|
| Grade 8
|
| Mathematics, Approved 2003
|
| 2
| Algebra |
|
|
|
|
| |
|
Standard
Benchmark
Indicator | Description | Lesson Plans | Thinkfinity | Resources |
|
2
|
The student uses algebraic concepts in a variety of situations.
|
|
|
|
|
2.1
|
The student recognizes, describes, extends, develops, and explains the general rule of a pattern in a variety of situations.
|
|
|
|
|
2.1.A1
|
generalizes numerical patterns using algebra and then translates between the equation, graph, and table of values resulting from the generalization (2.4.A1d-e, j) ($), e.g., water is billed at $1.00 per 1,000 gallons, plus a basic fee of $10 per month for being connected to the water district.
|
|
|
|
|
2.1.K1
|
identifies, states, and continues a pattern presented in various formats including numeric (list or table), algebraic (symbolic notation), visual (picture, table, or graph), verbal (oral description), kinesthetic (action), and written using these attributes:
|
|
|
|
|
2.1.K1A
|
counting numbers including perfect squares, cubes, and factors and multiples with positive rational numbers (number theory) (2.4.K1a).
|
|
|
|
|
2.1.K1B
|
rational numbers including arithmetic and geometric sequences (arithmetic: sequence of numbers in which the difference of two consecutive numbers is the same, geometric: a sequence of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number) (2.4.K1a);
|
|
|
|
|
2.1.K1C
|
geometric figures (2.4.K1h);
|
|
|
|
|
2.1.K1D
|
measurements (2.4.K1a);
|
|
|
|
|
2.1.K1E
|
things related to daily life ($);
|
|
|
|
|
2.1.K1F
|
variables and simple expressions,
|
|
|
|
|
2.1.K2
|
generates and explains a pattern (2.4.K1a).
|
|
|
|
|
2.1.A2
|
recognizes the same general pattern presented in different representations [numeric (list or table), visual (picture, table, or graph), and written] (2.4.A1a, j) ($).
|
|
|
|
|
2.1.K3
|
generates a pattern limited to two operations (addition, subtraction, multiplication, division, exponents) when given the rule for the nth term (2.4.K1a), e.g., the nth term is n2+1, find the first 4 terms beginning with n = 1; the terms are 2, 5, 10, and 17.
|
|
|
|
|
2.1.K4
|
states the rule to find the nth term of a pattern using explicit symbolic notation (2.4.K1a), e.g., given 2, 5, 8, 11, …; find the rule for the nth term, the rule is 3n –1.
|
|
|
|
|
2.1.K5
|
describes the pattern when given a table of linear values and plots the ordered pairs on a coordinate plane (2.4.K1f-g), e.g., in the table below, the pattern could be described as the x-coordinates are increasing by three, while the y-coordinates are increasing by 6, or the x is doubled and one is added to find the y.
|
|
|
|
|
2.2
|
The student uses variables, symbols, real numbers, and algebraic expressions to solve equations and inequalities in a variety of situations.
|
|
|
|
|
2.2.A1
|
represents real-world problems using (2.4.A1d) ($):
|
|
|
|
|
2.2.A1A
|
 
variables, symbols, expressions, one- or two-step equations with rational number coefficients and constants, e.g., today John is 3.25 inches more than half his sister’s height. If J = John’s height, and S = his sister’s height, then J = 0.5S + 3.25.
|
|
|
|
|
2.2.A1B
|
one-step inequalities with rational number coefficients and constants, e.g., after Randy paid $38.50 for a watch, he did not have enough money to by a calculator for $5.50. Represent this situation with an inequality.
|
|
|
|
|
2.2.A1C
|
systems of linear equations with whole number coefficients and constants, e.g., two students collected the same amount of money for a walk-a-thon. One student received $5 per mile and a donation of $10, while the other student received $2 per mile and a donation of $20. How many miles did they walk?
|
|
|
|
|
2.2.K1
|
identifies independent and dependent variables within a given situation.
|
|
|
|
|
2.2.K2
|
simplifies algebraic expressions in one variable by combining like terms or using the distributive property (2.4.K1a), e.g., –3(x – 4) is the same as – 3 x + 12.
|
|
|
|
|
2.2.A2
|
solves real-world problems with two-step linear equations in one variable with rational number coefficients and constants and rational solutions intuitively, analytically, and graphically (2.4.A1e) e.g., Mike and Albert are friends, but Joe and Albert are not friends. Which of the following diagramps can be used to describe this situation? (Three dots labelled J, M, A: there is a line between J and A.) An otter slides down a river bank, hits his head on a rock, lies there in a stupor for a few seconds, then jumps into the water. While in the water, it dives to the bottom of the river, comes up for air, and climbs back onto the rock. Create a graph to show this activity.
|
|
|
|
|
2.2.A3
|
generates real-world problems that represent (2.4.A1d) ($):
|
|
|
|
|
2.2.A3A
|
one- or two-step linear equations, ($), e.g., given the equation 2x + 10 = 30, the problem could be I bought two shirts and a pair of pants for $10. How much was a shirt, if the total bill was $30?
|
|
|
|
|
2.2.A3B
|
one-step linear inequalities, e.g., write a real-world situation that represents the inequality x + 10 > 30. The problem could be: If you give me $10, I will have more than $30.
|
|
|
|
|
2.2.K3
|
solves (2.4.K1a,e) ($):
|
|
|
|
|
2.2.K3A
|
one- and two-step linear equations in one variable with rational number coefficients and constants intuitively and/or analytically;
|
|
|
|
|
2.2.K3B
|
one-step linear inequalities in one variable with rational number coefficients and constants intuitively, analytically, and graphically;
|
|
|
|
|
2.2.K3C
|
systems of given linear equations with whole number coefficients and constants graphically.
|
|
|
|
|
2.2.K4
|
knows and describes the mathematical relationship between ratios, proportions, and percents and how to solve for a missing monomial or binomial term in a proportion (2.4.K1c).
|
|
|
|
|
2.2.A4
|
explains the mathematical reasoning that was used to solve a real-world problem using one- or two-step linear equations and inequalities and discusses the advantages and disadvantages to various strategies that may have been used to solve the problem, (2.4.A1d) ($), e.g., given the inequality x + 10 > 30, subtract the same number from both sides of the inequality or graph as y1 = x + 10 and y = 30 and find on the graph where y1 is less than y2. The first method gives an exact answer; the second method is a visual representation and can be used to solve more difficult inequalities.
|
|
|
|
|
2.2.K5
|
represents and solves algebraically ($):
|
|
|
|
|
2.2.K5A
|
the number when a percent and a number are given,
|
|
|
|
|
2.2.K5B
|
what percent one number is of another number,
|
|
|
|
|
2.2.K5C
|
percent of increase or decrease, e.g., the price of a loaf of bread is $2.00. With a coupon, the cost is $1.00. What is the percent of decrease?
|
|
|
|
|
2.2.K6
|
evaluates formulas using substitution ($).
|
|
|
|
|
2.3
|
The student recognizes, describes, and analyzes constant, linear, and non linear relationships in a variety of situations.
|
|
|
|
|
2.3.A1
|
represents a variety of constant and linear relationships using written or oral descriptions of the rule, tables, graphs, and symbolic notation (2.4.A1d-f) ($).
|
|
|
|
|
2.3.K1
|
recognizes and examines constant, linear, and nonlinear relationships using various methods including mental math, paper and pencil, concrete objects, and graphing utilities or appropriate technology (2.4.K1a, e-g) ($).
|
|
|
|
|
2.3.K2
|
knows and describes the difference between constant, linear, and nonlinear relationships (2.4.K1g).
|
|
|
|
|
2.3.A2
|
interprets, describes, and analyzes the mathematical relationships of numerical, tabular, and graphical representations (2.4.A1j) ($).
|
|
|
|
|
2.3.A3
|
translates between the numerical, tabular, graphical, and symbolic representations of linear relationships with integer coefficients and constants (2.4.A1a), e.g., a fish tank is being filled with water with a 2-liter jug. There are already 5 liters of water in the fish tank. Therefore, you are showing how full the tank is as you empty 2-liter jugs of water into it. Y = 2x + 5 (symbolic) can be represented in a table (tabular) –
|
|
|
|
|
2.3.K3
|
explains the concepts of slope and x- and y-intercepts of a line (2.4.K1g).
|
|
|
|
|
2.3.K4
|
recognizes and identifies the graphs of constant and linear functions (2.4.K1g) ($).
|
|
|
|
|
2.3.K5
|
identifies ordered pairs from a graph, and/or plots ordered pairs using a variety of scales for the x- and y-axis (2.4.K1g).
|
|
|
|
|
2.4
|
The student generates and uses mathematical models to represent and justify mathematical relationships in a variety of situations.
|
|
|
|
|
2.4.A1
|
recognizes that various mathematical models can be used to represent the same problem situation. Mathematical models include:
|
|
|
|
|
2.4.A1A
|
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.A1-2, 1.2.A1-2, 1.3.A1-5, 1.4.A1, 2.1.A1, 3.1.A1, 3.2.A1-2, 3.3.A1, 3.4.A1-2) ($);
|
|
|
|
|
2.4.A1B
|
place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to model problem situations ($);
|
|
|
|
|
2.4.A1C
|
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 (3.2.A3) ($);
|
|
|
|
|
2.4.A1D
|
equations and inequalities to model numerical relationships (2.1.A2, 2.2.A1-2,2.3.A1,3.4.A2) ($);
|
|
|
|
|
2.4.A1E
|
function tables to model numerical and algebraic relationships (2.1.A2, 2.3.A2, 3.4.A2) ($);
|
|
|
|
|
2.4.A1F
|
coordinate planes to model relationships between ordered pairs and linear equations and inequalities (2.3.A1 3.4.A2) ($);
|
|
|
|
|
2.4.A1G
|
two- and three-dimensional geometric models (geoboards, dot paper, nets, or solids) and real-world objects to model perimeter, area, volume, surface area and properties of two- and three-dimensional figures (3.3.A3, 3.4.A2);
|
|
|
|
|
2.4.A1H
|
scale drawings to model large and small real-world objects (3.1.A1-2, 3.3.A4);
|
|
|
|
|
2.4.A1I
|
geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.A1-4);
|
|
|
|
|
2.4.A1J
|
frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, and histograms to describe, interpret, and analyze data (2.1.A1-2, 2.3.A2-3, 4.2.A1, 4.2.A3, 4.2.A1-3) ($);
|
|
|
|
|
2.4.A1K
|
Venn diagrams to sort data and to show relationships.
|
|
|
|
|
2.4.K1
|
knows, explains, and uses mathematical models to represent and explain mathematical concepts, procedures, and relationships. Mathematical models include:
|
|
|
|
|
2.4.K1A
|
process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate grids) to model computational procedures, algebraic relationships, and mathematical relationships and to solve equations (1.1.K1-6, 1.2.K1, 1.2.K3, 1.3.K1-2, 1.3.K4, 1.4.K1-2, 2.1.K1a-b, 2.1.K1d-e, 2.1.K2-4, 2.2.K2-3, 3.1.K9, 3.2.K1-4, 3.3.K1-4, 3.4.K4, 4.2.K4-5) ($);
|
|
|
|
|
2.4.K1B
|
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 ($);
|
|
|
|
|
2.4.K1C
|
fraction and mixed number models (fraction strips or pattern blocks) and decimal and money models (base ten blocks or coins) to compare, order, and represent numerical quantities (1.3.K3, 2.3.K6) ($):
|
|
|
|
|
2.4.K1D
|
factor trees to model least common multiple, greatest common factor, and prime factorization (1.4.K3);
|
|
|
|
|
2.4.K1E
|
equations and inequalities to model numerical relationships (2.2.K3, (3.4.K2) ($);
|
|
|
|
|
2.4.K1F
|
function tables to model numerical and algebraic relationships (2.1.K5, 3.4.K2) ($) ;
|
|
|
|
|
2.4.K1G
|
coordinate planes to model relationships between ordered pairs and linear equations and inequalities (2.1.K5, 2.3.K1-5, 3.4.K2-3) ($);
|
|
|
|
|
2.4.K1H
|
two- and three-dimensional geometric models (geoboards, dot paper, nets, or solids) and real-world objects to model perimeter, area, volume, surface area, and properties of two-and three-dimensional figures (2.1.K1c, 3.1.K1-6, 3.1.K8, 3.1.K10, 3.2.K5, 3.3.K4-5);
|
|
|
|
|
2.4.K1i
|
scale drawings to model large and small real-world objects (3.3.K3-4);
|
|
|
|
|
2.4.K1J
|
geometric models (spinners, targets, or number cubes), process models (coins, pictures, or diagrams), and tree diagrams to model probability (4.1.K1-5) ($);
|
|
|
|
|
2.4.K1K
|
frequency tables, bar graphs, line graphs, circle graphs, Venn diagrams, charts, tables, single and double stem-and-leaf plots, scatter plots, box-and-whisker plots, and histograms to organize and display data (4.2.K1, 4.2.K6) ($);
|
|
|
|
|
2.4.K1L
|
Venn diagrams to sort data and to show relationships (1.2.K2).
|
|
|
|
|
2.4.A2
|
determines if a given graphical, algebraic, or geometric model is an accurate representation of a given real-world situation ($). e.g., The graph of an otter that slides down a river bank, hits his head on a rock, lies there in a stupor for a few seconds, then jumps into the water. While in the water, it dives to the bottom of the river, comes up for air, and climbs back onto the rock. I have 2 more than 4 times the number of cookies that Joe has. If we let J be the number of cookies that Joe has, 4J + 2 models the number of cookies that I have. A spinner cut in fourths as a geometric model of a four situations with equal probability.
|
|
|
|
|
2.4.A3
|
uses the mathematical modeling process to analyze and make inferences about real-world situations ($).
|
|
|
|
|
|