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MATH INV Mathematics CCRS Standards and Alabama COS CCRS Standard 1. Critique ancient numeration systems and applications, including astronomy and the development and use of money and calendars. (Alabama) Standard ID Modeling Number/Quantity MI-NM.1 Determine relationships among mathematical achievements of ancient peoples, including the Sumerians, Babylonians, Egyptians, Mesopotamians, Chinese, Aztecs, and Incas. (Alabama) Explain origins of the Hindu-Arabic numeration system. (Alabama) Example: Perform addition and subtraction in both the Hindu-Arabic and the Roman numeration systems to compare place value and place holders. Franklin County Schools Evidence of Student Attainment Students: Create a presentation identifying and analyzing the mathematical achievements of ancient people. Compare and contrast Hindu-Arabic and Roman numeration systems. Teacher Vocabulary Knowledge Resources Numeration systems Place value/place holders Roman Numerals Hindu-Arabic Numerals Click below to access all ALEX resources aligned to this The significance standard. of place value. ALEX Resources Students know: MATH INV Mathematics CCRS Standards and Alabama COS 2. Analyze mathematical Modeling relationships in music to Number/Quantity interpret frequencies of MI-NM.2 musical notes and to compare mathematical structures of various musical instruments. (Alabama) Examples: Compare frequencies of notes exactly one octave apart on the musical scale; using frequencies and wave patterns of middle C, E above middle C, and G above middle C to explain why the C major chord is harmonious. Students: Demonstrate the relationship between math and music. Frequencies Octaves Pythagorean tuning Harmonious/harmonic tones Students know: Basic fractional concepts. Characteristics of periodic functions. Click below to access all ALEX resources aligned to this standard. ALEX Resources Determine lengths of strings necessary to produce harmonic tones as in Pythagorean tuning. (Alabama) 3. Use special numbers, including e, i, π; and the golden ratio, to solve application-based problems. (Alabama) Modeling Number/Quantity MI-NM.3 Students: Solve applicationbased problems involving e, i, π and the golden ratio. a. Identify transcendental numbers. Example: Calculate e to ten decimal places using a summation with 1/n! 4. Explain the development Modeling and uses of sets of Number/Quantity Franklin County Schools Students: Describe the e i π Real numbers Rational/irrational Transcendental numbers Golden ratio Summation Complex numbers Irrational numbers Factorial Students know: How to calculate factorials. The approximation for e and π. The definition of i. How to perform operations using imaginary numbers. Students know: Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX resources MATH INV numbers, including complex, real, rational, irrational, integer, whole, and natural numbers. (Alabama) Mathematics CCRS Standards and Alabama COS MI-NM.4 development and use of sets of numbers from natural to complex including the contributions of wellknown mathematicians. numbers Imaginary/complex numbers Gaussian plane (complex plane a. Analyze contributions to the number system by well-known mathematicians, including Archimedes, John Napier, René Descartes, Sir Isaac Newton, Johann Carl Friedrich Gauss, and Julius Wilhelm Richard Dedekind. (Alabama) Example: Plot solutions to the polynomial equation, x2 – 6x + 11 = 0, on the Gaussian plane. 5. Identify beginnings of algebraic symbolism and structure through the works of European mathematicians. (Alabama) a. b. Seeing Structure in Expressions Algebra MI-SSE.1 Create a Fibonacci sequence when given two initial integers. (Alabama) Investigate Tartaglia’s formula for solving cubic equations. (Alabama) 6. Explain the development and applications of logarithms, including contributions of John Napier, Henry Briggs, and the Bernoulli family. (Alabama) Seeing Structure in Expressions Algebra MI-SSE.2 Franklin County Schools Students: Summarize the works of European mathematicians focusing on the beginnings of algebraic symbolism and structure. Students: Outline the contributions of John Napier, Henry Briggs, and Bernoulli family to the development and application of logarithms aligned to this The characteristics standard. of each of the sets of numbers. ALEX Resources How to solve polynomial equations. Students know: Fibonacci sequence Cubic equations Exponentials Natural logarithms Click below to access Characteristics of all ALEX resources sequences. aligned to this How to identify a standard. cubic equation and that the solutions ALEX Resources are real or complex. Students know: Definition and basic application of logarithms and exponentials. Click below to access all ALEX resources aligned to this standard. ALEX Resources MATH INV 7. Justify the historical significance of the development of multiple perspectives in mathematics. (Alabama) Example: Relate the historical development of multiple perspectives to the works of Sir Isaac Newton and Gottfried Wilhelm von Leibniz in the foundations of calculus. a. b. Mathematics CCRS Standards and Alabama COS Seeing Structure in Expressions Algebra MI-SSE.3 Students: Cartesian coordinate system Critique historical significance of the development of multiple perspectives in mathematics. Click below to access all ALEX resources aligned to this The basic concepts standard. of the coordinate system. ALEX Resources Students know: Summarize the significance of René Descartes’ Cartesian coordinate system. (Alabama) Interpret the foundation of analytic geometry with regard to geometric curves and algebraic relationships. (Alabama) 8. Solve problems from non-Euclidean geometry, including graph theory, networks, topology, and fractals. (Alabama) Math Investigation Standard 8 Example Expressing Geometric Properties with Equations Geometry MI-GPE.1 Students: Will be able to solve problems from nonEuclidean geometry involving: Graph theory Networks Topology Fractals Topology Fractals Networks Non-Euclidean Geometry Euclidean Geometry Traversable Sierpinski’s triangle 9. Analyze works of visual art and architecture for mathematical relationships. (Alabama) Examples: Use Leonardo da Vinci’s Vitruvian Man to explore the golden ratio. Modeling with Geometry Geometry MI-MG.1 Students: Examine works of visual art and architecture and identify the mathematical relationships within them. Golden Ratio Tessellations Perspective Symmetry Franklin County Schools Click below to access all ALEX resources aligned to this The basic concepts standard. of Euclidean geometry. ALEX Resources Students know: Students know: Basic concepts of ratio and proportion. The types of symmetry and Click below to access all ALEX resources aligned to this standard. ALEX Resources MATH INV Mathematics CCRS Standards and Alabama COS Identify mathematical patterns in Maurits Cornelis Escher’s drawings, including the use of tessellations in art, quilting, paintings, pottery, and architecture. how to identify each type. a. Summarize the historical development of perspective in art and architecture. (Alabama) 10. Determine the mathematical impact of the ancient Greeks, including Archimedes, Eratosthenes, Euclid, Hypatia, Pythagoras, and the Pythagorean Society. (Alabama) Example: Use Euclid’s proposition to inscribe a regular hexagon within a circle. Modeling with Geometry Geometry MI-MG.2 Students: Investigate and analyze the impact of the ancient Greek mathematicians. Pythagorean Theorem Inscribe Regular Polygon Figurate Numbers Sequence Proofs Hypothesis a. Construct multiple proofs of the Pythagorean Theorem. (Alabama) b. Solve problems involving figurate numbers, including triangular and pentagonal numbers. (Alabama) Example: Write a sequence of the first 10 triangular numbers and hypothesize a formula for finding the nth triangular number. 11. Describe the development of mathematical tools and Students know: Modeling with Geometry Geometry Franklin County Schools Students: Summarize the use and Napier’s Bones Slide Rule Knotted Ropes How to construct proofs. The Pythagorean Theorem. How to generate a sequence. How to derive a formula for the nth term of a sequence. Students know: How to use a Click below to access all ALEX resources aligned to this standard. ALEX Resources Click below to access all ALEX resources aligned to this MATH INV Mathematics CCRS Standards and Alabama COS their applications. MI-MG.3 (Alabama) Examples: Use knotted ropes for counting; Napier’s bones for multiplication; a slide rule for multiplying and calculating values of trigonometric, exponential, and logarithmic functions; and a graphing calculator for analyzing functions graphically and numerically. development of mathematical tools. 12. Summarize the history of probability, including the works of Blaise Pascal; Pierre de Fermat; Abraham de Moivre; and PierreSimon, marquis de Laplace. (Alabama) Example: Discuss the impact of probability on gaming, economics, and insurance. Students: Are able to describe the historical development of probability. Making Inferences & Justifying Conclusions Statistics & Probability MI-IC.1 Franklin County Schools Trigonometric Function Exponential Function Logarithmic Function Probability Statistics Gaming Economics Insurance graphing standard. calculator. Concepts involving ALEX Resources trigonometric, exponential and logarithmic functions. Click below to access all ALEX resources aligned to this Concepts involving standard. simple probability. ALEX Resources Students know: