- Chapter 1: Adding Whole Numbers and Money Subtracting Whole Numbers and Money Fact Families, Part 1
- Chapter 10: Sequences Scales
- Chapter 100: Algebraic Addition of Integers
- Chapter 101: Ratio Problems Involving Totals
- Chapter 102: Mass and Weight
- Chapter 103: Perimeter of Complex Shapes
- Chapter 104: Algebraic Addition Activity
- Chapter 105: Using Proportions to Solve Percent Problems
- Chapter 106: Two-Step Equations
- Chapter 107: Area of Complex Shapes
- Chapter 108: Transformations
- Chapter 109: Corresponding Parts Similar Figures
- Chapter 11: Problems About Comparing Problems About Separating
- Chapter 110: Symmetry
- Chapter 111: Applications Using Division
- Chapter 112: Multiplying and Dividing Integers
- Chapter 113: Adding and Subtracting Mixed Measures Multiplying by Powers of Ten
- Chapter 114: Unit Multipliers
- Chapter 115: Writing Percents as Fractions, Part 2
- Chapter 116: Compound Interest
- Chapter 117: Finding a Whole When a Fraction is Known
- Chapter 118: Estimating Area
- Chapter 119: Finding a Whole When a Percent is Known
- Chapter 12: Place Value Through Trillions Multistep Problems
- Chapter 120: Volume of a Cylinder
- Chapter 13: Problems About Comparing Elapsed-Time Problems
- Chapter 14: The Number Line: Negative Numbers
- Chapter 15: Problems About Equal Groups
- Chapter 16: Rounding Whole Numbers Estimating
- Chapter 17: The Number Line: Fractions and Mixed Numbers
- Chapter 18: Average Line Graphs
- Chapter 19: Factors Prime Numbers
- Chapter 2: Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2
- Chapter 20: Greatest Common Factor (GCF
- Chapter 21: Divisibility
- Chapter 22: Equal Groups Problems with Fractions
- Chapter 23: Ratio Rate
- Chapter 24: Adding and Subtracting Fractions That Have Common Denominators
- Chapter 25: Writing Division Answers as Mixed Numbers Multiples
- Chapter 26: Using Manipulatives to Reduce Fractions Adding and Subtracting Mixed Numbers
- Chapter 27: Measures of a Circle
- Chapter 28: Angles
- Chapter 29: Multiplying Fractions Reducing Fractions by Dividing by Common Factors
- Chapter 3: Unknown Numbers in Addition Unknown Numbers in Subtraction
- Chapter 30: Least Common Multiple (LCM) Reciprocals
- Chapter 31: Areas of Rectangles
- Chapter 32: Expanded Notation More on Elapsed Time
- Chapter 33: Writing Percents as Fractions, Part 1
- Chapter 34: Decimal Place Value
- Chapter 35: Writing Decimal Numbers as Fractions, Part 1 Reading and Writing Decimal Numbers
- Chapter 36: Subtracting Fractions and Mixed Numbers from Whole Numbers
- Chapter 37: Adding and Subtracting Decimal Numbers
- Chapter 38: Adding and Subtracting Decimal Numbers and Whole Numbers Squares and Square Roots
- Chapter 39: Multiplying Decimal Numbers
- Chapter 4: Unknown Numbers in Multiplication Unknown Numbers in Division
- Chapter 40: Using Zero as a Placeholder Circle Graphs
- Chapter 41: Finding a Percent of a Number
- Chapter 42: Renaming Fractions by Multiplying by 1
- Chapter 43: Equivalent Division Problems Finding Unknowns in Fraction and Decimal Problems
- Chapter 44: Simplifying Decimal Numbers Comparing Decimal Numbers
- Chapter 45: Dividing a Decimal Number by a Whole Number
- Chapter 46: Writing Decimal Numbers in Expanded Notation Mentally Multiplying Decimal Numbers by 10 and by 100
- Chapter 47: Circumference Pi (
- Chapter 48: Subtracting Mixed Numbers with Regrouping, Part 1
- Chapter 49: Dividing by a Decimal Number
- Chapter 5: Order of Operations, Part 1
- Chapter 50: Decimal Number Line (Tenths) Dividing by a Fraction
- Chapter 51: Rounding Decimal Numbers
- Chapter 52: Mentally Dividing Decimal Numbers by 10 and by 100
- Chapter 53: Decimals Chart Simplifying Fractions
- Chapter 54: Reducing by Grouping Factors Equal to 1 Dividing Fractions
- Chapter 55: Common Denominators, Part 1
- Chapter 56: Common Denominators, Part 2
- Chapter 57: Adding and Subtracting Fractions: Three Steps
- Chapter 58: Probability and Chance
- Chapter 59: Adding Mixed Numbers
- Chapter 6: Fractional Parts
- Chapter 60: Polygons
- Chapter 61: Adding Three or More Fractions
- Chapter 62: Writing Mixed Numbers as Improper Fractions
- Chapter 63: Subtracting Mixed Numbers with Regrouping, Part 2
- Chapter 64: Classifying Quadrilaterals
- Chapter 65: Prime Factorization Division by Primes Factor Trees
- Chapter 66: Multiplying Mixed Numbers
- Chapter 67: Using Prime Factorization to Reduce Fractions
- Chapter 68: Dividing Mixed Numbers
- Chapter 69: Lengths of Segments Complementary and Supplementary Angles
- Chapter 7: Lines, Segments, and Rays Linear Measure
- Chapter 70: Reducing Fractions Before Multiplying
- Chapter 71: Parallelograms
- Chapter 72: Fractions Chart Multiplying Three Fractions
- Chapter 73: Exponents Writing Decimal Numbers as Fractions, Part 2
- Chapter 74: Writing Fractions as Decimal Numbers Writing Ratios as Decimal Number
- Chapter 75: Writing Fractions and Decimals as Percents, Part 1
- Chapter 76: Comparing Fractions by Converting to Decimal Form
- Chapter 77: Finding Unstated Information in Fraction Problems
- Chapter 78: Capacity
- Chapter 79: Area of a Triangle
- Chapter 8: Perimeter
- Chapter 80: Using a Constant Factor to Solve Ratio Problems
- Chapter 81: Arithmetic with Units of Measure
- Chapter 82: Volume of a Rectangular Prism
- Chapter 83: Proportions
- Chapter 84: Order of Operations, Part 2
- Chapter 85: Using Cross Products to Solve Proportions
- Chapter 86: Area of a Circle
- Chapter 87: Finding Unknown Factors
- Chapter 88: Using Proportions to Solve Ratio Word Problems
- Chapter 89: Estimating Square Roots
- Chapter 9: The Number Line: Ordering and Comparing
- Chapter 90: Measuring Turns
- Chapter 91: Geometric Formulas
- Chapter 92: Expanded Notation with Exponents Order of Operations with Exponents
- Chapter 93: Classifying Triangles
- Chapter 94: Writing Fractions and Decimals as Percents, Part 2
- Chapter 95: Reducing Rates Before Multiplying
- Chapter 96: Functions Graphing Functions
- Chapter 97: Transversals
- Chapter 98: Sum of the Angle Measures of Triangles and Quadrilaterals
- Chapter 99: Fraction-Decimal-Percent Equivalents
Saxon Math, Course 1 1st Edition - Solutions by Chapter
Full solutions for Saxon Math, Course 1 | 1st Edition
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.