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# Level 7 Achievement objectives

## Achievement objective M7-1

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• apply coordinate geometry techniques to points and lines.

### Indicators

• Uses algebra and geometry to link points and lines on the Cartesian plane.
• Uses geometric features such as parallel lines, perpendicular lines, collinear points, centre, radius and diameter of circles, and types of triangles to prove conjectures.

## Achievement objective M7-2

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs.

### Indicators

• Demonstrates understanding of functions and the relationships between the graph of functions, y=f(x), and the graphs of their transformations of the form y= Af(x-B) +C. Functions include:
• quadratics
• cubics
• polynomials of degree of 4 or more, but students are not be expected to factorise these
• simple piecewise
• exponential including the use of base e
• logarithmic y = loge(x) or y = logb(x)
• rectangular hyperbola y=1/x
• square root, f(x) = √ x
• absolute value, y = |x|
• Trig functions include sine, cosine and tangent with x in degrees or radians.
• Makes connections between representations, such as f(x) notation, tables, mapping, equations, words and graphs:
• Writes equations for graphs and vice versa.
• Explains effect on graph / equation of changing parameters.
• Understands effect of transformations in different representations.
• Demonstrates understanding of properties of functions, that is, one-to-one and many-to-one, and examples of relations that are not functions (for example, a circle).
• Understands the relationship between the graph of y = f(x) and the graphs of its transformations.
• Identifies and uses appropriate key features, that is, symmetry, period, amplitude, intercepts, maxima, minima, asymptotes, domain, and range.

## Achievement objective M7-3

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• use arithmetic and geometric sequences and series.

### Indicators

• Identifies and uses representations such as geometric patterns, coordinates, lists of terms, recursive formulae, general formulae, and sigma notation.
• Solves problems that can be modelled by arithmetic and geometric sequences and series.

## Achievement objective M7-4

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions.

### Indicators

• Solves problems that can be modelled by trigonometric relationships.
• Proves simple trigonometric identities making links to right-angled triangles.
• Uses the area formula for triangles and the sine and cosine rules to solve problems.
• Uses knowledge of right-angled triangles to find exact values of the trig ratios.
• Uses radian measure.

## Achievement objective M7-5

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• choose appropriate networks to find optimal solutions.

## Achievement objective M7-6

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• manipulate rational, exponential, and logarithmic algebraic expressions.

### Indicators

• Makes links between the concepts, properties, and manipulation of exponents and logarithms, and the inverse relationship between them.
• Solves problems that involve manipulating rational, exponential, and logarithmic algebraic expressions. Methods that could be used in solving problems include:
• simplifying algebraic expressions including rational expressions
• expanding and factorising
• manipulating expressions with exponents including fractional and negative exponents
• manipulating logarithms in order to solve exponential equations
• simplifying expressions including logarithms involving the power rule and also products and quotients.

## Achievement objective M7-7

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• form and use linear, quadratic, and simple trigonometric equations.

### Indicators

• Solves problems that can be modelled by a combination of linear or quadratics, simple trigonometric equations and interprets solutions in context.
• Uses completing the square and quadratic formula for solving quadratic equations.
• Links discriminant with nature of the solutions of a quadratic equation.
• Demonstrates understanding of the relationship between an equation and its solutions, for example, no, one, or two solutions for quadratic equations, multiple solutions for trigonometric equations.

## Achievement objective M7-8

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• form and use pairs of simultaneous equations, one of which may be non-linear.

### Indicators

• Solves problems that can be modelled by pairs of simultaneous equations where one is linear and the other equation can be:
• a quadratic, y = ax^2 + bx + c
• a hyperbola, y = a/(x+b) + c
• a circle, (x-h)^2 + (y-k)^2 = r^2
• Chooses appropriate methods and uses them to solve simultaneous equations.
• Interprets the existence and type of solution(s) in the context of the situation.

## Achievement objective M7-9

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• sketch the graphs of functions and their gradient functions and describe the relationship between these graphs.

### Indicators

• Sketches graphs of gradient functions of a variety of functions represented in various forms such as an equation or a graph (differentiable over the given domain).
• Sketches graphs of functions from the graph of the gradient function.
• Relates the gradient at a point on a curve to the instantaneous rate of change.
• Relates significant features of the functions such as:
• turning points on the original function and the zeros of the gradient function, or vice versa
• the increasing or decreasing of the original function and the sign of the gradient function, or vice versa.

## Achievement objective M7-10

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• apply differentiation and anti-differentiation techniques to polynomials.

### Indicators

• Develops the rules for differentiation of polynomials using understanding of gradient function and technology (graphics calculator or computer).
• Solves differentiation problems involving:
• rates of change
• finding points where gradient has a particular value
• finding the equation of the tangent to a curve at a point
• finding local maxima/minima of a function
• kinematics.
• Develops rules for anti-differentiation of polynomials using understanding of gradient function and technology (graphics calculator or computer).
• Solves anti-differentiation problems involving:
• finding a family of curves with a given gradient
• finding the constant of anti-differentiation
• simple polynomial differential equations (not including separation of variables).

## Achievement objective S7-1

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• Carry out investigations of phenomena, using the statistical enquiry cycle:
• A conducting surveys that require random sampling techniques, conducting experiments, and using existing data sets
• B evaluating the choice of measures for variables and the sampling and data collection methods used
• C using relevant contextual knowledge, exploratory data analysis, and statistical inference.

### Indicators

• Uses the statistical enquiry cycle to conduct surveys and experiments and to analyse existing data sets.
• Conducts surveys to find solutions to problems (or uses existing data sets):
• Poses survey questions, considering sources of variation, for example, what are the variables to be collected, how each variable will be measured.
• Designs, trials, and improves questionnaires using a range of appropriate questions types, checking the survey questions using, for example, desk review, conducting pilot surveys.
• Selects and uses appropriate sampling methods, for example, simple random, systematic, stratified, cluster, and quota.
• Evaluates sampling method used, for example, is a sample sufficiently large, randomly chosen, and representative of the population.
• Collects and manages data.
• Uses exploratory data analysis to explore features of the data:
• Uses appropriate statistical graphs and tables to explore the data and communicates relevant detail and overall distributions.
• Uses appropriate measures to communicate features of the data.
• Uses relevant contextual knowledge when communicating findings.
• Makes informal  statistical inferences.
• Communicates findings in a report which includes:
• relevant summary statistics, graphs and tables to support the findings of the survey
• quantitative and qualitative statements
• informal statistical inferences
• justified conclusions.
• Conducts experiments to find solutions to problems:
• Poses investigative questions about an experimental situation.
• Plans experiments:
• Considers sources of variation, for example, what are the variables to be collected, how each variable will be measured.
• Evaluates the choice of variables and measures used in the experiment.
• Selects and uses appropriate data collection and recording methods.
• Conducts the experiment and collects data.
• Communicates findings in a report which includes:
• relevant summary statistics, graphs and tables to support the findings of the experiment
• quantitative and qualitative statements
• suggestive statistical inferences
• justified conclusions.

## Achievement objective S7-2

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• Make inferences from surveys and experiments:
• A making informal predictions, interpolations, and extrapolations
• B using sample statistics to make point estimates of population parameters
• C recognising the effect of sample size on the variability of an estimate.

### Indicators

Note: this content is new to the statistics curriculum.

See CensusAtSchool under informal inference for further information and resources. At level 7, B, and C (below) are critical for students to advance to level 8.

A. Making informal predictions, interpolations, and extrapolations:

• Within the context of an investigation and statistical plots of observed data:
• Uses a scatterplot from a sample to make sensible predictions within the given data by plotting the trend informally by eye and showing likely variation band of response (y) values for a particular value of x, the explanatory variable.
• Informally extrapolates where appropriate and predicts the trend outside the given data values on a scatterplot and justifies using contextual understanding.

B. Using sample statistics to make point estimates of population parameters.

• Understands that the sample statistics can be used as point estimates of the population parameters, for example, sample medians and IQRs can be used as point estimates for population medians and IQRs, or sample proportions for population proportions when using categorical data.

C. Recognising the effect of sample size on the variability of an estimate:

• Within the context of an investigation and statistical plots of observed data:
• Finds informal confidence intervals for population medians.
• Plots sample data showing informal confidence intervals (median ± 1.5 IQR / √n) on boxplots.
• Uses an informal confidence interval to make an informal statistical inference about the population median from sample data plot.
• Makes a claim about whether one group has larger values than another group using informal confidence intervals for the population medians.
• Explains the connections among sample, population, sampling variability, sample size effect, informal confidence interval, and degree of confidence.

## Achievement objective S7-3

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• Evaluate statistically based reports:
• A interpreting risk and relative risk
• B identifying sampling and possible non-sampling errors in surveys, including polls.

### Indicators

Note: this content is new to the statistics curriculum.

A. Interpreting risk and relative risk:

• Calculates and interprets risk, selects baseline group, and calculates and interprets relative risk and writes a news clip reporting on findings.
• In a media article with text and/or a table, identifies absolute risk, baseline group and relative risk, and for relative risk, identifies the two groups being compared; identifies relevant missing information and justifies why it is important to include this information.

B. Identifying sampling and possible non-sampling errors in surveys, including polls:

• In a media report on a survey or a poll, identifies sampling error and explains the connection between sample, population, sampling variability, and sampling error.
• In a media report on a survey or a poll, identifies and evaluates, using critical questions, sampling methods and possible non-sampling errors such as self-selection, non response bias, behavioural considerations.

## Achievement objective S7-4

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

• Investigate situations that involve elements of chance:
• A comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions
• B calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.

### Indicators

A. Comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions:

• Describes and compares distributions and recognises when distributions have similar and different characteristics.
• Carries out experimental investigations of probability situations, understanding the ways a sample is likely to be representative of a population.
• Is beginning to use mean and standard deviation as sample statistics or as population parameters.
• Chooses an appropriate model to solve a problem.
• Uses theoretical distributions such as the normal distribution to solve probability problems.

B. Calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology:

• Uses two-way frequency tables to solve simple probability problems, including working informally with conditional probabilities.
• Constructs and interprets probability trees with probabilities and outcomes on branches to solve probability problems representing either a series of events in time and/or a series of decision-making points, as well as probability problems with and without replacement.
• Uses simple simulations to represent probability situations (using appropriate technology).
• Students investigate probabilities from situations involving real data.

Last updated September 17, 2018