Introduction: Below is a comparison of The California Mathematics Academic Content Standards and the Mathematics Student Learning Standards for the Los Angeles Unified School District. The LAUSD standards provide guidelines only for grades 3, 7, 9, and expectations for high school graduates.

The California Mathematics standards were approved by the California Board of Education in February 1998. The LAUSD mathematics standards were recommended by the Los Angeles Systemic Initiative and were unanimously approved (along with Science, History/Social Science, and Language Arts standards) by the Los Angeles Board of Education in the Spring of 1996 for implementation in the 1996/97 academic year.

Shortly after the adoption of the California Math Standards by the California Board of Education, LAUSD Superintendent of Schools, Ruben Zacarias, issued a memorandum stating that

"the LAUSD Standards include and go beyond the State Board standards."

"the high expectations for student achievement set forth by the [LAUSD] school board and the Superintendent will be met by implementing the standards-based curriculum recommended by the Los Angeles Systemic Initiative."

Mathematicians from five universities in the Los Angeles area challenged Zacarias' evaluation of the two sets of standards. They explained to the LAUSD Board of Education that exactly the opposite of Zacarias' assertions is true: the California Mathematics Standards are vastly superior to those of LAUSD. In broad terms, the LAUSD standards are so vague as to be almost meaningless. Use of calculators is required in the third grade, undermining the mastery of essential basic skills (whereas California's math standards will not allow calculators for examinations based on those standards in the elementary school grades). Many important and fundamental topics in mathematics are not even mentioned in the LAUSD standards.

An open letter in support of the California math standards was endorsed by more than 100 California mathematicians, including the chairs of the math departments at Stanford University, Caltech, UC Irvine, UC Riverside, Cal State Los Angeles, the vice president of the American Mathematical Society and a former president of the Mathematical Association of America. In a recent independent evaluation commissioned by the Fordham foundation, Prof. Raimi and Mr. Braden conducted a review of the mathematics standards for 46 states and the District of Columbia, as well as Japan. California's new board-approved mathematics standards received the highest score.

It is commonly agreed that the two most important things that students gain from the study of mathematics are the basic numeric tools needed for survival in modern society and an increased ability to make reasoned decisions. But the study of mathematics does more than this. The precision of thought and training in abstraction that one learns from mathematics are intellectual tools of profound importance.

This being so, it is reasonable to expect precision, clarity, and complete statements in the mathematics standards for LAUSD.

To illustrate the lack of these qualities, consider the LAUSD standard:

"3. Solve problems based on algebraic relationships and functions; explore the relationship between the symbolic mathematical form of a function (expressed in equalities or inequalities) and a two- or three- dimensional graph of that function."

It is difficult to make sense of standards like #3 above. The precision in communication, that mathematics seeks to achieve, requires a comparison of the California State Mathematics Standards with the Los Angeles Unified School District Standards not by what might be meant, but by what is actually stated.

With this in mind we now turn to a number of basic topics in the K-12 mathematics curriculum and compare the two documents directly. For each mathematical category listed below, all relevant LAUSD standards appear in the left column, and all relevant California standards appear in the right column. The most blatant short-coming of the LAUSD math standards requires no table for comparison; TRIGONOMETRY IS ENTIRELY MISSING FROM THE LAUSD MATH STANDARDS.

 

Number Line

Los Angeles

California  Grade 4, Number Sense  1.5 interpret different meanings for fractions including parts of a whole, parts of a set, indicated division of whole numbers and quantities (and measures) between whole numbers on a number line; and relate to simple decimals on a number line  1.8 use concepts of negative numbers (e.g., on a number line, in counting, in temperature, "owing")  1.9 identify the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line  Grade 4, Statistics, Data Analysis and Probability  1.1 formulate survey questions, systematically collect and represent data on a number line, and coordinate graphs, tables and charts  Grade 5, Number Sense  1.5 identify and represent positive and negative integers, decimals, fractions and mixed numbers on a number line  Grade 6, Number Sense  1.1 compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line  Grade 7 Number Sense  2.5 understand the meaning of the absolute value of a number, interpret it as the distance of the number from zero on a number line and determine the absolute value of real numbers

 

 

Development of Use of Negative Numbers

Los Angeles

California  Grade 4 Number Sense  1. Students understand place value of whole numbers and decimals to two decimal places, how these relate to simple fractions, and begin to work with negative numbers  1.8 use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature, "owing")  3. Students solve problems involving addition, subtraction, multiplication and division of whole numbers, including the addition and subtraction of negative numbers, and understand the relationships among the operations.  Grade 5 Number Sense  1. Students compute with very large and very small numbers, positive and negative numbers, decimals and fractions and understand the relationship between decimals, fractions and percents.  1.3 understand and compute squares and cubes of non-negative whole numbers; compute examples as repeated multiplication  1.5 identify and represent positive and negative integers, decimals, fractions and mixed numbers on a number line  2.1 add, subtract, multiply and divide with decimals and negative numbers and verify the reasonableness of the results  Grade 6 Number Sense  1.1 compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line  2.3 solve addition, subtraction, multiplication and division problems, including those arising in concrete situations that use positive and negative numbers and combinations of these operations  Grade 7 Number Sense  1.1 read, write and compare rational numbers in scientific notation (positive and negative powers of 10), approximate numbers using scientific notation  2.1 understand negative whole number exponents. Multiply and divide expressions involving exponents with a common base  2.2 interpret positive whole number powers as repeated multiplication and negative whole numbers as repeated division or multiplication by the multiplicative inverse. simplify and evaluate expressions that include exponents

 
 
 

Fractions Grades K-3

Los Angeles  Grade 3  37. Apply the basic operations (addition, subtraction, multiplication, and division) using whole numbers and simple fractions (halves, fourths); use rounding to the tens, hundreds, and thousands as an estimation strategy to check the reasonableness of results.

California  Grade 2 Number Sense  4. Students understand that fractions and decimals can refer to parts of a set and parts of a whole.  4.1 recognize, name and compare unit fractions up to 1/12  4.2 recognize fractions of a whole and parts of a group (e.g., 1/4th of a pie, 2/3'rds of 15 balls)  4.3 know that when all fractional parts are included, such as four-fourths, the result is equal to the whole  Grade 3 Number Sense  3. Students understand the relationship between whole numbers, simple fractions and decimals.  3.1 compare fractions represented by drawings or concrete materials to show equivalency, and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is more than 1/8)  3.2 add and subtract simple fractions (e.g. determine that 1/8 + 3/8 is the same as 1/2)  3.4 know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents 1/2 of a dollar, 75 cents is 3/4 of a dollar)

 
 
 

Fractions Grades 4-7

Los Angeles  Grade 7  25. Add, subtract, multiply, and divide using whole numbers, integers, primes, factors, multiples, fractions, decimals, rational numbers, exponents, and scientific notation; estimate and check the reasonableness of results.  33. Use deductive and inductive reasoning to solve mathematical problems; apply proportional reasoning to examine the relationships among fractions, decimals, and percents through examples involving rates, ratios, proportions, and scales.

California  Grade 4 Number Sense  1. Students understand place value of whole numbers and decimals to two decimal places, how these relate to simple fractions, and begin to work with negative numbers  1.5 interpret different meanings for fractions including parts of a whole, parts of a set, indicated division of whole numbers and quantities (and measures) between whole numbers on a number line; and relate to simple decimals on a number line  1.6 write tenths and hundredths in decimal and fraction notation and know fraction/decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or.50; 7/4 = 1 3/4 = 1.75)  1.7 write the fraction represented by a drawing of parts of a figure; represent a given fraction using drawings  1.9 identify the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line  Grade 5 Number Sense  1. Students compute with very large and very small numbers, positive and negative numbers, decimals and fractions and understand the relationship between decimals, fractions and percents.  1.2 interpret percents as part of a hundred; find decimal and percent equivalents for common fractions; explain why they represent the same value; and compute a given percent of a whole number  1.5 identify and represent positive and negative integers, decimals, fractions and mixed numbers on a number line  2. Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals.  2.3 solve simple problems including ones arising in concrete situations involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in simplest form  2.4 understand the concept of multiplication and division of fractions  2.5 compute and perform simple multiplication and division of fractions and apply these procedures to solving problems  Grade 5 Statistics, Data Analysis and Probability  5ST1.3 use fractions and percentages to compare data sets of different size  Grade 6 Number Sense  1. Students compare and order fractions, decimals, and mixed numbers. They solve problems involving fractions, ratios, proportions, and percentages.  1.1 compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line  2.1 solve problems involving addition, subtraction, multiplication and division of fractions and explain why a particular operation was used for a given situation  2.2 explain the meaning of multiplication and division of fractions and perform the calculations (e.g., 5/8 divided by 15/16 = 5/8 x 16/15 = 2/3)  2.4 determine the least common multiple and greatest common divisor of whole numbers. Use them to solve problems with fractions (e.g., to find a common denominator in order to add two fractions or to find the reduced form for a fraction)  Grade 7 Number Sense  1.2 add, subtract, multiply and divide rational numbers, integers, fractions and decimals and take rational numbers to whole number powers  1.3 convert fractions to decimals and percents and use these representations in estimation, computation and applications  1.5 know that every fraction is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions  2. Students use exponents, powers, and roots and use exponents in working with fractions.  2.2 add and subtract fractions using factoring to find common denominators  2.3 multiply, divide, and simplify fractions using exponent rules

 

 

Percent, Interest, Compound Interest

Los Angeles  Grade 7  33. Use deductive and inductive reasoning to solve mathematical problems; apply proportional reasoning to examine the relationships among fractions, decimals, and percents through examples involving rates, ratios, proportions, and scales.

California  Grade 5, Number Sense  1. Students compute with very large and very small numbers, positive and negative numbers, decimals and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers.  1.2 interpret percents as part of a hundred; find decimal and percent equivalents for common fractions; explain why they represent the same value; and compute a given percent of a whole number  Grade 5, Statistics, Data Analysis and Probability  1.3 use fractions and percentages to compare data sets of different size  Grade 6, Number Sense  1. Students compare and order fractions, decimals, and mixed numbers. They solve problems involving fractions, ratios, proportions, and percentages.  1.4 calculate given percentages of quantities and solve problems involving discounts at sales, interest earned and tips  Grade 6, Statistics, Data Analysis and Probability  3.3 represent probabilities as ratios, proportions, and decimals between 0 and 1, and percents between 0 and 100 and check that probabilities computed are reasonable; know how this is related to the probability of an event not occurring  Grade 7, Number Sense  1.3 convert fractions to decimals and percents and use these representations in estimation, computation and applications  1.6 calculate percent of increases and decreases of a quantity  1.7 solve problems that involve discounts, markups, commissions, profit and simple compound interest  Algebra I  15. Students apply algebraic techniques to rate problems, work problems, and percent mixture problems.

 

 

Lines and Linear Equations Grade K-3

Los Angeles  Grade 3  40. Use the geometric concepts of space and form to construct, describe, and compare the properties of one-, two-, and three-dimensional figures such as line segments, circles, simple polygons, and solids.

California  Grade 2, Statistics Data Analysis and Probability  2.1 recognize, describe, extend and explain how to get the next term in linear patterns (e.g., 4, 8, 12 ...; the number of ears on 1 horse, 2 horses, 3 horses, 4 horses)  Grade 3, Algebra and Functions  2.2 extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses can be calculated by counting by 4s or by multiplying the number of horses by 4)  Grade 3, Statistics, Data Analysis and Probability  1.3 summarize and display the results of probability experiments in a clear and organized way (e.g., use a bar graph or a line plot)  1.4 use the results of probability experiments to predict future events (e.g., use a line plot to predict the temperature forecast for the next day)

 
 
 

Lines and Linear Equations Grade 4-7

Los Angeles

California  Grade 4, Measurement and Geometry  2. Students use two-dimensional coordinate grids to represent points and graph lines and simple figures.  2.1 draw the points corresponding to linear relationships on graph paper (e.g., draw the first ten points for the equation y = 3x and connect them using a straight line)  2.2 understand that the length of a horizontal line segment equals the difference of the x-coordinates  2.3 understand that the length of a vertical line segment equals the difference of the y-coordinates  3. Students demonstrate understanding of plane and solid geometric objects. They use this knowledge to show relationships and solve problems.  3.1 identify lines that are parallel and perpendicular  Grade 5, Algebra and Functions  1.5 solve problems involving linear functions with integer values, write the equation, and graph the resulting ordered pairs of integers on a grid  Grade 5, Measurement and Geometry  2.1 measure, identify and draw angles, perpendicular and parallel lines, rectangles and triangles, using appropriate tools (e.g., straight edge, ruler, compass, protractor and drawing software)  Grade 6, Algebra and Functions  1. Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations and graph and interpret their results.  1.1 write and solve one-step linear equations in one variable  Grade 7, Algebra and Functions  3. Students graph and interpret linear and some non-linear functions.  3.1 graph functions of the form y = nx$2$ and y = nx$3$ and use in solving problems  3.2 plot the values from the volumes of a 3-D shape for various values of its edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and a varying length equilateral triangle base)  3.3 graph linear functions, noting that the vertical change (change in y-value) per unit horizontal change (change in x-value) is always the same and know that the ratio ("rise over run") is called the slope of a graph  3.4 plot values of the quantities whose ratio is always the same (cost vs. number of an item, feet vs. inches, circumference vs. diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities.  4. Students solve simple linear equations and inequalities over the rational numbers.  4.1 solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution(s) in terms of the context from which they arose and verify the reasonableness of the results  4.2 solve multi-step problems involving rate, average speed, distance and time, or direct variation  Grade 7, Measurement and Geometry  3.3 know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments, and, in some situations, empirically verify the Pythagorean Theorem by direct measurement  3.6 identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and how two or more objects are related in space (e.g., skew lines, the possible ways three planes could intersect)

 
 
 

Lines and Linear Equation Grades 8-12

Los Angeles

California  Algebra I  4. Students simplify expressions prior to solving linear equations and inequalities in one variable such as 3(2x-5) + 4(x-2) = 12.  5. Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable, with justification of each step.  6. Students graph a linear equation, and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4).  They are also able to sketch the region defined by linear inequality (e.g., sketch the region defined by 2x + 6y < 4).  7. Students verify that a point lies on a line given an equation of the line. Students are able to derive linear equations using the point-slope formula.  8. Students understand the concepts of parallel and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.  9. Students solve a system of two linear equations in two variables algebraically, and are able to interpret the answer graphically.  Students are able to use this to solve a system of two linear inequalities in two variables, and to sketch the solution sets.  Geometry  7. Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.  16. Students perform basic constructions with straightedge and compass such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.  17. Students prove theorems using coordinate geometry, including the midpoint of a line segment, distance formula, and various forms of equations of lines and circles.  Algebra II  2. Students solve systems of linear equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices.  Trigonometry  7. Students know that the tangent of the angle a line makes with the x-axis is equal to the slope of the line.

 

 

Quadratics

Los Angeles

California  Algebra 1  14. Students solve a quadratic equation by factoring or completing the square.  19. Students know the quadratic formula and are familiar with its proof by completing the square.  20. Students use the quadratic formula to find the roots of a second degree polynomial and to solve quadratic equations.  21. Students graph quadratic functions and know that their roots are the x-intercepts.  22. Students use the quadratic formula and/or factoring techniques to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.  23. Students apply quadratic equations to physical problems such as the motion of an object under the force of gravity.  25.3 Given a specific algebraic statement involving linear, quadratic or absolute value expressions, equations or inequalities, students determine if the statement is true sometimes, always, or never.  Algebra 2  8. Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.  9. Students demonstrate and explain the effect changing a coefficient has on the graph of quadratic functions. That is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b)2 + c.  10. Students graph quadratic functions and determine the maxima, minima, and zeros of the function.  16. Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.  17. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method of completing the square to put the equation into standard form and can recognize whether its graph is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.  Mathematical Analysis  5.1 Students can take a quadratic equation in two variables, put it in standard form by completing the square and using rotations and translations if necessary, determine what type of conic section the equation represents, and determine its geometric components (foci, asymptotes, etc.).  5.2 Students can take a geometric description of a conic section (e.g. the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6), and derive a quadratic equation representing it.

 

 

Exponents, Roots and Logarithms

Los Angeles  Grade 7  25. Add, subtract, multiply, and divide using whole numbers, integers, primes, factors, multiples, fractions, decimals, rational numbers, exponents, and scientific notation; estimate and check the reasonableness of results.

California  Grade 5 Number Sense  1.4 determine the prime factors of all numbers through 50 and write numbers as the product of their prime factors using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23 x 3);  Grade 7 Number Sense  2. Students use exponents, powers, and roots and use exponents in working with fractions.  2.1 understand negative whole number exponents. Multiply and divide expressions involving exponents with a common base  2.3 multiply, divide, and simplify fractions using exponent rules  2.4 use the inverse relationship between raising to a power and root extraction for perfect square integers; and, for integers which are not square, determine without a calculator, the two integers between which its square root lies, and explain why  Grade 7 Algebra and Functions  2.1 interpret positive whole number powers as repeated multiplication and negative whole numbers as repeated division or multiplication by the multiplicative inverse. simplify and evaluate expressions that include exponents  2.2 multiply and divide monomials; extend the process of taking powers and extracting roots to monomials, when the latter results in a monomial with an integer exponent  Algebra 1  2. Students understand and use such operations as taking the opposite, reciprocal, raising to a power, and taking a root.  This includes the understanding and use of the rules of exponents.  Algebra 2  7. Students add, subtract, multiply, divide, reduce and evaluate rational expressions with monomial and polynomial denominators, and simplify complicated fractions including fractions with negative exponents in the denominator.  11. Students prove simple laws of logarithms.  11.1 Students understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.  11.2 Students judge the validity of an argument based on whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.  13. Students use the definition of logarithms and the product formula for logs to translate between logarithms in any bases.  14. Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and identify their approximate values.  15. Students determine if a specific algebraic statement involving rational expressions, radical expressions, logarithmic or exponential functions, is sometimes true, always true, or never true.  Note: Additional standards involving exponential distributions and functions include:   Probability and Statistics standard 4   Probability and Statistics-Advanced standard 7   Calculus standard 4.4

 
 

Triangles Grades K-3

Los Angeles  (There is no mention of triangles. See polygons below)

California  Kindergarten Measurement and Geometry  2.1 identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone)  Kindergarten Statistics, Data Analysis and Probability  1.2 identify, describe and extend simple patterns involving shape, size, or color such as circle, triangle, or red, blue  Grade 1 Measurement and Geometry  2.1 identify triangles, rectangles, squares and circles, including the faces of three-dimensional objects  Grade 2 Measurement and Geometry  2.1 describe and classify plane and solid geometric shapes (e.g., circle, triangle, square, rectangle, sphere, pyramid, cube, rectangular prism) according to the number and shape of faces, edges and vertices  2.2 put shapes together and take them apart to form other shapes (e.g., two congruent right triangles can form a rectangle)  Grade 3 Measurement and Geometry  2.2 identify attributes of triangles, (e.g., two equal sides for the isosceles triangle, three equal sides for the equilateral triangle, right angle for the right triangle)

 
 
 

Triangles Grades 4-7

Los Angeles  (There is no mention of triangles. See polygons below)

California  Grade 4 Measurement and Geometry  3.7 know the definition of different triangles (equilateral, isosceles, scalene)  Grade 5 Measurement and Geometry  1.1 derive and use the formula for the area of right triangles and of parallelograms by comparing with the area of rectangles (i.e., two of the same triangles make a rectangle with twice the area; a parallelogram is compared to a rectangle with the same area found by cutting and pasting a right triangle)  2.1 measure, identify and draw angles, perpendicular and parallel lines, rectangles and triangles, using appropriate tools (e.g., straight edge, ruler, compass, protractor and drawing software)  2.2 know that the sum of the angles of any triangle is 180 degrees and the sum of the angles of any quadrilateral is 360 degrees and use this information to solve problems  Grade 6 Algebra and Functions  3.1 use variables in expressions describing geometric quantities, e.g., P = 2w + 2l, A = 1/2 bh, C = (pi) d which give the perimeter of a rectangle, area of a triangle, and circumference of a circle, respectively  Grade 6 Measurement and Geometry  2.2 use the properties of complimentary and supplementary angles and of the angles of a triangle to solve problems involving an unknown angle  2.3 draw quadrilaterals and triangles given information about them (e.g., a quadrilateral having equal sides but no right angles, a right isosceles triangle)  Grade 7 Algebra and Functions  3.2 plot the values from the volumes of a 3-d shape for various values of its edge lengths (e.g., cubes with varying edge lengths or a triangle prism with a fixed height and a varying length equilateral triangle base)  Grade 7 Measurement and Geometry  2.1 routinely use formulas for finding the perimeter and area of basic two-dimensional figures and for the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, cones and circular cylinders  3.3 know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments.

 
 
 

Triangles Grades 8-12

Los Angeles  (There is no mention of triangles. See polygons below)

California  Geometry  5. Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles.  6. Students know and are able to use the triangle Inequality Theorem.  10. Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.  12. Students find and use measures of sides, interior and exterior angles of triangles and polygons to classify figures and solve problems.  15. Students use the Pythagorean Theorem to determine distance and find missing lengths of sides of right triangles.  18. Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them, (e.g., tan(x) = sin(x)/cos(x), (sin (x))2 + (cos (x))2 = 1).  19. Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.  20. Students know and are able to use angle and side relationships in problems with special right triangles such as 30-60-90 triangles and 45-45-90 triangles.  Trigonometry  12. Students use trigonometry to determine unknown sides or angles in right triangles.  14. Students determine the area of a triangle given one angle and the two adjacent sides.  Note: Additional standards involving trigonometry include:   Trigonometry standards 3.2 8, 9, 10, 11, and 19   Mathematical Analysis standard 2   Calculus standards 4.4, 17, 18, and 20

 

 

Polygons

Los Angeles  Grade 3  40. Use the geometric concepts of space and form to construct, describe, and compare the properties of one-, two-, and three-dimensional figures such as line segments, circles, simple polygons, and solids.

California  Grade 3 Measurement and Geometry  1.3 find the perimeter of a polygon with integer sides  2.1 identify and describe and classify polygons (including pentagons, hexagons and octagons)  Grade 6 Number Sense  1.3 use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, [understanding it as multiplication of both sides of an equation by a multiplicative inverse.]  Geometry  10. Students compute areas of polygons including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.  12. Students find and use measures of sides, interior and exterior angles of triangles and polygons to classify figures and solve problems.  13. Students prove relationships between angles in polygons using properties of complementary, supplementary, vertical, and exterior angles.  21. Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

 

 

Any Further Topics Relating to Linear Algebra

Los Angeles

California  LINEAR ALGEBRA The general goal in this discipline is that students learn the techniques of matrix manipulation so as to be able to solve systems of linear equations in any number of variables.  Linear Algebra is most often combined with another subject, such as Trigonometry, Mathematical Analysis, or Pre-Calculus.  1. Students solve simultaneous linear equations in any number of variables using Gauss-Jordan elimination.  2. Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.  3. Students reduce rectangular matrices to row echelon form.  4. Students perform addition on matrices and vectors.  5. Students perform matrix multiplication, multiply vectors by matrices and by scalars.  6. Students demonstrate understanding that linear systems are either inconsistent (no solutions), have exactly one solution, or have infinitely many solutions.  7. Students demonstrate understanding of the geometric interpretation of vectors and vector addition (via parallelograms) for vectors in the plane and in three dimensional space.  8. Students interpret the solution sets of systems of equations geometrically. For example the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two by two system is interpreted as the intersection of a pair of lines in the plane.  9. Students demonstrate understanding of the notion of the inverse to a square matrix, and apply it to solve systems of linear equations.  10. Students compute the determinants of 2 by 2 and 3 by 3 matrices, and are familiar with their geometric interpretations as area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in 2-dimensional and 3-dimensional spaces.  11. Students know that a square matrix is invertible if, and only if, its determinant is non-zero. They can compute the inverse to 2 by 2 and 3 by 3 matrices using row reduction methods or Cramer's rule.  12. Students compute the scalar (dot) product of two vectors in n-dimensional space, and know that perpendicular vectors have zero dot product.

 

 

Topics Which Either do not Exist or Have Nothing to do with Mathematics

Los Angeles  11. Analyze how inventions, discoveries, and events influence the development of mathematical theories and how mathematics continues to respond to changing societal, cultural, and technological forces.  23. Analyze the influences that historical events, scientific discoveries, and social changes have had and continue to have on the development of mathematical theories.  35. Compare and describe the number systems and mathematical concepts, for example, the Pythagorean Theorem, developed in different civilizations, historical periods, and cultures.  47. Compare the use of various number systems (for example, Hindu-Arabic, Roman, tally, etc.) from different historical periods.

California

 

Remarks: The two standards 33 and 40 in the LAUSD mathematics standards together seem to bear the brunt of the content requirements there. Note that they each appear twice in the lists above, and in three of the four areas where they appear, they are the ONLY STANDARD from the LAUSD standards which mentions the topic.

It is perhaps also worth noting that standard 40 appears in the third grade standards and standard 33 in the seventh grade standards. In particular, this must minimize the level at which 40 applies. So we may reasonably infer that the expected background for students in LAUSD in lines and linear equations as well as their knowledge of such figures as polygons is expected to be negligible. Of course, while it could be argued that polygons are not central to applications of mathematics in the world and in the workplace, the same CANNOT be said for lines and linear equations. These are BASIC. Moreover, standard 33 only refers to line segments in geometric figures. So it can fairly be said that there is absolutely no discussion of linear equations, systems of linear equations, or related topics in these entire standards. It is hard to find the words to describe this situation.

Similarly, having a discussion of the essential topics of percent, interest, and compound interest confined to a single standard would imply, at best, minimal competence with these concepts. But these concepts are likely to influence every one of the major financial transactions for each of these students throughout their lives.

It is hard to find parallel statements by which to compare the LA Standards and the CA Standards point-for-point. In general, the LA Standards are too vague to admit comparison with anything. And sometimes they are downright foolish, as in my first example.

LA Grade 7:

"(35) Compare and describe the number systems and mathematical concepts, for example, the Pythagorean Theorem, developed in different civilizations, historical periods, and cultures."

Here is the nearest thing to a pair for comparison, for Grade 7:

 

Los Angeles  28. Identify, describe, compare, and classify geometric figures; apply geometric properties and relationships to solve problems; and use geometric concepts as a means to describe the physical world.  [At the Grade 7 level this is about all there is on geometry, except for the historical-philosophical item (35) Compare and describe the number systems and mathematical concepts, for example, the Pythagorean Theorem, developed in different civilizations, historical periods, and cultures.]

California  2. Students compute the perimeter, area and volume of common geometric objects and use these to find measures of less common objects; they know how perimeter, area, and volume are affected under changes of scale.  2.1 routinely use formulas for finding the perimeter and areas of basic two-dimensional figures and for the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and circular cylinders  2.2 estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking them up into more basic geometric objects  2.3 compute the length of the perimeter, the surface area of the faces, and the volume of a 3-D object built from rectangular solids. They understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor  2.4 relate the changes in measurement under change of scale to the units used (e.g., square inches, cubic feet) and to conversions between units (1 square foot = 12 square inches, 1 cubic inch = 2.54 cubic centimeters)  3. Students know the Pythagorean Theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures.  3.1 identify and construct basic elements of geometric figures, (e.g., altitudes, midpoints, diagonals, angle bisectors and perpendicular bisectors; and central angles, radii, diameters and chords of circles) using compass and straight-edge  3.2 understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections  3.3 know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and lengths of other line segments, and, in some situations, empirically verify the Pythagorean Theorem by direct measurement  3.4 demonstrate an understanding of when two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures  3.5 construct two-dimensional patterns for three-dimensional models such as cylinders, prisms and cones  3.6 identify elements of three-dimensional geometric objects (e.g., diagonals of rectangular solids) and how two or more objects are related in space (e.g., skew lines, the possible ways three planes could intersect)

It is possible that LA believes its Standard (28) quoted above includes all the things detailed in the new CA Standards, but I don't believe it, and belief is not a good test of a contract anyway.

Other particularly vague items in the LA Standards, that should be held up to scorn, and which could, with effort, be found to correspond with some rather extensive and particular listings in the CA Standards, are:

Graduate Level, items 8,9,10

8. Investigate the relationship between mathematical models and real-life problems by using hands-on materials and/or current technology such as calculators and computer modeling.

9. Make and test conjectures (inductive and deductive), construct simple arguments, validate solutions, and apply conclusions to various real-world situations.

10. Make connections among related mathematical concepts and apply these concepts to other content areas and the world of work.

21. Apply inductive and deductive reasoning and problem-solving strategies such as analysis of patterns, properties, relations of number systems, to validate solutions and apply conclusions to various real-world situations.

22. Make connections among mathematical concepts and apply these concepts to other content areas and to real-life situations.

16. Use inductive and deductive reasoning and concepts of coordinate and transformational geometry to analyze geometric relationships, validate formal and informal proofs, and solve problems in geometric relationships such as congruency and similarity.

This item fails to specify whether Euclid's axiom system, and deductive proof, are intended anywhere. The phrase "deductive and inductive reasoning" is borrowed from NCTM and means nothing.

29. Apply a variety of discrete structures (series, sequences, matrices, and tree diagrams) to find possible combinations and arrangements in a problem situation.

46. Make connections among mathematical concepts and relate them to concepts in other content areas and in daily life.

3. Solve problems based on algebraic relationships and functions; explore the relationship between the symbolic mathematical form of a function (expressed in equalities or inequalities) and a two- or three-dimensional graph of that function.

15. Identify patterns, functions, and other algebraic relationships including inequalities; use tables, graphs, and equations to model functional relationships in real-life situations; apply knowledge of functions to analyze and interpret problems, predict solutions, and create algebraic algorithms to solve problems."

Fundamentally, the writing here is so sloppy and imprecise that it is impossible to give the standards any reasonable meaning within the context of mathematics.

The vacuousness of the LAUSD math standards facilitates poor achievement of LAUSD students in mathematics. Since these standards can mean whatever a particular reader wants them to mean, they are not standards at all. They serve only to protect poor achievement in mathematics -- the status quo for LAUSD.