Kate Collins Middle School Pre-Algebra Grade 6 1
1 - Real Number System How are the real numbers related? *some numbers can appear in more than one subset *the attributes of one subset can be contained in whole or in part in another subset *SETS: natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers Describe each subset of the set of real numbers and include examples and non-examples Describe orally and in writing the relationships among the sets Identify the subsets of the real number system to which a given number belongs. Determine whether a given number is a member of a particular subset of the real number system, and explain why. Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams Recognize that the sum or product of two rational numbers is rational Recognize that the sum of a rational number and an irrational number is irrational Students need additional practice illustrating the relationships among the subsets of the real number system. Examples: Which are NOT integers? Which fall ONLY in the rational subset? Which subset(s) does 4 fall within? Abs val distance not (-) center of universe idea, all directions from sun. N, S, E, W, all positive directions, different locations. 2 Recognize that the product of a nonzero rational number and an irrational number is irrational SOL 8.2: describe orally and in writing the relationships between the subsets of the real number system. SOL 7.1d determine square roots SOL 6.3c identify and describe absolute value of integers. 8/14-8/18
1 Applications of Operations with Rational Numbers: Multiplying and Dividing Fractions How are the real numbers related? *some numbers can appear in more than one subset *the attributes of one subset can be contained in whole or in part in another subset *SETS: natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers 3 Describe each subset of the set of real numbers and include examples and non-examples Describe orally and in writing the relationships among the sets Identify the subsets of the real number system to which a given number belongs. Determine whether a given number is a member of a particular subset of the real number system, and explain why. Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams Recognize that the sum or product of two rational numbers is rational Recognize that the sum of a rational number and an irrational number is irrational Recognize that the product of a nonzero rational number and an irrational number is irrational 8/21 to 9/1 6.4 demonstrate multiple representations of multiplication and division of fractions. 6.6 a) multiply and divide fractions and mixed numbers; and b) estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.
2 - Comparing and Ordering Numbers What is a rational number? *any number that can be written in fraction form How does the different ways rational numbers can be represented help us compare and order rational numbers? *numbers can be represented as decimals, fractions, percents, and in scientific notation. *useful to convert numbers to one representation to compare and order When are numbers written in scientific notation? *to represent very large and very small numbers Ordering in ascending order. Ordering in descending order. Other Examples: Which number would go in the space and keep the order from Greatest to Least? What do you know about: 50%, 0.5,, 5 x 10 4 What does a negative exponent mean when the base is 10? *represents a number between 0 and 1 Recognize powers of 10 with negative exponents by examining patterns Recognize a number greater than 0 in scientific notation Write a number greater than 0 in scientific notation Order no more than 3 numbers greater than 0 written in scientific notation. Compare and determine equivalent relationships between numbers larger than 0 written in scientific notation. Compare and order no more than 5 fractions, decimals, percents and numbers written in scientific notation using positive and negative exponents. SOL 6.5 investigate and describe concepts of positive exponents and perfect squares. SOL 6.2 8.1: b) compare and order decimals, fractions, percents, and numbers written in scientific notation. SOL 7.1: a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals, percents, and numbers written in scientific notation; 9/4 to 9/15
3 - Simplifying Expressions Add, subtract, multiply and divide integers. Demonstrate that the sums, differences, products and quotients of integers are either positive, zero, or negative through the use of patterns and models. Solve practical problems involving addition, subtraction, multiplication and division with integers What is the role of the order of operations? *prescribes the order to use to simplify expressions containing more than one operations *ensures there s only one correct answer Why is it important to apply properties of operations when simplifying expressions? *helps with understanding mathematical relationships Identify the properties of operations used in simplifying expressions Apply the properties of operations to simplify expressions Simplify numerical expressions containing: *exponents (base is a rational number, exponent is a positive whole number) *fractions, decimals, integers and square roots of perfect squares *grouping symbols (no more than 2 embedded grouping symbols Evaluate expressions for given replacement values *no more than 3 replacements per expression *exponents are positive and limited to whole numbers less than 4 *square roots are limited to perfect squares Apply the order of operations to evaluate formulas *positive exponents *square roots limited to perfect squares Notes: Grouping symbols include parentheses, brackets, braces, absolute value, division/fraction bar, and radical symbol. Use of GEMDAS or Order of Operations Pyramid. Use of fractions in expressions for challenge. 5
(continued) Compute simple interest and new balance earned in an investment or on a loan for a given number of years. Substitute values for variables in given formulas. *I = prt determine the value of I given p, r, and t. SOL 6.8 evaluate whole number numerical expressions, using the order of operations. SOL 7.3: a) model addition, subtraction, multiplication, and division of integers; b) add, subtract, multiply, and divide integers. 6 SOL 7.13: a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b) evaluate algebraic expressions for given replacement values of the variables. SOL 7.16: apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero. SOL 8.1: 9/18 to 9/22 a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers SOL 8.4 apply the order of operations to evaluate algebraic expressions for given replacement values of the variables
4 - Proportional Relationships Write a proportion given the relationship of equality between two ratios. Solve a proportion to find a missing term. Apply proportions to solve practical problems, including scale drawings. *denominators no greater than 12 and decimals no less than tenths. 7 What is a percent? *a special ratio with a denominator of 100 Compute a discount or markup and the resulting sale price for one discount or markup What is the difference between percent increase and percent decrease? *both percents of change measuring the percent a quantity increases or decreases *percent increase = growing change *percent decrease = lessening change Compute the percent increase or decrease for a one-step equation found in a real life situation. Solve practical problems by using computation procedures for whole numbers, integers, fractions, percents, ratios and proportions. Some involve application of a formula.
(continued) Solve problems involving tips, tax and discounts. *only one percent computation per problem Maintain a checkbook and check registry for five or fewer transactions. How do polygons that are similar compare to polygons that are congruent? *congruent polygons have the same size and shape ratio 1:1 *similar polygons have the same shape and corresponding angles are congruent, but the lengths of the corresponding sides are proportional Write proportions to express the relationships between the lengths of corresponding sides of similar figures. Calculation of markup amount and selling price. Calculation of discount amount and sale price. Calculation of tip and total paid. Calculation of tax and total cost. 8 SOL 6.1 describe and compare data, using ratios, and will use appropriate notations, such as a, a to b, and a:b. b SOL 7.4: solve single-step and multistep practical problems, using proportional reasoning. SOL 8.3: a) solve practical problems involving rational numbers, percents, ratios, and proportions; and b) determine the percent increase or decrease for a given situation. 9/25-10/13
5 - Probability How are the probabilities of dependent and independent events similar? Different? *dependent when the outcome of one influenced by the outcome of the other *independent when the outcome of one is influenced by the outcome of the other Determine the probability of a compound event containing no more than 2 events. Compare the outcomes of events with and without replacement Determine the probability of no more than 3 independent events. Determine the probability of no more than 2 dependent events without replacement. Understand various ways independent events can be described ( one die, two dice, one spinner, a bag of marbles with replacement). Dependent probability the 2 nd event is effected by the 1 st event (without replacement, set aside, etc. ) Examples: 20% chance of rain Monday, 40% chance of rain Tuesday. What is probability it will NOT rain Monday and Tuesday. Given a Dependent probability scenario, then asking what is the probability of the 2 nd event occurring. 9 SOL 6.16 a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events. SOL 7.9 investigate and describe the difference between the experimental probability and theoretical probability of an event. SOL 7.10: determine the probability of compound events, using the Fundamental (Basic) Counting Principle. SOL 8.12: determine the probability of independent and dependent events with and without replacement. 10/16-10/27
6 - Equations and Inequalties Identify properties of operations used to solve an equation from among: commutative property (addition and multiplication) associative property (addition and multiplication) distributive property identity properties (addition and multiplication) inverse properties (addition and multiplication) multiplicative property of zero Combining like terms Solve 2 to 4 step linear equations in one variable Methods of solving 2-4 step linear equations should include: using concrete materials pictorial representations, paper and pencil illustrating the steps performed. 10 How is the solution to an inequality different from that of a linear equation? *an inequality has more than one value for the variable that makes the inequality true Identify a numerical value that satisfies the inequality. For challenge, use fraction coefficients and constants. How are the procedures for solving equations and inequalities the same? *same with one exception when an inequality is multiplied or divided on both sides by a negative number. Solve two-step inequalities in one variable by showing the steps and using algebraic sentences. Graph solutions to two-step linear inequalities on a number line. SOL 7.15: a) solve one-step inequalities in one variable; and b) graph solutions to inequalities on the number line. SOL 8.15: a) solve multistep linear equations in one variable with the variable on one and two sides of the equation; b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation. 11/6 12/8
7 - Linear Relationships What are the different ways to represent the relationship between two sets of numbers? *word sentences, equations, tables of values, graphs or illustrated pictorially Describe and represent relations and functions, using tables, graphs, words, and rules. Given one representation, represent it another form. Graph in a coordinate plane ordered pairs that represent a relation. Construct a table of ordered pairs by substituting values for x in a linear equation to find values for y. Plot in the coordinate plane ordered pairs (x,y) from a table. Connect the ordered pairs to form a straight line (a continuous function). What types of real life situations can be represented with linear equations? *any situation with a constant rate Given equations, words, tables, graphs students sort to find the matching words to equation to table to graph. Students need to be able to identify multiple ways to represent a table (e.g. by graph, by equation, by words). DIXI to help remember (Domain, Input, X-coordinate, Independent Variable) ROYD to help remember (Range, Output, Y-coordinate, Dependent Variable) 11 Interpret the unit rate of the proportional relationship graphed as the slope of the graph. Compare two different proportional relationships in different ways.
(continued) What are the similarities and differences among the terms domain, range, independent variable and dependent variable? *value of the dependent variable changes as the independent variable changes *domain = set of all input values for the independent variable *range = set of all possible values for the dependent variable Apply the following algebraic terms appropriately: domain, range, independent variable and dependent variable. 12 Determine the independent variable of a relationship. Determine the dependent variable of a relationship. Determine the domain of a function. Determine the range of a function. Identify examples of domain, range, independent variable and dependent variable. SOL 8.14: make connections between any two representations (tables, graphs, words, and rules) of a given relationship. SOL 8.16: graph a linear equation in two variables. SOL 8.17: identify the domain, range, independent variable, or dependent variable in a given situation. 1/9to 1/19
8 - Statistics Collect, organize and interpret a data set of no more than 20 items using scatterplots. Interpret a set of data points in a scatterplot as having a positive relationship, a negative relationship, or no relationship. What are the inferences that can be drawn from sets of data points having a positive relationship, a negative relationship, and no relationship? *positive = values of the two variables are increasing *negative = as the value of the independent variable increases, the value of the dependent variable decreases Predict from the trend an estimate of the line of best fit with a drawing. Review bar, circle, histogram, picture, and line graphs making predictions, inferences, comparisons. Scatterplots As x increases, what does y do? The x-axis has the Independent variable, the y-axis has the dependent variable. Students interpret the meaning of the relationship shown on the scatterplot. (Ex. Taller mothers tend to have taller daughters) As well as make predictions using the line of best fit. 13 Why do we estimate a line of best fit for a scatterplot? *helps in making interpretations and predictions about the situation modeled in the data set. SOL 8.13: a) make comparisons, predictions, and inferences, using information displayed in graphs; and b) construct and analyze scatterplots. 6.14 The student, given a problem situation, will a) construct circle graphs; b) draw conclusions and make predictions, using circle graphs; and c) compare and contrast graphs that present information from the same data set. 6.15 a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose. 7.11 The student, given data in a practical situation, will a) construct and analyze histograms; and b) compare and contrast histograms with other types of graphs presenting information from the same data set. 1/22 1/26
9 - Angle Relationships Measure angles of less than 360 using appropriate tools. How are vertical, adjacent, complementary and supplementary angles related? *adjacent = two non-overlapping angles that share a common side and a common vertex *vertical = will always be nonadjacent angles and have the same measurement *supplementary = sum of the angle measurements is 180 *complementary = sum of the angle measurements is 90 *supplementary and complementary angles may or may not be adjacent. 14 Identify and describe the relationships between angles formed by two intersecting lines. Identify and describe the relationship between pairs of angles that are vertical. Identify and describe the relationship between pairs of angles that are adjacent. Identify and describe the relationship between pairs of angles that are supplementary. Identify and describe the relationship between pairs of angles that are complementary. Use the relationships among supplementary, complementary, vertical and adjacent angles to solve practical problems. SOL 6.12 determine congruence of segments, angles, and polygons. SOL 7.6 determine whether plane figures quadrilaterals and triangles are similar and write proportions to express the relationships between corresponding sides of similar figures. SOL 8.6: a) verify by measuring and describe the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles; and b) measure angles of less than 360. 1/29 2/9
10 Compare and Contrast Quadrilaterals use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Compare and contrast attributes of the following quadrilaterals: parallelogram, rectangle, square, rhombus, and trapezoid. Identify the classification(s) to which a quadrilateral belongs, using deductive reasoning and inference. A quadrilateral is a closed plane (two-dimensional) figure with four sides that are line segments. A parallelogram is a quadrilateral whose opposite sides are parallel and opposite angles are congruent. A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are the same length and bisect each other. A square is a rectangle with four congruent sides whose diagonals are perpendicular. A square is a rhombus with four right angles. A rhombus is a parallelogram with four congruent sides whose diagonals bisect each other and intersect at right angles. A trapezoid is a quadrilateral with exactly one pair of parallel sides. A trapezoid with congruent, nonparallel sides is called an isosceles trapezoid. Quadrilaterals can be sorted according to common attributes, using a variety of materials. A chart, graphic organizer, or Venn diagram can be made to organize quadrilaterals according to attributes such as sides and/or angles. Why can some quadrilaterals be classified in more than one category? Every quadrilateral in a subset has all of the defining attributes of the subset. For example, if a quadrilateral is a rhombus, it has all the attributes of a rhombus. However, if that rhombus also has the additional property of 4 right angles, then that rhombus is also a square 15 6.13 describe and identify properties of quadrilaterals. 7.7 compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. 1/29 2/9cont
11 - Perfect Squares and Pythagorean Theorem How does the area of a square relate to the square of a number? *area determines the perfect square number *if it is not a perfect square, area provides a means for estimation Define a perfect square Identify the perfect squares from 0 to 400 Identify the two consecutive whole numbers between which the square root of a given whole number from 0 to 400 lies. Why do numbers have both positive and negative roots? *the square root of a number is any number which when multiplied by itself equals the number Graph If a scalene the triangle on the has number sides line that (as measure a way 12 to show cm, 8 the cm, two and consecutive 15 cm, could whole it also numbers a right the triangle? square root of a number falls between). A triangle has sides that measure 5 in, 3 in, and 4 in. Is it a right triangle? 16 Find the square roots of a given whole number from 0 to 400. *use the symbol to ask for the positive root *use the symbol to ask for the negative root (continued) How can the area of squares generated by the legs and the hypotenuse of a right triangle be used to verify the Pythagorean Theorem? *the area of the square with one side equal to the measure of the hypotenuse = the sum of the areas of the squares with one side each equal to the measures of the legs of the triangle Identify the parts of a right triangle (hypotenuse and legs)
Verify the Pythagorean Theorem, using diagrams, concrete materials and measurement. Verify a triangle is a right triangle given the measure of its three sides. Find the measure of a side of a right triangle, given the measures of the other two sides. Solve practical problems involving right triangles by using the Pythagorean Theorem. 17 SOL 8.5: a) determine whether a given number is a perfect square; and b) find the two consecutive whole numbers between which a square root lies. SOL 8.10: a) verify the Pythagorean Theorem; and b) apply the Pythagorean Theorem. 2/12 2/16
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12 - Composite 2D Figures How does knowing the areas of polygons assist in calculating the areas of composite figures? *composite figures can be subdivided into triangles, rectangles, squares, trapezoids, semi-circles *the area of a composite figure = the sum of the subdivided areas Subdivide a figure into triangles, rectangles, squares, trapezoids and semicircles. Estimate the area of subdivisions and combine to determine the area of the composite figure. 19 Apply perimeter, circumference and area formulas to solve practical problems. Also, determine the area and/or perimeter of shapes inside of shapes. SOL 6.9 make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system. SOL 6.10 SOL 8.11: a) define pi (π) as the ratio of the circumference of a circle to its diameter; b) solve practical problems involving circumference and area of a circle, given the diameter or radius; c) solve practical problems involving area and perimeter; and d) describe and determine the volume and surface area of a rectangular prism. solve practical area and perimeter problems involving composite plane figures. 3/5 3/16
13 - Orthographic Projections How does knowledge of the two-dimensional figures inform work with three-dimensional objects? *a three-dimensional object can be represented as a two-dimensional model with views of the object from different perspectives. Identify three-dimensional models given a two-dimensional perspective Construct three-dimensional models given the top or bottom, side and front views 20 SOL 8.9: construct a three-dimensional model, given the top or bottom, side, and front views.. 4/16-4/20
14 - Surface Area and Volume Describe the two-dimensional figures that result from slicing threedimensional figures parallel to the base. How does the volume of a three-dimensional figure differ from its surface area? *volume = amount a container holds *surface area = sum of the area on the surfaces of the figure Distinguish between situations that are applications of surface area and those that are applications of volume. 21 How are the formulas for the volume of prisms and cylinders similar? *both find the area of the base and multiply it by the height Solve practical problems that require finding the volume of a prism. Solve practical problems that require finding the volume of a cylinder. How does the volume of a rectangular prism change when one of the attributes is increased? *when one dimension is changed by a factor greater than 1, the volume is increased by that same factor Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor *factors of 2, 3, 5 or 10 Compare and contrast the volume and surface area of a prism with a given set of attributes Investigate and compute the volume of a prisms (rectangular or triangular), cylinders, cones, and pyramids (square or triangular) using concrete objects, nets, diagrams and formulas.
Investigate and compute the surface area of a prisms (rectangular or triangular), cylinders, cones, and pyramids (square or triangular) by finding the sum of the areas of the faces using concrete objects, nets, diagrams and formulas. How are the formulas for the volume of cones and pyramids similar? *volume of a cone = and height times the volume of a cylinder with the same size base 22 *volume of a pyramid = and height times the volume of a prism with the same size base Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a scale factor. Solve practical problems involving volume and surface area of prisms, cylinders, cones and pyramids. SOL 7.5: a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing one measured attribute of a rectangular prism affects its volume and surface area. SOL 8.7: a) investigate and solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids; and b) describe how changing one measured attribute of a figure affects the volume and surface area. 3/19-3/23
15 - Transformations How does the transformation of a figure affect the size, shape and position of that figure? *translations, rotations and reflections do not change the size or shape, but do change the location *dilations do NOT change the shape, dilations by a scale factor other than 1 produces a proportional figure *rotations and reflections change the orientation of the image. Identify the type of transformation in a given example. Identify practical applications of transformations. *tiling, fabric, wallpaper designs, art and scale drawings Demonstrate the translation of a polygon on a coordinate grid. *identify the coordinates of the image Demonstrate the reflection of a polygon over the vertical or horizontal axis on a coordinate grid. *identify the coordinates of the image Demonstrate 90, 180, 270, and 360 clockwise and counterclockwise rotations of a figure on a coordinate grid. *center of rotation will be limited to the origin. *identify the coordinates of the image Demonstrate the dilation of a polygon from a fixed point on a coordinate grid. *scale factors of,, 2, 3 or 4 *identify the coordinates of the image *center of dilation will be the origin SOL 7.8: The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations by graphing in the coordinate plane. SOL 8.8: Students need to practice demonstrating transformations in the coordinate plane and showing knowledge of the specific transformation rules. a) apply transformations to plane figures; and b) identify applications of transformations. 4/19 4/13 23