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### Session 5: 10-02-12 AM (Gr. 6)

CCSS 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

(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 lesson-design practices are most effective with our students?

From Kanold and Larson, 2012

• Do you have a rubust process for engaging in work related to the Teach-Learn-Assess Cycle?  If so, what does it look like?
• What do you do when students do not demonstrate understanding (or need enrichment)?

>> Unpacking the Grades 6-8 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 1-3: 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

## A.  Apply and extend previous understandings of arithmetic to algebraic expressions.

• 6.EE.A.1 - Write and evaluate numerical expressions involving whole-number 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 numbersFor example, express the calculation “Subtract y from 5” as 5 – y.
• 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.
Driscoll, M. (1999). Fostering Algebraic Thinking: A guide for Teachers Grades 6-10. 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 real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 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 expressionsFor 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.
• 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 one-variable equations and inequalities.
• Sample Formative Assessment Items:

## B.  Reason about and solve one-variable 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 real-world 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 real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which pq 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 real-world 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 real-world 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 Misconceptions

## A.  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 real-life and mathematical problems using numerical and algebraic expressions and equations.

• 7.EE.B.3 - Solve multi-step real-life 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 real-world 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 pq, 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?
• 7.EE.B.4b - Solve word problems leading to inequalities of the form px + q > r or px + q < r, where pq, 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, 32 × 3–5 = 3–3 = 1/33 = 1/27.
• 8.EE.A.2 - Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = 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/on-city-rooftops-scrappy-green-spaces-in-bloom.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 whole-number 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 108 and the population of the world as 7 times 109, 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 distance-time graph to a distance-time 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 non-vertical 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 = aa = 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.
• 8.EE.C.8c - Solve real-world 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 Unit-by-Unit Actions of Mathematics Collaborative Teams
• Identify the high-leverage 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?