Ireland

YT-6.1 Statistics and Probability

• YT-6.1.1 Counting

  • YT-6.1.1.1 list outcomes of an experiment

    • Compound events: find the number of outcomes (T-FF.3)
    • Compound events: find the number of outcomes (5Y-FF.3)
    • Introduction to probability (6Y-T.1)

  • YT-6.1.1.2 apply the fundamental principle of counting

    • Counting principle (T-FF.8)
    • Counting principle (5Y-FF.6)
    • Combinations and permutations (6Y-T.3)
    • Find probabilities using combinations and permutations (6Y-T.4)

• YT-6.1.2 Concepts of probability

  • YT-6.1.2.1 decide whether an everyday event is likely or unlikely to occur

  • YT-6.1.2.2 recognise that probability is a measure on a scale of 0-1 of how likely an event is to occur

    • Theoretical probability (T-FF.1)
    • Experimental probability (T-FF.2)
    • Theoretical and experimental probability (5Y-FF.2)
    • Calculate probabilities of events (5Y-FF.4)
    • Introduction to probability (6Y-T.1)

  • YT-6.1.2.3 use set theory; discuss experiments, outcomes, sample spaces

    • Introduction to probability (6Y-T.1)

  • YT-6.1.2.4 use the language of probability to discuss events, including those with equally likely outcomes

    • Identify independent and dependent events (T-FF.4)
    • Identify independent events (5Y-FF.10)
    • Introduction to probability (6Y-T.1)

  • YT-6.1.2.5 estimate probabilities from experimental data

    • Experimental probability (T-FF.2)
    • Theoretical and experimental probability (5Y-FF.2)

  • YT-6.1.2.6 recognise that, if an experiment is repeated, there will be different outcomes and that increasing the number of times an experiment is repeated generally leads to better estimates of probability

  • YT-6.1.2.7 associate the probability of an event with its long run relative frequency

    • Experimental probability (T-FF.2)
    • Theoretical and experimental probability (5Y-FF.2)

• YT-6.1.3 Outcomes of random processes

  • YT-6.1.3.1 construct sample spaces for two independent events

    • Identify independent events (6Y-T.6)
    • Independence and conditional probability (6Y-T.8)

  • YT-6.1.3.2 apply the principle that in the case of equally likely outcomes the probability is given by the number of outcomes of interest divided by the total number of outcomes (examples using coins, dice, spinners, urns with coloured objects, playing cards, etc.)

    • Theoretical probability (T-FF.1)
    • Calculate probabilities of events (5Y-FF.4)
    • Calculate probabilities of events (6Y-T.2)

• YT-6.1.4 Statistical reasoning with an aim to becoming a statistically aware consumer

  • YT-6.1.4.1 engage in discussions about the purpose of statistics and recognise misconceptions and misuses of statistics

    • Identify biased samples (T-GG.3)
    • Identify biased samples (5Y-GG.3)

  • YT-6.1.4.2 discuss populations and samples

    • Identify biased samples (T-GG.3)
    • Identify biased samples (5Y-GG.3)
    • Identify biased samples (6Y-V.1)

  • YT-6.1.4.3 decide to what extent conclusions can be generalised

    • Identify biased samples (6Y-V.1)

  • YT-6.1.4.4 work with different types of data: categorical: nominal or ordinal numerical: discrete or continuous in order to clarify the problem at hand

    • Interpret histograms (T-HH.1)
    • Create histograms (T-HH.2)
    • Interpret stem-and-leaf plots (T-HH.3)
    • Interpret box-and-whisker plots (T-HH.4)
    • Interpret a scatter plot (T-HH.5)
    • Scatter plots: line of best fit (T-HH.6)
    • Interpret histograms (5Y-HH.1)
    • Create histograms (5Y-HH.2)
    • Interpret stem-and-leaf plots (5Y-HH.3)
    • Interpret box-and-whisker plots (5Y-HH.4)
    • Interpret a scatter plot (5Y-HH.5)
    • Scatter plots: line of best fit (5Y-HH.6)
    • Identify discrete and continuous random variables (6Y-U.1)

• YT-6.1.5 Finding, collecting and organising data

  • YT-6.1.5.1 clarify the problem at hand

  • YT-6.1.5.2 formulate one (or more) questions that can be answered with data

  • YT-6.1.5.3 explore different ways of collecting data

  • YT-6.1.5.4 generate data, or source data from other sources including the internet

  • YT-6.1.5.5 select a sample (Simple Random Sample)

  • YT-6.1.5.6 recognise the importance of representativeness so as to avoid biased samples

    • Identify biased samples (T-GG.3)
    • Identify biased samples (5Y-GG.3)

  • YT-6.1.5.7 design a plan and collect data on the basis of above knowledge

• YT-6.1.6 Representing data graphically and numerically

  • YT-6.1.6.G Graphical

    • YT-6.1.6.G.1 select appropriate graphical or numerical methods to describe the sample (univariate data only)

    • YT-6.1.6.G.2 evaluate the effectiveness of different displays in representing the findings of a statistical investigation conducted by others

    • YT-6.1.6.G.3 use stem and leaf plots and histograms (equal intervals) to display data

      • Interpret histograms (T-HH.1)
      • Create histograms (T-HH.2)
      • Interpret stem-and-leaf plots (T-HH.3)
      • Interpret histograms (5Y-HH.1)
      • Create histograms (5Y-HH.2)
      • Interpret stem-and-leaf plots (5Y-HH.3)

  • YT-6.1.6.N Numerical

    • YT-6.1.6.N.1 use a variety of summary statistics to describe the data

      • YT-6.1.6.N.1.a central tendency: mean, median, mode

        • Mean, median, mode and range (T-GG.1)
        • Mean, median, mode and range (5Y-GG.1)

      • YT-6.1.6.N.1.b variability: range

        • Mean, median, mode and range (T-GG.1)
        • Mean, median, mode and range (5Y-GG.1)

• YT-6.1.7 Analysing, interpreting and drawing inferences from data

  • YT-6.1.7.1 recognise how sampling variability influences the use of sample information to make statements about the population

  • YT-6.1.7.2 use appropriate tools to describe variability, drawing inferences about the population from the sample

    • Quartiles (T-GG.2)
    • Mean absolute deviation (T-GG.4)
    • Variance and standard deviation (T-GG.5)
    • Quartiles (5Y-GG.2)
    • Mean absolute deviation (5Y-GG.4)
    • Variance and standard deviation (5Y-GG.5)

  • YT-6.1.7.3 interpret the analysis

  • YT-6.1.7.4 relate the interpretation to the original question

• YT-6.1.8 Synthesis and problem-solving skills

  • YT-6.1.8.1 explore patterns and formulate conjectures

  • YT-6.1.8.2 explain findings

  • YT-6.1.8.3 justify conclusions

  • YT-6.1.8.4 communicate mathematics verbally and in written form

  • YT-6.1.8.5 apply their knowledge and skills to solve problems in familiar and unfamiliar contexts

  • YT-6.1.8.6 analyse information presented verbally and translate it into mathematical form

  • YT-6.1.8.7 devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.

    • Interpret histograms (T-HH.1)
    • Create histograms (T-HH.2)
    • Interpret stem-and-leaf plots (T-HH.3)
    • Interpret histograms (5Y-HH.1)
    • Create histograms (5Y-HH.2)
    • Interpret stem-and-leaf plots (5Y-HH.3)

YT-6.2 Geometry and Trigonometry

• YT-6.2.1 Synthetic geometry

  • YT-6.2.1.1 perform constructions 18, 19, 20

• YT-6.2.2 Co-ordinate geometry

  • YT-6.2.2.1 use slopes to show that two lines are

    • YT-6.2.2.1.a parallel

      • Slopes of parallel and perpendicular lines (T-F.22)

    • YT-6.2.2.1.b perpendicular

      • Slopes of parallel and perpendicular lines (T-F.22)

  • YT-6.2.2.2 recognise the fact that the relationship ax + by + c = 0 is linear

    • Write equations in standard form (T-F.14)
    • Standard form: find x- and y-intercepts (T-F.15)
    • Standard form: graph an equation (T-F.16)

  • YT-6.2.2.3 solve problems involving slopes of lines

    • Find the slope of a graph (T-F.2)
    • Find the slope from two points (T-F.3)
    • Slope-intercept form: write an equation from a table (T-F.9)
    • Slope-intercept form: write an equation from a word problem (T-F.10)
    • Slopes of parallel and perpendicular lines (T-F.22)
    • Write the equation of a linear function (6Y-A.5)

• YT-6.2.3 Trigonometry

  • YT-6.2.3.1 use the theorem of Pythagoras to solve problems (2D only)

    • Pythagoras' Theorem (T-AA.1)
    • Converse of Pythagoras' theorem (T-AA.2)
    • Pythagoras' Theorem and its converse (5Y-U.1)

• YT-6.2.4 Transformation geometry, enlargements

  • YT-6.2.4.1 investigate enlargements paying attention to

    • YT-6.2.4.1.a centre of enlargement

      • Dilations: graph the image (T-X.4)
      • Dilations: find the coordinates (T-X.5)
      • Dilations: scale factor and classification (T-X.6)
      • Dilations and parallel lines (T-X.7)

    • YT-6.2.4.1.b scale factor k, where 0 < k < 1, k > 1 k ∈ Q

      • Dilations: graph the image (T-X.4)
      • Dilations: find the coordinates (T-X.5)
      • Dilations: scale factor and classification (T-X.6)
      • Dilations and parallel lines (T-X.7)
      • Similarity ratios (T-Z.2)
      • Similarity ratios (5Y-Y.2)

    • YT-6.2.4.1.c area

      • Areas of similar figures (T-Z.9)
      • Areas of similar figures (5Y-Y.11)

  • YT-6.2.4.2 solve problems involving enlargements

    • Dilations: graph the image (T-X.4)
    • Dilations: find the coordinates (T-X.5)
    • Dilations: scale factor and classification (T-X.6)
    • Dilations and parallel lines (T-X.7)
    • Identify similar figures (T-Z.1)
    • Similarity ratios (T-Z.2)
    • Similarity statements (T-Z.3)
    • Side lengths and angle measures in similar figures (T-Z.4)
    • Similar triangles and indirect measurement (T-Z.5)
    • Perimeters of similar figures (T-Z.6)
    • Similarity rules for triangles (T-Z.7)
    • Similar triangles and similarity transformations (T-Z.8)
    • Areas of similar figures (T-Z.9)
    • Introduction to similar solids (T-CC.7)
    • Surface area and volume of similar solids (T-CC.8)
    • Trigonometric ratios in similar right triangles (5Y-U.5)
    • Identify similar figures (5Y-Y.1)
    • Similarity ratios (5Y-Y.2)
    • Similarity statements (5Y-Y.3)
    • Side lengths and angle measures in similar figures (5Y-Y.4)
    • Similar triangles and indirect measurement (5Y-Y.5)
    • Perimeters of similar figures (5Y-Y.6)
    • Similarity rules for triangles (5Y-Y.7)
    • Similar triangles and similarity transformations (5Y-Y.8)
    • Similarity of circles (5Y-Y.9)
    • Triangle Proportionality Theorem (5Y-Y.10)
    • Areas of similar figures (5Y-Y.11)
    • Area and perimeter of similar figures (5Y-Z.9)
    • Introduction to similar solids (5Y-CC.7)
    • Surface area and volume of similar solids (5Y-CC.8)

• YT-6.2.5 Synthesis and problem-solving skills

  • YT-6.2.5.1 explore patterns and formulate conjectures

  • YT-6.2.5.2 explain findings

  • YT-6.2.5.3 justify conclusions

  • YT-6.2.5.4 communicate mathematics verbally and in written form

  • YT-6.2.5.5 apply their knowledge and skills to solve problems in familiar and unfamiliar contexts

    • Congruence statements and corresponding parts (T-V.1)
    • Solve problems involving corresponding parts (T-V.2)
    • Identify congruent figures (T-V.3)
    • SSS and SAS Theorems (T-W.1)
    • ASA and AAS Theorems (T-W.2)
    • SSS, SAS, ASA and AAS Theorems (T-W.3)
    • Congruency in isosceles and equilateral triangles (T-W.5)
    • Hypotenuse-Leg Theorem (T-W.6)
    • Classify triangles (T-Y.1)
    • Triangle Angle-Sum Theorem (T-Y.2)
    • Midsegments of triangles (T-Y.3)
    • Triangles and bisectors (T-Y.4)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (T-Y.5)
    • Angle-side relationships in triangles (T-Y.6)
    • Triangle Inequality Theorem (T-Y.7)
    • Midsegments of triangles (5Y-X.1)
    • Triangles and bisectors (5Y-X.2)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (5Y-X.3)
    • Angle-side relationships in triangles (5Y-X.4)
    • Triangle Inequality Theorem (5Y-X.5)

  • YT-6.2.5.6 analyse information presented verbally and translate it into mathematical form

  • YT-6.2.5.7 devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.

    • Congruence statements and corresponding parts (T-V.1)
    • Solve problems involving corresponding parts (T-V.2)
    • Identify congruent figures (T-V.3)
    • SSS and SAS Theorems (T-W.1)
    • ASA and AAS Theorems (T-W.2)
    • SSS, SAS, ASA and AAS Theorems (T-W.3)
    • Congruency in isosceles and equilateral triangles (T-W.5)
    • Hypotenuse-Leg Theorem (T-W.6)
    • Classify triangles (T-Y.1)
    • Triangle Angle-Sum Theorem (T-Y.2)
    • Midsegments of triangles (T-Y.3)
    • Triangles and bisectors (T-Y.4)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (T-Y.5)
    • Angle-side relationships in triangles (T-Y.6)
    • Triangle Inequality Theorem (T-Y.7)
    • Midsegments of triangles (5Y-X.1)
    • Triangles and bisectors (5Y-X.2)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (5Y-X.3)
    • Angle-side relationships in triangles (5Y-X.4)
    • Triangle Inequality Theorem (5Y-X.5)

YT-6.3 Number

• YT-6.3.1 Number systems

  • YT-6.3.1.1 recognise irrational numbers and appreciate that R≠Q

    • Identify rational and irrational numbers (T-A.3)
    • Identify rational and irrational numbers (5Y-A.3)

  • YT-6.3.1.2 revisit the operations of addition, multiplication, subtraction and division in the following domains:

    • YT-6.3.1.2.a N of natural numbers and represent these numbers on a number line

    • YT-6.3.1.2.b Z of integers and represent these numbers on a number line

      • Number lines (T-A.1)
      • Add, subtract, multiply and divide integers (T-B.1)
      • Lengths of segments on number lines (T-R.2)
      • Number lines (5Y-A.1)

    • YT-6.3.1.2.c Q of rational numbers and represent these numbers on a number line

      • Number lines (T-A.1)
      • Add and subtract rational numbers (T-B.4)
      • Multiply and divide rational numbers (T-B.5)
      • Number lines (5Y-A.1)

    • YT-6.3.1.2.d R of real numbers and represent these numbers on a number line

  • YT-6.3.1.3 appreciate that processes can generate sequences of numbers or objects

    • Evaluate variable expressions for number sequences (T-K.4)
    • Write variable expressions for arithmetic sequences (T-K.5)
    • Write variable expressions for geometric sequences (T-K.6)
    • Evaluate formulas for sequences (5Y-EE.5)
    • Write a formula for an arithmetic sequence (5Y-EE.6)
    • Write a formula for a geometric sequence (5Y-EE.7)

  • YT-6.3.1.4 investigate patterns among these sequences

    • Identify arithmetic and geometric sequences (T-K.1)
    • Arithmetic sequences (T-K.2)
    • Geometric sequences (T-K.3)
    • Number sequences: mixed review (T-K.7)
    • Classify formulas and sequences (5Y-EE.1)
    • Find terms of an arithmetic sequence (5Y-EE.2)
    • Find terms of a geometric sequence (5Y-EE.3)
    • Sequences: mixed review (5Y-EE.9)
    • Identify a sequence as explicit or recursive (6Y-S.3)

  • YT-6.3.1.5 use patterns to continue the sequence

    • Arithmetic sequences (T-K.2)
    • Geometric sequences (T-K.3)
    • Find terms of an arithmetic sequence (5Y-EE.2)
    • Find terms of a geometric sequence (5Y-EE.3)
    • Find terms of a sequence (6Y-S.1)
    • Find terms of a recursive sequence (6Y-S.2)

  • YT-6.3.1.6 generate rules/formulae from those patterns

    • Write variable expressions for arithmetic sequences (T-K.5)
    • Write variable expressions for geometric sequences (T-K.6)
    • Write a formula for an arithmetic sequence (5Y-EE.6)
    • Write a formula for a geometric sequence (5Y-EE.7)
    • Find a recursive formula (6Y-S.4)
    • Find recursive and explicit formulas (6Y-S.5)
    • Convert a recursive formula to an explicit formula (6Y-S.6)
    • Convert an explicit formula to a recursive formula (6Y-S.7)
    • Convert between explicit and recursive formulas (6Y-S.8)

  • YT-6.3.1.7 develop decimals as special equivalent fractions strengthening the connection between these numbers and fraction and place value understanding

    • Convert between decimals and fractions (T-A.2)
    • Convert between decimals and fractions (5Y-A.2)
    • Write a repeating decimal as a fraction (6Y-S.18)

  • YT-6.3.1.8 consolidate their understanding of factors, multiples, prime numbers in N

    • Prime factorisation (T-A.5)
    • Prime factorisation (5Y-A.5)

  • YT-6.3.1.9 express numbers in terms of their prime factors

    • Prime factorisation (T-A.5)
    • Prime factorisation (5Y-A.5)

  • YT-6.3.1.10 appreciate the order of operations, including brackets

    • Evaluate numerical expressions involving integers (T-B.2)
    • Evaluate numerical expressions involving rational numbers (T-B.6)

  • YT-6.3.1.11 express non-zero positive rational numbers in the form a x10ⁿ, where n ∈ N and 1 ≤ a < 10 and perform arithmetic operations on numbers in this form

    • Convert between standard and scientific notation (T-I.1)
    • Compare numbers written in scientific notation (T-I.2)
    • Multiply numbers written in scientific notation (T-I.3)
    • Divide numbers written in scientific notation (T-I.4)
    • Convert between standard and scientific notation (5Y-C.1)
    • Compare numbers written in scientific notation (5Y-C.2)
    • Multiply numbers written in scientific notation (5Y-C.3)
    • Divide numbers written in scientific notation (5Y-C.4)

• YT-6.3.2 Indices

  • YT-6.3.2.1 solve problems using the rules for indices (where a ∈ Q, p, q ∈ N; a ≠ 0):

    • YT-6.3.2.1.a a to the p power x a to the q power = a to the (p+q) power

      • Multiplication with exponents (T-H.4)
      • Multiplication and division with exponents (T-H.6)

    • YT-6.3.2.1.b a to the p power/a to the q power = a to the (p-q) power p>q

      • Division with exponents (T-H.5)
      • Multiplication and division with exponents (T-H.6)

    • YT-6.3.2.1.c (a to the p power) to the q power = a to the pq power

      • Power rule (T-H.7)

    • YT-6.3.2.1.d a° = 1

      • Evaluate expressions using properties of exponents (T-H.8)

• YT-6.3.3 Arithmetic

  • YT-6.3.3.1 check a result by considering whether it is of the right order of magnitude and by working the problem backwards; round off a result

  • YT-6.3.3.2 make and justify estimates and approximations of calculations; calculate percentage error and tolerance

    • Greatest possible error (T-DD.3)
    • Minimum and maximum area and volume (T-DD.4)
    • Percent error (T-DD.5)
    • Percent error: area and volume (T-DD.6)

  • YT-6.3.3.3 calculate average rates of change (with respect to time)

    • Rate of travel: word problems (T-EE.2)

  • YT-6.3.3.4 solve problems involving

    • YT-6.3.3.4.a finding depreciation (reducing balance method)

      • Exponential growth and decay: word problems (T-J.4)
      • Exponential growth and decay: word problems (5Y-S.12)

    • YT-6.3.3.4.b costing: materials, labour and wastage

    • YT-6.3.3.4.c metric system; change of units; everyday imperial units (conversion factors provided for imperial units)

      • Convert rates and measurements (T-DD.1)

  • YT-6.3.3.5 make estimates of measures in the physical world around them

• YT-6.3.4 Length, area and volume

  • YT-6.3.4.1 select and use suitable strategies to find

    • YT-6.3.4.1.a length of the perimeter and the area of the following plane figures: parallelogram, trapezium, and figures made from combinations of these

      • Area of triangles and quadrilaterals (T-U.3)
      • Area and perimeter in the coordinate plane I (T-U.4)
      • Area and perimeter in the coordinate plane II (T-U.5)
      • Area of compound figures (T-U.7)
      • Area of triangles and quadrilaterals (5Y-Z.3)
      • Area and perimeter in the coordinate plane I (5Y-Z.4)
      • Area and perimeter in the coordinate plane II (5Y-Z.5)
      • Area of compound figures (5Y-Z.7)

    • YT-6.3.4.1.b surface area and volume of the following solid figures: cylinder, right cone, right prism and sphere

      • Introduction to surface area and volume (T-CC.1)
      • Surface area of prisms and cylinders (T-CC.2)
      • Surface area of pyramids and cones (T-CC.3)
      • Volume of prisms and cylinders (T-CC.4)
      • Volume of pyramids and cones (T-CC.5)
      • Surface area and volume of spheres (T-CC.6)
      • Introduction to surface area and volume (5Y-CC.1)
      • Surface area of prisms and cylinders (5Y-CC.2)
      • Surface area of pyramids and cones (5Y-CC.3)
      • Volume of prisms and cylinders (5Y-CC.4)
      • Volume of pyramids and cones (5Y-CC.5)
      • Surface area and volume of spheres (5Y-CC.6)

  • YT-6.3.4.2 use the Trapezoidal Rule to approximate area

  • YT-6.3.4.3 investigate the nets of prisms (polygonal bases), cylinders and cones

• YT-6.3.5 Synthesis and problem-solving skills

  • YT-6.3.5.1 explore patterns and formulate conjectures

  • YT-6.3.5.2 explain findings

  • YT-6.3.5.3 justify conclusions

  • YT-6.3.5.4 communicate mathematics verbally and in written form

  • YT-6.3.5.5 apply their knowledge and skills to solve problems in familiar and unfamiliar contexts

    • Congruence statements and corresponding parts (T-V.1)
    • Solve problems involving corresponding parts (T-V.2)
    • Identify congruent figures (T-V.3)
    • SSS and SAS Theorems (T-W.1)
    • ASA and AAS Theorems (T-W.2)
    • SSS, SAS, ASA and AAS Theorems (T-W.3)
    • Congruency in isosceles and equilateral triangles (T-W.5)
    • Hypotenuse-Leg Theorem (T-W.6)
    • Classify triangles (T-Y.1)
    • Triangle Angle-Sum Theorem (T-Y.2)
    • Midsegments of triangles (T-Y.3)
    • Triangles and bisectors (T-Y.4)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (T-Y.5)
    • Angle-side relationships in triangles (T-Y.6)
    • Triangle Inequality Theorem (T-Y.7)
    • Midsegments of triangles (5Y-X.1)
    • Triangles and bisectors (5Y-X.2)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (5Y-X.3)
    • Angle-side relationships in triangles (5Y-X.4)
    • Triangle Inequality Theorem (5Y-X.5)

  • YT-6.3.5.6 analyse information presented verbally and translate it into mathematical form

  • YT-6.3.5.7 devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.

    • Congruence statements and corresponding parts (T-V.1)
    • Solve problems involving corresponding parts (T-V.2)
    • Identify congruent figures (T-V.3)
    • SSS and SAS Theorems (T-W.1)
    • ASA and AAS Theorems (T-W.2)
    • SSS, SAS, ASA and AAS Theorems (T-W.3)
    • Congruency in isosceles and equilateral triangles (T-W.5)
    • Hypotenuse-Leg Theorem (T-W.6)
    • Classify triangles (T-Y.1)
    • Triangle Angle-Sum Theorem (T-Y.2)
    • Midsegments of triangles (T-Y.3)
    • Triangles and bisectors (T-Y.4)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (T-Y.5)
    • Angle-side relationships in triangles (T-Y.6)
    • Triangle Inequality Theorem (T-Y.7)
    • Midsegments of triangles (5Y-X.1)
    • Triangles and bisectors (5Y-X.2)
    • Identify medians, altitudes, angle bisectors and perpendicular bisectors (5Y-X.3)
    • Angle-side relationships in triangles (5Y-X.4)
    • Triangle Inequality Theorem (5Y-X.5)

YT-6.4 Algebra

• YT-6.4.1 Expressions

  • YT-6.4.1.1 evaluate expressions given the value of the variables

    • Evaluate variable expressions involving integers (T-B.3)
    • Evaluate variable expressions involving rational numbers (T-B.7)
    • Roots of integers (5Y-N.1)
    • Roots of rational numbers (5Y-N.2)
    • Find roots using a calculator (5Y-N.3)
    • Nth roots (5Y-N.4)
    • Evaluate rational expressions I (5Y-P.2)
    • Evaluate rational expressions II (5Y-P.3)

  • YT-6.4.1.2 expand and simplify expressions

    • Evaluate expressions using properties of exponents (T-H.8)
    • Simplify radical expressions (T-P.1)
    • Simplify radical expressions involving fractions (T-P.2)
    • Multiply radical expressions (T-P.3)
    • Add and subtract radical expressions (T-P.4)
    • Simplify radical expressions using the distributive property (T-P.5)
    • Simplify radical expressions: mixed review (T-P.6)
    • Simplify complex fractions (T-Q.2)
    • Simplify rational expressions (T-Q.3)
    • Multiply and divide rational expressions (T-Q.4)
    • Divide polynomials (T-Q.5)
    • Add and subtract rational expressions (T-Q.6)
    • Simplify radical expressions with variables I (5Y-N.5)
    • Simplify radical expressions with variables II (5Y-N.6)
    • Multiply radical expressions (5Y-N.7)
    • Divide radical expressions (5Y-N.8)
    • Add and subtract radical expressions (5Y-N.9)
    • Simplify radical expressions using the distributive property (5Y-N.10)
    • Simplify radical expressions using conjugates (5Y-N.11)
    • Simplify rational expressions (5Y-P.4)
    • Multiply and divide rational expressions (5Y-P.5)
    • Add and subtract rational expressions (5Y-P.6)

• YT-6.4.2 Solving equations

  • YT-6.4.2.1 select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to equations of the form:

    • YT-6.4.2.1.a f(x) = g(x) with f(x) = ax+b, g(x) = cx+d where a, b, c, d ∈ Q

      • Solve equations: complete the solution (T-D.7)
      • Is (x, y) a solution to the simultaneous equations? (T-G.1)
      • Solve simultaneous equations by graphing (T-G.2)
      • Solve simultaneous equations by graphing: word problems (T-G.3)
      • Find the number of solutions to simultaneous equations by graphing (T-G.4)
      • Find the number of solutions to simultaneous equations (T-G.5)
      • Solve simultaneous equations using substitution (T-G.6)
      • Solve simultaneous equations using substitution: word problems (T-G.7)
      • Solve simultaneous equations using elimination (T-G.8)
      • Solve simultaneous equations using elimination: word problems (T-G.9)
      • Is (x, y) a solution to the simultaneous equations? (5Y-G.1)
      • Solve simultaneous equations by graphing (5Y-G.2)
      • Solve simultaneous equations by graphing: word problems (5Y-G.3)
      • Find the number of solutions to simultaneous equations (5Y-G.4)
      • Solve simultaneous equations using substitution (5Y-G.5)
      • Solve simultaneous equations using substitution: word problems (5Y-G.6)
      • Solve simultaneous equations using elimination (5Y-G.7)
      • Solve simultaneous equations using elimination: word problems (5Y-G.8)
      • Solve simultaneous equations using any method (5Y-G.9)
      • Solve simultaneous equations using any method: word problems (5Y-G.10)
      • Solve simultaneous equations by graphing (6Y-I.1)
      • Solve simultaneous equations by graphing: word problems (6Y-I.2)
      • Classify simultaneous equations (6Y-I.3)
      • Solve simultaneous equations using substitution (6Y-I.4)
      • Solve simultaneous equations using substitution: word problems (6Y-I.5)
      • Solve simultaneous equations using elimination (6Y-I.6)
      • Solve simultaneous equations using elimination: word problems (6Y-I.7)

    • YT-6.4.2.1.b f(x) = 0 with f(x) = ax² + bx + c where b² ≥ 4ac; a, b, c ∈ Z and interpret the results

      • Solve a quadratic equation using square roots (T-N.3)
      • Solve a quadratic equation using the zero product property (T-N.4)
      • Solve a quadratic equation by factorising (T-N.5)
      • Complete the square (T-N.6)
      • Solve a quadratic equation by completing the square (T-N.7)
      • Solve a quadratic equation using the quadratic formula (T-N.8)
      • Using the discriminant (T-N.9)
      • Solve a quadratic equation using square roots (5Y-L.6)
      • Solve a quadratic equation using the zero product property (5Y-L.7)
      • Solve a quadratic equation by factorising (5Y-L.8)
      • Complete the square (5Y-L.9)
      • Solve a quadratic equation using the quadratic formula (5Y-L.10)
      • Using the discriminant (5Y-L.11)
      • Solve a quadratic equation using square roots (6Y-C.5)
      • Solve a quadratic equation by factorising (6Y-C.6)
      • Solve a quadratic equation by completing the square (6Y-C.7)
      • Solve a quadratic equation using the quadratic formula (6Y-C.8)
      • Using the discriminant (6Y-C.9)

• YT-6.4.3 Inequalities

  • YT-6.4.3.1 select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to inequalities of the form:

    • YT-6.4.3.1.a g(x) ≤ k, g(x) ≥ k,

      • Graph inequalities (T-E.1)
      • Write inequalities from graphs (T-E.2)
      • Identify solutions to inequalities (T-E.3)
      • Solve one-step linear inequalities: addition and subtraction (T-E.4)
      • Solve one-step linear inequalities: multiplication and division (T-E.5)
      • Solve one-step linear inequalities (T-E.6)
      • Graph solutions to one-step linear inequalities (T-E.7)
      • Solve two-step linear inequalities (T-E.8)
      • Graph solutions to two-step linear inequalities (T-E.9)
      • Solve advanced linear inequalities (T-E.10)
      • Graph solutions to advanced linear inequalities (T-E.11)
      • Graph a linear inequality in one variable (5Y-E.1)
      • Write inequalities from graphs (5Y-E.3)
      • Solve linear inequalities (5Y-E.5)
      • Graph solutions to linear inequalities (5Y-E.6)

    • YT-6.4.3.1.b g(x) < k, g(x) > k, where g(x) = ax + b and a, b, k ∈ Z

      • Graph inequalities (T-E.1)
      • Write inequalities from graphs (T-E.2)
      • Identify solutions to inequalities (T-E.3)
      • Solve one-step linear inequalities: addition and subtraction (T-E.4)
      • Solve one-step linear inequalities: multiplication and division (T-E.5)
      • Solve one-step linear inequalities (T-E.6)
      • Graph solutions to one-step linear inequalities (T-E.7)
      • Solve two-step linear inequalities (T-E.8)
      • Graph solutions to two-step linear inequalities (T-E.9)
      • Solve advanced linear inequalities (T-E.10)
      • Graph solutions to advanced linear inequalities (T-E.11)
      • Graph a linear inequality in one variable (5Y-E.1)
      • Write inequalities from graphs (5Y-E.3)
      • Solve linear inequalities (5Y-E.5)
      • Graph solutions to linear inequalities (5Y-E.6)

• YT-6.4.4 Complex Numbers

  • YT-6.4.4.1 Ireland has no content for this topic at this Level

• YT-6.4.5 Synthesis and problem-solving skills

  • YT-6.4.5.1 explore patterns and formulate conjectures

    • Identify arithmetic and geometric sequences (T-K.1)
    • Arithmetic sequences (T-K.2)
    • Geometric sequences (T-K.3)
    • Evaluate variable expressions for number sequences (T-K.4)
    • Write variable expressions for arithmetic sequences (T-K.5)
    • Write variable expressions for geometric sequences (T-K.6)
    • Number sequences: mixed review (T-K.7)
    • Classify formulas and sequences (5Y-EE.1)
    • Find terms of an arithmetic sequence (5Y-EE.2)
    • Find terms of a geometric sequence (5Y-EE.3)
    • Evaluate formulas for sequences (5Y-EE.5)
    • Write a formula for an arithmetic sequence (5Y-EE.6)
    • Write a formula for a geometric sequence (5Y-EE.7)
    • Sequences: mixed review (5Y-EE.9)

  • YT-6.4.5.2 explain findings

  • YT-6.4.5.3 justify conclusions

  • YT-6.4.5.4 communicate mathematics verbally and in written form

  • YT-6.4.5.5 apply their knowledge and skills to solve problems in familiar and unfamiliar contexts

    • Solve simultaneous equations by graphing: word problems (T-G.3)
    • Solve simultaneous equations using substitution: word problems (T-G.7)
    • Solve simultaneous equations using elimination: word problems (T-G.9)
    • Solve simultaneous equations by graphing: word problems (5Y-G.3)
    • Solve simultaneous equations using substitution: word problems (5Y-G.6)
    • Solve simultaneous equations using elimination: word problems (5Y-G.8)
    • Solve simultaneous equations using any method: word problems (5Y-G.10)

  • YT-6.4.5.6 analyse information presented verbally and translate it into mathematical form

    • Solve linear equations: word problems (T-D.10)
    • Write linear functions to solve word problems (T-F.11)
    • Solve simultaneous equations by graphing: word problems (T-G.3)
    • Solve simultaneous equations using substitution: word problems (T-G.7)
    • Solve simultaneous equations using elimination: word problems (T-G.9)
    • Solve linear equations: word problems (5Y-D.2)
    • Write a linear inequality: word problems (5Y-E.4)
    • Solve simultaneous equations by graphing: word problems (5Y-G.3)
    • Solve simultaneous equations using substitution: word problems (5Y-G.6)
    • Solve simultaneous equations using elimination: word problems (5Y-G.8)
    • Solve simultaneous equations using any method: word problems (5Y-G.10)
    • Exponential growth and decay: word problems (5Y-S.12)

  • YT-6.4.5.7 devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.

YT-6.5 Functions

• YT-6.5.1 Functions

  • YT-6.5.1.1 recognise that a function assigns a unique output to a given input

    • Identify linear functions (T-F.1)
    • Identify functions (5Y-F.2)
    • Find values using function graphs (5Y-F.4)
    • Complete a table for a function graph (5Y-F.5)
    • Identify functions (6Y-A.2)
    • Evaluate functions (6Y-A.7)
    • Find values using function graphs (6Y-A.8)
    • Complete a table for a function graph (6Y-A.9)

  • YT-6.5.1.2 graph functions of the form

    • YT-6.5.1.2.a ax where a ∈ Q, x ∈ R

      • Find the slope of a linear function (6Y-A.3)

    • YT-6.5.1.2.b ax+b where a, b ∈ Q, x ∈ R

      • Slope-intercept form: graph an equation (T-F.6)
      • Complete a table and graph a linear function (T-F.12)
      • Graph a horizontal or vertical line (T-F.18)
      • Graph a linear function (5Y-F.7)
      • Find the slope of a linear function (6Y-A.3)
      • Graph a linear function (6Y-A.4)

    • YT-6.5.1.2.c ax² + bx + c where a, b, c ∈ Z, x ∈ R

      • Graph a quadratic function (5Y-L.4)
      • Graph a quadratic function (6Y-C.3)
      • Match quadratic functions and graphs (6Y-C.4)

    • YT-6.5.1.2.d a2^x where a ∈ N, x ∈ R

      • Match exponential functions and graphs (T-J.2)
      • Match exponential functions and graphs (5Y-S.3)
      • Domain and range of exponential and logarithmic functions (6Y-F.1)
      • Identify linear and exponential functions (6Y-F.13)
      • Exponential functions over unit intervals (6Y-F.14)

    • YT-6.5.1.2.e a3^x where a ∈ N, x ∈ R

      • Match exponential functions and graphs (T-J.2)
      • Match exponential functions and graphs (5Y-S.3)
      • Domain and range of exponential and logarithmic functions (6Y-F.1)
      • Identify linear and exponential functions (6Y-F.13)
      • Exponential functions over unit intervals (6Y-F.14)

  • YT-6.5.1.3 interpret equations of the form f(x) = g(x) as a comparison of the above functions

    • Classify simultaneous equations (6Y-I.3)

  • YT-6.5.1.4 use graphical methods to find approximate solutions to f(x) = 0, f(x) = k, f(x) = g(x) where f(x) and g(x) are of the above form

    • Solve simultaneous equations by graphing (6Y-I.1)
    • Solve simultaneous equations by graphing: word problems (6Y-I.2)

• YT-6.5.2 Calculus

  • YT-6.5.2.1 Ireland has no content for this topic at this Level

• YT-6.5.3 Synthesis and problem-solving skills

  • YT-6.5.3.1 explore patterns and formulate conjectures

  • YT-6.5.3.2 explain findings

  • YT-6.5.3.3 justify conclusions

  • YT-6.5.3.4 communicate mathematics verbally and in written form

    • Identify linear functions (T-F.1)
    • Identify functions (5Y-F.2)

  • YT-6.5.3.5 apply their knowledge and skills to solve problems in familiar and unfamiliar contexts

    • Solve simultaneous equations using substitution: word problems (T-G.7)
    • Solve simultaneous equations using elimination: word problems (T-G.9)
    • Solve simultaneous equations using substitution: word problems (5Y-G.6)
    • Solve simultaneous equations using elimination: word problems (5Y-G.8)
    • Solve simultaneous equations using any method: word problems (5Y-G.10)

  • YT-6.5.3.6 analyse information presented verbally and translate it into mathematical form

    • Solve simultaneous equations using substitution: word problems (T-G.7)
    • Solve simultaneous equations using elimination: word problems (T-G.9)
    • Solve simultaneous equations using substitution: word problems (5Y-G.6)
    • Solve simultaneous equations using elimination: word problems (5Y-G.8)
    • Solve simultaneous equations using any method: word problems (5Y-G.10)

  • YT-6.5.3.7 devise, select and use appropriate mathematical models, formulae or techniques to process information and to draw relevant conclusions.