Durham Public Schools 2012-2013

Unit Overview: Instructional Time: 3 weeks Quarter MACROBUTTON HTMLDirectOne MACROBUTTON HTMLDirect Two MACROBUTTON HTMLDirect Three MACROBUTTON HTMLDirect Four Unit Theme: Rational Number Relationships Depth of Knowledge: Level 3, Strategic Thinking

Unit Summary: Students of sixth grade mathematics often come to middle school lacking a firm grasps on the concept of positive and negative numbers. This unit is critical for the development of this understanding and serves as a vital springboard for future work with negative numbers. The unit is divided into three main components: (1) fundamental understanding of positive and negative whole numbers (2) opposite numbers, (3) distance and absolute value.

Critical Area of Concentration: (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; North Carolina Information and Technology Essential Standards: 6.SI.1.1 Analyze resources in terms of their reliability (which can be determined by currency, creditability, or authority, depending on the topic or purpose. 6.SI.1.2 Analyze content for relevance to the assigned task 6.TT.1.1 Select appropriate technology tools to gather data and information (e.g. Web-based resources, e-books, online communication tools, etc.) 6.TT.1.2 Select appropriate technology tools to organize data and information (e.g., word processor, database, spreadsheet, graphic organizer, audio and visual recording, online collaboration tools, etc.) 6.TT.1.3 Select appropriate technology tools to present data and information effectively (multimedia, audio and visual recording, online collaboration tools, etc.) 6.RP.1 Implement a research process collaboratively 6.SE.1.1 Apply ethical behavior (copyright, not plagiarizing, proper netiquette) when using resources. Common Core State Standards: The Number System: Apply and extend previous understandings of numbers to the system of rational numbers 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite. c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. d. Distinguish comparisons of absolute value from statements about order.

Essential Questions:  Why are there negative numbers?  How do we compare and contrast numbers?  Are there more rational numbers than integers?  Is distance ever negative? Enduring Understandings:  Negative numbers can be used to represent quantities less than zero or quantities with an associated direction such as debt, elevations below sea level, low temperatures, moving backward in time, or an object slowing down.  Communication and collaboration with others is more efficient and accurate using rational numbers. I Can Statements:  I can identify an integer and its opposite.  I can use integers to represent quantities in real world situations.  I can identify a rational number using a number line.  I can order rational numbers on a number line.  I can locate 0 on a number line.  I can recognize and locate positive and negative integers on a number line.  I can understand that the “opposite” of the “opposite of a number”, is the number itself.  I can identify absolute value on a number line.  I can calculate absolute value.  I can interpret statements of inequality about two numbers on a number line.  I can write, interpret, and explain statements of order for rational numbers in real-world contexts.  I can interpret absolute value for a positive or negative quantity in a real-world situation.  I can distinguish comparisons of absolute value from statements about order and apply to real-world situations.  I can solve real world problems by graphing points in all four quadrants of the coordinate plane. Vocabulary: Absolute value Rational number Inequality Magnitude Integers Opposite Sum Positive numbers Negative numbers Value Coordinates X-axis Coordinate Pair X-coordinate Y-axis Origin Quadrants Ordered pair Coordinate Plane Graph Y-coordinate Coordinate System Transdisciplinary Connections: Common Core State Standards for English Language Arts: Language: Conventions of Standard English L.6.1 Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. L.6.2 Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.

Writing: Text Types and Purposes W.6.1 Write arguments to support claims with clear reasons and relevant evidence. W.6.6 Use technology, including the Internet, to produce and publish writing as well as to interact and collaborate with others; demonstrate sufficient command of keyboarding skills to type a minimum of three pages in a single sitting.

North Carolina Essential Standards for Social Studies: Geography and Environmental Literacy 6.G.2.2 Construct maps, charts and graphs to explain data about geographic phenomena. Evidence of Learning (Formative Assessment): Summative Assessment:  Math Journal or Math Blog Self-Reflections  Visitor Brochure  Checkbook registers  End of Unit Assessment  Weather Report Presentation  Durham Public Schools’ Small Goal Assessment Unit Implementation: A Positive or Negative Experience (CCSS: 6.NS.5) Using the number line, introduce this lesson by having students compare whole number vs. integer thinking. Have students define what a whole number is (counting numbers w/o zero), then discuss if 9 will always be greater than 4. Introduce the integers -9 and -4. Ask students if the rule still applies. Have students discuss and debate. Have a student record the responses on the board based on “mathematical reasoning.” Re-introduce the number line banner or projected number line (this can be designed using PowerPoint or Online Interactive Number Line. Have them examine the number line with specific attention to its structure – numbers to the right of another number are always greater and numbers to the left of another number are always less, regardless of the sign. Once students have grasped this concept, present them with the following real-world application: “It is -9º F on a November day in Juno, Alaska at 8:00 a.m. By 10:00 a.m., it is -4º F. Have the temperatures in Juno dropped or risen? Is it colder at 10:00 a.m. or 8:00 a.m.?” Show an example of a thermometer. Ask students to relate the thermometer to the number line.

Next, introduce/remind students of the role of zero on the number line. Ask students if zero is positive? Negative? Have them discuss their reasoning. Remind them that in a real-world context zero can represent ideas contextually. Offer them the following examples as various representations of zero, positive and negative number: (1) Sea-level when measuring elevation, (2) Temperature (above/below zero), (3) Blood Pressure (high/correct/low) In pairs, have them Think-Pair-Share and create pictorial representation of each situation.

Throughout the week, continue reinforcing the concept of positive/negative numbers and zero by having students participate in the following activities: (1) Positive and Negative Integers: A Card Game: Students practice addition subtraction of positive and negative integers. Black cards represent positive integers, red cards represent negative integers. After the game, students write about it in their math journals. To add a twist, have them explain the rules of the game in their own words. (2) Let’s Play Football!: Have students make a football field from a number line and play with a token or chip as a football. As an artifact for this week, have students create their own checkbooks. Give each student a starting balance. Allow them to “buy” items throughout the week (computer time, pencils, etc.) using their debit card (an index card that they have designed). You may also offer “deposits” for extra “credit” or completion of in-class assignments. Have them record their purchases and journal about their experience in their math journal or math blog. Sample Check register.

Connected Mathematics Lesson: CMP2 Unit – Bits and Pieces II, Investigation 2: ACE 51 Revealing Reflections (CCSS: 6.NS.6) Building on the previous weeks concepts, remind students that number lines are an effective tool for thinking about positive and negative integers and their relationship to zero. Separate students into pairs. Have one student draw a horizontal number line, the other a vertical number line. Using chips (or similar manipulative), have student cover zero in one color, and use the other color to cover rational numbers and their opposites (as you call them out). In their pairs have them discuss and record similarities and differences in their math journal. Have them describe what the “opposite” of number is, in their own words.

The charge model assigns a negative or positive value to two different objects within a set. (Usually, this is included in a manipulative kit as red and yellow counters). The actual object being used is not as important as the distinction that one item represents positive values and the other negative values. Separate students into groups of four. Using a sandwich bag or baby food jar (or similar container), use jellybeans, Skittles, M&M’s, etc. of two different colors. Present a container to a group, and have them determine the final “charge” of the container. By eliminating zero pairs (1 green Skittle = +1 and 1 red Skittle = -1), the students can determine the final charge of the container by counting the residual. For example, 3 green Skittles = +3. Have students record their findings on a chart using MS Word or Google Docs.

Connected Mathematics Lessons:  CMP2 Unit – Bits and Pieces I; Investigations 1,2,3,4  CMP2 Unit – Bits and Pieces II; Investigations 1,2,3,4  CMP2 Unit - Bits and Pieces III; Investigations 1,2,3,4  CMP2 Unit – Bits and Pieces II; Investigation 2: ACE 51  CMP2 Unit – CC Transition Kit; CC Investigation 3 It’s Absolute! (CCSS: 6.NS.7) Review with students the position of numbers on the number line (Week 1 lesson). Introduce inequalities by having the number line serve as a visual aid. Highlight the arrow ends at the ends of the number line. Show them that the arrow end on the left represents the less than symbol, the arrow end on the right represents the greater than symbol. By questioning, have them determine that numbers further right on the line are greater and numbers further left are smaller. Have them read inequalities from left to right, and determine their value by noticing their position on the number line. For example, ask students “Is -3 is less than, greater than, or equal to -7?” Have them plot both numbers. Students should realize that -3 > -7 because -3 is further right than -7 on the number line. Reverse the question. “Is -7 less than, greater than, or equal to -3?” Students should look at the number line, start with -7, and look at it’s relationship to -3. Since it’s further left, the answer would be -7< -3.

Have students practice writing inequalities by completing using integers, fractions, and decimals.

Next, have students interpret and write statements of order. As students are able to make statements based on relative position on a number line as addressed above, they can continue to build on the concept by contextualizing those numbers in a given situation. Begin with asking students, “Will -7º feel colder than -3º? How do you know?” For students having difficulties visualizing, use the thermometer from Week One’s lesson, and place it on its side. Help students see that temperatures are growing greater (or warmer) the further right (or up) you travel on the thermometer/number line. Temperatures grow colder the further left (or down) you travel on the thermometer/number line. As an activity, (linking the concept from yesterday), have students pick a city and state from a map and follow the weather report from The Weather Channel for the 10-day forecast. Have students create weather reports, utilizing the vocabulary introduced from this lesson and visuals from the number line to describe the change in temperature throughout the week. A variation of this task is to have students give forecasts from various countries. Students may create their weather report using Prezi! To help with creativity, have students refer to (but not copy) the weather experience from EdHeads Weather Center. Students may also make videocasts using digital camcorders or iPads.

Lastly, help students along with the whole number association of quantity. Students may conceptualize quantity by the number and not the distance from zero. For some, -7 “feels” like more than -3 because 7 is greater than 3. To help students with this concept, use the vocabulary “debt” and “owe” to signify negative numbers. Offer the following example: “Simone owes her mother $6 for girl scout cookies, while Jessica owes her mother $3. Who owes more money? How do you know?” Now show this same scenario showing negative numbers in a t-chart. Simone is represented by -6 and Jessica is represented by -3. Then have students graph this relationship on a number line. Explain to students that while -6 is “less” than -3, Simone’s debt is more or “greater” than Jessica’s. Ask students for their observations as to why this is true. This example introduces the concept of absolute value. Continue this by illustrating the absolute value symbol as a “car wash.” Explain to students that “Clean” is positive; “dirty” is negative. No matter which way the car goes into the car wash, it comes out clean! For this example, it works great to have car magnets with negative and positive numbers attached to them. Place the cars between the absolute value brackets. For example:

absolute value brackets -9

The car has a value of -9. So when it goes in the “car wash” it will come out “clean,” or a positive number. No matter whether a number starts out “clean” or “dirty” it comes out of the absolute value operation “clean”. Include issue of negative outside absolute brackets: if you “dirty” the car after the car wash, it will always be “dirty”.

Connected Mathematics Lesson:  CMP2 Unit – Bits and Pieces I; Investigation 1,2,3  CMP2 Unit – Bits and Pieces II; Investigation 2: ACE 51  CMP2 Unit – Bits and Pieces III; Investigation 1: ACE 58  CMP2 Unit – CC Transition Kit, CC Inv 3 Welcome to Our School (CCSS: 6.NS.5, 6, 7, 8) This week’s lesson is a culminating activity. Students will revisit coordinate pairs, interpreting a coordinate, and using absolute value while creating their own map using a coordinate plane. Have students create a brochure for visitors to the school. Include in this brochure a map for a visual. Have them choose 3 to 4 nearby sites, and have them graph those on a coordinate plane, assigning each landmark a coordinate pair. Remind students that whenever the first or second coordinates share the same number, they are on the same line. (or in this case, on the same side of the street or on the same block). For example, The Subway nearest to the school has coordinate (7W, 5S), meaning it is 7 miles west of the school, and 5 miles south of the school (or 7 blocks west, 5 blocks south). The Nail Salon is (7W, 9S). The students should be able to see that both sites are 7 miles west of the school, but one landmark is further south (or negative) of the other. Students may utilize Google Maps to determine direction and mileage (in terms of distance), however encourage students to explore nearby attractions with friends or family on foot (counting blocks and direction). Also remind students that the length of the line between two points can be determined by counting the spaces between the points. On the coordinate plane, point A is 5 units away from point B. Likewise, point B is 5 units away from point A. We wouldn’t speak of them being -5 units away, because distance is always positive. So have students discuss the absolute value between two points as “distance.

Students can utilize MS Word or Google Docs to work collaboratively. Students are to create brochure that shows a map of nearby attractions, their descriptions, and their position in relation to the school. The resulting project may be presented using MS Publisher or Apple Pages.

Connected Mathematics Lessons:  CMP2 Unit – Covering and Surrounding; Investigation 2  CMP2 Unit – Data About Us; Investigation 2  CMP2 Unit – CC Transition Kit; CC Investigation 3

Supportive Unit Resources: (Please note that these are resources that can be used to supplement instruction before or during a lesson.) Scaffolding Option 1: Scaffolding Option 2: Scaffolding Option 3: Intervention Maintenance Extension Instructional Review of Number Types Multisensory Teaching: Math Guide Self Lesson on Operations Activities: (CCSS 6.NS.5) Positive and Negative Integers (Parts 1 & 2) on Integers Foundational Knowledge for Integers http://www.resourceroom.net/math/integers.as Online Lesson p (CCSS 6.NS.5 & 6) http://www.purplemath.com/modules/numt (CCSS 6.NS.5) Students may move ahead and practice ypes.htm Birthday Party Invitation (R.S.V.P) addition, subtraction, division and For this activity, give students the task of multiplication of positive/negative integers. (CCSS: 6.NS.6) writing directions for a birthday party, and http://www.mathguide.com/lessons/Integers. include a map. Give the following scenario: Coordinate Plane Lesson Video and “Suppose you are having a birthday party and html Follow-up Worksheets - Video a friend you have invited has asked you for introduces coordinates and graphing, and directions from the school to your house. You Books: how it is used in the real world. Use the tell them it is five blocks away. Is this enough Treasure Island by Robert Louis Stevenson handout as an additional resource for information for them to find your house? is a classic story of a search for pirate students while they watch the video. Design a map that better explains the distance treasure. Stevenson based the story on a from your house from their house.” http://mathcrush.com/graph_worksheets.h map drawn by his son, Lloyd. tml Books: Math Connections: Students can read this (CCSS: 6.NS.4, 5) Hottest, Coldest, Highest, Deepest story and create connections using Classifying Numbers (Video Lesson) by Steve Jenkins coordinates and mapping activities. http://www.phschool.com/atschool/acade ISBN-10: 9780618494880 Summary: Jenkins describes various wonders my123/english/academy123_content/wl- The Phantom Tollbooth by Norton Juster book-demo/ph-144s.html of the world such as the deepest lake, hottest/coldest temperatures, deepest spot in ISBN-10: 0394815009 the ocean, etc. Each natural wonder includes Summary: Everything is number-oriented in (CCSS: 6.NS.7) data, inset maps, diagrams, and comparisons. Milo’s adventures to Digitopolis. He meets Finding Absolute Value (Video Lesson) half a boy who is 0.58 a member of the http://www.phschool.com/atschool/acade average family—a mother, father and 2.58 Math Connections: Students can use this book children. The boy explains that he is the my123/english/academy123_content/wl- to explore integers in the real world. book-demo/ph-145s.html only one who can drive the 0.3 of a car—the average family owns 1.3 cars. The Fly on the Ceiling by Julie Glass Less Than Zero by Stuart Murphy ISBN-10: 0679886079 Math Connections: This section of the story ISBN-10: 9780060001261 Summary: Rene Descartes, a French is an invitation to discuss decimal numbers Summary: A penguin named Perry needs mathematician and philosopher, revolutionized as related to averages. 9 clams to buy an ice scooter, but he has mathematics with the Cartesian Coordinate difficulty saving. Negative numbers are System. In this story, he invented a way to explored as Perry earns, spends, finds, keep track of his possessions by using a grid loses, and borrows clams. and coordinates. Descartes is recognized as the father of analytic geometry. Math Connections: Students can explore Math Connections: This book can be an integers by making a line graph to show invitation to learn when students begin to work how Perry’s clams increase and decrease with graphing points on the coordinate plane. as the story is read. Technology Integration: (Please note that these are resources that can be used to supplement instruction before or during a lesson.) Multimedia Integers: Understanding integers Ordering Positive and Negative Numbers Battleship! Activities: Given scenarios of increase/decrease in without the number line. Have students play the game online, then word problems, students can practice http://www.free-training-tutorial.com/negative- create their own game via coordinate identify positive and negative numbers. numbers/number-balls.html planes. Have students identify coordinate http://www.ixl.com/math/grade- pairs as “hits.” 6/understanding-integers Decimal Numbers: Inequalities with http://www.primarygames.com/puzzles/strat Decimals egy/battleship/index.htm Integers: Absolute value and opposite http://www.ixl.com/math/grade-6/inequalities- integers with-decimals Extra practice identifying absolute value. http://www.ixl.com/math/grade-6/absolute- Coordinate Graph Review – Students identify value-and-opposite-integers quadrants, coordinate pairs, and negative/positive numbers. http://www.ixl.com/math/grade-6/coordinate- graphs-review