Session 17 Complete

MMSTLC

Session 17.4 SVSU 12/04/08

OGAP Pancake Problem

Jamie is in charge of purchasing pancake mix for the Club’s Annual Breakfast fundraiser. He is using the ratio table below to determine the amount of mix to purchase.

Number of Pancakes

12

24

36

120

Cups of Pancake Mix

1 3/4

3 1/2

5 1/4

17 1/2

Milk

1 1/4

2 1/2

3 3/4

12 1/2

400

The club expects to make about 400 pancakes. •

How many cups of mix does Jamie need? Explain your reasoning.

•

Identify the structures of the problem, i.e., problem type, internal structure, multiplicative relationship, types of numbers, and representations.

OGAP Warm-up Task Marge Petit Consulting, MPC

12/08

A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR0227057) November 25, 2008 Version 5.0

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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 + ¼ = 1 ¼. 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

¾ - y = ½*. 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

www.michigan.gov/mde

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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., ½=4/8 and ¾ = 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 ¼ + x = 7/12.*

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 ; Michigan Department of Education Grade 4

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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. [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 Michigan Department of Education Grade 4

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decimal numbers. [Core]

ALGEBRA Calculate rates A.PA.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.PA.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]

Seventh Grade The main focus in grade seven is the algebra concept of linear relationships, including ideas Michigan Department of Education Grade 4

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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.PA.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.PA.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.PA.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.PA.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] Understand and represent linear functions A.PA.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.PA.07.07 Represent linear functions in the form y = x + b, y = mx, and y = mx + b, and Michigan Department of Education Grade 4

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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.PA.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.PA.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 πr2, volume of a sphere is πr3, 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.

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. Michigan Department of Education Grade 4

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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

MMSTLC

Session 17.5 SVSU 12/04/08

Scan Across a Unit and a Year Step 1: Review the table of contents of your mathematics program. Highlight the proportionality topics/contexts on the OGAP Proportionality Framework (use the copy attached) that are addressed in your mathematics program. Step 2: Select a MAJOR unit that focuses on developing proportional reasoning. Scan the unit and then highlight the structures evidenced in the problems across the unit. Indicate multiple hits on a structure with tic marks. Particularly look for: • • • • • • •

Mathematical topics and contexts Problem types Multiplicative relationships Internal structures For ratio problems – Referents (implied vs. explicit) Numbers Representations

Step 3: Given the GLECS at your grade level and the OGAP Framework answer the following questions. 1) What surprised you?

2) In what ways does your program support the GLECS at your grade level? In what ways does your program support the OGAP Framework Problem Structures?

3) In what ways does the unit (s) you reviewed provide opportunities for students to solve different types of problems with varying problem structures?

4) What gaps, if any, did you find between your program and the OGAP Framework Problem Structures?

• A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) November 2008 • Progam Review Task page 1 of 3

MMSTLC

Session 17.5 SVSU 12/04/08

STEP 4: 1) Join the other groups at your grade and program and complete the flip chart paper provided to you. Place your completed chart on the wall with the charts for other grades and your program.

(Note: We will come back to these analyses after you have analyzed the student work from the OGAP pre-assessment that you administered to your students. At that point you will know what strategies your students used to solve the problems and how problem structures did or did not affect their solution path.)

• A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) November 2008 • Progam Review Task page 2 of 3

Session 17.5 SVSU 12/04/08

• A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR0227057) November 2008 • Progam Review Task page 3 of 3

MMSTLC

Strand___________________

Strand___________________

Strand___________________

(These materials were created by the Vermont Mathematics Partnership funded by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057) © Vermont Institutes 2007) August 2007 V3

1

Strand___________________

Chapter 4 Adding it Up

Examples of the features you identified

How does this strand relate to the other strands?

Major features of this strand

Strand:

Session 5.1 - Strands of Mathematical Proficiency (NRC)

MMSTLC

Session 17.7 SVSU 12/4/08

Evidence in Student Work to Inform Instruction In this activity you wlll be using the OGAP Framework to help describe evidence in over 20 student solutions to problems that you have encountered in previous OGAP work. Questions to keep in mind: •

What is the solution strategy that the student used?

•

What is the evidence of that strategy?

•

What structure (s) in the problem facilitated the use of a proportional strategy or may have resulted in a student using either a transitional or non-proportional strategy?

•

What might your next instructional/assessment step be given the student solution? (e.g., what evidence of understanding can be built on? What else do you need to know to help make decisions about the next instructional step? What questions can you ask to build on understanding? What activities or models can be used?)

There are some underlying assumptions when asking about next instruction steps: •

A student solution usually provides evidence of understanding that can be built upon;

•

One might need to collect additional information about the student understanding as a part of the next step;

•

While you are identifying next instructional steps in response to one student response in this activity, these evidences are common across classrooms. So when you answer questions about individual pieces of student work in this activity think about this being an example of common errors across groups of students that can be applied to full classrooms of students.

•

Even when a student correctly solves a problem, there are instructional next steps to consider.

Important Note: The purpose of reviewing this work is NOT to spend time to reliably agree about the evidences, but to give us a way to describe the evidence that will inform instruction. A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

MMSTLC

Session 17.7 SVSU 12/4/08

A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

Uses y=mx

Sets up a proportion and uses cross products

Applies multiplicative relationship

Compares simplified fractions, rates, or ratios

Finds and applies unit rate to situation

•

•

•

•

•

•

Other

Misinterprets the meaning of the quantities

Remainders are not treated correctly

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Other

Uses incorrect ratio referent

Error in equation

Rounding errors

Computational error

Computational errors in student solutions:

•

•

•

•

Other

•

Underlying issues or concerns in student solutions:

No attempt

Not enough information to determine/lacks supporting evidence

Misunderstands vocabulary and related concept (e.g. ratio, similarity)

Solves a non-proportional situation proportionally

Uses whole number reasoning

Uses additive reasoning

•

•

•

•

•

•

The Vermont Mathematics Partnership is funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR0227057) November 21, 2008

Instructional Notes:

Units inconsistent or absent

Uses additive strategies rather than multiplicative strategy (e.g., uses repeated addition instead of multiplication)

Error in the application of cross products

•

•

•

•

Underlying Concerns/Errors

Makes an error in applying a multiplicative relationship

Makes a cross product error

Uses models

Finds equivalent fractions/ratios with an error

Builds up/down

Non-proportional Strategies Description of evidence to inform instruction: • Guesses or uses random application of numbers, operations, or strategies

OGAP Proportional Reasoning Item Analysis Sheet Transitional Proportional Strategies Description of evidence to inform instruction:

Underlying issues or concerns in student solutions:

• Other For ratio problems: • Applies the correct ratio referent

•

•

•

•

•

MMSTLC Session 17.8 Item Background: Proportional Strategies Description of evidence to inform instruction:

OGAP Proportional Reasoning Item Analysis Sheet

The Vermont Mathematics Partnership is funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR0227057) November 21, 2008

MMSTLC Session 17.8

MMSTLC

Session 17.8 SVSU 12/4/08

The OGAP Student Work Sort Process There are three steps to the OGAP Student Work Sort Process. For a single question: STEP 1: Review and then sort the work for the class into three piles consistent with the OGAP Proportionality Framework. Proportional Strategies

Transitional Proportional

Non-Proportional

STEP 2: Record the evidence on an OGAP Item Analysis Sheet by piles. We suggest starting with the Proportional Strategy pile of student work first and then repeat the process for each of the other piles. A) Record the strategy (you may want to sub sort the work first (e.g., All that use multiplicative relationships, or unit rate) by placing the students’ #s (in your case name, initials) that corresponds with the strategy. B) Record any underlying issues, errors, or misconceptions evidenced in the work by placing the students’ #s (in your case name, initials) that corresponds with the error et al.

Student 1

1

Student 1

STEP 3: In the “Instructional notes section or on the back make some quick notes about trends in the class or instructional ideas that you may have after reviewing the work. A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

1

These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

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These materials were created by the Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

9.2 Analyzing Pre-assessment Participant Directions Goals: • To gather evidence about strategies your students’ use when they solve proportionality problems to inform instruction and unit planning. • To gather evidence about any underlying issues, errors, or misconceptions found in student pre-assessments to inform instruction and unit planning. Materials Needed: • 5 OGAP Proportionality Item Analysis Sheets per person stapled together (9.4) • The Pre-assessment Analysis Directions(9.2) • Telling the Story (9.3) • Completed student pre-assessments Part I: Analyzing student work and collecting evidence on OGAP Item Analysis Sheets (2.5 hours) In General: You will analyze each item across all your students, NOT across a student. As with the practice in the last session you will NOT grade or score the student responses from the pre-assessment, but will collect descriptive information on the OGAP Item Analysis Sheet that will be used to inform instruction and unit planning. Suggested order for analyzing pre-assessments: Please analyze items in the order suggested below. You can see that we suggest first analyzing the rate/ratio comparison problems, then the missing value problems, and then other item types. Order 1 2 3 4 5

Grade 6 Raccoons Car traveled Bob’s Shower Marbles Sherwood Forest

Item Type Rate comparison Rate comparison Missing value Ratio Qualitative

Grade 7 Big Horn Ranch Similarity Paul’s Dog Bob’s Shower Kim and Bob

Item Type Rate comparison Ratio comparison Missing value Missing value Non-proportional

As you analyze each item we suggest the following: 1) Make notes about the structures of the problem that might influence student solutions on the fist line of an OGAP Item Analysis Sheet.

A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The 1 Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)

9.2 Analyzing Pre-assessment Participant Directions

2) Complete the OGAP Sort and collect evidence in the OGAP Item Analysis Sheet. IMPORTANT: We suggest that you actually put the students’ initials on the item analysis sheets. That way you won’t loose important individual student data as you analyze items across the classroom of students. 3) Write comments on the “Instructional Notes” section of the OGAP Item Analysis Sheet before moving onto the analysis of the next item. Complete analysis of all five items in this way. Part II: Telling the Story After you complete the analysis of all the items in the pre-assessment address these three questions on the Telling the Story template (9.3). 1) What are some strategies evidenced in the student work that you can build upon? 2) What are some underlying issues or concerns evidenced in the student work? 3) What are some implications for instruction? You will use the information from this activity in the next session as you do unit planning.

Part III: Telling the story across grades 1) Return to your school level team. In a round robin have each teacher “Tell the Story” for the group of students that they analyzed their pre-assessments (about 5 minutes each). 2) Be prepared to discuss general observations, findings, and implications for your school.

A derivative OGAP product created for MMSTLC November 2008. Original materials were developed as a part of The 2 Vermont Mathematics Partnership funded by a grant provided by the US Department of Education (Award Number S366A020002) and the National Science Foundation (Award Number EHR-0227057)