Academic Language of Mathematics

ESOLALM: Academic Language of Mathematics

ESOLALM.A: Number System

ESOLALM.B: Expressions

ESOLALM.C: Exponents, Expressions, and Equations

ESOLALM.D: Equations

ESOLALM.E: Introduction to Functions

ESOLALM.F: Linear Functions

ESOLALM.G: Linear Models and Tables

ESOLALM.H: Quadratic Functions

ESOLALM.I: Exponential Functions

ESOLALM.A.1: use the academic language of mathematics to describe how to find common factors and multiples; compute fluently with multi-digit numbers and find common factors and multiples in real world scenarios, with scaffolding appropriate to the proficiency level

ESOLALM.B.2: use the academic language of mathematics in applying and extending previous understandings of arithmetic to algebraic expressions in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.C.3: use the academic language of mathematics in evaluating and solving expressions and equations using integer exponents in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.D.4: use the academic language of mathematics in analyzing and solving linear equations and inequalities in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.E.5: use the academic language of mathematics to describe, define, evaluate, and compare functions in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.F.6: use the academic language of mathematics to describe the connections between proportional relationships, lines, and linear equations in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.F.7: use the academic language of mathematics to describe, define, evaluate, and compare functions with scaffolding appropriate to the proficiency level

ESOLALM.G.8: use the academic language of mathematics when investigating and describing patterns of association in bivariate data in real world scenarios with scaffolding appropriate to the proficiency level

ESOLALM.H.9: use the academic language of mathematics in graphing, interpreting, and factoring quadratic and exponential functions with scaffolding appropriate to the proficiency level

ESOLALM.I.10: use the academic language of mathematics to interpret exponential functions with scaffolding appropriate to the proficiency level

ESOLALM.A.1.a: add, subtract, multiply, and divide multi-digit decimals fluently using the standard algorithm for each operation

ESOLALM.A.1.b: find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100; find the greatest common factor of 2 whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor; (e.g., express 36 + 8 as 4(9 + 2)); apply the least common multiple of the two whole numbers less than or equal to 12 to solve real world problems

ESOLALM.A.1.c: show that a number and its opposite that have a sum of 0 are additive inverses; describe situations in which opposite quantities combine to make 0; for example, your bank account balance is -$25.00; you deposit $25.00 into your account; the net balance is $0.00

ESOLALM.A.1.d: apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram

ESOLALM.A.1.e: apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers

ESOLALM.A.1.f: solve real-world and mathematical problems involving the four operations with rational numbers

ESOLALM.B.2.a: write and evaluate numerical expressions involving whole-number exponents by applying order of operations

ESOLALM.B.2.b: write, read, and evaluate expressions in which letters stand for numbers.

ESOLALM.B.2.c: write expressions that record operations with numbers and with letters standing for numbers (e.g., express the calculation “subtract y from 5” as 5 - y.)

ESOLALM.B.2.d: identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity (e.g., describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms)

ESOLALM.B.2.e: apply the properties of operations to generate equivalent expressions (e.g., apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y)

ESOLALM.B.2.f: identify when two expressions are equivalent (e.g., when the two expressions name the same number regardless of which value is substituted into them)

ESOLALM.C.3.a: apply and know the properties of integer exponents to generate equivalent numerical expressions

ESOLALM.C.3.b: rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)

ESOLALM.D.4.a: solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solution; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., { x \ 2 < x = 10})

ESOLALM.D.4.b: solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms

ESOLALM.E.5.a: understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]

ESOLALM.E.5.b: evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context

ESOLALM.E.5.c: construct a function to model a linear relationship between two quantities; determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph; interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values

ESOLALM.E.5.d: describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g. where the function is increasing or decreasing, linear or nonlinear); sketch a graph that exhibits the qualitative features of a function that has been described verbally

ESOLALM.F.6.a: graph proportional relationships, interpreting the unit rate as the slope of the graph; compare two different proportional relationships represented in different ways (e.g. compare a distance-time graph to a distance-time equation to determine which of the two moving objects has greater speed)

ESOLALM.F.6.b: determine the meaning of slope by using similar right triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive and graph linear equations in slope intercept form y = mx + b

ESOLALM.F.7.a: compare properties of two functions each represented among verbal, tabular, graphic and algebraic representations of functions (e.g. given a linear function represented by a table of values and a linear function represented by an algebraic equation, determine which function has a greater rate of change)

ESOLALM.F.7.b: interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear (e.g. the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1); (2, 4); and (3, 9) which are not on a straight line)

ESOLALM.G.8.a: construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities; describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association with scaffolding appropriate to the proficiency level

ESOLALM.G.8.b: know that straight lines are widely used to model relationships between two quantitative variables; for scatter plots that suggest linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line

ESOLALM.G.8.c: apply the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting slope and intercept (e.g. in a linear model for a Biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height)

ESOLALM.H.9.a: graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)

ESOLALM.H.9.b: interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship

ESOLALM.H.9.c: choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (e.g., reveal the zeros, minimum, or maximum)

ESOLALM.H.9.d: factor any quadratic expression to reveal the zeros of the function it defines

ESOLALM.I.10.a: graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior

ESOLALM.I.10.b: interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship