--- title: "Maths and Numeracy Toolkit" url: "https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy/toolkit" school: "Bishop Hogarth Catholic Education Trust" --- Admissions Consultation BHCET schools [Click Here](https://bhcet.org.uk/admissions-consultation) # Maths and Numeracy Toolkit Foundational Knowledge & Skills ## Maths & Numeracy Toolkit Detailed guidance to support the teaching and development of foundational Maths and Numeracy knowledge across Years 7, 8 and 9. **Jump to a year group**Year 7 Year 8 Year 9 [Back to Maths & Numeracy Overview](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) ## Year 7 Foundations Number ### Understand and use place value accurately #### Description Place value refers to the value of a digit depending on its position in a number (e.g. ones, tens, hundreds, tenths). - Recognise the value of each digit in whole and decimal numbers - Read, write, and interpret numbers correctly - Compare and order numbers using understanding of place value #### Cross-Curricular Examples - Geography: Interpreting population figures (e.g. 2.3 million vs 230,000) - History: Ordering dates and chronological order - Technology: Reading measurements in design specifications - Art: Converting measurements and scaling dimensions accurately #### Teaching Model **Teacher Modelling** - Use place value grids and place value counters to break down numbers and what each digit represents - Use correct vocabulary and language, especially after the decimal point (0.47 is nought point four seven, NOT nought point forty seven) **Student Practice** - Identify the value of a digit in a number - Place a set of numbers in ascending/descending order - Place a number at the appropriate position on a number line **Scaffolding Strategies** - Printed place value grids - Printed digit cards **Rehearsal** - Students explain numbers verbally and in written form **Aim:** Students can interpret and represent numbers confidently. They can compare and order numbers accurately. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Number ### Develop fluency with the four operations #### Description Students perform addition, subtraction, multiplication, and division accurately and efficiently, including in problem solving contexts. Fluency includes: - Recall of key facts (times tables, number bonds) - Choosing appropriate operations - Performing written and mental methods #### Cross-Curricular Examples - Geography: Calculating appropriate averages - PE: Totalling scores and finding averages - Science: Calculating results (e.g. totals, differences, averages) - Computer Science: Arithmetic in algorithms #### Teaching Model **Teacher Modelling** - Demonstrate efficient written and mental methods of multiplication, division, addition and subtraction – see school methods’ policy for preferred/consistent methods **Student Practice** - Mixed-operation questions increasing in complexity - Blocked practice of each skill - routine/drill to secure the skill for each operation - Questions with subject specific context **Scaffolding Strategies** - Provide blank, pre-drawn grids - Partly completed solutions for pupils to fill in the blanks- backwards fading **Rehearsal** - Students explain why a method is appropriate - Use of mini whiteboards - Spaced practice – use of starters and other assessment - Subject specific exam practice **Aim:** Students can perform calculations accurately and efficiently and apply operations confidently in different contexts. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Number ### Complete calculations in the correct order (BIDMAS) #### Description BIDMAS refers to the order in which mathematical operations are performed: Brackets, Indices, Division/Multiplication (which have the same weighting and should be performed in the order they are seen), Addition/Subtraction (which have the same weighting and should be performed in the order they are seen). - Follow the correct sequence when solving calculations - Recognise that changing the order changes outcomes - Apply the operations to make a number sentence produce the given answer #### Cross-Curricular Examples - Science: Substituting into any formula - Technology: Multi-step calculations in design - Geography: Compound calculations (e.g. averages) - Other subjects: Any structured multi-step problem #### Teaching Model **Teacher Modelling** - Model each stage of working, highlighting which operation you are using each time - Write each line of working explicitly until you get to an answer - Multiple examples will be required, including those where subtraction appears before addition and multiplication appears before division **Student Practice** - Blocked practice of increasing difficult problems - Mixed practice with a variety of difficulties - Substitution into formulae questions - Subject specific questions **Scaffolding Strategies** - BIDMAS visual reminders - Partly completed solutions for pupils to fill in the blanks **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can evaluate expressions accurately and avoid common sequencing errors. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Number ### Understand and calculate with directed numbers #### Description Directed numbers are values above (positive) and below (negative) zero. - Place directed numbers on a number line - Perform calculations, involving positive and negative numbers and all four operations - Understand real-world contexts (e.g. temperature) #### Cross-Curricular Examples - Geography: Temperatures below zero - Science: Charge or energy changes - PE: Score differences - Other subjects: Any context involving deficits or decreases #### Teaching Model **Teacher Modelling** - Use number lines and double sided counters to demonstrate calculations involving addition and subtraction - Use double sided counters to demonstrate calculations involving multiplication and division - Be explicit with use of number lines in particular – make sure pupils draw them for each question **Student Practice** - Blocked practice of each skill, focusing solely on the calculations - Mixed practice of each skill - Subject specific contextual problems (temperature, scoring) **Scaffolding Strategies** - Visual representations (number lines, double sided counters) - Printed and laminated number lines for students to use - Partly completed solutions for pupils to fill in the blanks **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can accurately calculate with directed numbers and apply understanding in real-life contexts. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Algebra ### Understand the meaning of substitution #### Description Substitution involves replacing variables (letters) with numbers and then completing the resulting calculations. - Identify variables - Replace them correctly - Evaluate expressions and formulae #### Cross-Curricular Examples - Science: Substituting into formulae - Technology: Inputs into calculations - Computer Science: Variables in code - PE: Replacing one player on a team for another – helps with mathematical definition when seen/discuss in other areas #### Teaching Model **Model** - Model each stage of working, clearly highlighting which variable you are substituting each time - Write each line of working explicitly until you get to a solution - Multiple examples will be required, including those where subtraction appears before addition and multiplication appears before division **Student Practice** - Basic questions substituting one variable - Gradually increase difficulty by adding multiple operations - Continue to increase difficulty by adding multiple variables - Subject specific substituting into formulae **Scaffolding Strategies** - Use symbols (hearts, smiley faces etc) as variables - Use physical representations such counters to represent variables and physically move them to substitute with numbers - Partly completed calculations to allow pupils to fill in the blanks **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can substitute into algebraic expressions in different subjects. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Algebra ### Write and simplify algebraic sentences #### Description Writing algebraic sentences refers to writing a sentence or instruction using variables (letters) rather than words. Simplifying is combining like terms (e.g. 2x + 3x = 5x). #### Cross-Curricular Examples - Science: Expressing relationships between variables - Computer Science: Symbolic representation - Other subjects: Simplifying a paragraph into key points #### Teaching Model **Teacher Modelling** - Assign variables a letter - Simplify wording using mathematical symbols (eg and = +) - Collect like terms by highlighting each variable and the sign before it - Show each line of working out clearly **Student Practice** - Matching tasks - Blocked practice of single variables - Blocked practice of multiple variables - Mixed practice involving mixed variables **Scaffolding Strategies** - Underline variables - Colour-code like terms - Partly completed solutions where pupils fill in blanks **Rehearsal** - Students verbalise the translation process with their partner or teacher - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers **Aim:** Students can represent problems algebraically and simplify expressions accurately. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Statistics ### Accurately draw a range of charts and representations #### Description Students create graphs (bar charts, pie charts, line graphs) accurately using correct scales and labels. #### Cross-Curricular Examples - Geography: Climate graphs - PE: Fitness data - Psychology: Survey data - Science: Data representation and experiment results #### Teaching Model **Teacher Modelling** - Using visualiser, construct charts step-by-step (see departmental methods policy for more detail) - Be explicit with each step - Label axes and keys clearly where appropriate **Student Practice** - Create charts from given data - Collect data in class and create appropriate charts - Subject specific questions **Scaffolding Strategies** - Templates – axes printed - Printed checklists with step by step process - Partly completed examples that pupils complete **Rehearsal** - Students explain what their chart shows and how they created it verbally - Students practice choosing appropriate graphs for the data they have - Regular review with questions interleaved into other areas of the curriculum - Subject specific exit tickets **Aim:** Students can present data clearly and accurately and select appropriate chart types. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Statistics ### Understand the importance of totals (mean) #### Description The mean is the total amount shared equally between a number of people or things. Pupils often know this as a process ‘adding them up and dividing by how many there are’. While this is the process of calculating mean, it is important that pupils know that they are finding the total amount in order to share it evenly. - Calculate totals correctly - Understand effect of values on averages – ie if you add another value that is significantly larger than the current mean, the total increases and so does the mean #### Cross-Curricular Examples - Science: Calculating mean from a set of data from an experiment - Geography: Mean temperatures - PE: Average performance eg batting averages in cricket - All subjects: Average scores in assessments #### Teaching Model **Teacher Modelling** - Clearly demonstrate calculating the total amount (of runs scored/temperatures taken) - Divide the total amount by ‘how many there are’ (ie number of innings the batsman has had/number of days the temperature was recorded) - Explicitly explain that the value represents that this is what, if everything was evenly spread, would be the number of runs in each innings/the temperature on each day etc. - Emphasise that the mean can be a decimal **Student Practice** - Use real datasets to ground practice in real life context - Collect data such as height from each other and find real class averages **Scaffold** - Step-by-step calculation prompts - Use of physical representations such as counters to show how the total is divided equally **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can calculate the mean reliably and interpret its meaning in context. They understand the importance of calculating the total and that the mean means the total has been shared equally. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Proportion and Rates of Change ### Understand what is meant by a percentage #### Description A percentage is a value that has scaled to be out of 100. Using percentages allows a more efficient way of comparing data than, for example, a fraction. - Convert between fractions, decimals, percentages - Interpret percentages in context #### Cross-Curricular Examples - Geography: Population percentages - Science: Concentrations - PE: Success rates - All subjects: Scores in assessments #### Teaching Model **Teacher Modelling** - Without a calculator: Use equivalent fractions to convert to a fraction with a denominator of 100 - Once out of 100, the numerator is the percentage - With a calculator: Convert from a fraction to a decimal (numerator divided by denominator) - Convert from a decimal to a percentage by multiplying by 100 **Student Practice** - Convert between fractions, decimals, percentages in subject context - Use percentage conversion to compare statistics in context - Convert scores in assessments **Scaffolding Strategies** - Partly completed solutions that pupils can fill in the blanks **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can interpret percentages confidently and convert to percentages to allow more efficient comparisons. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Geometry ### Understand and apply properties of 2D and 3D shapes #### Description Students identify shapes based on their properties (sides, angles, faces). #### Cross-Curricular Examples - Art: Shape in design - Technology: 3D models - Science: Molecular shapes #### Teaching Model **Teaching Model** - Identify and describe properties explicitly **Student Practice** - Sort and classify shapes based on their properties - Matching activities - Describing shapes to partners using only properties **Scaffolding Strategies** - Property tables and diagrams - Partially completed tables for students to fill in the blanks **Rehearsal** - Students justify classifications verbally - Regularly interleave questions on shape properties **Aim:** Students can classify shapes accurately and use the correct vocabulary associated to their properties. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Numerical Communication ### Show organised working out #### Description Students present clear, step-by-step calculations. This includes: - Logical sequencing - Legible layout #### Cross-Curricular Examples - All subjects: Pupils organise any mathematical working out in a structured and ordered manner #### Teaching Model **Teacher Modelling** - Always provide clear, structured examples using a visualiser where possible - Ensure pupils follow your clear modelling - Write working out line by line, working down the page - Highlight answers clearly **Student Practice** - Subject specific examples - Teachers consistently remind students of expectations of working out **Scaffolding Strategies** - Provide writing frames (step-by-step structure) - Partially completed written solutions that pupils complete **Rehearsal** - Students explain their method - Pupils can talk staff and pupils through the stages of working out **Aim:** Students can present solutions clearly and make their thinking easy to follow. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Calculator Skills ### Begin to use a calculator successfully #### Description Students do not use calculators in primary school. It is important to remember this as we try to develop their skills in using them across different subjects. #### Cross-Curricular Examples - Science: Calculations in experiments - Geography: Data handling - Technology: Measurements and calculations #### Teaching Model **Teacher Modelling** - Demonstrate correct input using visualiser - Be explicit with each input - Be aware of different types of calculators and their functionality (see maths staff for advice) **Student Practice** - Simple guided calculations - Subject specific calculations **Scaffolding Strategies** - Step-by-step prompts - Keep returning to live modelling whenever appropriate **Rehearsal** - Students explain what they input and why - Calculators used at every opportunity in lessons **Aim:** Students can use a calculator reliably for basic tasks and develop confidence in checking answers. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) ## Year 8 Development Number ### Understand fractions and develop fluency with the four operations with fractions #### Description Fractions represent parts of a whole and can be written as a numerator over a denominator. - Understand equivalent fractions - Add, subtract, multiply, and divide fractions - Apply fraction operations in context #### Cross-Curricular Examples - Geography: Interpreting proportions of land use - History: Analysing fractions of populations affected by events - Technology: Measuring materials (e.g. ¾ length) - Art: Scaling designs proportionally - Music: Fractional note values (e.g. half, quarter notes) - English: Analysing statistics in texts - PE: Recording performance ratios - Biology/Chemistry/Physics: Ratios in experiments #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Addition/Subtraction: Convert to common denominators - Addition/Subtraction: Denominator stays the same - Addition/Subtraction: Add or subtract the numerators - Addition/Subtraction: Simplify answer if possible - Multiplication: Do not need common denominators (common misconception) - Multiplication: Multiply numerators - Multiplication: Multiply denominators - Multiplication: Simplify answer if possible - Division: Do not need common denominators (common misconception) - Division: Multiply first fraction by the reciprocal of the second fraction - Division: Simplify answer if possible **Student Practice** - Blocked practice of each skill - Interleaved practice of all skills - Subject specific practice including problems in context **Scaffolding Strategies** - Visual models (fraction bars, diagrams) - Physical representations - Partially completed solutions that pupils complete - Visual reminders of processes **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can accurately calculate with fractions in different contexts, select the correct operation independently and explain their reasoning clearly. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Algebra ### Re-arrange formulae #### Description Rearranging formulae involves changing the subject of a formula by using inverse operations. #### Cross-Curricular Examples - Science: Rearranging scientific (e.g. speed = distance/time) #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Identify inverse operations and their order - Show clearly that the formula remains balanced by performing the same operation to each side of the equals sign **Student Practice** - Simple, one step formulae to emphasise use of inverse operations - Increase difficulty of practice by introducing formulae with more operations - Further practice with more complex operations such as square roots - Use subject specific formulae **Scaffolding Strategies** - Physical and visual representations (use pictures rather than letters for variables) - Visual reminders to always balance the formulae and use inverse operations - Partially completed solutions for pupils to complete **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can confidently change the subject of a formula across multiple subjects. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Algebra ### Solve equations using inverse operations #### Description Solving equations means finding the value of the unknown variable(s). Inverse operations are opposite operations (e.g. addition ↔ subtraction). #### Cross-Curricular Examples - Science: Solving formula-based problems - Technology: Calculating unknown values - Computer Science: Debugging variable outputs #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Identify inverse operations and their order - Show clearly that the equation remains balanced by performing the same operation to each side of the equals sign **Student Practice** - Blocked practice of one step equations - Blocked practice of multi-step equations - Interleaved practice of a mix of equations - Subject specific practice including problems in context **Scaffolding Strategies** - Visual models such as function machines - Physical representations such as algebra tiles - Partially completed solutions that pupils complete - Visual reminders to always balance the equations and use inverse operations **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can solve equations independently across multiple subjects. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Statistics ### Use averages and measures of spread to analyse and interpret data #### Description Averages include mean, median, and mode. Different averages are more suitable for comparing particular data. Measures of spread (range and interquartile range) show variation in data and how consistent it is. #### Cross-Curricular Examples - Geography: Comparing climate data - Science: Analysing experimental results - PE: Evaluating performance consistency - Psychology: Interpreting study results #### Teaching Model **Teacher Modelling** - Step-by-step worked examples of appropriate averages and measures of spread - Mean – see Y7 statistics for detailed information - Mode – most common value in a list - Median – middle value in an ordered list - Range – distance between biggest and smallest value (biggest – smallest) - Model using averages and range to compare datasets - Explicitly explain when one average is more suitable than another (median may be a more suitable average to use than the mean if an outlier increases or decreases the mean superficially) **Student Practice** - Blocked practice of each skill - Interleaved practice of all skills - Subject specific practice including problems in context - Practice on comparing distributions using averages for justification - Practice to include pupils choosing a particular average to compare data and justifying why **Scaffolding Strategies** - Use of counters to support calculating the mean - Physical representations or datasets to help pupils sort into order - Partially completed solutions that pupils complete - Visual reminders of processes for calculating each average and measure of spread **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems and justifications on choice of average with partners or teacher **Aim:** Students can calculate averages and range and use them to compare datasets. They can also choose the most appropriate average to use for comparing a dataset. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Proportion and Rates of Change ### Learn how to scale up and scale down using multiplicative relationships #### Description Scaling involves multiplying or dividing quantities while maintaining proportional relationships. For example, if 5 t-shirts cost £40, 15 of the same t-shirts would cost £120. #### Cross-Curricular Examples - Geography: Map scales - Technology: Model making and use of scales - Art: Enlarging/reducing images #### Teaching Model **Teacher Modelling** - Step-by-step worked examples, highlighting that both values need to increase or decrease by the same scale factor (eg multiply both by 4 or divide both by 9 etc) - Explicitly show both values increasing or decreasing by the same scale factor - Use subject specific examples – recipes/maps/percentages/prices **Student Practice** - Practice scaling numbers in more familiar examples such as the price of 1 sweet is X so how much would 8 sweets cost - Move to more subject specific practice but keep referring back to scenarios pupils are more familiar with **Scaffolding Strategies** - Visual models – use pictures to show values increasing/decreasing - Physical representations of problems using counters and cards - Partially completed solutions that pupils complete - Visual reminders of processes and examples **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can apply scaling across subjects and maintain proportional relationships. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Geometry ### Understand and apply angle facts #### Description Angle facts include: - Angles on a straight line = 180° - Angles around a point = 360° - Vertically opposite angles are equal - Angles in triangles = 180° - Students should apply these rules to solve problems. #### Cross-Curricular Examples - Technology: Designing structures - Art: Perspective drawing - Physics: Angles in forces or reflection #### Teaching Model **Teacher Modelling** - Step-by-step worked examples of relevant angle fact and finding unknown angles - Highlight the importance of writing the angle fact as part of their reasoning **Student Practice** - Blocked practice of each relevant angle fact - Interleaved practice of all relevant angle facts - Subject specific practice including problems in context **Scaffolding Strategies** - Partially completed solutions that pupils complete - Visual reminders of angle facts **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can solve angle problems with justification. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Numerical Communication ### Present accurate, ordered working out #### Description Students present solutions logically with: - Correct sequencing - Clear notation - Answers clearly highlighted #### Cross-Curricular Examples - Science: Recording experiments and solving problems clearly - Technology: Design calculations - Computer Science: Structured algorithms #### Teaching Model **Teacher Modelling** - All examples should be modelled in a way we would expect students to **Student Practice** - Accurate, ordered working out is expected in all student practice **Scaffolding Strategies** - Provide templates - Partially completed solutions, set out correctly, that pupils complete - Visual reminders of expectations **Aim:** Students communicate solutions clearly and produce work that is easy to follow and assess. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Calculator Skills ### Successfully use a calculator, knowing how to check answers are reasonable #### Description Students should know how to use their calculator efficiently, but also check that the output they get is reasonable using non calculator estimation. #### Cross-Curricular Examples - Science: Verifying results - Geography: Checking data calculations - Technology: Measurement calculations #### Teaching Model **Teacher Modelling** - Estimate answers first by rounding to 1 significant figure and doing a mental calculation - Use visualiser to model calculator input - Check how reasonable calculator answer is by comparing to estimate **Student Practice** - Variety of different calculations to complete - Paired work on estimation - Pupils check each other’s calculations and estimates **Scaffolding Strategies** - Provide estimates for initial calculations and gradually reduce - Partially completed estimations that pupils complete - Visual reminders to estimate → calculate → check **Rehearsal** - Verbalise solutions to problems with partners or teacher **Aim:** Students use calculators effectively and critically evaluate how appropriate their answers are. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) ## Year 9 Application Number ### Apply knowledge of number topics in contexts including assessing how reasonable an answer is #### Description Students draw on number skills (operations, fractions, decimals, percentages) in real-world contexts. This requires them to select appropriate methods and solve multi-step problems then evaluate whether answers are realistic. #### Cross-Curricular Examples - Geography: Population growth calculations - Science: Applying formulae in experiments - Business/Law: Financial calculations - PE: Performance tracking #### Teaching Model **Teacher Modelling** - Model multi-step contextual problems - Be explicit with your choice of operation and thought process behind the order in which you tackle each problem - At the end of each model, refer back to the question to check answer is reasonable **Student Practice** - Problem solving, real world problems are included as often as possible in lessons where appropriate **Scaffolding Strategies** - Partially completed solutions that pupils complete - Visual reminders of how to tackle such as checklists and reminders **Rehearsal** - Mini whiteboards to review systematically - Group problems to solve together, explaining each step to your group - Verbalise solutions to problems with partners or teacher **Aim:** Students solve more complex problems independently and evaluate and refine answers. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Algebra ### Apply knowledge of algebra topics (including connections with graphs) in contexts including assessing how reasonable an answer is #### Description Students use their build up algebra knowledge in real-world problems including interpreting graphs and relationships. They will then assess whether solutions are reasonable. #### Cross-Curricular Examples - Science: Graphing relationships (e.g. speed-time) - Geography: Trend graphs - Computer Science: Variable relationships #### Teaching Model **Teacher Modelling** - Explicitly link problem solving to algebraic methods (including using graphs) - Model how to check answers are reasonable **Student Practice** - A variety of problems that can be solved using algebra, including graphing methods **Scaffolding Strategies** - Partially completed solutions that pupils complete - Visual reminders of how to tackle problems using algebra **Rehearsal** - Group work and discussion about algebra problems with emphasis on the methods used to solve - Verbalise solutions to problems with partners or teacher **Aim:** Students use algebra to solve problems and interpret and explain graphical relationships. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Statistics ### Present and interpret data and charts accurately #### Description Build on work from Y7 and Y8. Pupils should: - Select appropriate charts - Present data clearly - Interpret patterns, trends, and anomalies #### Cross-Curricular Examples - Geography: Data interpretation - Science: Experimental findings - PE: Performance statistics #### Teaching Model **Teacher Modelling** - Explicitly model the most appropriate graphs to use in a given context - Reinforce methods for calculating statistics from Y7 and Y8 - Reinforce how to use statistics to draw conclusions and compare datasets **Student Practice** - Real datasets - Pupils collect their own data and present and interpret it accurately **Scaffolding Strategies** - Visual models - Verbal cues on which type of graph or chart may be most appropriate - Partially completed solutions that pupils complete - Visual reminders of processes for producing graphs, charts and statistics **Rehearsal** - Subject specific review activities such as exam questions, exit tickets or starter activities - Verbalise solutions to problems with partners or teacher **Aim:** Students can present data effectively and draw accurate conclusions. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Proportion and Rates of Change ### Understand proportional reasoning #### Description Proportional reasoning involves understanding relationships between quantities and scaling consistently. Deeper understanding of scaling up and down from Y8. #### Cross-Curricular Examples - Science: Formula relationships - Geography: Population density - Technology: Design scaling #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Explicitly show and discuss the relationship between each part of the question **Student Practice** - Multiple problems across different contexts **Scaffolding Strategies** - Visual models such as number lines and printed tables - Partially completed solutions that pupils complete - Visual reminders of scaling up **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Subject specific exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students understand the impact of proportion when solving problems. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Proportion and Rates of Change ### Use the most efficient method to solve a problem #### Description Students select the most appropriate strategy (mental, written, calculator, algebraic). Focus is on efficiency and accuracy. #### Cross-Curricular Examples - Science: Choosing methods for calculations - Computer Science: Efficient algorithms - Technology: Optimising processes #### Teaching Model **Teacher Modelling** - Model multiple solution strategies to problems - Explicitly highlight the benefits and drawbacks of each strategy - Discuss the most efficient strategy for each modelled problem - Emphasise that the best strategy will differ depending on the problem **Student Practice** - Multiple problems that could be solved using different strategies - Students not only solve the problems, but also describe why they have chosen the strategy they have used **Scaffolding Strategies** - Partially completed solutions that pupils complete - Visual reminders of potential numerical strategies that could be used to solve each problem **Rehearsal** - Mini whiteboards to review systematically - Group work solving multiple problems, with emphasis on discussion around strategies that have been used - Verbalise solutions to problems with partners or teacher **Aim:** Students understand there are a range of ways to solve the same problem and can work efficiently and accurately when solving problems. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Geometry ### Reason geometrically #### Description Reasoning geometrically is being able to explain and justify why you have come to an answer about a geometric problem rather than just being able to complete the process of getting to the answer. #### Cross-Curricular Examples - Art: Perspective reasoning - Technology: Structural design - Physics: Angles and forces with reasoning #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Model structured reasoning, writing reasons for answers at all times **Student Practice** - Justification tasks – pupils must write their reasons rather than just giving answers - Exercises where the answers are written out and pupils have to add the reasons alongside **Scaffolding Strategies** - Sentence starters - Partially completed solutions that pupils complete - Visual reminders of geometric rules such as angle rules **Rehearsal** - Mini whiteboards to review systematically - Subject specific formative assessment such as starters, exit tickets and mini quizzes - Verbalise solutions to problems with partners or teacher **Aim:** Students construct clear geometric arguments and explain reasons for answers. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Numerical Communication ### Present a structured, coherent argument with accurate syntax #### Description Students communicate mathematical reasoning clearly using: - Logical structure - Correct terminology - Clear justification #### Cross-Curricular Examples - English: Structured argument writing - Science: Writing conclusions #### Teaching Model **Teacher Modelling** - Step-by-step worked examples at all times - Clear, explicit reasoning is modelled **Student Practice** - Multi-step problems that require clear working - ‘Show that’ questions where the answer is given and pupils need to show why - Spot the mistake questions - Questions where pupils have to say why another person’s answer is wrong **Scaffolding Strategies** - Partially completed solutions that pupils complete - Visual reminders of methods required to answer questions - Checklist of how to tackle multi step problems **Rehearsal** - Subject specific formative assessment such as exam questions, starter activities, exit tickets - Verbalise solutions to problems with partners or teacher **Aim:** Students can communicate their reasoning verbally and in written form. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) Calculator Skills ### Develop an understanding of some specific calculator functions #### Description Students use advanced functions such as: - Powers and roots - Trigonometric ratios - Memory functions #### Cross-Curricular Examples - Science: Complex calculations - Geography: Data analysis - Technology: Engineering calculations #### Teaching Model **Teacher Modelling** - Step-by-step worked examples - Visualiser explicitly shows calculator input **Student Practice** - Blocked practice of each advanced skill - Interleaved practice of all advanced skills - Subject specific practice including problems in context **Scaffolding Strategies** - Printed calculator which is labelled to support pupils - Partially completed solutions that pupils complete - Visual reminders of process of inputting into calculator **Rehearsal** - Mini whiteboards to review systematically - Multiple choice questions with common misconceptions as incorrect answers - Verbalise solutions to problems with partners or teacher **Aim:** Students can build up confidence using calculators and can use them for more complex problems. Back to top ↑[Back to overview →](https://bhcet.org.uk/foundational-knowledge-and-skills/maths-and-numeracy) ![Bishop Hogarth Catholic Education Trust](https://files.schudio.com/bishop-hogarth-catholic-education-trust/images/logo/Bishop_Hogarth_and_Hexham_Logo_new_(4).png) ### Bishop Hogarth Catholic Education Trust The Headlands Darlington Co. 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