Unit 3: Equal or Not Key Standards Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Key Standards Quarterly Math Content Overview M8A1. Students will use algebra to represent, analyze, and solve problems. Represent a given situation using algebraic expressions or equations in one variable. Simplify and evaluate algebraic expressions. Solve algebraic equations or inequalities in one variable, including those involving absolute values. Solve equations involving several variables for one variable in terms of the others. Interpret solutions in problem contexts. These are the Key Standards for the unit. There are also Related Standards that should be addressed in the Framework unit found at www.georgiastandards.org. Atlanta Public Schools

Unit 3: Equal or Not Key Standards Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Key Standards Quarterly Math Content Overview M8A2. Students will understand and graph inequalities in one variable. Represent a given situation using an inequality in one variable. Use the properties of inequality to solve inequalities. Graph the solution of an inequality on a number line. Interpret solutions in problem contexts. These are the Key Standards for the unit. There are also Related Standards that should be addressed in the Framework unit found at www.georgiastandards.org. Atlanta Public Schools

GPS Math Process Standards Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview P1. 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. P2. 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. As stated on the APS Curriculum Maps, ALL math instruction should include the Process Standards. We should consider the Process Standards the interactive, rigorous, and higher-level ways that students learn the Content Standards. That is why the Process Standards are first in this professional learning session. To save time, move through these next 3 slides quickly highlighting the top-level of each process standard. Atlanta Public Schools

GPS Math Process Standards Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview P3. 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. Highlight the top-level of each process standard. Atlanta Public Schools

GPS Math Process Standards Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview P4. 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. P5. 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. Highlight the top-level of each process standard. Atlanta Public Schools

Unit 3: Equal or Not Concepts & Skills to Maintain Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Concepts & Skills to Maintain Quarterly Math Content Overview Proportional reasoning Evaluate algebraic expressions Solve one and two step equations Operations with rational numbers Properties of equality These are the concepts and skills to maintain for this unit. Students should have mastered these concepts during kindergarten through second grade. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Atlanta Public Schools

Unit 3: Equal or Not Enduring Understandings Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Enduring Understandings Quarterly Math Content Overview Algebraic expressions, equations and inequalities are used to represent relationships between numbers. Absolute value is used to represent distances between numbers. Graphs can be used to represent all of the possible solutions to a given situation. Many problems encountered in everyday life can be solved using equations or inequalities. These are the Enduring Understandings that students should have at the end of the unit. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Atlanta Public Schools

Unit 3: Equal or Not Concepts & Skills for this unit Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Concepts & Skills for this unit Quarterly Math Content Overview use algebraic expressions, equations, or inequalities in one variable to represent a given situation; simplify and evaluate algebraic expressions, including those with exponents; solve and interpret algebraic equations and inequalities in one variable, including those with absolute values; and graph the solution of an equation or inequality on a number line. These are the new concepts and skills for this unit. It is critical that every student master these concepts and skills prior to moving to the next grade-level because they are essential to success as they progress in school and life. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Atlanta Public Schools

Unit 3: Equal or Not Misconceptions and Errors Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Misconceptions and Errors Quarterly Math Content Overview There is no relationship between solving equations and solving inequalities. The absolute value of a number is always its opposite. To find the value of a number with an exponent, multiply the base by the exponent. These are a few common misconceptions and/or error patterns that students sometimes have regarding the concepts and skills in the unit. You will uncover other misconceptions as you assess knowledge prior to and while teaching the unit. Can you think of any others you have experienced with your students. If misconceptions and error patterns are not directly addressed in our instruction, students may remember rote procedures and rules for a short period of time. However, students whose misconceptions are directly addressed, will not only retain the concept, but will show gains in moving on to higher-levels of learning with the concept. See the next slide for the related research. Today’s task will give an opportunity for us to focus on common misconceptions. Atlanta Public Schools

Unit 3: Equal or Not Terms & Symbols Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Terms & Symbols Quarterly Math Content Overview Absolute Value: The distance a number is from zero on the number line. Examples: |-4| = 4 and |3| = 3 Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation. Additive Inverse: Two numbers that when added together equal 0. Example, 3.2 and -3.2 Algebraic Expression: A mathematical phrase involving at least one variable. Expressions can contain numbers and operation symbols. Equation: A mathematical sentence that contains an equals sign. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Have each table group select a reporter whose birthday is closest to today. Give them 3 minutes to discuss and be ready to report out: What strategies do you use as an instructor for math terms and math symbol acquisition and long-term retention? Have table reporters share out their strategies for math vocabulary acquisition, daily use, and retention. Either you or someone from the group chart the strategies up front (perhaps in Word on your computer or on chart paper) for the entire class. If you chart in Word, the file can be sent to all the participants afterwards – this is your choice as a facilitator. Atlanta Public Schools

Unit 3: Equal or Not Terms & Symbols Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Terms & Symbols Quarterly Math Content Overview Evaluate an Algebraic Expression: To perform operations to obtain a single number or value. Inequality: A mathematical sentence that contains the symbols >, <, ≥, or ≤. Inverse Operation: Pairs of operations that undo each other. Examples: Addition and subtraction are inverse operations and multiplication and division are inverse operations. Like Terms: Monomials that have the same variable raised to the same power. In other words, only coefficients of terms can be different. Linear Equation in One Variable: an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a  0 This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Have each table group select a reporter whose birthday is closest to today. Give them 3 minutes to discuss and be ready to report out: What strategies do you use as an instructor for math terms and math symbol acquisition and long-term retention? Have table reporters share out their strategies for math vocabulary acquisition, daily use, and retention. Either you or someone from the group chart the strategies up front (perhaps in Word on your computer or on chart paper) for the entire class. If you chart in Word, the file can be sent to all the participants afterwards – this is your choice as a facilitator. Atlanta Public Schools

Unit 3: Equal or Not Terms & Symbols Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Terms & Symbols Quarterly Math Content Overview Multiplication Property of Equality: For real numbers a, b, and c (c ≠ 0), if a + b, then ac = bc. In other words, multiplying both sides of an equation by the same number produces an equivalent expression. Multiplicative Inverses: Two numbers that when multiplied together equal 1. Example: 4 and ¼. Solution: the value or values of a variable that make an equation a true statement Solve: Identify the value that when substituted for the variable makes the equation a true statement. Variable: A letter or symbol used to represent a number. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Have each table group select a reporter whose birthday is closest to today. Give them 3 minutes to discuss and be ready to report out: What strategies do you use as an instructor for math terms and math symbol acquisition and long-term retention? Have table reporters share out their strategies for math vocabulary acquisition, daily use, and retention. Either you or someone from the group chart the strategies up front (perhaps in Word on your computer or on chart paper) for the entire class. If you chart in Word, the file can be sent to all the participants afterwards – this is your choice as a facilitator. Atlanta Public Schools

Unit 3: Equal or Not Framework Unit Tasks Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Framework Unit Tasks Quarterly Math Content Overview Expanding Space Station Secret Codes Acting Out Making the Grade Number Tricks Go Fish! These are the tasks from today’s focus unit. Today we will focus on the Making the Grade task. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Atlanta Public Schools

Unit 3: Equal or Not Evidence of Learning Q2 Grade 8 Mathematics 2009-2010 Unit 3: Equal or Not Evidence of Learning Quarterly Math Content Overview use algebraic expressions, equations, or inequalities in one variable to represent a given situation; simplify and evaluate algebraic expressions, including those with exponents; solve and interpret algebraic equations and inequalities in one variable, including those with absolute values; and graph the solution of an equation or inequality on a number line. This is a guide for the minimum evidence of learning that would be considered as mastery for standards in this unit. This content is directly from the GaDOE GPS math unit. GPS Math Framework Units for every grade level can be found at www.georgiastandards.org. Atlanta Public Schools

Model Lesson Unit 3: Equal or Not Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Model Lesson Unit 3: Equal or Not Number Tricks Let’s look at a lesson that deals with the current quarter’s content. If you decide to give it a try with your students, we would love to hear from you about what worked and how you might make it even better the next time you teach it. We could then share our findings with your fellow grade level math teachers across the district. We would also like to showcase APS student work and request samples of various levels of understandings be sent to the math & science department: 3rd floor of the CLL Building at 130 Trinity Avenue, Atlanta 30303. Atlanta Public Schools

Pre-lesson Reflective Teacher Questions Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview What is the lesson about? What prior knowledge do you think the students have? What unique considerations need to be included when planning for this group of students? Pass out the GPS Task You should role-play the Pre-lesson Reflective Teacher Model Questions by having someone from the class ask you the questions. Suggestions for role-playing the Reflective Teacher Model: What is the lesson about? This lesson/task is asking students to solve inequalities in one variable, including those involving absolute values. Students will use their prior knowledge of solving equations and their knowledge of computing with negative rational numbers. What prior knowledge do you think the students have? Students in my class have been working with understanding the meaning of absolute value as the distance from zero on a number line. In Grade 7, they worked with placing positive and negative numbers on a number line. What unique considerations need to be included when planning for this group of students? Like many other classes I have had, I have several students still thinking the absolute value of a number is always its opposite. Students need to remember and use concepts and skills previously taught in Grade 7 when computing with negative numbers. Today I am going to be anxious to see if they remember our expectations for regrouping themselves once I give the instruction to do so. Atlanta Public Schools

Pre-lesson Reflective Teacher Questions Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview What manipulatives or tools can be used for conceptual modeling? What do you already know through pre-assessments or other formative assessments about their misconceptions and/or error patterns related to this concept? How do you think they will do? You should role-play the Pre-lesson Reflective Teacher Model Questions by having someone from the class ask you the questions. Suggestions for role-playing the Reflective Teacher Model: What manipulatives or tools can be used for conceptual modeling? The task itself lends to the use of algeblocks and/or algebra tiles could help students still struggling. What do you already know through pre-assessments or other formative assessments about their misconceptions and/or error patterns related to this concept? As I said in previously, some of my students are still having difficulty remembering how to compute with negative numbers. Atlanta Public Schools

Engage: Opening (Warm-Up) Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Do the following sequence of operations in order:   b. What did you get for your final number? c. Check with your partner, what did that person get for their final number? d. Everyone should have the same number. What number is that? e. Why did everyone end with the same number? f. How does this trick work? 1) Write down any number. (This is your ‘start’ number.) 2) Add to it the number that comes before it. 3) Add 11. 4) Divide by 2. 5) Subtract your start number. Atlanta Public Schools

Standards/Elements M8A1. Students will use algebra to represent, analyze, and solve problems. Represent a given situation using algebraic expressions or equations in one variable. Simplify and evaluate algebraic expressions. Interpret solutions in problem contexts.

Explore: Opening (Mini-Review) Sentence Algebraic Equation A number increased by nine is fifteen. y + 9 = 15 Twice a number is eighteen. 2n = 18 Four less than a number is twenty. x - 4 = 20 A number divided by six is eight.

Explore: Opening (Mini-Review) Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Sentence Algebraic Equation Twice a number, decreased by twenty-nine, is seven. Thirty-two is twice a number increased by eight. The quotient of fifty and five more than a number is ten. Twelve is sixteen less than four times a number. This can be a think pair share activity Atlanta Public Schools

Explore/Explain: Opening (Mini-Lesson) Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Now try this one: Do you feel lucky? Why or why not? Explain what made this trick work. 1) Take the number of your birth month. 2) Add 32. 3) Add the difference between your birth month number and 12. 4) Divide by 4. 5) Add 2.   This is your Lucky Number! The teacher should walk students through how to write an algebraic expression for this trick . Atlanta Public Schools

Elaborate: Station Activities (Work Period) Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Practice Plaza Teacher Directed The Shop The Teacher Directed station is for level one students Practice Plaza is for level two and three students Everyone should go to the shop Atlanta Public Schools

Evaluate: Closing It is time to show what you know… Today we … Q2 Grade 8 Mathematics 2009-2010 Evaluate: Closing Quarterly Math Content Overview It is time to show what you know… Today we … Tomorrow we will… Homework Choice 1 or Homework Choice 2 Lesson Summary-Closing Instructor Notes Have each group quickly share their results with at least one other group. Bring the large group together and ask for no more than three different possibilities for each amount. Be sure each person sharing justifies their results and explains their thinking. As explanations are made, facilitate the understanding and categorizations for the different strategies used. Lead the group into a discussion of how this task could be differentiated. (They should be led to understand working in groups is a legitimate method of differentiating along with additional strategies.) The teacher should ensure that any misconceptions are questioned, addressed, and that formative feedback or further questions to explore are provided to individual group approaches. If time permits, review the slides with student work and discuss misconceptions and ways to address them. Refer to the second point on the slide. As a teacher, lesson timing is an art to be practiced and perfected over time! Atlanta Public Schools

Evaluate: Closing (Exit Ticket) Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview Choose one of the 5 prompts to include in your Math Journal or Thinking Map as your Exit Ticket. I feel I really understood… I am unsure about… I am curious to learn more about…. Today’s lesson left me wondering about…. The thing I will remember most about this lesson is ….. because…. I continue to struggle with… because This is one strategy that can be used to incorporate writing in a mathematics classroom. Atlanta Public Schools

Session Reflection Questions Q2 Grade 8 Mathematics 2009-2010 Quarterly Math Content Overview 1. Which strategies in this session are you most likely to use? 2. How would you adapt the strategies in this session to suit your needs? 3. What else would you like to know? As part of this discussion, encourage participants to fill out their session survey form. Atlanta Public Schools