Mathematics and the World We Live In  Gateway Algebra

Gateway Mathematics

Standard Number:

1.0  Number and Operations

Performance Indicators

Reporting As documented through state assessment  -

State:

Category

A

A

A

A

A

A

NS

NS

NS

AE

NS

NS

At Level 1, the student is able to

·          select the best estimate for the coordinate of a given point on a number line (only rational);

·          identify the opposite of a rational number;

·          determine the square root of a perfect square less than 169;

·          use exponents to simplify a monomial written in expanded form;

·          apply order of operations when computing with integers using no more than two sets of grouping symbols and exponents 1 and 2;

·          select a reasonable solution for a real-world division problem in which the remainder must be considered.

·

A

A

A

A

A

A

NS

NS

AE

NS

NS

NS

#### At Level 2, the student is able to

·          order a given set of rational numbers (both fraction and decimal notations);

·          identify the reciprocal of a rational number;

·          add and subtract algebraic expressions;

·          multiply two polynomials with each factor having no more than two terms;

·          use estimation to determine a reasonable solution for a tedious arithmetic computation;

·          select ratios and proportions to represent real-world problems (e.g., scale drawings, sampling, etc.).

A

RW

#### At Level 3, the student is able to

·          apply the concept of slope to represent rate of change in a real-world situation.

Performance Indicators

As documented through teacher observation -

Teacher:

At Level 1, the student is able to

·          connect a variety of real-world situations to integers;

·          use manipulatives to represent commutative and associative properties of addition and multiplication;

·          investigate alternate algorithms that show the relationship of division to subtraction and multiplication to addition;

·          analyze prime and composite numbers;

·          compare and contrast the GCF and LCM of a set of numbers;

·          refine strategies for estimating whole numbers, fractions, and percentages.

#### At Level 2, the student is able to

·          probe the relationships among various subsets of the real number system;

·          compare and contrast the GCF and LCM of a set of algebraic expressions;

·          construct a number line to describe the absolute value of a number as distance from zero;

·          model operations using real-world situations and physical representations;

·          perform operations on matrices using appropriate technology (addition, subtraction, and scalar multiplication);

·          explore various representations of absolute value.

#### At Level 3, the student is able to

·          research the history of prime numbers and their uses;

·          scrutinize approximate values of real numbers such as pi and the square root of two.

 Standard Number: 2.0 Algebra Performance Indicators Reporting As documented through state assessment  - State: Category A A A A A SSG AE AE AE EI At Level 1, the student is able to ·          extend a geometric pattern; ·          extend a numerical pattern; ·          translate a verbal expression into an algebraic expression; ·          evaluate a first degree algebraic expression given values for one or more variables; ·          solve one- and two-step linear equations using integers (with integral coefficients and constants). A   A A A   A   A   A   A   A   A A A A A EI   EI GG EI   EI   EI   GG   GG   AE   GG RW EI GG EI At Level 2, the student is able to ·          select the algebraic notation which generalizes the pattern represented by data in a given table; ·          translate a verbal sentence into an algebraic equation; ·          select the graph that represents a given linear function expressed in slope-intercept form; ·          solve multi-step linear equations (more than two steps, variables on only one side of the equation); ·          solve multi-step linear equations (more than two steps, with variables on both sides of the equation); ·          solve multi-step linear equations (more than two steps, with one set of parentheses on each side of the equation); ·          select the linear graphs that models the given real-world situation described in a narrative (no data set given); ·          select the linear graph that models the given real-world situation described in a tabular set of data; ·          evaluate an algebraic expression given values for one or more variables using grouping symbols and/or exponents less than four; ·          determine the slope from the graph of a linear equation (no labeled points); ·          apply the concept of rate of change to solve real-world problems; ·          select the appropriate graphical representation of a given linear inequality; ·          select the nonlinear graph that models the given real-world situation or vice versa; ·          identify the graphical representation of the solution to a one-variable inequality on a number line.

A

A

A

RW

GG

GG

#### At Level 3, the student is able to

·          solve multi-step linear inequalities in real-world situations;

·          recognize the graphical transformation that occurs when coefficients and/or constants of the corresponding linear equations are changed;

·          determine the domain and/or range of a function represented by the graph of real-world situations,

*         Select the system of equations that could be used to solve a given real-world problem.

*         Find the solution to a quadratic equation given in standard form (integral solutions and a leading coefficient of one).

*         Select the solution to a quadratic equation given solutions represented in graphical form (integral solutions and a leading coefficient of one).

*         Select one of the factors (x + 3) of a quadratic equation (integral solutions and a leading coefficient of one).

*         Select the discriminant of a quadratic equation (integral solutions and a leading coefficient of one).

*         Recommended by the 2003 committee as additional state performance indicators. Additional state performance indicators will begin to be assessed during 2005-2006.

Performance Indicators

As documented through teacher observation -

Teacher:

At Level 1, the student is able to

·          analyze rational number patterns;

·          describe in writing the pattern for real-world data listed in a function table.

At Level 2, the student is able to

·          produce an equation to describe the relationship between data sets;

·          explore patterns including Pascal's Triangle and a Fibonacci sequence;

·          solve a system of two linear equations using the graphing, elimination, and substitution methods;

·          defend the selection of a method for solving a system of equations;

·          represent algebraic expressions and operations using manipulatives;

·          model the steps for solving simple linear equations using manipulatives;

·          write an equation that symbolically expresses a problem solving situation;

·          justify correct results of algebraic procedures;

·          distinguish between a function and other relationships.

At Level 3, the student is able to

·          analyze "families of functions" using technology.

 Standard Number: 3.0 Geometry Performance Indicators Reporting As documented through state assessment  - State: Category A GG At Level 1, the student is able to ·          identify ordered pairs in the coordinate plane. A   A SSG   SSG At Level 2, the student is able to ·          apply the given Pythagorean theorem to a real life problem illustrated by a diagram (no radicals in answer); ·          apply proportion and the concepts of similar triangles to find the length of a missing side of a triangle. · A SSG At Level 3, the student is able to ·          calculate the distance between two points given the Pythagorean theorem and the distance formula. Performance Indicators As documented through teacher observation - Teacher: At Level 1, the student is able to ·         describe real-world uses of geometric formulas and relationships; ·         discuss issues related to estimating areas of irregular-shaped figures for real-world uses (i.e., fencing, painting, laying carpet, or purchasing wallpaper or border). At Level 2, the student is able to ·          explain how to determine if a triangle is a right triangle when given the measurements of all three sides; ·          illustrate the Pythagorean theorem by measuring the length, width, and diagonals of rectangular objects; design area models to illustrate the Pythagorean theorem. At Level 3, the student is able to ·          determine the height of an object that is difficult to measure by using the properties of similar triangles.

Standard Number:

4.0 Measurement

Performance Indicators

Reporting As documented through state assessment  -

State:

Category

A

A

A

SSG

RW

SSG

#### At Level 1, the student is able to

·          estimate the area of irregular geometric figures on a grid;

·          calculate rates involving cost per unit to determine the best buy (no more than three samples);

·          apply the given formula to determine the area or perimeter of a rectangle.

A

SSG

At Level 2, the student is able to

·          apply the given formula to find the area of a circle, the circumference of a circle, or the volume of a rectangular solid.

A

AE

##### At Level 3, the student is able to

·          select the area representation for a given product of two one-variable binomials with positive constants and coefficients.

Performance Indicators

As documented through teacher observation -

Teacher:

#### At Level 1, the student is able to

·          justify the selection of a unit of measure in specific situations;

·          defend estimates of the perimeter and/or area of rectangles and triangles;

·          discover and explain formulas used to compute area and volume.

##### At Level 2, the student is able to

·          describe the procedure for determining the area of a composite shape in a real-world situation;

·          generalize area formulas using manipulatives for a parallelogram, a triangle, and a trapezoid;

·          defend an estimate for the volume of a container;

·          relate the volume of a container to its shape;

·          analyze precision, accuracy, tolerance, and approximate error in measurement situations.

At Level 3, the student is able to

·          discover the dimensions of a rectangle when given its area and the relationship between two adjacent sides;

·          describe how changes in the dimensions of figures affect perimeter, area, and volume.

 Standard Number: 5.0 Data Analysis and Probability Performance Indicators Reporting As documented through state assessment  - State: Category A   A A RW   RW RW At Level 1, the student is able to ·          determine the mean (average) of a given set of real-world data (no more than five two-digit numbers); ·          interpret bar graphs representing real-world data; ·          interpret circle graphs (pie charts) representing real-world data. A A A GG GG RW At Level 2, the student is able to ·          choose the matching linear graph given a set of ordered pairs; ·          make a prediction from the graph of a real-world linear data set; ·          determine the median for a given set of real-world data (even number of data). A RW At Level 3, the student is able to ·          apply counting principles of permutations or combinations in real-world situations. Performance Indicators As documented through teacher observation - Teacher: At Level 1, the student is able to ·          design a strategy for collecting real-world data for a scientific investigation; ·          collect and organize real-world data. At Level 2, the student is able to ·          graph real-world data using a variety of representations; ·          debate the selection of a graphical representation which best describes specific data; ·          model situations to determine theoretical and experimental probabilities; ·          judge the validity of claims made in probabilistic situations; ·          defend the sampling method chosen to conduct a survey. At Level 3, the student is able to ·          debate possible conclusions that can be supported by given data; ·          make predictions from real-world data using a line of best fit.

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