Linden Community Schools

STRAND II: Geometry and Measurement

Content Standard 1: Shape and Shape Relationships-Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes.

Grade Level: 8^th Grade

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 2 Generalize the characteristics of shapes and apply their generalizations to classes of shapes.	G.SR.08.03 Understand the definition of a circle; know and use the formulas for circumference and area of a circle to solve problems. G.SR.08.05 Solve applied problems involving areas of triangles, quadrilaterals, and circles. G.SR.08.07 Understand the concept of surface area and find the surface area of prisms, cones, spheres, pyramids, and cylinders.	Explore physical models of shapes to determine characteristics. Find the surface area of prisms, cones, spheres, pyramids, and cylinders using given formulas. Find circumference and area of a circle.	Draw three different shapes and describe what they have in common. Students will find the area of a given shape, when given dimensions.
Benchmark 4 Construct familiar shapes using coordinates, appropriate tools (including technology), sketching and drawing two- and three-dimensional shapes.	G.SR.08.08 Sketch a variety of two-dimensional representations of three-dimensional solids, including: orthogonal views (top, front, and side), picture views (projective or isometric) and nets, and use such two-dimensional representations to help solve problems.	Given a three dimensional shape, students will draw a two-dimensional net, and be able to identify the top, front, and side of the shape.	A parallelogram can be drawn many different ways. Complete the following 2 parts. Show all of your work. A. Using your ruler, draw a parallelogram with at least one right angle. B. Identify 3 other characteristics of your parallelogram.
Benchmark 5 Combine, dissect and transform shapes.	G.SR.08.04 Find area and perimeter of complex figures by subdividing them into basic shapes (quadrilaterals, triangles, circles).	Divide irregular shapes into basic shapes to find the area and perimeter.	Example Assessment Question: Given an irregular 6-sided polygon, students will find the area and perimeter.
Benchmark 6 Generalize about the common properties of similar, congruent, parallel and perpendicular shapes and verify their generalizations informally.	G.TR.08.09 Understand the definition of dilation from a point in the plane and relate it to the definition of similar polygons. G.TR.08.10 Understand and use reflective and rotational symmetries of two- dimensional shapes and relate them to transformations to solve problems.	Looking at two shapes, determine whether they are similar or congruent. Students will find corresponding sides and angles of similar figures. In this logo, which transformation of the red (dark) part will produce the blue (light) part? A. A translation, then a reflection. B. A reflection, then a translation. C. A 180° clockwise rotation. D. A 90° clockwise rotation. Answer C	Assessment example: Students will determine if two shapes are similar and determine the scale factor.
Benchmark 7 Use shape, shape properties and shape relationships to describe the physical world and to solve problems.	G.GS.08.01 Understand at least one proof of the Pythagorean Theorem; use the Pythagorean Theorem and its converse to solve applied problems, including perimeter, area, and volume problems. G.SR.08.05 Solve applied problems involving areas of triangles, quadrilaterals, and circles. G.SR.08.08 Sketch a variety of two-dimensional representations of three-dimensional solids, including: orthogonal views (top, front, and side), picture views (projective or isometric) and nets, and use such two-dimensional representations to help solve problems.	Find the diagonal of a rectangle using the Pythagorean Theorem. Use isometric dot paper to draw 2-dimensional representations of 3-dimensional objects. Construct 3-dimensional models to correspond to a drawing or diagram. Solve area problems involving triangles, quadrilaterals, and circles.	Tyrone is decorating his room. He has glow-in-the-dark poster board that measures 72 cm x 56 cm. What is the area of the biggest triangle he can make from this poster board? A. 4032 cm^2. B. 2016 cm^2. C. 1008 cm^2. D. 720 cm^2. Answer B

Content Standard 2: Position -Students identify locations of objects, identify location relative to other objects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object.

Grade Level: 8^th Grade

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 2 Locate and describe objects in terms of their orientation and relative position, including coincident, collinear, parallel, perpendicular; differentiate between fixed (e.g., N-S-E-W) and relative (e.g., right-left) orientations; recognize and describe examples of bilateral and rotational symmetry.	G.TR.08.10 Understand and use reflective and rotational symmetries of two- dimensional shapes and relate them to transformations to solve problems.	Create a figure with rotational symmetry (e.g., 30°, 60°, 90°, 120°, 150° or 180°).	Create or find a figure with bilateral or rotational symmetry and explain the type of symmetry it has. If bilateral, show the lines of symmetry and if rotational, state the degrees of rotation.
Benchmark 3 Describe translations, reflections, rotations and dilations using the language of transformations, and employ transformations to verify congruence of figures.	G.TR.08.09 Understand the definition of a dilation from a point in the plane and relate it to the definition of similar polygons. G.TR.08.10 Understand and use reflective and rotational symmetries of two- dimensional shapes and relate them to transformations to solve problems.	Select a figure to transform and write the algorithm (directions) for a transformation in 3 or 4 steps. Exchange the directions with another student. Follow the directions for the transformation. Did the directions result in the intended transformation? Students will use scale factors to create similar figures. Students will use scale factors to determine the distance between two cities on a map.	Students should engage in activities in which they see that one example of how dilation may be clarified is by indicating in parentheses “made larger.” Students should know transformation vocabulary and know what specific transformations look like, by going from words to pictures/pictures to words. List the steps that transform the line segment in Quadrant 1 to its image in Quadrant 3. Answer: One possible answer is to reflect over the y-axis and translate 7 units down.
Benchmark 4 Locate the position of points or objects described by two or more conditions; locate all the points (locus) that satisfy a given condition.	G.LO.08.02 Find the distance between two points on the coordinate plane, using the distance formula; recognize that the distance formula is an application of the Pythagorean Theorem.	Students will find the distance between two points on graph paper, by using the Pythagorean theorem.	Students will be able to place at least five different points equidistance from a given point on a coordinate graph.

Linden Community Schools

STRAND IV: Sense and Numeration

Content Standard 1: Concepts and of Numbers-Students experience and measuring activities to develop intuitive sense about numbers, develop understanding about properties of numbers, understand the need for and existence of different sets of numbers, and investigate properties of special numbers.

Grade Level: 8^th Grade Math

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 1 Develop an understanding of and rational numbers and represent rational numbers in both fraction and decimal form.	N.ME.08.03 Understand that in decimal form, rational numbers either terminate or eventually repeat, and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals, e.g., 0.1= 1/9; 0.3= ÿ .	Convert between fractions and decimals. Notice that some decimals will terminate and some will repeat.	Compare fractions and decimals and place them on a number line.
Benchmark 2 Extend their understanding of numeration systems to include decimal numeration, scientific numeration and non-decimal numeration systems.	N.ME.08.01 Understand the meaning of a square root of a number and its connection to the square whose area is the number; understand the meaning of a cube root and its connection to the volume of a cube. N.ME.08.02 Understand meanings for zero and negative integer exponents.	Find the area of a square and its relationship to the sides. Find the volume of a cube and its relationship to the sides. Find the outcome of increasing and decreasing exponents.	Given the area of a square, students will find the side length. Given the volume of a cube, students will find the side length.

Content Standard 2: Representation and Uses of Numbers-Students recognize that numbers are used in different ways such as , measuring, ordering and estimating, understand and produce multiple representations of a number, and translate among equivalent representations.

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 1 Give geometric representations of fractions, prime and composite numbers, triangular and numbers, and other number concepts; represent rational numbers and on the number line.	N.ME.08.03 Understand that in decimal form, rational numbers either terminate or eventually repeat, and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals, e.g., 0.1= 1/9; 0.3= ÿ .	See Strand IV, benchmark 1	See Strand IV, benchmark 1
Benchmark 2 Recognize equivalent representations of a , especially fractions, decimals and percents, and translate freely among representations.	N.ME.08.03 Understand that in decimal form, rational numbers either terminate or eventually repeat, and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals, e.g., 0.1= 1/9; 0.3= ÿ .	Write the fraction in lowest terms that is equivalent to a given decimal.	Test
Benchmark 4 Develop and refine strategies for estimating quantities, including fractional quantities, and evaluate the reasonableness and appropriateness of their estimates.	N.ME.08.04 Understand that irrational numbers are those that cannot be expressed as the quotient of two integers, and cannot be represented by terminating or repeating decimals; approximate the position of familiar irrational numbers, e.g. ÿ2 , ÿ3 , ÿ) on the number line. N.FL.08.05 Estimate and solve problems with square roots and cube roots using calculators. N.FL.08.06 Find square roots of perfect squares and approximate the square roots of non-perfect squares by locating between consecutive integers, e.g. ÿ between 11 and 12.	Estimate the square root of a given number by using perfect squares. Explain the difference between rational and irrational numbers. Discuss perfect squares are rational numbers, however all other square roots are irrational.	Given a list of numbers, identify as rational or irrational. Approximate the square root of a number using perfect squares.
Benchmark 5 Select appropriate representations for numbers, including and rational numbers, in order to simplify and solve problems.	N.FL.08.05 Estimate and solve problems with square roots and cube roots using calculators. 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.MR.08.11 Solve problems involving ratio units such as miles per hour, dollars per pound, or persons per square mile.	Students will find discounted prices and price mark-ups. Students will find compound interest and monthly payments on loans. Decide what their next test score would need to be in order to raise their grade to a given percent using weighted percentages. Solve problems involving ratio units such as miles per hour, dollars per pound, or persons per square mile.	Given regular prices, students will find the sale price when given a percent discount. Given different loan amounts, students will use compound interest to find monthly payments. Students will average their grades, given weighted percentages. Students will find unit rates.

Content Standard 3: Number Relationships-Students investigate relationships such as equality, inequality, inverses, factors and multiples, and represent and very large and very small numbers.

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 1 and order and rational numbers using relations of equality and inequality.	N.ME.08.03 Understand that in decimal form, rational numbers either terminate or eventually repeat, and that calculators truncate or round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals, e.g., 0.1= 1/9; 0.3= ÿ . N.ME.08.04 Understand that irrational numbers are those that cannot be expressed as the quotient of two integers, and cannot be represented by terminating or repeating decimals; approximate the position of familiar irrational numbers, e.g. ÿ2 ÿ3, on the number line.	Compare rational and irrational numbers using inequality symbols.	Given a list a numbers, students will use inequality symbols to compare.
Benchmark 2 Express a numerical comparison as ratios and rates.	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. 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.	Students will find the amount of increase and the new total after the percent increase. Graph linear and nonlinear relationships and explain how slope represents a rate of change.	Find the total cost of a meal after adding tax and tip. Graph linear and nonlinear relationships and explain how slope represents a rate of change.
Benchmark 3 Distinguish between prime and composite numbers; identify factors, multiples, common factors and multiples, and relatively prime numbers; and apply divisibility tests to numbers.	N.ME.08.01 Understand the meaning of a square root of a number and its connection to the square whose area is the number; understand the meaning of a cube root and its connection to the volume of a cube. N.ME.08.02 Understand meanings for zero and negative integer exponents.	Develop a procedure for finding common factors, common multiples, greatest common factors, and least common multiples. Develop strategies for determining whether a number is prime.	Given sets of numbers, find the GCF and LCM. Write out the factors of a given number.
Benchmark 4 Explain the meaning of powers and roots of numbers and use calculators to compute powers and roots.	A.RP.08.01 Identify and represent linear functions, quadratic functions, and other simple functions including inverse functions (y = k/x), cubics (y = ax³) roots, (y =ÿx ), and exponentials (y = xª , x > 0), using tables, graphs, and equations.	Write numbers with exponents in expanded form. (Ex. 5^2= 5 X 5) Use tables to graph linear, quadratic, and other simple functions. Use calculators to compute powers and square roots.	Find the square roots and powers of given numbers. Make a graph, given an equation involving an exponent.
Benchmark 5 Apply their understanding of in solving problems.	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.MR.08.11 Solve problems involving ratio units such as miles per hour, dollars per pound, or persons per square mile.	Students should engage in activities in which they recognize number relationships including equality, inequality, inverses, factor, multiples. Students will be asked to compare numbers to each other using differences, ratios, rates, etc. Expect applied problem solving questions requiring more than one step; read carefully.	Solve problem like: Ann swam 54 laps in 45 minutes. If she continues at this rate, how many laps would you expect her to complete in 1 hour?

Linden Community Schools

STRAND V: Numerical and Algebraic Operations and Analytical Thinking

Content Standard 2: Algebraic and Analytic Thinking-Students analyze problems to determine an appropriate process for solution and use algebraic notations to model or represent problems.

Grade Level: 8^th Grade Math

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 1 Read and write algebraic expressions; develop original examples expressed verbally and algebraically; simplify expressions and translate between verbal and algebraic expressions; and solve linear equations and 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.08.12 Solve linear inequalities in one and two variables, and graph the solution sets.	Given a story problem, students will write the expression that is given. Given an expression, students will describe a real life example. Graph inequalities and show the solution set.	Students will match an expression with a corresponding story problem. Given a linear inequality, students will graph the solution set.
Benchmark 3 Solve linear equalities and inequalities using algebraic and geometric methods, and use the context of the problem to interpret and explain their solutions.	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 fr om 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.	Use number lines to model solutions to linear equations and inequalities in one variable. (For example, the solution to 2x + 3 > 7 and the graph of its solution set is shown.)	Kellogg Middle School is selling T-shirts as a fundraiser. The graph below shows prices charged by two different companies. One company charges less per T-shirt, but has a one-time initial setup charge. The other company has no setup charge. Use the graph to answer the following questions: How much does Company A charge per t-shirt? [NOTE: $5.00.] How many T-shirts would have to be purchased to make Company B more economical? [NOTE: 7 or more; at 6 both companies have same price. What inequality are you solving?] How much is the initial setup charge for Company B? [NOTE: $20.00.]

Linden Community Schools

STRAND VI: Probability and Discrete Mathematics

Content Standard I: Probability -Students develop an understanding of the notion of certainty and of probability as a of the of likelihood that can be assigned to a given event based on the knowledge available, and make critical judgments about claims that are made in probabilistic situations.

Grade Level: 8^th Grade

Benchmark	Performance Description	Recommended Activities	Recommended Assessments
Benchmark 3 Conduct experiments and give examples to illustrate the difference between dependent and independent events.	D.PR.08.04 Apply the Basic Counting Principle to find total number of outcomes possible for independent and dependent events, and calculate the probabilities using organized lists or tree diagrams. D.PR.08.06 Understand the difference between independent and dependent events, and recognize common misconceptions involving probability, e.g., Alice rolls a 6 on a die three times in a row; she is just as likely to roll a 6 on the fourth roll as she was on any previous roll.	Given a scenario, students will find the number of outcomes using the Basic Counting Principle and show them using a tree diagram. Given multiple events students will determine whether the events are independent or dependent of each other.	Students will find the number of different lunch combinations that are available.
Benchmark 4 Explain the difference between probabilities determined from experiments or chance events (empirical) and probabilities derived mathematically (theoretical), and explain how the empirical probability changes for a large of trials.