indiana academic standards for mathematics – finite standards resource guide … ·...

Indiana Academic Standards for Mathematics – Finite Standards Resource Guide Document

This Teacher Resource Guide, revised in July 2018, provides supporting materials to help educators successfully implement the Indiana Academic Standards for Finite. This resource guide is provided to help ensure all students meet the rigorous learning expectations set by the academic standards. Use of this guide and the resources on the web page is optional – teachers should decide which resources will work best for their students. However, all guidance contained in this document and on the website has been chosen to best support effective teaching practices and promote the Mathematics Process Standards.

With an increased emphasis on content area literacy, academic vocabulary has been noted. Best practices should be utilized when teaching students academic vocabulary. Please see the Literacy Framework and the Science and Technical Subjects Content Area Literacy Standards for examples of best practices.

Examples have been removed from the document as they tend to limit interpretation and classroom application. Rather, success criteria, in the form of “I can” statements, have been included. According to Hattie (2017), success criteria is specific, concrete and measurable, describing what success looks like when a learning goal is reached. Additionally, success criteria contributes to teacher clarity, which has a 0.75 effect size! An effect size of 0.40 reportedly indicates one year of growth. Utilizing success criteria in the classroom allows students to monitor their own learning and increases motivation (Hattie, p. 57). It is important to note that the success criteria provided here are not intended to be limiting. Teachers may have additional success criteria for their students.

Guidance around vertical articulation has been provided in the last two columns. Knowing what was expected of students at previous grade levels will help teachers connect new learning to prior knowledge. Additionally, understanding what a student will be expected to learn in the future provides the teacher a context for the current learning. This information is not exhaustive; rather it is provided to give teachers a quick understanding of how the work builds from previous grade levels into subsequent courses. The Indiana Department of Education (IDOE) math team recommends teachers further study this vertical articulation to situate their course objectives in the broader math context.

If you have any questions, please do not hesitate to reach out to the IDOE math team. Contact information for the Elementary and Secondary Math Specialists can be found on the website: https://www.doe.in.gov/standards/mathematics. If you have suggested resources for the website, please share those as well.

Hattie, J., Fisher, D., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. (2017). Visible learning for mathematics: What works best to optimize student learning, grades K-12. Thousand Oaks, CA: Corwin Mathematics.

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https://www.doe.in.gov/standards/mathematics


Sets Finite Mathematics Standards Success Criteria Academic Vocabulary Looking Back

MA.FM.S.1: Know and use the concepts of sets, elements, and subsets.

I can identify and describe sets, elements, and subsets.

I can apply the concepts of sets, elements, and subsets to describe problems.

Set

Element

Subset

Determine the nature and number of elements in a finite sample space to model the outcomes of real-world events. (MA.AA.DSP.7)

MA.FM.S.2: Perform operations on sets (union, intersection, complement, cross product) and illustrate using Venn diagrams.

I can identify the union and intersection of sets.

I can identify the complement of sets.

I can find the cross product of sets.

I can illustrate sets using Venn diagrams.

I can use Venn diagrams to perform operations on sets.

Union

Intersection

Complement

Cross product

Venn diagram


Matrices Finite Mathematics Standards Success Criteria Academic Vocabulary Looking Back

MA.FM.MA.1: Add, subtract, and multiply matrices of appropriate dimensions (i.e. up to 3x3 matrices). Multiply matrices by

I can add and subtract matrices with dimensions up to 3x3.

Matrices

Scalar

The mastery of thisstandard requires

strong computationskills.

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scalars. Calculate row and column sums for matrix equations.

I can multiply matrices with dimensions up to 3x3.

I can multiply matrices by scalars.

I can calculate row and column sums for matrix equations.

MA.FM.MA.2: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers.

I can explain the role that the zero matrix plays in matrix addition and connect it to the role of 0 in addition of real numbers.

I can relate multiplication with the identity matrix with matrices to the multiplication with 1 in real numbers.

Zero matrix

Null matrix

Identity matrix

Apply the order of operations and properties of operations including the identity and inverse properties to evaluate numerical expressions with nonnegative rational numbers. (MA.6.C.6)

Apply the properties of operations including identity and inverse properties to create equivalent linear expressions. (MA.7.AF.1)

MA.FM.MA.3: Understand the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

I can determine if a square matrix has a multiplicative inverse.

I can determine if a square matrix has a determinant.

I can find the determinant of a square matrix where applicable.

Determinant

Square matrix

Multiplicative inverse

Apply the order of operations and properties of operations including the identity and inverse properties to evaluate numerical expressions with nonnegative rational numbers. (MA.6.C.6)

Apply the properties of operations including

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identity and inverse properties to create equivalent linear expressions. (MA.7.AF.1)

MA.FM.MA.4: Solve problems represented by matrices using row-reduction techniques and properties of matrix multiplication, including identity and inverse matrices.

I can use row-reduction techniques to solve problems represented by matrices.

I can apply the properties of matrix multiplication to solve problems represented by matrices.

I can determine when the use of the identity matrix or the inverse matrix is needed when solving problems involving matrices.

Row-reduction techniques

Properties of matrix multiplication

Identity matrix

Inverse matrix

Solve a system of equations consisting of a linear equation and a quadratic equation in two variables. (MA.AII.SE.1)

Solve systems of two or three linear equations in two or three variables algebraically and using technology. (MA.AII.SE.2)

Represent real-world problems using a system of linear equations in three variables and solve such problems with and without technology. (MA.AII.SE.3)

Represent real-world problems using a system of linear equations and/or inequalities in two or three variables. Solve such systems graphically or with matrices, as appropriate to the system, with technology. (MA.AA.LF.7)

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MA.FM.MA.5: Use matrices to solve real-world problems that can be modeled by a system of equations (i.e. up to three linear equations) in two or three variables using technology.

I can use matrices to solve real-world problems that can be modeled by a system of equations in two variables with technology.

I can solve real-world problems modeled by a system of equations in three variables using matrices and technology.

Matrices

System of equations

Solve systems of two or three linear equations in two or three variables algebraically and using technology. (MA.AII.SE.2)

Represent real-world problems using a system of linear equations in three variables and solve such problems with and without technology. (MA.AII.SE.3)

Represent real-world problems using a system of linear equations and/or inequalities in two or three variables. Solve such systems graphically or with matrices, as appropriate to the system, with technology. (MA.AA.LF.7)

MA.FM.MA.6: Build and use matrix representations to model polygons, transformations, and computer animations.

I can build matrix representations to model polygons, transformations, and computer animations.

I can use matrix representations to model polygons, transformations, and computer animations.

Matrices

Polygons

Transformation

Computer animation

Scalar

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. (MA.8.GM.6)

Use geometric descriptions of rigid motions to transform figures and to predict and

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describe the results of translations, reflections and rotations on a given figure. (MA.G.TR.1)

Networks Finite Mathematics Standards Success Criteria Academic Vocabulary Looking Back

MA.FM.N.1: Use networks, traceable paths, tree diagrams, Venn diagrams, and other pictorial representations to find the number of outcomes in a problem situation.

I can use networks to find the number of outcomes in a problem situation.

I can use traceable paths to find the number of outcomes in a problem situation.

I can use tree diagrams, Venn diagrams, and other pictorial representations to find the number of outcomes in a problem situation.

Network

Traceable path

Tree diagram

Venn diagram

Outcomes

Represent sample spaces of compound events (independent and dependent) using methods, such as organized lists, tables, and tree diagrams. (MA.8.DSP.5)

MA.FM.N.2: Optimize networks in different ways and in different contexts by finding minimal spanning trees, shortest paths, and Hamiltonian paths including real-world problems.

I can optimize networks for real-world problems by finding minimal spanning trees.

I can optimize networks for real-world problems by finding the shortest path.

Minimal spanning tree

Edge-weighted graph

Hamiltonian path

Traceable path

Shortest path

Represent sample spaces of compound events (independent and dependent) using methods, such as organized lists, tables, and tree diagrams. (MA.8.DSP.5)

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I can optimize networks for real-world problems by finding Hamiltonian paths.

MA.FM.N.3: Use critical-path analysis in the context of scheduling problems and interpret the results.

I can use critical-path analysis for scheduling problems.

I can interpret the results from a critical-path analysis in the context of scheduling problems.

Critical-path analysis State, use, and examine the validity of the converse, inverse, and contrapositive of conditional and bi-conditional statements. (MA.G.LP.3)

Develop geometric proofs, including direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry. (MA.G.LP.4)

MA.FM.N.4: Construct and interpret directed and undirected graphs, decision trees, networks, and flow charts that model real-world contexts and problems.

I can construct and interpret directed and undirected graphs to model real-world contexts and problems.

I can construct and interpret decision trees to model real-world contexts and problems.

I can construct and interpret networks to model real-world contexts and problems.

I can construct and interpret flow charts to model real-world contexts and problems.

Directed graph

Undirected graph

Edge-weighted graph

Decision trees

Network

Flow chart

State, use, and examine the validity of the converse, inverse, and contrapositive of conditional and bi-conditional statements. (MA.G.LP.3)

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MA.FM.N.5: Use graph-coloring techniques to solve problems.

I can use graph-coloring techniques to solve problems.

Graph-coloring techniques

Vertex coloring

Edge coloring

State, use, and examine the validity of the converse, inverse, and contrapositive of conditional and bi-conditional statements. (MA.G.LP.3)

Develop geometric proofs, including direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry. (MA.G.LP.4)

MA.FM.N.6 Construct vertex-edge graph models involving relationships among a finite number of elements. Describe a vertex-edge graph using an adjacency matrix. Use vertex-edge graph models to solve problems in a variety of real-world settings.

I can construct vertex-edge graph models to show the relationships among a finite number of elements.

I can use an adjacency matrix to describe a vertex-edge graph.

I can solve a variety of real-world problems using vertex-edge graphs.

Vertex-edge graph

Finite

Elements

Adjacency matrix


Optimization Finite Mathematics Standards Success Criteria Academic Vocabulary Looking Back

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MA.FM.O.1: Use bin-packing techniques to solve problems of optimizing resource usage.

I can use bin-packing techniques to optimize resource usage.

Bin-packing

MA.FM.O.2: Use geometric and algebraic techniques to solve optimization problems with and without technology.

I can use geometric techniques to solve optimization problems with and without technology.

I can use algebraic techniques to solve optimization problems with and without technology.

Optimization Apply geometric methods to solve design problems. (MA.G.TS.6)

Understand composition of functions and combine functions by composition. (MA.AII.F.2)

MA.FM.O.3: Use the Simplex method to solve optimization problems with and without technology.

I can solve optimization problems using the Simplex method without technology.

I can solve optimization problems using the Simplex method with technology.

Simplex method Represent real-world problems using a system of linear equations in three variables and solve such problems with and without technology. Interpret the solution and determine whether it is reasonable. (MA.AII.SE.3) & (MA.AA.LF.7)

Probability Finite Mathematics Standards Success Criteria Academic Vocabulary Looking Back

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MA.FM.P.1: Use Markov chains to solve problems with and without technology.

I can use Markov chains to solve problems with technology.

I can use Markov chains to solve problems without technology.

Markov chains

Random variables

Conditionally independent

State space

Game theory

Genetics

Understand dependent and independent events, and conditional probability. (MA.AII.DSP.5)

Understand the multiplication counting principle, permutations, and combinations. (MA.AII.DSP.6)

MA.FM.P.2: Understand and use the addition rule to calculate probabilities for mutually exclusive and non-mutually exclusive events.

I can determine if events are mutually exclusive.

I can state and apply the addition rule for calculating probabilities for mutually exclusive events.

I can give and use the addition rule for calculating probabilities for non-mutually exclusive events.

Addition rule

Mutually exclusive events

Non-mutually exclusive events

Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

Understand and use the addition rule to calculate probabilities for mutually exclusive and non-mutually exclusive events. (MA.PS.P.1)

MA.FM.P.3: Understand and use the multiplication rule to calculate probabilities for independent and dependent events. Understand that two events A and B are independent if the probability of A and B occurring together is

I can state and apply the multiplication rule to calculate probabilities for independent and dependent events.

I can define independent events as those in which the probability of the

Multiplication rule

Independent events

Dependent events

Understand dependent and independent events, and conditional probability; apply these concepts to calculate probabilities. (MA.AII.DSP.5)

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the product of their probabilities, and use this characterization to determine if they are independent.

event happening together is the product of their individual probabilities.

Use a two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. (MA.PS.DA.6)

Understand and use the multiplication rule to calculate probabilities for independent and dependent events. (MA.PS.P.2)

MA.FM.P.4: Understand the multiplication counting principle, permutations, and combinations; use them to solve real-world problems. Use simulations with and without technology to solve counting and probability problems.

I can state the multiplication counting principle and when it is applicable.

I can define and apply the concept of permutations and combinations in order to solve real-world problems.

I can use technology to perform simulations in order to solve counting and probability problems.

I can perform simulations to solve counting and probability problems.

Multiplication counting principle

Permutations

Combinations

Develop the multiplication counting principle and apply it to situations with a large number of outcomes. (MA.8.DSP.6)

Understand the multiplication counting principle, permutations, and combinations; apply these concepts to calculate probabilities. (MA.AII.DSP.6)

Understand and use the multiplication rule to calculate probabilities for independent and dependent events. (MA.PS.P.3)

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MA.FM.P.5: Calculate the probabilities of complementary events.

I can identify complementary events.

I can calculate the probabilities of complementary events.

Complementary event Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

Calculate the probabilities of complementary events. (MA.PS.P.4)

MA.FM.P.6: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

I can calculate the expected value of a random variable.

I can interpret an expected value of a random variable as the mean of its probability distribution.

Probability distribution

Random variable

Expected value

Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (MA.PS.P.5)

MA.FM.P.7: Analyze decisions and strategies using probability concepts. Analyze probabilities to interpret odds and risk of events.

I can analyze decision and strategies using probability concepts.

I can analyze probabilities to infer odds and risk of events.

Odds

Risk

Analyze decisions and strategies using probability concepts. Analyze probabilities to interpret odds and risk of events. (MA.PS.P.6)

MA.FM.P.8: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.

I can describe subsets of a sample space using characteristics of the outcomes.

I can describe subsets of a sample space using unions, intersections, or complements of other events.

Subset

Sample space

Outcome

Union

Understand and use appropriate terminology to describe independent, dependent, complementary, and mutually exclusive events. (MA.8.DSP.4)

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Intersection

Complement

MA.FM.P.9: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

I can create a probability distribution for a random variable for which theoretical probabilities can be calculated.

I can find the expected value of a random variable given its probability distribution.

Probability distribution

Random variable

Theoretical probability

Expected value

Record multiple observations random events and construct empirical models of the probability distributions. (MA.AII.DSP.4)

Compute and interpret the expected value of random variables. (MA.PS.P.9)

MA.FM.P.10: Use the relative frequency of a specified outcome of an event to estimate the probability of the outcome and apply the law of large numbers in simple examples.

I can apply the law of large numbers in simple problems.

I can use relative frequency of outcomes to estimate the probability of those outcomes.

Relative frequency

Outcome

Law of large numbers

Construct a theoretical model and apply the law of large numbers to show the relationship between a theoretical model and its empirical model. (MA.AII.DSP.4)

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