# LEAP Assessment Guide, Mathematics Grade 8

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1 This guide includes: LEAP Assessment Guide, Mathematics Grade 8 Purpose of Assessment Guide Introduction to LEAP Overview of Mathematics Task Types and Reporting Categories Design of LEAP Mathematics Assessments Assessable Content LEAP Test Administration Policies Resources Appendix I. Purpose of Assessment Guide This document is designed to assist Louisiana educators in understanding the LEAP mathematics assessment for grade 8, which will be administered in spring II. Introduction to LEAP All students in grades 3 8 will take the LEAP ELA and mathematics assessments. In order for Louisiana to maintain comparability between assessments administered in spring 2015 and spring 2016, a percentage of the items (not more than 49.9%) for the LEAP assessments comes from the Partnership for Assessment of Readiness for College and Careers (PARCC). PARCC is a group of states working together to develop high-quality assessments. The remaining percentage of items for the LEAP assessments comes from the College and Career Readiness Item Bank belonging to Data Recognition Corporation, winner of the LEAP mathematics and ELA test development contract. The LEAP assessments will offer the following: Consistency with the rigor and types of questions used in the spring 2015 Louisiana assessments Measurement of the full range of Louisiana content standards in ELA and mathematics Ability to measure the full range of student performance, including the performance of high- and low-performing students LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 1

7 Fill-in-the-Blank Grids The grid for grade 8 has a column for entering and shading a bubble when the answer is negative. See the example in the grid on the left below. Students will write the number in the boxes at the top of the grid. Numbers are entered without commas. Students will then shade the bubble in the column that corresponds to the entry in the top row. The recommended method for entry of the digits and a decimal point (if needed) is to start in column 2 as shown in the two examples on the left. Blank spaces within the answer are not allowed. Equivalent forms of numbers, such as 0.75 or for.75 as shown in the second example, are accepted providing that the response fits within any rounding limits that may be required by the question. For example, if a question requires the response to be rounded to the hundredths place, 0.75 or.75 would be accepted as correct, whereas would not be. Enter digit or decimal point starting in column 2. Do not skip boxes when entering the number. Use only if answer is a negative number. Note: Should a student mistakenly start in a column other than column 2, the entry will be scored as correct under the following conditions: There are no spaces within the answer. The answer fits within the remaining columns. The negative symbol is used when the answer is a negative number. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 7

9 Pointer tool Sticky Note tool Measurement tools Highlighter tool Magnifying tool Equation Builder Cross-Off tool Line Guide Help tool Calculator Mathematics Reference Sheet All students taking the computer-based tests should work through the Online Tools Training to practice using the online tools so they are well prepared to navigate the online testing system. Permitted Testing Materials The chart that follows summarizes the tools and resources for the grade 8 mathematics assessment. Provided (by vendor or part of online system) 1 8 -inch and centimeter ruler and mathematics reference sheet ASSESSMENT RESOURCES/TOOLS FOR GRADE 8 Required (provided by school) Scratch paper (lined, graph, or unlined) Other Allowable (may be used, not required) Yellow highlighter, protractor, tracing paper, reflection tools, straight edge, and compass Provided tools are sent by the test vendor to the districts for the districts to distribute during testing; districts and students may not substitute their own tools for provided tools. Required tools must be supplied by the school and distributed to all testers during testing. Schools may give or permit students to bring allowable tools. If schools permit students to bring their own allowable tools, tools must be given to the test administrator prior to testing to ensure that the tools are appropriate for testing (e.g., tools do not have any writing on them). LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 9

10 Grade 8 ruler provided on the LEAP paper-based mathematics assessment (not actual size): Grade 8 rulers provided on the LEAP computer-based mathematics assessment (not actual size): To ensure accurate measurement, the size of the computer-based ruler, along with the object being measured, varies depending on the computer monitor s resolution. To practice with the computer-based ruler and protractor, please visit the Online Tools Training. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 10

11 Calculators The LEAP mathematics test allows a scientific calculator in grade 8 during Sessions 2 and 3. Calculators are not allowed during Session 1 of the test. For students with the approved accommodation, a scientific calculator is allowed during all test sessions. The student should use the calculator they have regularly used throughout the school year in their classroom and are most familiar with, provided their regular-use calculator is not outside the boundaries of what is allowed. The following table includes calculator information by session for both general testers and testers with approved accommodations for calculator use. UPDATE: Clarification of Calculator Policy Test Mode PBT CBT Session Session 1 Session 2 Session 3 Session 1 Session 2 Session 3 Testers Not allowed Scientific, hand-held Scientific, hand-held Not allowed Scientific, online available may also have hand-held Scientific, online available may also have hand-held Testers with approved accommodation for calculator use Scientific, hand-held Scientific, hand-held Scientific, hand-held Scientific, hand-held Scientific, online available may also have hand-held Scientific, online available may also have hand-held Additional information for testers with approved accommodations for calculator use: Students may also use a hand-held four-function calculator in addition to the scientific calculator, provided the accommodation is documented. The four-function calculator may have square root, percent, memory, and +/- keys. If a student needs an adaptive calculator (e.g., large key, talking), the student may bring his or her own or the school may provide one, as long as it is specified in his or her approved IEP or 504 Plan. Additionally, schools must adhere to the following guidance regarding calculators: Calculators with Computer Algebra System (CAS) features are not allowed. Tablet, laptop (or PDA), or phone-based calculators are not allowed. Students are not allowed to share calculators within a testing session. Test administrators must confirm that memory on all calculators has been cleared before and after the testing sessions. Calculators with QWERTY keyboards are not permitted. If schools or districts permit students to bring their own hand-held calculators, test administrators must confirm that the calculators meet all the requirements as defined above. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 11

12 Reference Sheets Students in grade 8 will be provided a reference sheet with the information below. Grade 8 Reference Sheet 1 inch = 2.54 centimeters 1 meter = inches 1 mile = 5280 feet 1 mile = 1760 yards 1 mile = kilometers 1 kilometer = 0.62 mile 1 pound = 16 ounces 1 pound = kilogram 1 kilogram = 2.2 pounds 1 ton = 2000 pounds 1 cup = 8 fluid ounces 1 pint = 2 cups 1 quart = 2 pints 1 gallon = 4 quarts 1 gallon = liters 1 liter = gallon 1 liter = 1000 cubic centimeters Triangle A = 1 2 bh Cylinder V = πr 2 h Parallelogram A = bh Sphere V = 4 3 πr3 Circle A = πr 2 Cone V = 1 3 πr2 h Circle C = ππ or C = 2ππ Pythagorean Theorem a 2 + b 2 = c 2 General Prisms V = Bh Requisite Knowledge Students in grade 8 will be required to know relative sizes of measurement units within one system of units. Therefore, the following requisite knowledge is necessary for the grade 8 assessments and is not provided in the reference sheet. 1 meter = 100 centimeters 1 meter = 1000 millimeters 1 kilometer = 1000 meters 1 kilogram = 1000 grams 1 gram = 1000 milligrams 1 liter = 1000 milliliters 1 foot = 12 inches 1 yard = 3 feet 1 day = 24 hours 1 minute = 60 seconds 1 hour = 60 minutes Area formulas for rectangles For more information about accessibility and accommodations, please refer to the LEAP Accessibility Features and Accommodations Overview. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 12

14 VIII. Appendix Assessable Content for Sub-Claim A (Major Content) Sub-Claim A: Major Content 8.EE.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, = 3-3 = 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. 8.EE.A.3 Use numbers expressed in the form of a single digit times an integer 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. 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. 8.EE.B Understand the connections between proportional relationships, lines, and linear equations. 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. 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. 8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.7 Solve linear equations in one variable. a. 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). b. 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. a. 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. b. 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. c. 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. 8.F.A Define, evaluate, and compare functions. 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 1 1 Function notation is not required for grade 8. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 14

15 8.F.A.2 8.F.A.3 8.G.A 8.G.A.1 8.G.A.2 8.G.A.3 8.G.A.4 8.G.A.5 8.G.B 8.G.B.6 8.G.B.7 8.G.B.8 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. Understand congruence and similarity using physical models, transparencies, or geometry software. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Understand and apply the Pythagorean Theorem. Explain a proof of the Pythagorean Theorem and its converse. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. UPDATE: 8.F.B moved from Subclaim A to Sub-claim B in accordance with recent updates to PARCC Model Content Frameworks LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 15

16 Assessable Content for Sub-Claim B (Additional and Supporting Content) Sub-Claim B: Additional and Supporting Content 8.NS.A Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 8.NS.A.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 8.F.B Use functions to model relationships between quantities. 8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. 8.SP.A Investigate patterns of association in bivariate data. 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 16

17 Assessable Content for Sub-Claim C (Reasoning Applications) Sub-Claim C: Reasoning Applications Base reasoning on the principle that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Content Scope: Knowledge and skills articulated in 8.EE.B.6 - Tasks require students to derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intersecting the vertical axis at b. 8.EE.C.8a Given an equation or system of equations, present the solution steps as a logical argument that concludes with the set of solutions (if any). Content Scope: Knowledge and skills articulated in 8.EE.C.7a, 8.EE.C.7b, 8.EE.C.8b - Tasks may have three equations, but students are only required to analyze two equations at a time. Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. Content Scope: Knowledge and skills articulated in 8.F.A.3 Tasks require students to justify whether a given function is linear or nonlinear. 8.G.A.2, 8.G.A.4 8.G.A.5 7.RP.A, 7.NS.A, 7.EE.A Tasks may have scaffolding. 2 Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as = = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions. Content Scope: Knowledge and skills articulated in 8.EE.C.8c Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content Scope: Knowledge and skills articulated in 8.EE.B.6 8.G.A.2, 8.G.A.4 8.G.B - Some of tasks require students to use the converse of the Pythagorean Theorem. 2 Scaffolding in a task provides the student with an entry point into a pathway for solving a problem. In unscaffolded tasks, the student determines his/her own pathway and process. Both scaffolded and unscaffolded tasks will be included in reasoning and modeling items. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 17

18 Assessable Content for Sub-Claim D (Modeling Applications) Sub-Claim D: Modeling Applications Solve multi-step contextual word problems with degree of difficulty appropriate to Grade 8, requiring application of knowledge and skills articulated in Sub-claim A 3. Tasks may have scaffolding. 2 Solve multi-step contextual problems with degree of difficulty appropriate to grade 8, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.A.3, 7.EE, 7.G, and 7.SP.B. Tasks may have scaffolding. 2 Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique yields valuable results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional relationships to a phenomenon that is obviously nonlinear or statistical in nature) requiring knowledge and skills articulated in Sub-claim A 3. Tasks may have scaffolding. 2 Reasoned estimates: Use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity requiring knowledge and skills articulated in Sub-claim A 3. Tasks may have scaffolding. 2 3 Standards 8.EE.C.7a, 8.G.A.5 and 8.G.B.6 are not assessable in Modeling. LEAP Assessment Guide, Mathematics Grade 8 Posted: February 17, 2016 Page 18

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