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 Riemannian Geometry (PDF)
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M →...
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from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere S3 in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (S3, ·) into a Lie group with neutral element e = (1
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere <span class="highlight">S3</span> in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (<span class="highlight">S3</span>, ·) of (H∗, ·). They are both non-abelian. We shall now introduce some of the classical real and complex matrix Lie groups. As a reference on this topic we recommend the wonderful book: A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser (2002). Example 2.31. Let Nil3 be the subset of R3×3 given by Nil3 = { 1 x z0 1 y 0 0 1
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, <span class="highlight">S3</span> and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), <span class="highlight">S3</span> ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=41
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=41
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere <span class="highlight">S3</span> in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (<span class="highlight">S3</span>, ·) into a Lie group with neutral element e = (1
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
(− � , � )→ O(n) is a geodesic if and only if γt · γ̈ = γ̈t · γ. Exercise 7.3. Find a proof for Proposition 7.23. Exercise 7.4. Find a proof for Corollary 7.24. Exercise 7.5. For the real parameter θ ∈ (0, pi/2) define the 2- dimensional torus T 2θ by T 2θ = {(cos θeiα, sin θeiβ) ∈ <span class="highlight">S3</span>| α, β ∈ R}. Determine for which θ ∈ (0, pi/2) the torus T 2θ is a minimal submanifold of the 3-dimensional sphere <span class="highlight">S3</span> = {(z1, z2) ∈ C2| |z1|2 + |z2|2 = 1}. Exercise 7.6. Find a proof for Corollary 7.27. Exercise 7.7. Determine the totally
Extron Electronics: Selecting A Scan Converter
Extron Electronics - Selecting a Scan Converter Extron Logo Login | Sign up S3 Support Hotline800.633.9876 ProductsTrainingMarketsTechnologiesCompanyDownload
Karbos Guide: Illustrated Guide to Video Cards
video chip sets. You just have to know which ones! Regardless of whether it is replaceable or integrated, the video adapter consists of three components: A video chip set of some brand (ATI, Matrox, Nvidia, S3, Intel, to name some of the better known). The video...
Colorado Schools Unit: Science of the People
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environmen...
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29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (S3) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
37Goals 2000 Partnership for Educating Colorado Students Lesson 9: Dr. Eloy Rodriquez What will students be learning? OUTCOME(S) Students understand the process of scientific investigation and design, conduct, communicate about, and evaluate such investigations. (S1) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students understand that science involves a particular way of
science_pub2003
Vermont Department of Education (Science Grade Expectations) S35 Science — Physical Science: Vermont Standards and Evidences—Nuclear Change 7.12 b, bb, bbb, e, ee, eee, 1.19 Grades 5-6 Grades 3-4 Properties of Matter No S3-4:17 at this level No S3-4:18 at this level Pr...
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Vermont Department of Education (Science Grade Expectations) S35 Science — Physical Science: Vermont Standards and Evidences—Nuclear Change 7.12 b, bb, bbb, e, ee, eee, 1.19 Grades 5-6 Grades 3-4 Properties of Matter No S3-4:17 at this level No S3-4:18 at this level Properties of Matter No S5-6:17 at this level No S5-6:18 at this level Physical Science
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=5
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
+ Assessment (GEs) + Learning Opportunity + Teaching Opportunity = Curriculum How do we read a GE? S5-6:14 Students demonstrate their understanding of Physical Change by…• Predicting the effect of heating and cooling on the physical state and themass of a substance. Science Concepts:a. Energy is required to transform the physical state of a substance from solidto liquid to gas, while conserving mass. Physical changes are reversible. <span class="highlight">S3</span>-4:14 Students demonstrate their understanding of Physical Change by
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=12
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S11 Science — Scientific Inquiry: Vermont Standards and Evidences—Scientific Questioning 7.1 a, aa, aaa; 2.1 a, b, c, d; Predicting and Hypothesizing 7.1 b, bb, bbb; Designing Experiments 7.1 c, cc, 7.2 a, aa, b, bb; 2.2 a, aa, aaa, b, bb, 3.10 Grades 3-4 Grades 5-6 Scientific Questioning <span class="highlight">S3</span>-4:1 Students demonstrate their understanding of SCIENTIFIC QUESTIONING by… • Identifying at least one variable that affects a system
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Vermont Department of Education (Science Grade Expectations) S15 Science — Grades 3-4 Grades 5-6 Scientific Inquiry: Vermont Standards and Evidences—Conducting Experiments 7.1 c, cc, i; 7.2 c, cc, d; 2.2 e, ee, f, g; Rep- resenting Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd, 3.10, 3.11, 3.12, 3.13 Conducting Experiments <span class="highlight">S3</span>-4:4 Students demonstrate their ability to CONDUCT EXPERI- MENTS by… • Referring to and
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=20
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S19 Science — Scientific Inquiry: Vermont Standards and Evidences—Representing Data and Analysis 7.1 c, cc, d, dd, ddd, e, f, g; 1.17a, aa, aaa, b, bb, bbb, c, cc, ccc, d, dd; 1.18, 1.20; 2.2c, cc, d, dd Grades 3-4 Grades 5-6 Representing Data and Analysis <span class="highlight">S3</span>-4:6 Students demonstrate their ability to ANALYZE DATA by… • Interpreting patterns or trends in data. AND • Relating data to the original question and prediction
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=24
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S23 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:9 Students demonstrate their understanding of the Properties of Matter by… • Investigating and measuring how the total weight of the parts of a substance, no matter how they are
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=28
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S27 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:12 Students demonstrate their understanding of the States of Matter by… • Identifying , describing and comparing the properties of selected solids, liquids and gases. Science
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=32
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Vermont Department of Education (Science Grade Expectations) S31 Science — Physical Science: Vermont Standards and Evidences—Energy and Energy Transformation 7.12 b, bb, bbb; Chemical Change 7.12 b, bb, bbb, e, ee, eee, 2.2, 3.10, 3.11 [See also connection with S: 48 Universe, Earth, Environment—Weather] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:14 Students demonstrate their understanding of Physical Change by… • Investigating and explaining what happens to liquids in open containers
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education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S35 Science — Physical Science: Vermont Standards and Evidences—Nuclear Change 7.12 b, bb, bbb, e, ee, eee, 1.19 Grades 5-6 Grades 3-4 Properties of Matter No <span class="highlight">S3</span>-4:17 at this level No <span class="highlight">S3</span>-4:18 at this level Properties of Matter No S5-6:17 at this level No S5-6:18 at this level Physical Science
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Vermont Department of Education (Science Grade Expectations) S39 Science — Physical Science: Vermont Standards and Evidences—Motion 7.12 d, dd, ddd, 2.2a, aa, aaa, 3.11 Grades 3-4 Grades 5-6 Motion No <span class="highlight">S3</span>-4:19 at this grade level Teachers may review Grades 1-2 Motion Concepts. No <span class="highlight">S3</span>-4:20 at this grade level Motion S5-6:19 Students demonstrate their understanding of Motion by… • Measuring and calculating speed (the distance an object moves over a measured amount
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=44
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S43 Science — Grades 3-4 Grades 5-6 Physical Science: Vermont Standards and Evidences—Force 7.12 d, dd, ddd, 2.2a, aa, aaa, 3.11 Force <span class="highlight">S3</span>-4:21 Students demonstrate their understanding of Force by… • Investigating and describing how different amounts of force can change the direction and speed of an object in motion. Science Concepts: a. Changes in speed or direction of motion are caused by forces. b. The greater the force
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=48
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S47 Science — Grades 3-4 Grades 5-6 Physical Science: Vermont Standards and Evidences—Energy 7.12 e, ee, eee, f, ff, fff; Magnetism 7.12 e, ee, eee, f, ff, fff Energy and Energy Transformation No <span class="highlight">S3</span>-4:23 at this grade level Teachers may review Grades 1-2 Heat Energy Concepts. <span class="highlight">S3</span>-4:24 Students demonstrate their understanding of Electrical En- ergy by… • Building complete. circuits , drawing diagrams of these elec
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=52
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
the magnetic or electric effect it produces. Science Concepts: a. Moving electrical charges [electricity] produce magnetic force [magnetism] (i.e., electromagnet, motor). b. Moving magnets produce electricity (e.g., generator). No S5-6:27 at this grade level Energy and Energy Transformation No <span class="highlight">S3</span>-4:26 at this level No <span class="highlight">S3</span>-4:27 at this level Grades 3-4 Grades 5-6 Physical Science
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=56
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
reflected, trans- mitted or some combination. c. Objects can be seen only when light waves are emitted from or reflected off the object and enter into the eye. S5-6:29 Students demonstrate their understanding of Sound En- ergy by… • Generating a sound and identifying the path of vibration from the source to the ear. Science Concepts: a. Sound is produced by vibrations in materials that set up wavelike disturbances that spread away from the source. Energy and Energy Transformation <span class="highlight">S3</span>-4:28 Students
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b. All cells are enclosed in a membrane that allows materi- als to pass into and out of the cell. c. Most cells are microscopic. Survival of Organisms and Cells <span class="highlight">S3</span>-4:30 Students demonstrate their understanding of Structure and Function–Survival Requirements by… • Identifying how the physical structure/characteristic of an organism allows it to survive and defend itself (e.g., The coloring of a fiddler crab allows it to camouflage itself in the sand and grasses of its environment so that it will
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Vermont Department of Education (Science Grade Expectations) S63 Science — Grades 5-6 Grades 3-4 Life Cycles and Reproduction <span class="highlight">S3</span>-4:31 Students demonstrate their understanding of Reproduction by… • Investigating and describing a variety of plant and animal life cycles. Science Concept: a. Although all organisms have common stages of develop- ment, details of a life cycle are different for different organ- isms. Life Cycles and Reproduction No S5-6:31 at this grade level
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Vermont Department of Education (Science Grade Expectations) S67 Science — Grades 3-4 Grades 5-6 Cell and Tissue Differentiation No <span class="highlight">S3</span>-4:32 at this grade level Chemical Reactions within Cells No <span class="highlight">S3</span>-4:33 at this grade level Cell and Tissue Differentiation S5-6:32 Students demonstrate their understanding of how Differen- tiation by… • Explaining the relationship between cell, tissue, organ and system. AND • Observing plant or animal tissue and explaining how
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Vermont Department of Education (Science Grade Expectations) S71 Science — Grades 5-6 Grades 3-4 Interdependence within Ecosystems <span class="highlight">S3</span>-4:34 Students demonstrate their understanding of Energy Flow in an Ecosystem by… • Identifying the source of energy for the survival of organisms. Science Concept: a. Energy derived from food is needed for all organisms (plants and animals) to stay alive and grow. <span class="highlight">S3</span>-4:35 Students demonstrate their understanding of Food Webs in an Ecosystem by
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Vermont Department of Education (Science Grade Expectations) S75 Science — Interdependence within Ecosystems <span class="highlight">S3</span>-4:36 Students demonstrate their understanding of Equilibrium in an ecosystem by… • Explaining how one organism depends upon another or- ganism to survive. Science Concept: a. Organisms interact with one another in various ways besides providing food (e.g., Many plants depend on ani- mals for carrying their pollen to other plants for fertiliz- ing their flowers). No <span class="highlight">S3</span>-4
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Vermont Department of Education (Science Grade Expectations) S79 Science — Classification of Living Things <span class="highlight">S3</span>-4:38 Students demonstrate their understanding of Classification of Organisms by… • Describing and sorting plants and animals into groups based on structural similarities and differences (e.g., All pine, spruce and evergreen trees have similar leaf struc- tures; Spiders have eight legs, and insects have six). Science Concept: a. The great variety of living things can be sorted
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education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S83 Science — Heredity <span class="highlight">S3</span>-4:40 Students demonstrate their understanding of Human He- redity by… • Identifying similarities that are inherited from a biological parent. Science Concepts: a. Some similarities between children and parents such as eye color, are inherited. Heredity S5-6:40 Students demonstrate their understanding of Human He- redity by… • Identifying that an offspring’s traits are determined by combining
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Vermont Department of Education (Science Grade Expectations) S87 Science — Body Systems <span class="highlight">S3</span>-4:41 Students demonstrate their understanding of Human Body Systems by… • Showing connections between external and internal body structures and how they help humans survive, Science Concepts: a. There are external and internal structures that provide for the survival needs of human organisms. - Skin protects the body from harmful substances and other organisms and from drying out. - The
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Vermont Department of Education (Science Grade Expectations) S95 Science — Patterns of Human Development No <span class="highlight">S3</span>-4:43 at this level Teacher may review Grade 1-2 Patterns of Human Devel- opment Concepts. Patterns of Human Development S5-6:43 Students demonstrate their understanding of the Patterns of Human Development by… • Drawing/diagramming/modeling the life span of humans in a timeline highlighting major points in the cycle (e.g., one cell grows into a many-celled embryo, composed of
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Vermont Department of Education (Science Grade Expectations) S99 Science — Solar System <span class="highlight">S3</span>-4:44 Students demonstrate their understanding Characteristics of the Solar System by… • Creating a model of the planets and their correct order from the sun. AND • Drawing or building and then explaining a model of the earth rotating on its axis in relation to the sun and moon (i.e., day and night). Science Concepts: a. The earth is one of several planets that orbit the sun, and the moon orbits
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Vermont Department of Education (Science Grade Expectations) S103 Science — Scale, distances, star formation, theories, instrumen- tation <span class="highlight">S3</span>-4:45 Students demonstrate their understanding of Processes and Change over Time within Systems of the Universe by… • Identifying similar star patterns/or groups from night pho- tographs of the same location at different times of the years. AND • Comparing (similarities) between the sun and stars. Science Concepts: a. Stars are like the sun, but so
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Vermont Department of Education (Science Grade Expectations) S111 Science — Forces and Changes on the Earth’s Surface <span class="highlight">S3</span>-4:47 Students demonstrate their understanding of Processes and Change over Time within Earth Systems by… • Building models that simulate deposits of sediments (e.g., a stream table. AND • Investigating local land forms and comparing them with models created in the classroom. Science Concept: a. Waves, wind, water and ice shape and reshape the earth’s
GRADE 4
and Functions Concept 4: Analysis of Change Analyze how changing the values of one quantity corresponds to change in the values of another quantity. In Grade 4, students make predictions based on changes in data over time. Performance Objectives Process Integration Explanations and Examples St...
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and Functions Concept 4: Analysis of Change Analyze how changing the values of one quantity corresponds to change in the values of another quantity. In Grade 4, students make predictions based on changes in data over time. Performance Objectives Process Integration Explanations and Examples Students are expected to: PO 1. Identify the change in a quantity over time and make simple predictions. Connections: M04-S1C3-02, M04-S2C1- 02, M04-S3-C1-02, SS04-S5C5-01 M04-S5C2-05
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http://www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade4.pdf#page=26
www.ade.state.az.us/standards/math/Articulated08/Gradeleveldocs/MathGrade...
and Functions Concept 4: Analysis of Change Analyze how changing the values of one quantity corresponds to change in the values of another quantity. In Grade 4, students make predictions based on changes in data over time. Performance Objectives Process Integration Explanations and Examples Students are expected to: PO 1. Identify the change in a quantity over time and make simple predictions. Connections: M04-S1C3-02, M04-S2C1- 02, M04-<span class="highlight">S3</span>-C1-02, SS04-S5C5-01 M04-S5C2-05
Microsoft Word - 3969.doc
represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-S3) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences afte...
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represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-S3) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences after stories are read aloud (PK-LL-L1) (PK-LL-S1) (ELA-1-E6) 11. Orally express thoughts about characters or events in a story (PK-LL-S1) (PK-LL-S2) (PKS-LL-R2) (ELA-1-E6) Standard 7: 12. Demonstrate understanding
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http://www.doe.state.la.us/lde/uploads/3906.pdf#page=1
www.doe.state.la.us/lde/uploads/3906.pdf#page=1
represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-<span class="highlight">S3</span>) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences after stories are read aloud (PK-LL-L1) (PK-LL-S1) (ELA-1-E6) 11. Orally express thoughts about characters or events in a story (PK-LL-S1) (PK-LL-S2) (PKS-LL-R2) (ELA-1-E6) Standard 7: 12. Demonstrate understanding
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http://www.doe.state.la.us/lde/uploads/3906.pdf#page=2
www.doe.state.la.us/lde/uploads/3906.pdf#page=2
E1) 22. Carry on a conversation about a topic, thought, or idea from the classroom, home, or community (PK-LL-S1) (PK-LL-<span class="highlight">S3</span>) (ELA-4-E1) 23. Repeat an instruction given orally (PK-LL-S1) (ELA-4-E2) 24. Follow one- and two-step verbal and nonverbal directions (PK-LL-L2) (ELA-4-E2) 25. Retell part of a favorite story (PK-LL-R2) (ELA-4-E3) 26. Speak about life experiences or topics of interest (PK-LL-<span class="highlight">S3</span>) (ELA-4-E4) 27. Actively participate in role-playing, creative dramatics, finger plays, nursery
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find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD S3: SaMPLeS,...
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find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD S3: SaMPLeS, SurveyS, anD exPeriMentS Students understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias
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http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
HIGH SCHOOL MATHEMATICS CONTENT EXPECTATIONS page 3 of 19 11/07 MICHIGAN DEPARTMENT OF EDUCATION STANDARDS (and number of core expectations in each standard) A1: Expressions, Equations, and Inequalities (16) A2: Functions (16) A3: Families of Functions (27) G1: Figures and Their Properties (29) G2: Relationships Between Figures (10) G3: Transformations of Figures in the Plane (5) S1: Univariate Data- Examining Distributions (9) S2: Bivariate Data— Examining Relationships (6) <span class="highlight">S3</span>: Samples
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http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=20
www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=20
find the least squares regression line by estimating visually and by calculating the equation of the regression line. Interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions. StanDarD <span class="highlight">S3</span>: SaMPLeS, SurveyS, anD exPeriMentS Students understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias
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