The Common Core in Montana

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Skills available for Montana high school math standards

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Montana Common Core Standards: Algebra I Montana Common Core Standards: Algebra I
Montana Common Core Standards: Algebra II Montana Common Core Standards: Algebra II
Montana Common Core Standards: All High School Pathways Montana Common Core Standards: All High School Pathways
Montana Common Core Standards: Geometry Montana Common Core Standards: Geometry
Montana Common Core Standards: Mathematical Practices Montana Common Core Standards: Mathematical Practices
Montana Common Core Standards: Mathematics I Montana Common Core Standards: Mathematics I
Montana Common Core Standards: Mathematics II Montana Common Core Standards: Mathematics II
Montana Common Core Standards: Mathematics III Montana Common Core Standards: Mathematics III
Montana Content Standards: Algebra I with Probability Montana Content Standards: Algebra I with Probability
Montana Content Standards: Algebra II with Statistics Montana Content Standards: Algebra II with Statistics
Montana Content Standards: Geometry with Data Analysis Montana Content Standards: Geometry with Data Analysis
Montana Content Standards: Integrated Math I Montana Content Standards: Integrated Math I
Montana Content Standards: Integrated Math II Montana Content Standards: Integrated Math II
Montana Content Standards: Integrated Math III Montana Content Standards: Integrated Math III

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MT.HS.PLUS.NUM Core Plus Number and Quantity Standards

MT.HS.PLUS.NUM.REAS Numeric Reasoning
- MT.HS.PLUS.NUM.REAS.1 Extend the properties of exponents to rational exponents, including converting between exponential and radical form.
- MT.HS.PLUS.NUM.REAS.2 Understand there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b as real numbers by:
  - MT.HS.PLUS.NUM.REAS.2.a Adding, subtracting, multiplying, and dividing complex numbers.
  - MT.HS.PLUS.NUM.REAS.2.b Finding the conjugate of a complex number.
    - Complex conjugates (A2-K.3)

MT.HS.PLUS.ALG Core Plus Algebraic and Functional Reasoning Standards

MT.HS.PLUS.ALG.FUN Functions, Expressions, and Inequalities
- MT.HS.PLUS.ALG.FUN.1 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(k + x) for specific values of k (both positive and negative).
- MT.HS.PLUS.ALG.FUN.2 Understand the relationship between a function and its inverse.
MT.HS.PLUS.ALG.POLY Polynomial Functions
- MT.HS.PLUS.ALG.POLY.1 Understand polynomials are created by multiplying linear factors.
- MT.HS.PLUS.ALG.POLY.2 Understand that polynomial expressions can be represented in both factored and standard form, and the specific features of each form by:
  - MT.HS.PLUS.ALG.POLY.2.a Choosing the form strategically based on given information and intended use when writing an expression.
  - MT.HS.PLUS.ALG.POLY.2.b Converting between factored and standard form symbolically and using representations.
  - MT.HS.PLUS.ALG.POLY.2.c Interpreting the relationship between the factored form of the expression and the zeros of the function.
- MT.HS.PLUS.ALG.POLY.3 Graph polynomial functions with and without the use of technology, by identifying zeros, relative maxima and minima, and end behavior.
- MT.HS.PLUS.ALG.POLY.4 Solve quadratic equations that have complex solutions, and understand why the solutions form a conjugate pair.
  - Solve a quadratic equation using the quadratic formula (A2-O.6)
MT.HS.PLUS.ALG.EXP Exponential and Logarithmic Functions
- MT.HS.PLUS.ALG.EXP.1 Understand logarithmic functions as the inverse of exponential functions.
- MT.HS.PLUS.ALG.EXP.2 Understand why e is defined as the natural base.
  - Convert between natural exponential and logarithmic form (A2-AA.2)
  - Evaluate natural logarithms (A2-AA.5)
- MT.HS.PLUS.ALG.EXP.3 Understand that exponential and logarithmic functions can be represented using multiple forms by:
  - MT.HS.PLUS.ALG.EXP.3.a Expressing exponential functions in the form f(x) = ab^x and f(x) = Pe^rt.
    - Solve exponential equations by rewriting the base (A2-CC.1)
    - Continuously compounded interest: word problems (A2-CC.5)
  - MT.HS.PLUS.ALG.EXP.3.b Expressing logarithmic functions in base 10 and base e.
    - Solve exponential equations using common logarithms (A2-CC.2)
    - Solve exponential equations using natural logarithms (A2-CC.3)
- MT.HS.PLUS.ALG.EXP.4 Graph logarithmic and exponential functions with and without the use of technology by identifying intercepts, asymptotes, and end behavior.
  - Graph exponential functions (A2-Z.4)
  - Graph logarithmic functions (A2-BB.2)
- MT.HS.PLUS.ALG.EXP.5 Solve exponential and logarithmic equations using inverse operations with and without the use of technology.
  - Solve logarithmic equations I (A2-CC.6)
  - Solve logarithmic equations II (A2-CC.7)
MT.HS.PLUS.ALG.TRIG Trigonometric Functions
- MT.HS.PLUS.ALG.TRIG.1 Understand how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers by:
  - MT.HS.PLUS.ALG.TRIG.1.a Defining the sine and cosine functions in terms of the unit circle.
  - MT.HS.PLUS.ALG.TRIG.1.b Defining the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine.
    - Find tangent, cotangent, secant, and cosecant ratios using the unit circle (A2)
- MT.HS.PLUS.ALG.TRIG.2 Understand and use the radian measure of an angle, and convert between degree and radian measures.
- MT.HS.PLUS.ALG.TRIG.3 Graph trigonometric functions with and without the use of technology by:
  - MT.HS.PLUS.ALG.TRIG.3.a Graphing sine and cosine functions, identifying period, midline, and amplitude.
  - MT.HS.PLUS.ALG.TRIG.3.b Graphing tangent functions, identifying period and asymptotes.
- MT.HS.PLUS.ALG.TRIG.4 Solve trigonometric equations with and without the use of technology.
  - Solve trigonometric equations: sin, cos, tan (A2-JJ.9)
  - Solve trigonometric equations (A2-JJ.10)
- MT.HS.PLUS.ALG.TRIG.5 Apply the Law of Sines and the Law of Cosines to find unknown measurements in non-right triangles.
MT.HS.PLUS.ALG.MOD Modeling
- MT.HS.PLUS.ALG.MOD.1 Model situations in context with polynomial, exponential, logarithmic, and trigonometric functions. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities by:
  - MT.HS.PLUS.ALG.MOD.1.a Determining if a set of data is best modeled by a polynomial, exponential, logarithmic, or trigonometric function or none, and explaining why.
  - MT.HS.PLUS.ALG.MOD.1.b Understanding that there are contexts where solutions may not lie on the curve.
- MT.HS.PLUS.ALG.MOD.2 Interpret the coefficients in a polynomial, exponential, logarithmic, and trigonometric model in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.
  - Determine the period and amplitude of a periodic function: graphs (A2)
- MT.HS.PLUS.ALG.MOD.3 Use and interpret units correctly in modeling situations.
- MT.HS.PLUS.ALG.MOD.4 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in modeling situations.

MT.HS.PLUS.DATA Core Plus Data and Reasoning Standards

MT.HS.PLUS.DATA.NORM Normal Distribution
- MT.HS.PLUS.DATA.NORM.1 Determine if a data set is normally distributed.
- MT.HS.PLUS.DATA.NORM.2 Use technology to find the mean and standard deviation of a normally distributed data set and apply the empirical rule to estimate population percentages.
- MT.HS.PLUS.DATA.NORM.3 Estimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.
  - Find z-values (A2-RR.4)
MT.HS.PLUS.DATA.DES Experimental Design
- MT.HS.PLUS.DATA.DES.1 Describe the purposes of and differences among sample surveys, experiments, and observational studies and explain how randomization relates to each.
  - Experiment design (A2-SS.9)
- MT.HS.PLUS.DATA.DES.2 Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect by:
  - MT.HS.PLUS.DATA.DES.2.a Explaining the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.
    - Experiment design (A2-SS.9)
  - MT.HS.PLUS.DATA.DES.2.b Evaluating where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.
    - Identify biased samples (A2-SS.1)
    - Experiment design (A2-SS.9)
- MT.HS.PLUS.DATA.DES.3 Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic by:
  - MT.HS.PLUS.DATA.DES.3.a Simulating a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.
    - Distributions of sample means (A2-RR.6)
  - MT.HS.PLUS.DATA.DES.3.b Demonstrating that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.
    - Standard deviation of a population (A2-SS.2)
    - Standard deviation of a sample (A2-SS.3)
MT.HS.PLUS.DATA.INF Statistical Inference Using Simulation
- MT.HS.PLUS.DATA.INF.1 Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.
  - Identify biased samples (A2-SS.1)
- MT.HS.PLUS.DATA.INF.2 Estimate a population parameter from a representative sample by:
  - MT.HS.PLUS.DATA.INF.2.a Understanding why the sample statistic is the best estimate for the associated population parameter.
    - Identify biased samples (A2-SS.1)
    - Experiment design (A2-SS.9)
  - MT.HS.PLUS.DATA.INF.2.b Understanding that sampling variability introduces uncertainty in the estimate, and account for the uncertainty with a confidence interval by:
    - MT.HS.PLUS.DATA.INF.2.b.i Using resampling with replacement from an observed sample to produce a sampling distribution.
    - MT.HS.PLUS.DATA.INF.2.b.ii Verifying that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.
      - Interpret confidence intervals for population means (A2-SS.8)
    - MT.HS.PLUS.DATA.INF.2.b.iii Verifying that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.
      - Interpret confidence intervals for population means (A2-SS.8)
    - MT.HS.PLUS.DATA.INF.2.b.iv Creating and interpreting a 95% confidence interval based on an observed mean from a sampling distribution.
      - Find confidence intervals for population means (A2-SS.6)
      - Interpret confidence intervals for population means (A2-SS.8)
- MT.HS.PLUS.DATA.INF.3 Use data from a randomized experiment to test the hypothesis that two groups are equal by:
  - MT.HS.PLUS.DATA.INF.3.a Interpreting the difference or ratio between the group means as the observed effect between the groups.
    - Analyze the results of an experiment using simulations (A2-SS.10)
  - MT.HS.PLUS.DATA.INF.3.b Understanding that an observed effect may be due to randomization and using a randomization test (repeatedly reshuffling the observed data into new groups) to determine the probability that an observed effect is due to randomization alone.
    - Experiment design (A2-SS.9)

The Common Core in Montana

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MT.HS.PLUS.NUM Core Plus Number and Quantity Standards

MT.HS.PLUS.NUM.REAS Numeric Reasoning

MT.HS.PLUS.NUM.REAS.1 Extend the properties of exponents to rational exponents, including converting between exponential and radical form.

MT.HS.PLUS.NUM.REAS.2 Understand there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b as real numbers by:

MT.HS.PLUS.NUM.REAS.2.a Adding, subtracting, multiplying, and dividing complex numbers.

MT.HS.PLUS.NUM.REAS.2.b Finding the conjugate of a complex number.

MT.HS.PLUS.ALG Core Plus Algebraic and Functional Reasoning Standards

MT.HS.PLUS.ALG.FUN Functions, Expressions, and Inequalities

MT.HS.PLUS.ALG.FUN.1 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(k + x) for specific values of k (both positive and negative).

MT.HS.PLUS.ALG.FUN.2 Understand the relationship between a function and its inverse.

MT.HS.PLUS.ALG.POLY Polynomial Functions

MT.HS.PLUS.ALG.POLY.1 Understand polynomials are created by multiplying linear factors.

MT.HS.PLUS.ALG.POLY.2 Understand that polynomial expressions can be represented in both factored and standard form, and the specific features of each form by:

MT.HS.PLUS.ALG.POLY.2.a Choosing the form strategically based on given information and intended use when writing an expression.

MT.HS.PLUS.ALG.POLY.2.b Converting between factored and standard form symbolically and using representations.

MT.HS.PLUS.ALG.POLY.2.c Interpreting the relationship between the factored form of the expression and the zeros of the function.

MT.HS.PLUS.ALG.POLY.3 Graph polynomial functions with and without the use of technology, by identifying zeros, relative maxima and minima, and end behavior.

MT.HS.PLUS.ALG.POLY.4 Solve quadratic equations that have complex solutions, and understand why the solutions form a conjugate pair.

MT.HS.PLUS.ALG.EXP Exponential and Logarithmic Functions

MT.HS.PLUS.ALG.EXP.1 Understand logarithmic functions as the inverse of exponential functions.

MT.HS.PLUS.ALG.EXP.2 Understand why e is defined as the natural base.

MT.HS.PLUS.ALG.EXP.3 Understand that exponential and logarithmic functions can be represented using multiple forms by:

MT.HS.PLUS.ALG.EXP.3.a Expressing exponential functions in the form f(x) = abx and f(x) = Pert.

MT.HS.PLUS.ALG.EXP.3.b Expressing logarithmic functions in base 10 and base e.

MT.HS.PLUS.ALG.EXP.4 Graph logarithmic and exponential functions with and without the use of technology by identifying intercepts, asymptotes, and end behavior.

MT.HS.PLUS.ALG.EXP.5 Solve exponential and logarithmic equations using inverse operations with and without the use of technology.

MT.HS.PLUS.ALG.TRIG Trigonometric Functions

MT.HS.PLUS.ALG.TRIG.1 Understand how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers by:

MT.HS.PLUS.ALG.TRIG.1.a Defining the sine and cosine functions in terms of the unit circle.

MT.HS.PLUS.ALG.TRIG.1.b Defining the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine.

MT.HS.PLUS.ALG.TRIG.2 Understand and use the radian measure of an angle, and convert between degree and radian measures.

MT.HS.PLUS.ALG.TRIG.3 Graph trigonometric functions with and without the use of technology by:

MT.HS.PLUS.ALG.TRIG.3.a Graphing sine and cosine functions, identifying period, midline, and amplitude.

MT.HS.PLUS.ALG.TRIG.3.b Graphing tangent functions, identifying period and asymptotes.

MT.HS.PLUS.ALG.TRIG.4 Solve trigonometric equations with and without the use of technology.

MT.HS.PLUS.ALG.TRIG.5 Apply the Law of Sines and the Law of Cosines to find unknown measurements in non-right triangles.

MT.HS.PLUS.ALG.MOD Modeling

MT.HS.PLUS.ALG.MOD.1 Model situations in context with polynomial, exponential, logarithmic, and trigonometric functions. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities by:

MT.HS.PLUS.ALG.MOD.1.a Determining if a set of data is best modeled by a polynomial, exponential, logarithmic, or trigonometric function or none, and explaining why.

MT.HS.PLUS.ALG.MOD.1.b Understanding that there are contexts where solutions may not lie on the curve.

MT.HS.PLUS.ALG.MOD.2 Interpret the coefficients in a polynomial, exponential, logarithmic, and trigonometric model in context. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

MT.HS.PLUS.ALG.MOD.3 Use and interpret units correctly in modeling situations.

MT.HS.PLUS.ALG.MOD.4 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in modeling situations.

MT.HS.PLUS.DATA Core Plus Data and Reasoning Standards

MT.HS.PLUS.DATA.NORM Normal Distribution

MT.HS.PLUS.DATA.NORM.1 Determine if a data set is normally distributed.

MT.HS.PLUS.DATA.NORM.2 Use technology to find the mean and standard deviation of a normally distributed data set and apply the empirical rule to estimate population percentages.

MT.HS.PLUS.DATA.NORM.3 Estimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.

MT.HS.PLUS.DATA.DES Experimental Design

MT.HS.PLUS.DATA.DES.1 Describe the purposes of and differences among sample surveys, experiments, and observational studies and explain how randomization relates to each.

MT.HS.PLUS.DATA.DES.2 Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect by:

MT.HS.PLUS.DATA.DES.2.a Explaining the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.

MT.HS.PLUS.DATA.DES.2.b Evaluating where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.

MT.HS.PLUS.DATA.DES.3 Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic by:

MT.HS.PLUS.DATA.DES.3.a Simulating a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.

MT.HS.PLUS.DATA.DES.3.b Demonstrating that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.

MT.HS.PLUS.DATA.INF Statistical Inference Using Simulation

MT.HS.PLUS.DATA.INF.1 Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.

MT.HS.PLUS.DATA.INF.2 Estimate a population parameter from a representative sample by:

MT.HS.PLUS.DATA.INF.2.a Understanding why the sample statistic is the best estimate for the associated population parameter.

MT.HS.PLUS.DATA.INF.2.b Understanding that sampling variability introduces uncertainty in the estimate, and account for the uncertainty with a confidence interval by:

MT.HS.PLUS.DATA.INF.2.b.i Using resampling with replacement from an observed sample to produce a sampling distribution.

MT.HS.PLUS.DATA.INF.2.b.ii Verifying that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.

MT.HS.PLUS.DATA.INF.2.b.iii Verifying that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.

MT.HS.PLUS.DATA.INF.2.b.iv Creating and interpreting a 95% confidence interval based on an observed mean from a sampling distribution.

MT.HS.PLUS.DATA.INF.3 Use data from a randomized experiment to test the hypothesis that two groups are equal by:

MT.HS.PLUS.DATA.INF.3.a Interpreting the difference or ratio between the group means as the observed effect between the groups.

MT.HS.PLUS.DATA.INF.3.b Understanding that an observed effect may be due to randomization and using a randomization test (repeatedly reshuffling the observed data into new groups) to determine the probability that an observed effect is due to randomization alone.

MT.HS.PLUS.NUM.REAS.2 Understand there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b as real numbers by:

MT.HS.PLUS.ALG.EXP.3.a Expressing exponential functions in the form f(x) = ab^x and f(x) = Pe^rt.