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Kansas Math Standards - 7th Grade
MathScore aligns to the Kansas Math Standards for 7th Grade. The standards appear below along with the MathScore topics that match. If you click on a topic name, you will see sample problems at varying degrees of difficulty that MathScore generated. When students use our program, the difficulty of the problems will automatically adapt based on individual performance, resulting in not only true differentiated instruction, but a challenging game-like experience.
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Standard 1: Number and Computation
Benchmark 1: Number Sense . The student demonstrates number sense for rational numbers, the irrational number pi, and simple algebraic expressions in one variable in a variety of situations.KBI-1 knows, explains, and uses equivalent representations for rational numbers and simple algebraic expressions including integers, fractions, decimals, percents, and ratios; integer bases with whole number exponents; positive rational numbers written in scientific notation with positive integer exponents; time; and money (2.4.K1ac) ($), e.g., 253,000 is equivalent to 2.53 x 105 or x + 5x is equivalent to 6x. (Positive Number Line , Number Line , Compare Integers , Fractions to Decimals , Decimals To Fractions , Percentages , Proportions 1 , Simplifying Algebraic Expressions , Ratios , Exponent Basics , Scientific Notation )
KBI-2 compares and orders rational numbers and the irrational number pi (2.4.K1a) ($). (Compare Mixed Values , Compare Mixed Values 2 )
KBI-3 explains the relative magnitude between rational numbers and between rational numbers and the irrational number pi (2.4.K1a). (Number Line )
KBI-4 knows and explains what happens to the product or quotient when (2.4.K1a):
a. a whole number is multiplied or divided by a rational number greater than zero and less than one,
b. a whole number is multiplied or divided by a rational number greater than one,
c. a rational number (excluding zero) is multiplied or divided by zero.
KBI-5 explains and determines the absolute value of rational numbers (2.4.K1a). (Absolute Value 1 , Absolute Value 2 )
AI-1 generates and/or solves real-world problems using (2.4.A1a) ($):
a. ▴equivalent representations of rational numbers and simple algebraic expressions, e.g., you are in the mountains. Wilson Mountain has an altitude of 5.28 x 103 feet. Rush Mountain is 4,300 feet tall. How much higher is Wilson Mountain than Rush Mountain?
b. fraction and decimal approximations of the irrational number pi, e.g., Mary measured the distance around her 48-inch diameter circular table to be 150 inches. Using this information, approximate pi as a fraction and as a decimal. (Circle Area , Circle Circumference )
AI-2 determines whether or not solutions to real-world problems using rational numbers, the irrational number pi, and simple algebraic expressions are reasonable (2.4.A1a) ($), e.g., a sweater that cost $15 is marked 1/3 off. The cashier charged $12. Is this reasonable?
Benchmark 2: Number Systems and Their Properties . The student demonstrates an understanding of the rational number system and the irrational number pi; recognizes, uses, and describes their properties; and extends these properties to algebraic expressions in one variable.
KBI-4 knows and explains the relationships between natural (counting) numbers, whole numbers, integers, and rational numbers using mathematical models (2.4.K1a,k), e.g., number lines or Venn diagrams. (Positive Number Line , Number Line , Compare Integers )
KBI-5 classifies a given rational number as a member of various subsets of the rational number system (2.4.K1a,k), e.g., - 7 is a rational number and an integer.
KBI-6 names, uses, and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($):
a. commutative properties of addition and multiplication (changing the order of the numbers does not change the solution);
b. associative properties of addition and multiplication (changing the grouping of the numbers does not change the solution);
c. distributive property [distributing multiplication or division over addition or subtraction, e.g., 2(4 - 1) = 2(4) - 2(1) = 8 - 2 = 6];
d. substitution property (one name of a number can be substituted for another name of the same number), e.g., if a = 2, then 3a = 3 x 2 = 6. (Associative Property 1 , Associative Property 2 , Commutative Property 1 , Commutative Property 2 , Distributive Property , Distributive Property 2 , Variable Substitution , Variable Substitution 2 )
KBI-7 uses and describes these properties with the rational number system and demonstrates their meaning including the use of concrete objects (2.4.K1a) ($):
a. identity properties for addition and multiplication (additive identity - zero added to any number is equal to that number; multiplicative identity - one multiplied by any number is equal to that number);
b. symmetric property of equality (if 7 + 2x = 9 then 9 = 7 + 2x);
c. zero property of multiplication (any number multiplied by zero is zero);
d. addition and multiplication properties of equality (adding/multiplying the same number to each side of an equation results in an equivalent equation);
e. additive and multiplicative inverse properties. (Every number has a value known as its additive inverse and when the original number is added to that additive inverse, the answer is zero, e.g., +5 + -5 = 0. Every number except 0 has a value known as its multiplicative inverse and when the original number multiplied by its inverse, the answer will be 1, e.g., 8 x 1/8 =1.) (Inverse Equations 1 , Inverse Equations 2 )
KBI-5 recognizes that the irrational number pi can be represented by approximate rational values, e.g., 22/7 or 3.14. (Circle Area , Circle Circumference )
AI-1 generates and/or solves real-world problems with rational numbers and the irrational number pi using the concepts of these properties to explain reasoning (2.4.K1a) ($):
a. commutative and associative properties of addition and multiplication, e.g., at a delivery stop, Sylvia pulls out a flat of eggs. The flat has 5 columns and 6 rows of eggs. Express how to find the number of eggs in 2 ways.
b. distributive property, e.g., trim is used around the outside edges of a bulletin board with dimensions 3 ft by 5 ft. Explain two different methods of solving this problem.
c. substitution property, e.g., V = IR [Ohm's Law: voltage (V) = current (I) x resistance (R)] If the current is 5 amps (I = 5) and the resistance is 4 ohms (R = 4), what is the voltage?
d. symmetric property of equality, e.g., Sam took a $15 check to the bank and received a $10 bill and a $5 bill. Later Sam took a $10 bill and a $5 bill to the bank and received a check for $15. $15 = $10 + $5 is the same as $10 + $5 = $15.
e. additive and multiplicative identities, e.g., Bob and Sue each read the same number of books. During the week, they each read 5 more books. Compare the number of books each read. Let b=the number of books Bob read and s=the number of books Sue read, then b+5=s+5
f. zero property of multiplication, e.g., Jenny was thinking of two numbers. Jenny said that the product of the two numbers was 0. What could you deduct from this statement? Explain your reasoning.
g. addition and multiplication properties of equality, e.g., the total price (P) of a car, including tax (T), is $14, 685. 33. If the tax is $785.42, what is the sale price of the car (S)?
h. additive and multiplicative inverse properties, e.g., if 5 candy bars cost $1.00, what does one candy bar cost? Explain your reasoning. (Unit Cost , Distance, Rate, and Time )
AI-2 analyzes and evaluates the advantages and disadvantages of using integers, whole numbers, fractions (including mixed numbers), decimals, or the irrational number pi and its rational approximations in solving a given real-world problem (2.4.K1a, e.g., in the store everything is 25% off. When calculating the discount, which representation of 25% would you use and why?
Benchmark 3: Estimation . The student uses computational estimation with rational numbers and the irrational number pi in a variety of situations.
KBI-1 estimates quantities with combinations of rational numbers and/or the irrational number pi using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.K1a) ($).
KBI-2 N uses various estimation strategies and explains how they were used to estimate rational number quantities and the irrational number pi (2.4.K1a) ($). (Rounding Large Numbers , Decimal Rounding )
KBI-3 recognizes and explains the difference between an exact and approximate answer (2.4.K1a).
KBI-4 determines the appropriateness of an estimation strategy used and whether the estimate is greater than (overestimate) or less than (underestimate) the exact answer and its potential impact on the result (24.K1a).
KBI-5 knows and explains why the fraction (22/7) or decimal (3.14) representation of the irrational number pi is an approximate value (2.4.K1c).
AI-1 adjusts original rational number estimate of a real-world problem based on additional information (a frame of reference) (2.4.A1a) ($), e.g., estimate the weight of a bookshelf of books. Then weigh one book and adjust your estimate.
AI-2 estimates to check whether or not the result of a real-world problem using rational numbers, the irrational number pi, and/or simple algebraic expressions is reasonable and makes predictions based on the information (2.4.A1a), e.g., a goat is staked out in a pasture with a rope that is 7 feet long. The goat needs 200 square feet of grass to graze. Does the goat have enough pasture? If not, how long should the rope be?
AI-3 determines a reasonable range for the estimation of a quantity given a real-world problem and explains the reasonableness of the range (2.4.A1a), e.g., how long will it take your teacher to walk two miles? The range could be 25-35 minutes.
AI-4 determines if a real-world problem calls for an exact or approximate answer and performs the appropriate computation using various computational methods including mental math, paper and pencil, concrete objects, and/or appropriate technology (2.4.A1a) ($), e.g., Kathy buys items at the grocery store priced at $32.56, $12.83, $6.99, and 5 for $12.49 each. She has $120 with her to pay for the groceries. To decide if she can pay for her items, does she need an exact or an approximate answer? (Estimated Multiply Divide Word Problems )
Benchmark 4: Computation . The student models, performs, and explains computation with rational numbers, the irrational number pi, and first-degree algebraic expressions in one variable in a variety of situations.
KBI-1 computes with efficiency and accuracy using various computational methods including mental math, paper and pencil, concrete objects, and appropriate technology (2.4.K1a-c) ($).
KBI-2 performs and explains these computational procedures (2.4.K1a):
a. ▴N adds and subtracts decimals from ten millions place through hundred thousandths place;
b. ▴N multiplies and divides a four-digit number by a two-digit number using numbers from thousands place through thousandths place;
c. ▴N multiplies and divides using numbers from thousands place through thousandths place by 10; 100; 1,000; .1; .01; .001; or single-digit multiples of each, e.g., 54.2 ÷ .002 or 54.3 x 300;
d. ▴N adds, subtracts, multiplies, and divides fractions and expresses answers in simplest form;
e. N adds, subtracts, multiplies, and divides integers;
f. N uses order of operations (evaluates within grouping symbols, evaluates powers to the second or third power, multiplies or divides in order from left to right, then adds or subtracts in order from left to right) using whole numbers;
g. simplifies positive rational numbers raised to positive whole number powers;
h. combines like terms of a first degree algebraic expression. recognizes, describes, and uses different ways to express computational procedures, e.g., 5 - 2 = 5 + (-2) or 49 x 23 = (40 x 23) + (9 x 23) or 49 x 23 = (49 x 20) + (49 x 3) or 49 x 23 = (50 x 23) - 23. finds prime factors, greatest common factor, multiples, and the least common multiple (2.4.K1d). (Order Of Operations , Division with Remainders , Long Division with Remainders , Multiply By Multiples Of 10 , Prime Factoring , Prime Factoring 2 , Greatest Common Factor , Least Common Multiple , Fraction Addition , Fraction Subtraction , Fraction Multiplication , Fraction Division , Decimal Addition , Decimal Subtraction , Decimal Multiplication , Decimal Division , Integer Addition , Integer Subtraction , Positive Integer Subtraction , Integer Multiplication , Integer Division , Integer Equivalence , Simplifying Algebraic Expressions , Exponent Basics , Exponents Of Fractional Bases , Negative Exponents Of Fractional Bases )
KBI-5 ▴ finds percentages of rational numbers (2.4.K1a,c) ($), e.g., 12.5% x $40.25 = n or 150% of 90 is what number? (For the purpose of assessment, percents will not be between 0 and 1.) (Percent of Quantity )
AI-1 generates and/or solves one- and two-step real-world problems using these computational procedures and mathematical concepts (2.4.A1a) ($):
a. ■ addition, subtraction, multiplication, and division of rational numbers with a special emphasis on fractions and expressing answers in simplest form, e.g., at the candy store, you buy 3/4 of a pound of peppermints and ½ of a pound of licorice. The cost per pound for each kind of candy is $3.00. What is the total cost of the candy purchased?
b. addition, subtraction, multiplication, and division of rational numbers with a special emphasis on integers, e.g., the high temperatures for the week were: -40, 100, -10, 00, 70, 30, and -50. What is the mean temperature for the week?
c. first degree algebraic expressions in one variable, e.g., Jenny rents 3 videos plus $20 of other merchandise. Barb rents 5 videos plus $15 of other merchandise. Represent the total purchases of Jenny and Barb using V as the price of a video rental.
d. percentages of rational numbers, e.g., if the sales tax is 5.5%, what is the sales tax on an item that costs $36?
e. approximation of the irrational number pi, e.g., what is the approximate diameter of a 400-meter circular track? (Making Change , Unit Cost , Fraction Word Problems , Fraction Word Problems 2 , Purchases At Stores , Restaurant Bills , Commissions , Integers In Word Problems , Compound Interest , Algebraic Sentences 2 , Algebraic Sentences )
Standard 2: Algebra
Benchmark 1: Patterns . The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.KBI-1 identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:
a. ▴ counting numbers including perfect squares, cubes, and factors and multiples (number theory) (2.4.K1a);
b. ▴ positive rational numbers including arithmetic and geometric sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number) (2.4.K1a), e.g., 2, ., 1/8, 1/32, ...;
c. geometric figures (2.4.K1f);
d. measurements (2.4.K1a);
e. things related to daily life (2.4.K1a) ($), e.g., tide, moon cycle, or temperature. (Patterns: Numbers , Patterns: Shapes , Function Tables , Function Tables 2 )
KBI-2 generates a pattern (2.4.K1a).
KBI-3 extends a pattern when given a rule of one or two simultaneous changes (addition, subtraction, multiplication, division) between consecutive terms (2.4.K1a), e.g., find the next three numbers in a pattern that starts with 3, where you double and add 1 to get the next number; the next three numbers are 7, 15, and 31. (Function Tables , Function Tables 2 )
KBI-4 ▴ ■ states the rule to find the nth term of a pattern with one operational change (addition or subtraction) between consecutive terms (2.4.K1a), e.g., given 3, 5, 7, and 9; the nth term is 2n +1. (This is the explicit rule for the pattern.) (Function Tables , Function Tables 2 )
AI-1 generalizes a pattern by giving the nth term using symbolic notation (2.4.A1a,f), e.g., given the following, the nth term is 2n. 1 person has 2 eyes 2 people have 4 eyes 3 people have 6 eyes 4 people have 8 eyes . . . . . . n people have 2n eyes (Function Tables , Function Tables 2 )
AI-2 recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a,f,j-k) ($). X Y 1 2 2 4 3 6 4 8 . . . . . . n 2n
Benchmark 2: Variable, Equations, and Inequalities . The student uses variables, symbols, rational numbers, and simple algebraic expressions in one variable to solve linear equations and inequalities in a variety of situations.
KBI-1 knows and explains that a variable can represent a single quantity that changes (2.4.K1a), e.g., daily temperature.
KBI-2 knows, explains, and uses equivalent representations for the same simple algebraic expressions (2.4.K1a), e.g., x + y + 5x is the same as 6x + y. (Simplifying Algebraic Expressions )
KBI-3 shows and explains how changes in one variable affects other variables (2.4.A1a), e.g., changes in diameter affects circumference. (Area And Volume Proportions , Independent and Dependent Variables )
KBI-4 explains the difference between an equation and an expression. (Algebraic Terms )
KBI-5 solves (2.4.K1a,e) ($):
a. one-step linear equations in one variable with positive rational coefficients and solutions, e.g., 7x = 28 or x + 3/4 = 7 or x/3 = 5;
b. two-step linear equations in one variable with counting number coefficients and constants and positive rational solutions;
c. one-step linear inequalities with counting numbers and one variable, e.g., 3x > 12. (Single Variable Equations , Single Variable Equations 2 , Single Variable Inequalities )
KBI-6 explains and uses the equality and inequality symbols (=, ≠, <, ≤, >, ≥) and corresponding meanings (is equal to, is not equal to, is less than, is less than or equal to, is greater than, is greater than or equal to) to represent mathematical relationships with rational numbers (2.4.K1a) ($). (Compare Mixed Values 2 )
KBI-7 ▴ knows the mathematical relationship between ratios, proportions, and percents and how to solve for a missing term in a proportion with positive rational number solutions and monomials (2.4.K1a,c) ($), e.g., 5/6 = 2/x. (Proportions 1 )
KBI-8 ▴ evaluates simple algebraic expressions using positive rational numbers (2.4.K1c) ($), e.g., if x = 3/2, y = 2, then 5xy + 2 = 5(3/2)(2) + 2 = 17. (Variable Substitution , Variable Substitution 2 )
AI-1 ▴ represents real-world problems using variables and symbols to write linear expressions, one- or two-step equations (2.4.A1e) ($), e.g., John has three times as much money as his sister. If M is the amount of money his sister has, what is the equality that represents the amount of money that John has? To represent the problem situation, J = 3M could be written. (Algebraic Word Problems , Algebraic Sentences 2 , Algebraic Sentences )
AI-2 solves real-world problems with one- or two-step linear equations in one variable with whole number coefficients and constants and positive rational solutions intuitively and analytically (2.4.A1e) ($), e.g., Kim has read 5 more than twice the number of pages as Hank. Kim has read 15 pages. How many pages has Hank read? To solve analytically, write 2h + 5 = 15. Then find the answer. (Algebraic Word Problems )
AI-3 generates real-world problems that represent one- or two-step linear equations (2.4.A1e) ($), e.g., given the equation x + 10 = 30, the problem could be: Two items cost $30.00. If one item costs $10.00, what is the cost of the other item?
AI-4 explains the mathematical reasoning that was used to solve a realworld problem using a one- or two-step linear equation (2.4.A1e) ($), e.g., Kim has read 5 more than twice the number of pages as Hank. Kim has read 15 pages. How many pages has Hank read? To solve, write 2h + 5 = 15. Let h=number of pages Hank read. Then to find the answer subtract 5 from both sides of the equation. 2h = 10, then divide both sides of the equation by 2, so h = 5.
Benchmark 3: Functions . The student recognizes, describes, and analyzes constant and linear relationships in a variety of situations.
KBI-1 recognizes constant and linear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a,e-g) ($). (Determining Slope )
KBI-2 finds the values and determines the rule through two operations using a function table (input/output machine, T-table) (2.4.K1f). (Function Tables , Function Tables 2 )
KBI-3 demonstrates mathematical relationships using ordered pairs in all four quadrants of a coordinate plane (2.4.K1g). (Ordered Pairs , Translations and Reflections )
KBI-4 describes and/or gives examples of mathematical relationships that remain constant (2.4.K1e-g) ($), e.g., you will get $10.00 to do a job, no matter how long it takes for you to do it.
AI-1 represents a variety of constant and linear relationships using written or oral descriptions of the rule, tables, graphs, and when possible, symbolic notation (2.4.A13-g,k) ($), e.g., the relationship between cars and their wheels (written) becomes a table: Cars | Wheels 1 | 4 .. (1,4) 2 | 8 .. (2,8) 10 | 40 .. (10,40) . | . . | . . | . n | 4n and then the ordered pairs of (1,4), (2,8), (10,40), and (n,4n) can be graphed. (Function Tables , Function Tables 2 )
AI-2 interprets, describes, and analyzes the mathematical relationships of numerical, tabular, and graphical representations, including translations between the representations (2.4.A1k) ($). (Determining Slope )
Benchmark 4: Models . The student generates and uses mathematical models to represent and justify mathematical relationships found in a variety of situations.
KBI-1 knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships. Mathematical models include:
a. process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-5, 1.2.K1-4, 1.3.K1- 4, 1.4.K1-2, 1.4.K5, 2.1.K1a-b, 2.1.K1e. 2.1.K2-4, 2.2.K1-3, 2.2.K5-6, 2.3.K1, 3.1.K9, 3.2.K1-3, 3.2.K9, 3.3.K1-4, 3.4.K1, 4.2.K4-6) ($);
b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures (1.1.K1, 1.4.K2) ($);
c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.K1, 1.3.K5, 1,4,K2, 2.2.K7-8, 4.1.K3) ($):
d. factor trees to find least common multiple and greatest common factor and to model prime factorization (1.4.K4);
e. equations and inequalities to model numerical relationships (2.2.K5, 2.3.K1, 23.K4) ($);
f. function tables to model numerical and algebraic relationships (2.3.K1-2, 2.3.K4) ($);
g. coordinate planes to model relationships between ordered pairs and linear equations (2.3.K1, 2.3.K3-4, 3.4.K2-4) ($);
h. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two- and three-dimensional (2.1.K1c, 3.1.K1, 3.1.K3-8, 3.1.K10, 3.2.K1-2, 3.2.K4-8, 3.2.K10);
i. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.K1, 4.1.K4) ($);
j. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to organize and display data (4.2.K1) ($);
k. Venn diagrams to sort data and show relationships (1.2.K1- 2). (Positive Number Line , Number Line , Compare Integers , Percentage Pictures , Ordered Pairs , Stem And Leaf Plots , Tally and Pictographs , Bar Graphs , Line Graphs , Function Tables , Function Tables 2 )
AI-1 recognizes that various mathematical models can be used to represent the same problem situation. Mathematical models include:
a. process models (concrete objects, pictures, diagrams, flowcharts, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, mathematical relationships, and problem situations and to solve equations (1.1.A1, 1.2.A1-2, 1.3.A1-4, 1.4.A1, 2.1.A1-2, 3.1.A1, 3. 2.A1a, 3.2.A1d, 3.2.A1f, 3.2.A2, 3.3.A1. 4.2.A4) ($);
b. place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations (1.1.A1a, 1.1.A2a, 1.2.A12, 1.3A1.2, 1.4.A3a-e, 2.2.A3) ($);
c. fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.1.A1b, 1.1.A2b, 1.2.A1-2, 1.3.A1-2) ($);
d. factor trees to find least common multiple and greatest common factor and to model prime factorization (1.4.K5)
e. equations and inequalities to model numerical relationships (2.2.A1-4, 2.3.A1, 3.2.A1e). ($)
f. function tables to model numerical and algebraic relationships (2.1.A1-2, 2.3.A1) ($);
g. coordinate planes to model relationships between ordered pairs and linear equations (2.3.A1, 3.4.A1) ($);
h. two- and three-dimensional geometric models (geoboards, dot paper, nets or solids) to model perimeter, area, volume, and surface area, and properties of two- and three-dimensional models (3.1.A2-3, 3.2.A1b-c, 3.2.A1e, 3.3.A2, 3.4.A1);
i. scale drawings to model large and small real-world objects (3.3.A3);
j. geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.A1) ($);
k. frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single stem-and-leaf plots, scatter plots, and box-and-whisker plots to describe, interpret, and analyze data (2.1.A2, 2.3.A1-2, 4.1.A1-2, 4.2.A1-3) ($);
l. Venn diagrams to sort data and show relationships. (Positive Number Line , Number Line , Compare Integers , Percentage Pictures , Ordered Pairs , Stem And Leaf Plots , Tally and Pictographs , Bar Graphs , Line Graphs , Function Tables , Function Tables 2 )
AI-2 selects a mathematical model and justifies why some mathematical models are more accurate than other mathematical models in certain situations, e.g., recognizes that change over time is better represented through a line graph than through a table of ordered pairs.
AI-3 uses the mathematical modeling process to make inferences about real-world situations when the mathematical model used to represent the situation is given.
Standard 3: Geometry
Benchmark 1: Geometric Figures and Their Properties . The student recognizes geometric figures and compares their properties in a variety of situations.KBI-1 recognizes and compares properties of two- and three-dimensional figures using concrete objects, constructions, drawings, appropriate terminology, and appropriate technology (2.4.K1h).
KBI-2 classifies regular and irregular polygons having through ten sides as convex or concave.
KBI-3 ▴ identifies angle and side properties of triangles and quadrilaterals (2.4.K1h):
a. sum of the interior angles of any triangle is 180°;
b. sum of the interior angles of any quadrilateral is 360°;
c. parallelograms have opposite sides that are parallel and congruent;
d. rectangles have angles of 90°, opposite sides are congruent;
e. rhombi have all sides the same length, opposite angles are congruent;
f. squares have angles of 90°, all sides congruent;
g. trapezoids have one pair of opposite sides parallel and the other pair of opposite sides are not parallel. (Triangle Angles , Quadrilateral Angles , Quadrilateral Types )
KBI-4 identifies and describes (2.4.K1h):
a. the altitude and base of a rectangular prism and triangular prism,
b. the radius and diameter of a cylinder.
KBI-5 identifies corresponding parts of similar and congruent triangles and quadrilaterals (2.4.K1h). (Congruent And Similar Triangles , Proportions 2 )
KBI-6 uses symbols for right angle within a figure ( ), parallel (||), perpendicular (⊥), and triangle (Δ) to describe geometric figures(2.4.K1h).
KBI-7 classifies triangles as (2.4.K1h):
a. scalene, isosceles, or equilateral;
b. right, acute, obtuse, or equiangular. (Triangle Types )
KBI-8 determines if a triangle can be constructed given sides of three different lengths(2.4.K1h).
KBI-9 generates a pattern for the sum of angles for 3-, 4-, 5-, ... n-sides polygons (2.4.K1a). (Polygon Angles )
KBI-10 describes the relationship between the diameter and the circumference of a circle (2.4.K1h). (Circle Circumference )
AI-3 solves real-world problems by applying the properties of (2.4.A1a):
d. plane figures (regular and irregular polygons through 10 sides, circles, and semicircles) and the line(s) of symmetry; e.g., two guide wires are used to stabilize a tower. The wires with the ground form an isosceles triangle. The two base angles form 20o angles with the ground. What is the size of the vertex angle made where the wires meet on the tower?
e. solids (cubes, rectangular prisms, cylinders, cones, spheres, triangular prisms) emphasizing faces, edges, vertices, and bases; e.g., lace is to be glued on all of the edges of a cube. If one edge measures 34 cm, how much lace is needed?
AI-2 decomposes geometric figures made from (2.4.A1h):
c. regular and irregular polygons through 10 sides, circles, and semicircles;
d. nets (two-dimensional shapes that can be folded into threedimensional figures), e.g., the cardboard net that becomes a shoebox;
e. prisms, pyramids, cylinders, cones, spheres, and hemispheres.
AI-3 composes geometric figures made from (2.4.A1h):
c. regular and irregular polygons through 10 sides, circles, and semicircles;
d. nets (two-dimensional shapes that can be folded into threedimensional figures);
e. prisms, pyramids, cylinders, cones, spheres, and hemispheres.
Benchmark 2: Measurement and Estimation . The student estimates, measures, and uses measurement formulas in a variety of situations.
KBI-1 determines and uses rational number approximations (estimations) for length, width, weight, volume, temperature, time, perimeter, and area using standard and nonstandard units of measure (2.4.K1a) ($).
KBI-2 selects and uses measurement tools, units of measure, and level of precision appropriate for a given situation to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, area, and angle measurements (2.4.K1a) ($).
KBI-3 converts within the customary system and within the metric system (2.4.K1a). (Distance Conversion , Volume Conversion , Weight Conversion , Area and Volume Conversions )
KBI-4 ▴ knows and uses perimeter and area formulas for circles, squares, rectangles, triangles, and parallelograms (2.4.K1h). (Triangle Area , Parallelogram Area , Compare Rectangle Area and Perimeter , Circle Area )
KBI-5 finds perimeter and area of two-dimensional composite figures of circles, squares, rectangles, and triangles (2.4.K1h). (Perimeter and Area of Composite Figures )
KBI-6 ▴ uses given measurement formulas to find (2.4.K1h):
a. surface area of cubes,
b. volume of rectangular prisms. (Rectangular Solids )
KBI-7 finds surface area of rectangular prisms using concrete objects (2.4.K1h). (Rectangular Solids )
KBI-8 uses appropriate units to describe rate as a unit of measure (2.4.K1a), e.g., miles per hour.
KBI-9 finds missing angle measurements in triangles and quadrilaterals (2.4.K1h). (Triangle Angles , Quadrilateral Angles , Triangle Angles 2 )
AI-1 solves real-world problems by ($):
a. converting within the customary and metric systems (2.4.A1a), e.g., James added 30 grams of sand to his model boat that weighed 2 kg. With the sand included, what is the total weight of his boat in kg?
b. finding perimeter and area of circles, squares, rectangles, triangles, and parallelograms (2.4.A1h), e.g., the dimensions of a room are 22' x 12'. What is the total length of wallpaper border needed?
c. ▴ ■ finding perimeter and area of two-dimensional composite figures of squares, rectangles, and triangles (2.4.A1h), e.g., the front of a barn is rectangular in shape with a height of 10 feet and a width of 48 feet. Above the rectangle is a triangle that is 7 feet high with sides 25 feet long. What is the area of the front of the barn?
d. using appropriate units to describe rate as a unit of measure (2.4.A1a), e.g., a person traveled 20 miles in 10 minutes. What is the rate of travel? The answer could be 2 miles per minute or 120 miles per hour.
e. finding missing angle measurements in triangles and quadrilaterals (2.4.A1h), e.g., a fenced pasture is a quadrilateral with angles of 30o, 120o, and 90o degrees. What is the measure of the fourth angle?
f. applying various measurement techniques (selecting and using measurement tools, units of measure, and level of precision) to find accurate rational number representations for length, weight, volume, temperature, time, perimeter, and area appropriate to a given situation (2.4.A1a). (Perimeter and Area Word Problems )
AI-2 estimates to check whether or not measurements or calculations for length, width, weight, volume, temperature, time, perimeter, and area in real-world problems are reasonable and adjusts original measurement or estimation based on additional information (a frame of reference) (2.4.A1a) ($), e.g., students estimate the weight of their book in grams. Then the weight of their calculator is measured in grams. Students then adjust their estimate.
Benchmark 3: Transformational Geometry . The student recognizes and performs transformations on two- and threedimensional geometric figures in a variety of situations.
KBI-1 identifies, describes, and performs single and multiple transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on a two-dimensional figure (2.4.K1a).
KBI-2 identifies three-dimensional figures from various perspectives (top, bottom, sides, corners) (2.4.K1a).
KBI-3 draws three-dimensional figures from various perspectives (top, bottom, sides, corners) (2.4.K1a).
KBI-4 generates a tessellation (2.4.K1a).
AI-1 describes the impact of transformations [reflection, rotation, translation, reduction (contraction/shrinking), enlargement (magnification/growing)] on the perimeter and area of squares and rectangles (2.4.A1a); e.g., when a square is rotated, the perimeter and area stays the same, however, when the length of the sides of a square are tripled, the perimeter triples, and the area is 9 times bigger.
AI-2 investigates congruency and similarity of geometric figures using transformations (2.4.A1h). (Congruent And Similar Triangles )
AI-3 ▴ ■ determines the actual dimensions and/or measurements of a two-dimensional figure represented in a scale drawing (2.4.A1i).
Benchmark 4: Geometry From An Algebraic Perspective . The student relates geometric concepts to a number line and a coordinate plane in a variety of situations.
KBI-1 finds the distance between the points on a number line by computing the absolute value of their difference (2.4.K1a). (Line Segments , Absolute Value 1 )
KBI-2 uses all four quadrants of a coordinate plane to (2.4.K1g):
a. identify in which quadrant or on which axis a point lies when given the coordinates of a point,
b. plot points,
c. identify points,
d. list through five ordered pairs of a given line. (Ordered Pairs )
KBI-3 uses a given linear equation with whole number coefficients and constants and a whole number solution to find the ordered pairs, organize the ordered pairs using a T-table, and plot the ordered pairs on the coordinate plane (2.4.K1e-g).
KBI-4 examines characteristics of two-dimensional figures on a coordinate plane using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or other appropriate technology (2.4.A1g).
AI-1 represents and/or generates real-world problems using a coordinate plane to find (2.4.A1g-h):
a. perimeter of squares and rectangles; e.g., using a coordinate plane Jack started at the school (1, 2), traveled to the post office (3 ., 2), then went by the fire station (3 ., 3), then visited the park (1, 3), and finally returned to the school. Determine the distance Jack traveled.
b. circumference (perimeter) of circles, e.g., Jane jogs on a circular track with a radius of 100 feet. How far would she jog in one lap?
c. area of circles, parallelograms, triangles, squares, and rectangles; e.g., if the sprinkler head is centered at (3, 4) and we know it reaches point (3, - 2) on the coordinate plane, determine the area being watered.
Standard 4: Data
Benchmark 1: Probability . The student applies the concepts of probability to draw conclusions, generate convincing arguments, and make predictions and decisions including the use of concrete objects in a variety of situations.KBI-1 finds the probability of a compound event composed of two independent events in an experiment or simulation (2.4.K1i) ($). (Probability 2 , Object Picking Probability )
KBI-2 explains and gives examples of simple or compound events in an experiment or simulation having probability of zero or one.
KBI-3 uses a fraction, decimal, and percent to represent the probability of (2.4.K1c):
a. a simple event in an experiment or simulation;
b. a compound event composed of two independent events in an experiment or simulation. (Probability , Probability 2 , Object Picking Probability )
KBI-4 finds the probability of a simple event in an experiment or simulation using geometric models (2.4.K1i), e.g., using spinners or dartboards, what is the probability of landing on 2? The answer is ¼, .25, or 25%. (Probability )
AI-1 conducts an experiment or simulation with a compound event composed of two independent events including the use of concrete objects; records the results in a chart, table, or graph; and uses the results to draw conclusions and make predictions about future events (2.4.A1j-k). (Requires outside materials )
AI-2 analyzes the results of an experiment or simulation of a compound event composed of two independent events to draw conclusions, generate convincing arguments, and make predictions and decisions in a variety of real-world situations (2.4.A1j-k). (Requires outside materials )
AI-3 compares results of theoretical (expected) probability with empirical (experimental) probability in an experiment or situation with a compound event composed of two simple independent events and understands that the larger the sample size, the greater the likelihood that the experimental results will equal the theoretical probability(2.4.A1j). (Object Picking Probability )
AI-4 makes predictions based on the theoretical probability of a simple event in an experiment or simulation (2.4.A1j). (Batting Averages )
Benchmark 2: Statistics . The student collects, organizes, displays, and explains numerical (rational numbers) and non-numerical data sets in a variety of situations with a special emphasis on measures of central tendency.
KBI-1 ▴ organizes, displays, and reads quantitative (numerical) and qualitative (non-numerical) data in a clear, organized, and accurate manner including a title, labels, categories, and rational number intervals using these data displays (2.4.K1j) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single);
f. scatter plots;
g. box-and-whiskers plots. (Stem And Leaf Plots , Bar Graphs , Line Graphs )
KBI-2 selects and justifies the choice of data collection techniques (observations, surveys, or interviews) and sampling techniques (random sampling, samples of convenience, or purposeful sampling) in a given situation.
KBI-3 conducts experiments with sampling and describes the results.
KBI-4 determines the measures of central tendency (mode, median, mean) for a rational number data set (2.4.K1a) ($). (Mean, Median, Mode )
KBI-5 identifies and determines the range and the quartiles of a rational number data set (2.4.K1a) ($). (Stem And Leaf Plots )
KBI-6 identifies potential outliers within a set of data by inspection rather than formal calculation (2.4.K1a) ($), e.g., consider the data set of 1, 100, 101, 120, 140, and 170; the outlier is 1.
AI-1 uses data analysis (mean, median, mode, range) of a rational number data set to make reasonable inferences and predictions, to analyze decisions, and to develop convincing arguments from these data displays (2.4.A1k) ($):
a. frequency tables;
b. bar, line, and circle graphs;
c. Venn diagrams or other pictorial displays;
d. charts and tables;
e. stem-and-leaf plots (single);
f. scatter plots;
g. box-and-whiskers plots. (Mean, Median, Mode )
AI-2 explains advantages and disadvantages of various data displays for a given data set (2.4.A1k) ($).
AI-3 ▴ recognizes and explains (2.4.A1k):
a. ■ misleading representations of data;
b. the effects of scale or interval changes on graphs of data sets.
AI-4 determines and explains the advantages and disadvantages of using each measure of central tendency and the range to describe a data set (2.4.A1a) ($).
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