Session 3 Combined File

Overview of Session 3

What Practices Will Make Our Work Productive?

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Overview Consider Professional Practice Norms • Solve Trevor’s Problem • Interpret and Analyze Student Thinking • Reflect on Research Findings • Wrap-up

- Michigan Mathematics and Science Teacher Leadership Collaborative -

- Michigan Mathematics and Science Teacher Leadership Collaborative -

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Professional Practice Norms • Listening to and using others’ ideas • Keeping records of professional work • Adopting a tentative stance toward practice • Backing up claims with evidence and providing reasoning • Talking with respect yet engaging in critical analysis of teachers and students portrayed in video or cases. - Michigan Mathematics and Science Teacher Leadership Collaborative -

Trevor’s Problem • Trevor ordered the following numbers from smallest to largest. Is Trevor correct? Why or why not? • Trevor’s order .8 9% .55 - Michigan Mathematics and Science Teacher Leadership Collaborative -

Cognitive Demand of Trevor’s Problem Is Trevor correct? Why or why not? Trevor’s order is 0.8 9% .55

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Memorization task Procedure without connections Procedure with connections Doing mathematics

Focus Questions • What is the mathematics in the task? • How many solutions can you find that make sense to students? • How is the mathematics connected to the CLCEs?

- Michigan Mathematics and Science Teacher Leadership Collaborative - Michigan Mathematics and Science Teacher Leadership Collaborative -

Examining Student Work • Summarize the different ways in which the students worked on the task. • What do the students seem to know? What is the evidence? • Are there any misconceptions? Evidence? • Which solutions would you have a conversation about with your class? And, why? - Michigan Mathematics and Science Teacher Leadership Collaborative -

Findings from Research What is the best estimate of 12/13 + 7/8? – More than 1/2 U.S. 8th graders chose 19 or 21

Solve 4 + .3 = ? – 68% of 6th graders and 51 % of 6th and 8th graders claimed the answer was .7

- Michigan Mathematics and Science Teacher Leadership Collaborative -

Wrap-up • Reflection Questions: – What stood out for you today? – What do you really hope to learn?

• Assignment--Read the case “One Less”

- Michigan Mathematics and Science Teacher Leadership Collaborative -

Session 3 GVSU 7/28/08

Professional Practice Norms

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Listening to and using others’ ideas

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Keeping records of professional work

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Adopting a tentative stance toward practice—wondering versus certainty

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Backing up claims with evidence and providing reasoning

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Talking with respect yet engaging in critical analysis of teachers and students portrayed on the video or described in a case

Learning and Teaching Linear Functions: Video Cases for Mathematics Professional Development, 6-10 Seago, Mumme, and Braca Heinemann (2004)

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Session 3 GVSU 07/28/08

Fractions, Decimals, and Percents Mr. Clark, a 7th grade teacher, is preparing to begin a new school year. He knows that fractions, decimals, and percents make up a large portion of the 4th – 6th grade curriculum, and wants to see how much his incoming 7th graders understand about ordering and comparing rational numbers. Therefore, he decides to give his students the following OGAP item because it does not seem too intimidating for students at the beginning of the year and he hopes it will provide him with some useful baseline information.

Trevor ordered the following numbers from smallest to largest. Is Trevor correct? Why or why not? Trevor’s Order

.8

9%

.55

Answer the two questions on page 1 of the handout Analyzing and Working with Student Thinking.

The Vermont Mathematics Partnership

07/12/08

Why Isn’t It One Less?

Why Isn’t It One Less?

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4th GRADE LEVEL Math CONTENT EXPECTATIONS (Rational) NUMBER AND OPERATIONS

Read, interpret and compare decimal fractions N.ME.04.15 Read and interpret decimals up to two decimal places; relate to money and place value decomposition. N.ME.04.16 Know that terminating decimals represents fractions whose denominators are 10, 10 x 10, 10 x 10 x 10, etc., e.g., powers of 10. N.ME.04.17 Locate tenths and hundredths on a number line. N.ME.04.18 Read, write, interpret, and compare decimals up to two decimal places. N.MR.04.19 Write tenths and hundredths in decimal and fraction forms, and know the decimal equivalents for halves and fourths. * revised expectations in italics

Understand fractions N.ME.04.20 Understand fractions as parts of a set of objects. N.MR.04.21 Explain why equivalent fractions are equal, using models such as fraction strips or the number line for fractions with denominators of 12 or less, or equal to 100. N.MR.04.22 Locate fractions with denominators of 12 or less on the number line; include mixed numbers.* N.MR.04.23 Understand the relationships among halves, fourths, and eighths and among thirds, sixths, and twelfths. N.ME.04.24 Know that fractions of the form mn where m is greater than n, are greater than 1 and are called improper fractions; locate improper fractions on the number line.* N.MR.04.25 Write improper fractions as mixed numbers, and understand that a mixed number represents the number of “wholes” and the part of a whole remaining, e.g., 5/4 = 1 + ! = 1 !. N.MR.04.26 Compare and order up to three fractions with denominators 2, 4, and 8, and 3, 6, and 12, including improper fractions and mixed numbers.

Add and subtract fractions N.MR.04.27 Add and subtract fractions less than 1 with denominators through 12 and/or 100, in cases where the denominators are equal or when one denominator is a multiple of the other, e.g., 1/12 +5/12 = 6/12; 1/6 + 5/12 = 7/12; 3/10 – 23/100 = 7100 . * N.MR.04.28 Solve contextual problems involving sums and differences for fractions where one denominator is a multiple of the other (denominators 2 through 12, and 100).* N.MR.04.29 Find

the value of an unknown in equations such 1/8 + x = 5/8 or

! - y = "*. Multiply fractions by whole numbers N.MR.04.30 Multiply fractions by whole numbers, using repeated addition and area or array models.

Add and subtract decimal fractions N.MR.04.31 For problems that use addition and subtraction of decimals through hundredths, represent with mathematical statements and solve.* N.FL.04.32 Add and subtract decimals through hundredths.* Michigan Department of Education Grade 4

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Multiply and divide decimal fractions N.FL.04.33 Multiply and divide decimals up to two decimal places by a one-digit whole number where the result is a terminating decimal, e.g., 0.42 ÷ 3 = 0.14, but not 5 ÷ 3 = 1.6.

5th GRADE LEVEL Math CONTENT EXPECTATIONS Understand meaning of decimal fractions and percentages N.ME.05.08 Understand the relative magnitude of ones, tenths, and hundredths and the relationship of each place value to the place to its right, e.g., one is 10 tenths, one tenth is 10 hundredths. N.ME.05.09 Understand percentages as parts out of 100, use % notation, and express a part of a whole as a percentage.

Understand fractions as division statements; find equivalent fractions N.ME.05.10 Understand a fraction as a statement of division, e.g., 2 ÷ 3 = 2/3 , using simple fractions and pictures to represent. N.ME.05.11 Given two fractions, e.g., and , express them as fractions with a common denominator, but not necessarily a least common denominator, e.g., !=4/8 and " = 6/8 ; use denominators less than 12 or factors of 100.*

Multiply and divide fractions N.ME.05.12 Find the product of two unit fractions with small denominators using an area model.* N.MR.05.13 Divide a fraction by a whole number and a whole number by a fraction, using simple unit fractions.*

Add and subtract fractions using common denominators N.FL.05.14 Add and subtract fractions with unlike denominators through 12 and/or 100, using the common denominator that is the product of the denominators of the 2 fractions, e.g., 3/8 + 7/10; use 80 as the common denominator.*

Multiply and divide by powers of ten N.MR.05.15 Multiply a whole number by powers of 10: 0.01, 0.1, 1, 10, 100, 1,000; and identify patterns. N.FL.05.16 Divide numbers by 10!s, 100!s, 1,000!s using mental strategies. N.MR.05.17 Multiply one-digit and two-digit whole numbers by decimals up to two decimal places.

Solve applied problems with fractions N.FL.05.18 Use mathematical statements to represent an applied situation involving addition and subtraction of fractions.* N.MR.05.19 Solve contextual problems that involve finding sums and differences of fractions with unlike denominators using knowledge of equivalent fractions.* N.FL.05.20 Solve applied problems involving fractions and decimals; include rounding of answers and checking reasonableness.* N.MR.05.21 Solve for the unknown in equations such as # + x = 7/12.*

Michigan Department of Education Grade 4

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Express, interpret, and use ratios; find equivalences N.MR.05.22 Express fractions and decimals as percentages and vice versa. N.ME.05.23 Express ratios in several ways given applied situations, e.g., 3 cups to 5 people, 3 : 5, 3/5 ; recognize and find equivalent ratios.

Sixth Grade Work with number is essentially completed by the end of sixth grade, where students’ knowledge of whole numbers and fractions (ratios of whole numbers, with non-zero denominators) should be introduced to integers and rational numbers. All of the number emphasis is intended to lay a foundation for the algebra expectations that are included in grade six. Students should use variables, write simple expressions and equations, and graph linear relationships. In geometry, students continue to expand their repertoire about shapes and their properties. NUMBER AND OPERATIONS Multiply and divide fractions N.MR.06.01 Understand division of fractions as the inverse of multiplication, e.g., if 4/5 ÷ 2/3 = ! , then 2/3 • = 4/5, so = 4/5 • 3/2 = 12/10. [Core] N.FL.06.02 Given an applied situation involving dividing fractions, write a mathematical statement to represent the situation. [Core] N.MR.06.03 Solve for the unknown in equations such as: 1/4 ÷ = 1, 3/4 ÷ = 1/4, and 1/2 = 1 • . [Fut] N.FL.06.04 Multiply and divide any two fractions, including mixed numbers, fluently. [Core – NC] Represent rational numbers as fractions or decimals N.ME.06.05 Order rational numbers and place them on the number line. [Ext] N.ME.06.06 Represent rational numbers as fractions or terminating decimals when possible, and translate between these representations. [Ext] N.ME.06.07 Understand that a fraction or a negative fraction is a quotient of two integers, e.g., - 8/3 is -8 divided by 3. [Fut] Add and subtract integers and rational numbers N.MR.06.08 Understand integer subtraction as the inverse of integer addition. Understand integer division as the inverse of integer multiplication. [Fut] N.FL.06.09 Add and multiply integers between -10 and 10; subtract and divide integers using the related facts. Use the number line and chip models for addition and subtraction. [Fut – NC] N.FL.06.10 Add, subtract, multiply and divide positive rational numbers fluently. [Core – NC] Find equivalent ratios N.ME.06.11 Find equivalent ratios by scaling up or scaling down. [Core] Solve decimal, percentage and rational number problems N.FL.06.12 Calculate part of a number given the percentage and the number. [Ext – NC] N.MR.06.13 Solve contextual problems involving percentages such as sales taxes and tips. Michigan Department of Education Grade 4

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[Core] N.FL.06.14 For applied situations, estimate the answers to calculations involving operations with rational numbers. [Core] N.FL.06.15 Solve applied problems that use the four operations with appropriate decimal numbers. [Core]

ALGEBRA Calculate rates A.P A.06.01 Solve applied problems involving rates, including speed, e.g., if a car is going 50 mph, how far will it go in 3 1/2 hours? [Core] Understand the coordinate plane A.RP.06.02 Plot ordered pairs of integers and use ordered pairs of integers to identify points in all four quadrants of the coordinate plane. [Core] Use variables, write expressions and equations, and combine like terms A.FO.06.03 Use letters, with units, to represent quantities in a variety of contexts, e.g., y lbs., k minutes, x cookies. [Core] A.FO.06.04 Distinguish between an algebraic expression and an equation. [Ext] A.FO.06.05 Use standard conventions for writing algebraic expressions, e.g., 2x + 1 means “two times x, plus 1” and 2(x + 1) means “two times the quantity (x + 1).” [Fut] A.FO.06.06 Represent information given in words using algebraic expressions and equations. [Core] A.FO.06.07 Simplify expressions of the first degree by combining like terms, and evaluate using specific values. [Fut] Represent linear functions using tables, equations, and graphs A.RP.06.08 Understand that relationships between quantities can be suggested by graphs and tables. [Ext] A.P A.06.09 Solve problems involving linear functions whose input values are integers; write the equation; graph the resulting ordered pairs of integers, e.g., given c chairs, the “leg function” is 4c; if you have 5 chairs, how many legs?; if you have 12 legs, how many chairs? [Fut] A.RP.06.10 Represent simple relationships between quantities using verbal descriptions, formulas or equations, tables, and graphs, e.g., perimeter-side relationship for a square, distance-time graphs, and conversions such as feet to inches. [Fut] Solve equations A.FO.06.11 Relate simple linear equations with integer coefficients, e.g., 3x = 8 or x + 5 = 10, to particular contexts and solve. [Core] A.FO.06.12 Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution. [Core] A.FO.06.13 Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solutions. [Core] A.FO.06.14 Solve equations of the form ax + b = c, e.g., 3x + 8 = 15 by hand for positive integer coefficients less than 20, use calculators otherwise, and interpret the results. [Fut]

Michigan Department of Education Grade 4

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Seventh Grade The main focus in grade seven is the algebra concept of linear relationships, including ideas about proportional relationships. Students should understand the relationship of equations to their graphs, as well as to tables and contextual situation for linear functions. In addition, work in algebra extends into simplifying and solving simple expressions and equations. The main concept from geometry in grade seven is similarity of polygons, which also draws on ideas about proportion. Students apply their understanding of ratio in data-based situations. NUMBER AND OPERATIONS Understand derived quantities N.MR.07.02 Solve problems involving derived quantities such as density, velocity, and weighted averages. [Fut] Understand and solve problems involving rates, ratios, and proportions N.FL.07.03 Calculate rates of change including speed. [Core] N.MR.07.04 Convert ratio quantities between different systems of units, such as feet per second to miles per hour. [Core] N.FL.07.05 Solve proportion problems using such methods as unit rate, scaling, finding equivalent fractions, and solving the proportion equation a/b = c/d; know how to see patterns about proportional situations in tables. [Core] Compute with rational numbers N.FL.07.07 Solve problems involving operations with integers. [Core] N.FL.07.08 Add, subtract, multiply, and divide positive and negative rational numbers fluently. [Core – NC] N.FL.07.09 Estimate results of computations with rational numbers. [Core – NC] ALGEBRA Understand and apply directly proportional relationships and relate to linear relationships A.P A.07.01 Recognize when information given in a table, graph, or formula suggests a directly proportional or linear relationship. [Fut] A.RP.07.02 Represent directly proportional and linear relationships using verbal descriptions, tables, graphs, and formulas, and translate among these representations. [Core] A.P A.07.03 Given a directly proportional or other linear situation, graph and interpret the slope and intercept(s) in terms of the original situation; evaluate y = mx + b for specific x values, e.g., weight vs. volume of water, base cost plus cost per unit. [Fut] A.P A.07.04 For directly proportional or linear situations, solve applied problems using graphs and equations, e.g., the heights and volume of a container with uniform crosssection; height of water in a tank being filled at a constant rate; degrees Celsius and degrees Fahrenheit; distance and time under constant speed. [Core] A.P A.07.05 Recognize and use directly proportional relationships of the form y = mx, and distinguish from linear relationships of the form y = mx + b, b non-zero; understand that in a directly proportional relationship between two quantities one quantity is a constant multiple of the other quantity. [Fut]

Michigan Department of Education Grade 4

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Understand and represent linear functions A.P A.07.06 Calculate the slope from the graph of a linear function as the ratio of “rise/run” for a pair of points on the graph, and express the answer as a fraction and a decimal; understand that linear functions have slope that is a constant rate of change. [Fut] A.P A.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and graph, interpreting slope and y-intercept. [Fut] A.FO.07.08 Find and interpret the x- and/or y-intercepts of a linear equation or function. Know that the solution to a linear equation of the form ax + b=0 corresponds to the point at which the graph of y = ax+ b crosses the x-axis. [Fut] Understand and solve problems about inversely proportional relationships A.P A.07.09 Recognize inversely proportional relationships in contextual situations; know that quantities are inversely proportional if their product is constant, e.g., the length and width of a rectangle with fixed area, and that an inversely proportional relationship is of the form y = k/x where k is some non-zero number. [Fut] A.RP.07.10 Know that the graph of y = k/x is not a line, know its shape, and know that it crosses neither the x- nor the y-axis. [Fut] Apply basic properties of real numbers in algebraic contexts A.P A.07.11 Understand and use basic properties of real numbers: additive and multiplicative identities, additive and multiplicative inverses, commutativity, associativity, and the distributive property of multiplication over addition. [Core] Combine algebraic expressions and solve equations A.FO.07.12 Add, subtract, and multiply simple algebraic expressions of the first degree, e.g., (92x + 8y) – 5x + y, or x(x+2) and justify using properties of real numbers. [Core] A.FO.07.13 From applied situations, generate and solve linear equations of the form ax + b = c and ax + b = cx + d, and interpret solutions. [Fut]

Solve problems N.MR.08.07 Understand percent increase and percent decrease in both sum and product form, e.g., 3% increase of a quantity x is x + .03x = 1.03x. N.MR.08.08 Solve problems involving percent increases and decreases. N.FL.08.09 Solve problems involving compounded interest or multiple discounts. N.MR.08.10 Calculate weighted averages such as course grades, consumer price indices, and sports ratings. N.FL.08.11 Solve problems involving ratio units, such as miles per hour, dollars per pound, or persons per square mile.* • revised expectations in italics

Understand the concept of non-linear functions using basic examples A.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inversely proportional relationships (y = k/x); cubics (y = ax3); roots (y = !x ); and exponentials (y = ax , a > 0); using tables, graphs, and equations.* A.PA.08.02 For basic functions, e.g., simple quadratics, direct and indirect variation, and population growth, describe how changes in one variable affect the others. A.PA.08.03 Recognize basic functions in problem context, e.g., area of a circle is "r2, volume of a sphere is "r3, and represent them using tables, graphs, and formulas. A.RP.08.04 Use the vertical line test to determine if a graph represents a function in one variable.

Michigan Department of Education Grade 4

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Understand and represent quadratic functions A.RP.08.05 Relate quadratic functions in factored form and vertex form to their graphs, and vice versa; in particular, note that solutions of a quadratic equation are the x-intercepts of the corresponding quadratic function. A.RP.08.06 Graph factorable quadratic functions, finding where the graph intersects the x-axis and the coordinates of the vertex; use words “parabola” and “roots”; include functions in vertex form and those with leading coefficient –1, e.g., y = x2 – 36, y = (x – 2)2 – 9; y = – x2; y = – (x – 3)2.

Recognize, represent, and apply common formulas A.FO.08.07 Recognize and apply the common formulas: (a + b)2 = a2 + 2 ab + b2 (a – b)2 = a2 – 2 ab + b2 (a + b) (a – b) = a2 – b2 ; represent geometrically. A.FO.08.08 Factor simple quadratic expressions with integer coefficients, e.g., x2 + 6x + 9, x2 + 2x – 3, and x2 – 4; solve simple quadratic equations, e.g., x2 = 16 or x2 = 5 (by taking square roots); x2 – x – 6 = 0, x2 – 2x = 15 (by factoring); verify solutions by evaluation. A.FO.08.09 Solve applied problems involving simple quadratic equations.

Understand solutions and solve equations, simultaneous equations, and linear inequalities A.FO.08.10 Understand that to solve the equation f(x) = g(x) means to find all values of x for which the equation is true, e.g., determine whether a given value, or values from a given set, is a solution of an equation (0 is a solution of 3x2 + 2 = 4x + 2, but 1 is not a solution). A.FO.08.11 Solve simultaneous linear equations in two variables by graphing, by substitution, and by linear combination; estimate solutions using graphs; include examples with no solutions and infinitely many solutions. A.FO.08.12 Solve linear inequalities in one and two variables, and graph the solution sets. A.FO.08.13 Set up and solve applied problems involving simultaneous linear equations and linear inequalities. * revised expectations in italics. Each expectation is labeled [Core], [Ext] (Extended Core), [Fut] (Future Core) or [NASL] (Not Assessed at the State Level); NC designates a Non-Calculator item

Michigan Department of Education Grade 4

www.michigan.gov/mde