are very important and time should be taken to make sure that students can justify their thinking. September 20, 2006 Page 27 of 35

28 Models should show that if you combine two sets of objects in which there are no unpaired objects in either group, there will still be no unpaired objects when the two sets are combined. Therefore, the sum is even. Formally, letting 2m represent one even integer and 2n represent another, we add to get 2m + 2n. Using the distributive property, 2m + 2n = 2(m+ n). 2(n + m) is divisible by 2 and therefore an even number. 4. What kind of number do you get when you add two odd numbers? Justify your answer two different ways, by using models and by using algebraic expressions. Using models, students should show two different sets of objects in which objects are paired with one object left over. Combining the two sets, the one object left over in one set may be paired with the one object left over in the other set to form a pair. Therefore, when the sets are combined, every object will be paired with another. Formally, two different odd numbers may be represented as 2m + 1 and 2n + 1. Adding these numbers, we get: September 20, 2006 Page 28 of 35

29 2m n + 1 = 2m + 2n by the commutative property for addition 2m + 2n = 2m + 2m +2 by the associative property of addition 2m + 2n + 2 = 2(m + n + 1) by the distributive property 2(m + n + 1) is even because it is divisible by 2. Consecutive Integers I The lengths of the sides of the triangle below are consecutive integers. 1. What are consecutive integers? Give examples. Consecutive integers are integers that are next to each other on the number line with no integers between them. They may increase or decrease. For example, 2, 3, and 4 are consecutive integers as are -10, -11, -12, and How can you represent the lengths of the sides of the triangle in terms of one variable? Explain your thinking. If you let x be the length of the shortest side, the lengths of the other two sides would be represented by x + 1 and x + 2. However, this is not the only representation. If x stands for the length of the longest side, the other two lengths would be represented as x 1 and x 2. x could also be the length of the middle side. Then the lengths of the other two sides would be represented as x + 1 and x -1. It is always good to have students see different representations when writing and solving equations to show that they get the same answers as long as they represent the relationships between quantities correctly. September 20, 2006 Page 29 of 35

30 3. If the perimeter of the triangle is 81 inches, what are the lengths of the sides? Justify your answer by writing and solving an equation. Using the first representation above, the equation would read: x +( x + 1) +( x + 2) = 81 3x + 3 = 81 3x = 78 x = 26 x + 1 = 27 x + 2 = 28 Thus the lengths of the sides of the triangle are 26 inches, 27 inches and 28 inches. The lengths of the sides are consecutive integers and add to give a perimeter of 81 inches. Using the last representation, the equation would read: (x 1) + x +( x + 1) = 81 3x = 81 x = 27 x 1 = 26 x + 1 = 28 Thus the side lengths are the same. Consecutive Integers II Suppose three consecutive integers have a sum of If x denotes the middle integer, how can you represent the other two in terms of x? September 20, 2006 Page 30 of 35

31 . Make sure students understand that, even though the integers that must be represented here are negative, if the middle number is represented by x, the smallest integer will still be represented by x 1 and the largest integer will be represented by x Write an equation in x that can be used to find the integers. (x 1) + x + (x + 1) = x = -195 x = -65 x 1 = -66 x + 1 = Show that the sum of of three. any three consecutive integers is always a multiple Suppose we let our consecutive integers be represented by x, (x + 1), and (x + 2). Adding we get: x + (x + 1) + (x + 2) = 3x + 3 by the commutative and associative properties for addition 3x + 3 = 3(x + 1) by the distributive property 3 is a factor of 3(x + 1). Therefore 3(x + 1) is a multiple of 3. A Poster This culminating task represents the level of depth, rigor and complexity expected of all 7 th grade students to demonstrate evidence of learning. September 20, 2006 Page 31 of 35

32 Unit Three Task: A POSTER You are to make and present a poster showing what you hav e learned from your study of positive and negative rational numbers. Choose a theme for your poster. Be creative! Choose four rational numbers. At least two of your numbers should be between 1 and 1, one of which should be written as a decimal and the other should be written as a fraction. Two of the numbers should be positive and two of the numbers should be negative. Make your poster using the rubric below: Comparing Use the >, <, and = to compare your negative numbers. Graph all four numbers on a number line. Absolute value Write the absolute value of each number and explain what is meant by absolute value. Number problems Create two addition problems; one using numbers with like signs and the other using numbers with different signs. Create two subtraction problems; one using numbers with like signs and the other using numbers with different signs. Create two multiplication problems; one using numbers with like signs and the other using numbers with different signs. Create two division problems; one using numbers with like signs and the other using numbers with different signs. Model three of your problems with different operational signs. September 20, 2006 Page 32 of 35

33 Three real-life problems with solutions Write three real-life problems involving rational numbers and solve to show their solutions. Use a different operation in each problem. Properties of real numbers Use two of your numbers to illustrate the commutative property of addition Rules and common misconceptions Use two of your numbers to illustrate the associative property of multiplication. Use your numbers to illustrate the distributive property. Give the additive inverse of one of your numbers. Give the multiplicative inverse of one of your numbers. List any rules you have found for computing with positive and negative numbers Give examples of common misconceptions students have when working with positive and negative numbers Suggestions for Classroom Use While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them. This task is intended to be completed during class time and not as an out of school assignment. Presentations Project September 20, 2006 Page 33 of 35

34 Contest Peer Review Display for parent night Place in portfolio Photographs Discussion, Suggestions and Possible Solutions Throughout this culminating activity, or one similar in rigor and depth, the teacher should monitor students for understanding and help students correct any obvious weaknesses. This task is extensive and students should not be expected to complete each part in one class period. Students should be allowed to represent their answers in a variety of ways on the poster to show their mastery of the concepts of this unit. Many teachers might also have students share their posters with verbal explanations to their peers during a classroom presentation. Tor assessing this task, teachers should determine whether the skills listed on the poster rubric are performed accurately and with consistency. Models, rules stated in the students own words, and explanations should all be included. The following sections are sample solutions using the numbers 0.5, -1/3, and +2/9. Compare Rational Numbers: -1/3 = < -1/3, Absolute value section: Students should demonstrate an understanding that absolute value is the distance on the number line between the selected number and zero. Using the four basic operations: Addition: Using the given numbers students could select /3 =-1/2 + -1/3 =-3/ /6 = -5/6. In this case, students would also show they have mastered using decimals and fractions interchangeably. September 20, 2006 Page 34 of 35

35 Subtraction: Students showing 2/9 (-0.5) = 4/18 + 9/18 = 13/18 would show mastery of subtraction operations with rational numbers. Multiplication: 2/9 * ( -1/3) = -2/27 Division: 2/9 + (-1/3) = 2/9 * -3/1 = -6/9 = -2/3 Different operational signs: Students should be able to show that (-1/3) could also be written as 0.5 1/3. Three real life problems with solutions: In this section, students will demonstrate how accurately they can apply the concepts of positive and negative rational numbers to real world situations. Teachers could expect to read problems involving game scores, temperatures, profit and loss, etc. Answers will vary. Properties: Commutative property for Addition: = Associative Property for Multiplication; (-1/2 * -1/3) * 2/9 = -1/2 * (-1/3 * 2/9) Distributive property: -1/2(1/4 + 2/9) = -1/2 * ¼ + -1/2 * 2/9 Additive Inverses (Opposites): -.5 and.5 Multiplicative Inverses: -1/2 and -2 Rules and Common Misconceptions: Students should state rules and properties in their own words. Having students identify common misunderstandings may be helpful in determining misunderstandings that they still have in dealing with rational numbers. September 20, 2006 Page 35 of 35