| Name |
Description |
| MA.912.AR.9.6: | Given a real-world context, represent constraints as systems of linear equations or inequalities. Interpret solutions to problems as viable or non-viable options.Clarifications:
Clarification 1: Instruction focuses on analyzing a given function that models a real-world situation and writing constraints that are represented as linear equations or linear inequalities. |
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| MA.912.AR.9.8: | Solve real-world problems involving linear programming in two variables. |
| MA.912.AR.10.1: | Given a mathematical or real-world context, write and solve problems involving arithmetic sequences.
Examples:
Tara is saving money to move out of her parent’s house. She opens the account with $250 and puts $100 into a savings account every month after that. Write the total amount of money she has in her account after each month as a sequence. In how many months will she have at least $3,000? |
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| MA.912.AR.10.2: | Given a mathematical or real-world context, write and solve problems involving geometric sequences.
Examples:
A bacteria in a Petri dish initially covers 2 square centimeters. The bacteria grows at a rate of 2.6% every day. Determine the geometric sequence that describes the area covered by the bacteria after 0,1,2,3… days. Determine using technology, how many days it would take the bacteria to cover 10 square centimeters. |
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| MA.912.AR.10.5: | Given a mathematical or real-world context, write a sequence using function notation, defined explicitly or recursively, to represent relationships between quantities from a written description. |
| MA.912.AR.10.6: | Given a mathematical or real-world context, find the domain of a given sequence defined recursively or explicitly. |
| MA.912.DP.4.1: | Describe events as subsets of a sample space using characteristics, or categories, of the outcomes, or as unions, intersections or complements of other events. |
| MA.912.DP.4.9: | Apply the addition and multiplication rules for counting to solve mathematical and real-world problems, including problems involving probability. |
| MA.912.DP.4.10: | Given a mathematical or real-world situation, calculate the appropriate permutation or combination. |
| MA.912.GR.2.4: | Determine symmetries of reflection, symmetries of rotation and symmetries of translation of a geometric figure.Clarifications:
Clarification 1: Instruction includes determining the order of each symmetry.Clarification 2: Instruction includes the connection between tessellations of the plane and symmetries of translations.
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| MA.912.LT.1.1: | Apply recursive and iterative thinking to solve problems. |
| MA.912.LT.1.2: | Solve problems involving recurrence relations.Clarifications:
Clarification 1: Instruction includes finding explicit or recursive equations for recursively defined sequences.
Clarification 2: Problems include fractals, the Fibonacci sequence, growth models and finite difference.
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| MA.912.LT.2.1: | Define and explain the basic concepts of Graph Theory.Clarifications:
Clarification 1: Basic concepts include vertex, edge, directed edge, undirected edge, path, vertex degree, directed graph, undirected graph, tree, bipartite graph, circuit, connectedness and planarity. |
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| MA.912.LT.2.2: | Solve problems involving paths in graphs.Clarifications:
Clarification 1: Instruction includes simple paths and circuits; Hamiltonian paths and circuits; and Eulerian paths and circuits. |
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| MA.912.LT.2.3: | Solve scheduling problems using critical path analysis and Gantt charts. Create a schedule using critical path analysis. |
| MA.912.LT.2.4: | Apply graph coloring techniques to solve problems.Clarifications:
Clarification 1: Problems include map coloring and committee assignments. |
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| MA.912.LT.2.5: | Apply spanning trees, rooted trees, binary trees and decision trees to solve problems.Clarifications:
Clarification 1: Instruction includes the use of technology to determine the number of possible solutions and generating solutions when a feasible number of possible solutions exists. |
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| MA.912.LT.3.1: | Define and explain the basic concepts of Election Theory and voting.Clarifications:
Clarification 1: Basic concepts include approval and preference voting, plurality, majority, runoff, sequential runoff, Borda count, Condorcet and other fairness criteria, dummy voters and coalition. |
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| MA.912.LT.3.2: | Analyze election data using election theory techniques. Explain how Arrow’s Impossibility Theorem may be related to the fairness of the outcome of the election. |
| MA.912.LT.3.3: | Decide voting power within a group using weighted voting techniques. Provide real-world examples of weighted voting and its pros and cons. |
| MA.912.LT.3.4: | Solve problems using fair division and apportionment techniques.Clarifications:
Clarification 1: Problems include fair division among people with different preferences, fairly dividing an inheritance that includes indivisible goods, salary caps in sports and allocation of representatives to Congress. |
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| MA.912.LT.4.1: | Translate propositional statements into logical arguments using propositional variables and logical connectives. |
| MA.912.LT.4.2: | Determine truth values of simple and compound statements using truth tables. |
| MA.912.LT.4.3: | Identify and accurately interpret “if…then,” “if and only if,” “all” and “not” statements. Find the converse, inverse and contrapositive of a statement.Clarifications:
Clarification 1: Instruction focuses on recognizing the relationships between an “if…then” statement and the converse, inverse and contrapositive of that statement.
Clarification 2: Within the Geometry course, instruction focuses on the connection to proofs within the course.
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| MA.912.LT.4.4: | Represent logic operations, such as AND, OR, NOT, NOR, and XOR, using logical symbolism to solve problems. |
| MA.912.LT.4.5: | Determine whether two propositions are logically equivalent. |
| MA.912.LT.4.6: | Apply methods of direct and indirect proof and determine whether a logical argument is valid. |
| MA.912.LT.4.7: | Identify and give examples of undefined terms; axioms; theorems; proofs, including proofs using mathematical induction; and inductive and deductive reasoning. |
| MA.912.LT.4.8: | Construct proofs, including proofs by contradiction.Clarifications:
Clarification 1: Within the Geometry course, proofs are limited to geometric statements within the course. |
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| MA.912.LT.4.9: | Construct logical arguments using laws of detachment, syllogism, tautology, contradiction and Euler Diagrams. |
| MA.912.LT.4.10: | Judge the validity of arguments and give counterexamples to disprove statements.Clarifications:
Clarification 1: Within the Geometry course, instruction focuses on the connection to proofs within the course. |
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| MA.912.LT.5.1: | Given two sets, determine whether the two sets are equivalent and whether one set is a subset of another. Given one set, determine its power set. |
| MA.912.LT.5.2: | Given a relation on two sets, determine whether the relation is a function, determine the inverse of the relation if it exists and identify if the relation is bijective. |
| MA.912.LT.5.3: | Partition a set into disjoint subsets and determine an equivalence class given the equivalence relation on a set. |
| MA.912.LT.5.4: | Perform the set operations of taking the complement of a set and the union, intersection, difference and product of two sets.Clarifications:
Clarification 1: Instruction includes the connection to probability and the words AND, OR and NOT. |
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| MA.912.LT.5.5: | Explore relationships and patterns and make arguments about relationships between sets using Venn Diagrams. |
| MA.912.LT.5.6: | Prove set relations, including DeMorgan’s Laws and equivalence relations. |
| MA.K12.MTR.1.1: | Actively participate in effortful learning both individually and collectively. Mathematicians who participate in effortful learning both individually and with others:
- Analyze the problem in a way that makes sense given the task.
- Ask questions that will help with solving the task.
- Build perseverance by modifying methods as needed while solving a challenging task.
- Stay engaged and maintain a positive mindset when working to solve tasks.
- Help and support each other when attempting a new method or approach.
Clarifications:
Teachers who encourage students to participate actively in effortful learning both individually and with others:
- Cultivate a community of growth mindset learners.
- Foster perseverance in students by choosing tasks that are challenging.
- Develop students’ ability to analyze and problem solve.
- Recognize students’ effort when solving challenging problems.
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| MA.K12.MTR.2.1: | Demonstrate understanding by representing problems in multiple ways. Mathematicians who demonstrate understanding by representing problems in multiple ways: - Build understanding through modeling and using manipulatives.
- Represent solutions to problems in multiple ways using objects, drawings, tables, graphs and equations.
- Progress from modeling problems with objects and drawings to using algorithms and equations.
- Express connections between concepts and representations.
- Choose a representation based on the given context or purpose.
Clarifications:
Teachers who encourage students to demonstrate understanding by representing problems in multiple ways: - Help students make connections between concepts and representations.
- Provide opportunities for students to use manipulatives when investigating concepts.
- Guide students from concrete to pictorial to abstract representations as understanding progresses.
- Show students that various representations can have different purposes and can be useful in different situations.
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| MA.K12.MTR.3.1: | Complete tasks with mathematical fluency. Mathematicians who complete tasks with mathematical fluency: - Select efficient and appropriate methods for solving problems within the given context.
- Maintain flexibility and accuracy while performing procedures and mental calculations.
- Complete tasks accurately and with confidence.
- Adapt procedures to apply them to a new context.
- Use feedback to improve efficiency when performing calculations.
Clarifications:
Teachers who encourage students to complete tasks with mathematical fluency:- Provide students with the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately.
- Offer multiple opportunities for students to practice efficient and generalizable methods.
- Provide opportunities for students to reflect on the method they used and determine if a more efficient method could have been used.
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| MA.K12.MTR.4.1: | Engage in discussions that reflect on the mathematical thinking of self and others. Mathematicians who engage in discussions that reflect on the mathematical thinking of self and others: - Communicate mathematical ideas, vocabulary and methods effectively.
- Analyze the mathematical thinking of others.
- Compare the efficiency of a method to those expressed by others.
- Recognize errors and suggest how to correctly solve the task.
- Justify results by explaining methods and processes.
- Construct possible arguments based on evidence.
Clarifications:
Teachers who encourage students to engage in discussions that reflect on the mathematical thinking of self and others:- Establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.
- Create opportunities for students to discuss their thinking with peers.
- Select, sequence and present student work to advance and deepen understanding of correct and increasingly efficient methods.
- Develop students’ ability to justify methods and compare their responses to the responses of their peers.
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| MA.K12.MTR.5.1: | Use patterns and structure to help understand and connect mathematical concepts. Mathematicians who use patterns and structure to help understand and connect mathematical concepts: - Focus on relevant details within a problem.
- Create plans and procedures to logically order events, steps or ideas to solve problems.
- Decompose a complex problem into manageable parts.
- Relate previously learned concepts to new concepts.
- Look for similarities among problems.
- Connect solutions of problems to more complicated large-scale situations.
Clarifications:
Teachers who encourage students to use patterns and structure to help understand and connect mathematical concepts:- Help students recognize the patterns in the world around them and connect these patterns to mathematical concepts.
- Support students to develop generalizations based on the similarities found among problems.
- Provide opportunities for students to create plans and procedures to solve problems.
- Develop students’ ability to construct relationships between their current understanding and more sophisticated ways of thinking.
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| MA.K12.MTR.6.1: | Assess the reasonableness of solutions. Mathematicians who assess the reasonableness of solutions: - Estimate to discover possible solutions.
- Use benchmark quantities to determine if a solution makes sense.
- Check calculations when solving problems.
- Verify possible solutions by explaining the methods used.
- Evaluate results based on the given context.
Clarifications:
Teachers who encourage students to assess the reasonableness of solutions:- Have students estimate or predict solutions prior to solving.
- Prompt students to continually ask, “Does this solution make sense? How do you know?”
- Reinforce that students check their work as they progress within and after a task.
- Strengthen students’ ability to verify solutions through justifications.
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| MA.K12.MTR.7.1: | Apply mathematics to real-world contexts. Mathematicians who apply mathematics to real-world contexts: - Connect mathematical concepts to everyday experiences.
- Use models and methods to understand, represent and solve problems.
- Perform investigations to gather data or determine if a method is appropriate.
• Redesign models and methods to improve accuracy or efficiency.
Clarifications:
Teachers who encourage students to apply mathematics to real-world contexts:- Provide opportunities for students to create models, both concrete and abstract, and perform investigations.
- Challenge students to question the accuracy of their models and methods.
- Support students as they validate conclusions by comparing them to the given situation.
- Indicate how various concepts can be applied to other disciplines.
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| ELA.K12.EE.1.1: | Cite evidence to explain and justify reasoning.Clarifications:
K-1 Students include textual evidence in their oral communication with guidance and support from adults. The evidence can consist of details from the text without naming the text. During 1st grade, students learn how to incorporate the evidence in their writing.2-3 Students include relevant textual evidence in their written and oral communication. Students should name the text when they refer to it. In 3rd grade, students should use a combination of direct and indirect citations. 4-5 Students continue with previous skills and reference comments made by speakers and peers. Students cite texts that they’ve directly quoted, paraphrased, or used for information. When writing, students will use the form of citation dictated by the instructor or the style guide referenced by the instructor. 6-8 Students continue with previous skills and use a style guide to create a proper citation. 9-12 Students continue with previous skills and should be aware of existing style guides and the ways in which they differ.
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| ELA.K12.EE.2.1: | Read and comprehend grade-level complex texts proficiently.Clarifications:
See Text Complexity for grade-level complexity bands and a text complexity rubric. |
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| ELA.K12.EE.3.1: | Make inferences to support comprehension.Clarifications:
Students will make inferences before the words infer or inference are introduced. Kindergarten students will answer questions like “Why is the girl smiling?” or make predictions about what will happen based on the title page.
Students will use the terms and apply them in 2nd grade and beyond. |
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| ELA.K12.EE.4.1: | Use appropriate collaborative techniques and active listening skills when engaging in discussions in a variety of situations.Clarifications:
In kindergarten, students learn to listen to one another respectfully.In grades 1-2, students build upon these skills by justifying what they are thinking. For example: “I think ________ because _______.” The collaborative conversations are becoming academic conversations. In grades 3-12, students engage in academic conversations discussing claims and justifying their reasoning, refining and applying skills. Students build on ideas, propel the conversation, and support claims and counterclaims with evidence.
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| ELA.K12.EE.5.1: | Use the accepted rules governing a specific format to create quality work.Clarifications:
Students will incorporate skills learned into work products to produce quality work. For students to incorporate these skills appropriately, they must receive instruction. A 3rd grade student creating a poster board display must have instruction in how to effectively present information to do quality work. |
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| ELA.K12.EE.6.1: | Use appropriate voice and tone when speaking or writing.Clarifications:
In kindergarten and 1st grade, students learn the difference between formal and informal language. For example, the way we talk to our friends differs from the way we speak to adults. In 2nd grade and beyond, students practice appropriate social and academic language to discuss texts. |
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| ELD.K12.ELL.MA.1: | English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. |
