Mathematics

Skills

• Recognise and name nine- and ten-digit numbers in base 10
• Develop a sense of flexible number composition by solving problems involving time of day and quantities of time
• Use commutativity, associativity and distributivity to solve calculations efficiently
• Use the definitions of factors and multiples to find common factors and multiples 
• Express an integer as a product of its factors 
• Conjecture and make generalised statements e.g.: 
  • Square numbers cannot be prime 
  • The common multiples of 5 and 4 are always multiples of 20 
• Solve problems involving factors and multiples in unfamiliar contexts 
• Form written calculations, function machines and worded descriptions correctly embedding the order of operations 
• Form and identify equivalent calculations based on distributivity, commutativity and the order of operations 
• Form and interpret expressions involving variables

Knowledge

• Understand the values of and relationship between of different place value columns in base 10
• Understand a range of notation for quantities of time and time of day
• Have an awareness of different numerical systems and their representations
• Make use of and generalise the commutative, associative and distributive properties
• Compare and contrast scaling, area, repeated addition and grouping/sharing models for multiplication and division
• Develop number sense and efficient calculation strategies 
• Make links between efficient calculation strategies and the axioms 
• Understand the terms factor and multiple; recognise and define prime, square and cube numbers 
• Interpret and create representations of integers that reveal their structure 
• Understand the equal priority of addition with subtraction and multiplication with division in written calculations 
• Understand that operations of equal priority can be evaluated in any order 
• Understand that written calculations follow rules of ‘syntax’ determining the order of operations 
• Understand the higher priority of multiplication with division over addition with subtraction in written calculations
• Interpret the order of operations from written calculations, function machines and worded descriptions 

Rationale

This module is fundamentally about asking students to think more deeply about the number system, and the structures of numbers in general. 

The module begins with the ‘Numbers and Numerals’ unit, in which learners experience a range of number and numeral systems to develop their understanding of the base 10 place value system from primary years.

In ‘Axioms and Arrays,’ learners develop their understanding of different models for multiplication and division. Learners also explore the axioms of number and which operations they can be applied to. Generalisations such as algebra are introduced to communicate and explain mathematical ideas.

Learners will then be introduced to factors, multiples and important sets of numbers such as prime numbers, square numbers and cube numbers. Once the fundamental concepts have been introduced students are given the opportunity to develop their understanding, conjecture, problem solve and generalise in a series of structured tasks.

Lastly, learners develop an explicit understanding of the order of operations based on the equal priority of addition with subtraction and multiplication with division. They will understand the higher priority of multiplication with division in written calculations and understand the use of brackets and vincula to manipulate this priority order.

Skills

• Draw and measure acute and obtuse angles reliable to the nearest degree 
• Estimate the size of a given angle 
• Use angle facts: angles at a point, angles at a point on a straight line, vertically opposite angles to find missing angles
• Generalisations and reason with unfamiliar angle situations
• Use angle facts around corresponding, alternate and co-interior angles to find missing angles 
• Find unknown angles. Form algebraic expressions. Solve for unknowns.
• Know and use the angle sum of triangles and quadrilaterals 
• Generalise results for properties of special types of triangles and quadrilaterals 
• Form and solve equations from contexts arising from properties of triangles and quadrilaterals 
• Construct triangles and quadrilaterals for given conditions using ruler, protractor and compasses 
• Recognise when two triangles are congruent 

 Knowledge

• Understand an angle as a measure of turn, and identify the features that determine the size of an angle
• Know the classification of different types of angles
• Recall and derive basic angle facts
• Define parallel and perpendicular lines 
• Classifying polygons by symmetry, regularity, intersection of diagonals, number of parallel sides 
• Classify triangles and quadrilaterals according to properties (angles, regularity, symmetry) 
• Explore constructions through use of dynamic geometry software 
• Become familiar with the different cases of minimum conditions for the construction of triangles 

 Rationale

This module has two purposes. First is to expose learners to a range of new knowledge and skills in 2-D geometry, which will provide the foundation for further work in Years 8 and 9. Second, to provide more contexts for students to practice the algebraic reasoning developed in Autumn term.

The first unit covers angles. Learners describe, classify and identify types of angles using clear vocabulary, and measure and draw angles accurately. Learners revise facts involving angles from experiences in primary school. Learners explore and clarify definitions of parallel lines and perpendicular lines, and use rules around corresponding, alternate and co-interior angles. Students being to formulate equations to show relationships between angles using the angle facts. 

Next learners, study the geometric properties of polygons before focusing more closely on triangles and quadrilaterals. Learners are encouraged to conjecture and prove. Special types of triangles and quadrilaterals are defined. Generalisations are expressed algebraically, and learners set up and solve equations involving special types of triangle and quadrilaterals.

Lastly, students construct triangles and quadrilaterals. The unit begins by introducing the anatomy of a circle. The minimum conditions for constructing triangles are explored and thereby informally introducing congruence. Students will construct/draw triangles using initial information and conversely will deduce properties of triangles from completed constructions. 

 Skills

• Read and write coordinates in all four quadrants, including non-integer coordinates 
• Solving geometric problems involving missing coordinates
• Finding the mid-point of a line segment or two points 
• Recognise and plot horizontal and vertical lines on a coordinate axis
• Finding the area of rectilinear shapes  
• Finding the area of 2-D shapes including triangles, and special quadrilaterals  
• Rearrange formulae to make a different subject 
• Reflection of an object in a mirror line
• Identify horizontal and vertical mirror lines and their equations 
• Rotation of an object using the centre of rotation 
• Translating shapes by a given number of units (positive or negative)
• Combining transformations and which combinations can be expressed as a single transformation 
• Simple enlargements with positive scale factors 

 Knowledge

• Understand the 2-D coordinate system and start to understand its continuous nature.
• Use the language of horizontal and vertical lines
• Know that area is the 2D space inside a shape
• Develop understanding of counting strategies in arrays to using similar strategies to calculate the area of shapes
• Generalise formulae for finding the area of 2-D shapes using the language of height, base, width, length etc. 
• Reason about generalised statements of the relationship between area and perimeter 
• Exploring the ratios of sides lengths within and between shapes produced by an object being enlarged by a given scale factor 
• Use the language of congruence and recognise which transformations produce congruent shapes 

 Rationale

In this module, learners start to apply what they have learned about algebra and geometry in the Cartesian grid. 

First learners study coordinates in all four quadrants. Learners will be familiar with coordinates from work at primary school and in other subjects. The tasks in this unit will give learners opportunities to apply their understanding from previous units including negative numbers and geometric properties of triangles and quadrilaterals. 

Next, the concept of area as a measurable quantity is introduced. This begins by revisiting arrays. Reasoning about calculating the area of shapes is built up by decomposing shapes and connecting to existing knowledge. The generalised expressions for finding the area of shapes are used to introduce rearranging formulae.

Lastly, learners are expected to consider how transformations acting on an object produce different images. Reflection and rotation are introduced through previous experience of line and rotational symmetry.  There is a focus on language and consideration of the amount of information required to perform each transformation. The transformations are sorted by whether they result in a congruent image. This is the foundation for exploring similarity and trigonometry later.