Grade 6CCSS Expressions and Equations: A Focus on Planning and Implementation
Location: Elkton High School, Room F107 Start/End: 9:00 a.m.  11:30 a.m. Presenters: Jon Wray and Bill Barnes
>> Welcome, Introductions, and Outcomes >> Survey: Seven Stages of Teacher Collaboration (Graham & Feriter, 2008)
Stage  Questions That Define This Stage
 Stage 1: Filling the time
 What exactly are we supposed to do as a team?
 Stage 2: Sharing personal practice
 What is everyone doing in his or her classroom for instruction, lesson planning and assessment?
 Stage 3: Planning, planning, planning
 What should we be teaching during this unit, and how do we lighten the load for each other?
 Stage 4: Developing common assessments
 How will we know if students learned the standards? What does mastery look like for the standards in this unit?
 Stage 5: Analyzing student learning
 Are students learning what they are supposed to be learning? Do we agree on student evidence of learning?
 Stage 6: Adapting instruction to student needs
 How can we adjust instruction to help those students struggling and those exceeding expectations?
 Stage 7: Reflecting on instruction
 Which lessondesign practices are most effective with our students?

From Kanold and Larson, 2012
>> The TeachAssessLearn Cycle
 Do you have a rubust process for engaging in work related to the TeachLearnAssess Cycle? If so, what does it look like?
 What do you do when students do not demonstrate understanding (or need enrichment)?
>> Unpacking the Grades 68 Expressions and Equations Progression Document (U of Arizona IME)
 As you review this document, what (topics, clusters, standards) represent strength areas in your planning, teaching, and assessing? Area(s) of need?
>> Learning Stations
 Stations 13: Do the Math, Teach the Math
 Station 4: Exploring Resources that Support the Strand
 Station 5: Illustrative Math Site for Sample Items
 Station 6: Create a lesson plan or experience
 Station 7: Innovative Brainstorming Session
 Station 8: Learning @ Manipulatives
>> CCSS: Grades 68 Expressions & Equations Domains, Clusters & Standards Note: Red text indicates new concepts/skills (to that grade level) from previous grade levels.
A. Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.A.1  Write and evaluate numerical expressions involving wholenumber exponents.
 6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.
 Sample Formative Assessment Items:
 6.EE.A.2a  Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
 Sample UDL Lesson: "Charades"
 6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
 Text Resource/Task:
Driscoll, M. (1999). Fostering Algebraic Thinking: A guide for Teachers Grades 610. Heinemann. p. 24. "Postage Stamp Problem" Math Task p. 24: This math task allows students to devise a strategy for determining what combinations of postage is possible with two given sets of stamps. This would allow students to think about how to write expressions with multiple variables and analyze each part of the expression.
 6.EE.A.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^{3} and A = 6 s^{2} to find the volume and surface area of a cube with sides of length s = 1/2.
 Sample Worthwhile Mathematical Task: "My First Fish Tank"
 NCTM Journal Article Lesson Ideas/Resources:
Ameis, J.A. (March 2011). "The Truth about PEMDAS." Mathematics Teaching in the Middle School. Vol. 16. No. 7. p. 414. This article shows how a hierarchy of operators triangle shapes conceptual understanding of the order of operations.
 6.EE.A.3  Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
 Sample UDL Lesson: "The Stock Market"
 Web Resources:
 6.EE.A.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve onevariable equations and inequalities.
 Sample Formative Assessment Items:
B. Reason about and solve onevariable equations and inequalities. 6.EE.B.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
 6.EE.B.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
 6.EE.B.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
 Sample Formative Assessment Items:
 Sample UDL Lesson: "Party Planning"
 Web Resource
 Additional Resource: "Equation Charades"  Rather than asking students to translate verbal equations into mathematical equations, have them act out the equation to a partner. If the sentence reads "the sum of 4 and m is equal to y", the student should act out 4+m=y. The partner would then translate this and they would compare the answer on the board to the card the actor has.
 6.EE.B.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
 Sample Formative Assessment Item: Fishing Adventures 1
 Sample UDL Lesson: "Hershey Park Inequalities"
 Additional Resource: "Inequality Charades": Rather than asking students to translate verbal inequalities into mathematical statements, have them act out the inequality to a partner. If the sentence reads "the sum of 4 and m is less than y", the student should act out 4+m<y. The partner would then translate this and compare the answer on the board to the card the actor has.
C. Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.C.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Grade 7 > Expressions & Equations Overview (Big Ideas), Enduring Understandings, Essential Questions, Common MisconceptionsA. Use properties of operations to generate equivalent expressions. 7.EE.A.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
 7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
 Sample Worthwhile Mathematical Task: "Shop Smart"
B. Solve reallife and mathematical problems using numerical and algebraic expressions and equations. 7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
 7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
 Sample Formative Assessment Item: "Fishing Adventures 2"
 7.EE.B.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
 Sample UDL Lesson: "Introducing TwoStep Equations"
 Web Resources:
 Text Resource/Task:
Driscoll, M. (1999). Fostering Algebraic Thinking: A guide for Teachers Grades 610. Heinemann. p. 22. "Golden Apples" Math Task p. 22: This math task allows students to develop a strategy to solve a story problem. This would be a nice introduction for connecting their strategy to constructing equations.
 7.EE.B.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Grade 8 > Expressions & Equations Overview (Big Ideas), Enduring Understandings, Essential Questions, Common Misconceptions
A. Expressions and Equations Work with radicals and integer exponents. 8.EE.A.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^{2} × 3^{–5} = 3^{–3} = 1/3^{3} = 1/27.
 8.EE.A.2  Use square root and cube root symbols to represent solutions to equations of the form x^{2} = p and x^{3} = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
 Disciplinary Literacy Resource:
Green, P. (2011, June 22). On city rooftops, scrappy green spaces in bloom. New York Times. Retrieved from http://www.nytimes.com/2011/06/23/garden/oncityrooftopsscrappygreenspacesinbloom.html?_r=1&pagewanted=all There are several stories about various rooftop gardens. You could assign each group a garden to read about and then share with the class to save time. Then, you could assign each group an area for their garden and stipulate that it must be square. Students could "shop" for astro (field) turf for their space and then buy square planters to fill their garden. Students should sketch their design and include specifications for the area of each plant.
 8.EE.A.3  Use numbers expressed in the form of a single digit times a wholenumber power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 10^{8} and the population of the world as 7 times 10^{9}, and determine that the world population is more than 20 times larger.
 Sample Formative Assessment Item: "Ant and Elephant"
 Sample UDL Lesson: Scientific Notation
 8.EE.A.4  Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
B. Understand the connections between proportional relationships, lines, and linear equations. Sample Formative Assessment Items (for use with Cluster 8.EE.B)
 8.EE.B.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed.
 Sample Formative Assessment Items:
 Sample Worthwhile Mathematical Task: "Drops in a Bucket"
 Sample UDL Lesson: "NFL Football and Direct Variation"
 8.EE.B.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
C. Analyze and solve linear equations and pairs of simultaneous linear equations. Sample Formative Assessment Items (for use with Cluster 8.EE.C): "Two Lines"
 8.EE.C.7  Solve linear equations in one variable.
 8.EE.C.7a  Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
 8.EE.C.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
 8.EE.C.8  Analyze and solve pairs of simultaneous linear equations.
 Sample Formative Assessment Items:
 8.EE.C.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
 Sample UDL Lesson: "Pizza Lesson"
 8.EE.C.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
 Sample Worthwhile Mathematical Task: "The Baseball Shop"  Student Resource Sheet
 Sample UDL Lesson 1: "Solving Systems with Substitution"
 Sample UDL Lesson 2: "Introduction to Linear Combination"
 Resources for Teachers:
 Web Resource:
 8.EE.C.8c  Solve realworld and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
>> Closing  How will we continue in our collaborative journey of continuous improvement? ("So what's it going to take to do this work?")
High Leverage UnitbyUnit Actions of Mathematics Collaborative Teams Identify the highleverage actions your collaborative team(s) currently practice(s) extremely well. Rate the current levels of implementation (0% = "never" to 100% = "consistent and an ongoing practice"). Go here for the survey or visit: http://tinyurl.com/dynwfhx
 How might you use this information to identify which actions should be your team's priorities during this year and next school year?
Survey Results Green = 3 highest actions reported Red = 3 lowest actions reported  Grade 6  Grade 7
 Grade 8
 Grades 68 Combined
 Teaching and Learning      1. The team designs and implements agreedon prior knowledge skills to be assessed and taught during each lesson of the unit. The collaborative teacher team reaches agreement for teaching and learning in the lessons and unit.
 63%  82%
 86%  76%  2. The team designs and implements agreedon lessondesign elements that ensure active student engagement with the mathematics. Students experience some aspect of the CCSS Mathematical Practices, such as Construct viable arguments and critique the reasoning of others or Attend to precision, within the daily lessons of every unit or chapter.
 68%  68%
 79%  72%  3. The team designs and implements agreedon lessondesign elements that allow for studentled summaries and demonstrations of learning the daily lesson.
 57%  57%
 67%  60%  4. The team designs and implements agreedon lessondesign elements that include the strategic use of tools—including technology—for developing student understanding.
 54%  66%
 72%  63%  Assessment Instruments and Tools
 Grade 6  Grade 7
 Grade 8
 Grades 68 Combined
 1. The team designs and implements agreedon common assessment instruments based on highquality exam designs. The collaborative team designs all unit exams, unit quizzes, final exams, writing assignments, and projects for the course.
 33%  64%
 76%  57%  2. The team designs and implements agreedon common assessment instrument scoring rubrics for each assessment in advance of the exam.  32%  55%  57%  47%  3. The team designs and implements agreedon common scoring and grading feedback (level of specificity to the feedback) of the assessment instruments to students.  37%  60%  46%  47%  Formative Assessment Feedback  Grade 6  Grade 7
 Grade 8
 Grades 68 Combined
 1. The team designs and implements agreedon adjustments to instruction and intentional student support based on both the results of daily formative classroom assessments and the results of student performance on unit or chapter assessment instruments such as quizzes and tests.  57%  71%  55%  60%  2. The team designs and implements agreedon levels of rigor for daily inclass prompts and common highcognitivedemand tasks used to assess student understanding of various mathematical concepts and skills. This also applies to variance in rigor and task selection for homework assignments and expectations for makeup work. This applies to depth, quality, and timeliness of teacher descriptive formative feedback on all student work.  63%  59%  50%  58%  3. The team designs and implements agreedon methods to teach students to selfassess and set goals. Selfassessment includes students using teacher feedback, feedback from other students, or their own selfassessments to identify what they need to work on and to set goals for future learning.
 53%  48%
 28%  44% 
