Big-picture Goals
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Enduring IdeasCourseContent TopicTaskIdeal Timing
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Clear communication in math requires specificity and knowing what is relevant information.AllAllHave students do the exercise of writing a "5" on Day 1 based on oral instructions. Then, put them in pairs to ask them to try and describe a mathematical object to each other without naming it (ie. play Taboo). Debrief about what good math communication looks like.Start of year, first day
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All of our learning benefits when we recognize that each person has something to contribute.AllAllIn the first few weeks of the year, make particular emphasis to give open tasks and start off with "I notice, I wonder..." in groups, and then post all questions on the board to discuss. Establish group norms and assign groups (change groups every few days). The group leader needs to make sure there is equal participation and that one person summarizes the connections between the problems before starting next day's assignment.Start of year (build positive culture), then reinforce on on-going basis
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On-going, honest reflection about our learning process improves our individual mastery. AllAllAt the start of each unit, give the students a list of big-idea questions (actually, more complete than Essential Questions) to be answered at their own pace during the unit, in their math journal. Students may turn the journals in at any point during the unit for a check-in, but teacher also collects regularly for feedback.Weekly written dialogue; distribute questions at start of unit
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Effective learning needs to be supported by specific and intentional learning strategies.AllAllAt the end of each unit, student needs to provide a self-assessment on how well they have done as a learner during the course of the unit. They can, for example, show their detailed notes, their extra homework practice, their flash cards, or their graphical organizers. If they feel that the strategy has not worked for them, they should seek to experiment with different strategies during the subsequent unit.End of each unit, collect along with their formal assessment
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Two ways to signifcantly improve retention and retrieval are: 1. Understanding how ideas connect to other ideas and 2. Repetition.AllAllAsk students to explain connections between each day's assignment to the previous tasks, in groups. At the start of class, use scaffolded warm-ups to instill repetition of the most important "automatic" skills.On-going / start and end of class periods
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Math is a creative discipline rooted in necessity.AllHistory of Math / Functions / Surface Area and Volume / Tiling / IntegralsProjects (Geometry: History of math in different cultures; tesselation; visual illusion; building a solid) (Algebra 2: function picture; PSA on sustainability or social justice) (Calculus: integral picture; PSA on sustainability; vase volume)One project each term, in each class
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Having a sensible process is more important than getting the correct answer. AllWord problems Asking students to estimate answers and to discuss in groups. On the test, give a couple of explanation problems where the students need to estimate an answer before calculating it.Start and end of each unit
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Growing / decreasing patterns are generally predictable and can be generalized using symbols.Geometry / Algebra 2Induction / Functions / Multiple RepresentationsStart of each unit, have students sort patterns to introduce new vocabulary terms. During a unit, have students generate their own examples. End of a unit, have students bring in outside examples or create concrete representations via manipulatives.On-going, as new vocabulary terms are introduced
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We can model everyday phenomena using math equations.AllFunctions / RegressionDistribute math readings where students need to answer in-line comprehension questions regarding applicability of current topics. Use similar contexts on subsequent worksheets.Start of each unit
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We categorize mathematical objects in order to help us see the similarities and major differences.AllFunction families / geometric object families / transformationsAt the start of new algebra topics, always have a sorting activity to ask the students to inductively group similar-seeming objects and to characterize them using their own words.Start of each unit / revisit during unit
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An (x, y) equation is the locus of all points that satisfy the given equation.Geometry / Algebra 2Geometric construction / Functions / Systems of EquationsBefore teaching a new form of equation, give the students a few points and ask them which ones belong to a given curve. In Geometry, ask what those points all have in common in relationship to other objects, in order to deepen algebra-geometry connection.Start of each new topic or skill
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We should always validate our own answers and ask others to justify theirs.AllSolving equationsStart every unit by teaching how to use technology to check or obtain an answer, before showing the students how to manually calculate it. During group work, assign one person to make sure that everybody is taking turns checking their answers. Early in each unit, after introducing key vocab terms and before algebra skills
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Analyzing units in a problem can help us to choose the correct procedures and to interpret the results.AllWord problems / Meaning of derivative and integral quantities / DimensionGive equations and word problems fill-in-the-blank activity at the start of a unit as warmups.Teach early in the year, then revisit as appropriate
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N distinct points help us to find a specific equation with N parameters.Algebra 2 / CalculusSystems of Equations / FunctionsIn Algebra 2, teach Systems of Equations as soon as possible (immediately following fractions review), so that we can revisit this in the context of all functions. In Calculus, review this at the start of the year and then assign a Functions Picture project to practice applying this.Middle of each unit
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Besides input and output, there are other interesting and analytical features to a graph.AllWord problems / Polynomials / Min and max / Intersections / Derivative / IntegralIn Calculus, give students graphs to intuitively describe / estimate before teaching the algebra. In Algebra 2, teach the graphing calculator skills associated with finding max, min, zeroes, and intersections. In Geometry, teach max and min skills using volume and surface area problems.Self-contained Units
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Mathematical objects can morph into different but equivalent forms to give us additional information.Algebra 2 / CalculusFunctions Students should be asked to generate their own examples in their math journal and explain the tradeoffs between each algebraic form.On-going (at home)
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Starting with a vanilla function equation, we can manipulate its invisible parameters to control its resulting shape and location.Algebra 2 / CalculusFunctions / TransformationsExplorations using technology (table and graph) should help to structure student discussion, followed by practicing by creating Function Pictures.Self-contained units, with end-of-year review in Algebra 2
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Breaking down a problem into smaller parts is a powerful way to approach complex or new situations.AllWord problems / Induction / FactorizationGive problems of the week / puzzles that help to illustrate this idea and connect this to the idea of solving equations by factorization.Weekly (at home), but we discuss only bi-weekly
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Through deductive geometry, we are able to find the sizes of objects indirectly.Geometry Theorems / Deduction / Proof / Common geometric formulasEmphasize the process for solving a problem deductively, starting with sketching and labeling a picture. Towards the end of a unit, have each student create their own multi-step problem involving 2 or 3 of the learned concepts to share with the class.On-going
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We need to prove the universality of certain ideas, in order to build newer knowledge upon them.Geometry / CalculusTherems / Deduction / Induction / Proof / Triangles, Quadrilaterals, and Circles / Limits of Difference QuotientTeach formal processes for proofs, and give students time to critique proofs with holes.Self-contained Units
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When mathematicians define a new object, they always do so in harmony with existing patterns and definitions.Algebra 2 Rules of Exponents / Negative and Fractional Exponents / Complex Numbers / Dimensional analysisWhen teaching rules, help students inductively see the pattern behind existing portions of the rules before asking them to derive the new definitions.Self-contained Units
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Just as it is possible to un-do individual math operations, it is also possible to un-do an entire function.Algebra 2 / CalculusInverse function / Domain / Range / TransformationThe idea of inverses is best revisited regularly, so every few units, come back to this idea and do some mixed practice in between topics. It will also help to reinforce/deepen already learned algebra skills. Inverses should also be used to introduce certain topics, such as radical and logarithmic functions.Periodically (once every couple of big topics)
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Some functions can "break" in the middle of the graph.Algebra 2 / CalculusDomain / Range / Rational functions / Radical functions / AsymptotesExplorations using technology (table and graph) should help to structure student discussion, so that they understand why functions become undefined at certain values and not others.Self-contained Units
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When we cannot possibly calculate something directly, we try and get infinitesimally close to it instead.CalculusLimits / Asymptotes / Definition of derivative and integralTowards the end of Calculus, once the students understand the practical implication of derivatives and integrals, come back and re-emphasize the theoretical aspects.End of year in Calculus
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Calculus, aka. understanding instantaneous rates of change, can be applied to analyzing a variety of real-world problems.CalculusDerivative / Integral word problemsWork sheets and a real world-based sustainability analysis project.On-going in Calculus
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