How to learn Tercer ciclo (5.º–6.º): Mathematics

Real Decreto 157/2022, enseñanzas mínimas de la Educación Primaria: Matemáticas (Mathematics), saberes básicos, Tercer ciclo (5.º–6.º)

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1. A. Sentido numérico · 2. Cantidad

Order of operations (age 10+): Look for and use mathematical structure: exploit the hierarchy of 2-D shapes to deduce properties; use order of operations and algebraic structure to simplify expressions; connect fraction–decimal–percentage equivalences; use ratio structure to solve proportion problems efficiently
Estimation to check answers to calculations: Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy
Reading and writing numbers (age 10+): Read, write, and compare decimals to thousandths using base-ten numerals, number names, and expanded form; compare using >, =, < based on place-value meaning
Percentages as Fractions: Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100%

Ratio (age 10+): Perform mental calculations including with mixed operations and large numbers, using strategies such as partitioning, compensation, and derived facts
Multi-step problems: choosing operations: Solve problems involving addition, subtraction, multiplication, and division, deciding which operations and methods to use and why; solve multi-step problems in contexts
Square and cube numbers: Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations
Fractions, Decimals & Percentages: Look for and use mathematical structure: exploit the relationship between fractions, decimals, and percentages; use factor pairs to simplify multiplication; apply angle facts to find unknowns; use properties of regular polygons systematically

Tenths (age 9+): Recognise and use thousandths; relate them to tenths, hundredths, and their decimal equivalents (e.g. 1/1000 = 0.001, 35/1000 = 0.035)
Decimals and fractions (age 10+): Associate a fraction with division and calculate decimal fraction equivalents for simple fractions (e.g. 3/8 = 0.375); recall and use equivalences between simple fractions, decimals, and percentages in different contexts
Using inverse operations: Recognise and use relationships between operations including inverse operations; use these relationships to check answers and simplify calculations

Proportional Reasoning Vocabulary: Know and use advanced vocabulary of multiplicative reasoning — direct proportion, inverse proportion, ratio, rate, unit rate, compound unit, scale factor — accurately in problem-solving contexts
Proportion: Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations

Unit Conversions: Convert freely between related standard units of measurement (time, length, area, volume/capacity, mass) using decimal notation to up to three decimal places where appropriate
Compound Units: Use compound units such as speed (distance ÷ time), unit pricing (cost ÷ quantity), and density (mass ÷ volume); solve problems involving compound units

Measuring length (age 7+): Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, metre sticks, and measuring tapes

Decimal place value: Convert among different-sized standard measurement units within a given system (e.g. 5 cm to 0.05 m) using decimal notation to up to three decimal places; convert between smaller and larger units of length, mass, volume, and time
Measurement Conversions: Solve problems involving the calculation and conversion of units of measure, using decimal notation and multi-step reasoning in real-world contexts
Estimating Angles: Estimate and compare acute, obtuse, and reflex angles in degrees; classify angles by type and order them by size

Classifying shapes by properties: Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties
Transformations on a grid: Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size

Scale and similar shapes (age 11+): Use scale factors to interpret and create scale diagrams and maps, calculating real-life distances from map measurements and vice versa
Plotting points in the first quadrant: Plot and read ordered pairs in the first quadrant of the coordinate plane; represent real-world and mathematical problems by graphing points and interpreting coordinate values in context

Coordinate Transformations: Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original
Shape patterns: Generate a number or shape pattern that follows a given rule; identify apparent features of the pattern not explicit in the rule and explain informally why they occur

Using Simple Formulae: Use simple formulae expressed in words or symbols to calculate values (e.g. perimeter = 2 × (length + width))

Real-World Maths Modelling: Model real-world problems involving scaling, unit conversion, area/perimeter, and percentage by selecting appropriate mathematical representations and interpreting results in context

Expressions & Equations Vocabulary: Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors; distinguish between an expression (no equals sign), an equation (equals sign), and an inequality (inequality sign)

Experimental probability: Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability
Pictograms and tally charts (age 11+): Construct and interpret frequency tables, bar charts, pie charts, pictograms, and vertical line charts for both categorical and grouped numerical data, choosing appropriate representations for the data type
Comparing measurements: Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers