# KS3 Numeracy Year 8 Algebra Through Numeracy Problem Solving

Teach all of the algebra content for Key Stage 3 with these problem-solving numeracy investigations. They will give students a numerical grounding for understanding KS3 algebra while helping them develop numeracy confidence and meta skills. Students will also learn resilience and tactics for approaching mathematical problem solving.

Each unit consist of three lessons. The first lesson provides teaching, activities, practice and homework materials to teach a piece of algebra. The second lesson provides a problem-solving, investigative activity involving an aspect of number work. This is designed to stimulate students’ logical reasoning skills. The third lesson draws on the algebra taught in lesson 1 to generalise the solution to the investigation.

There could not be a better context in which to teach all the algebra for KS3!

## Unit 1 Same as (free)

### Objectives

KS3 Programme of Study statements

• Model situations by translating them into algebraic expressions.
• Use and interpret algebraic notation.
• Substitute numerical values into expressions.
• Simplify and manipulate algebraic expressions to maintain equivalence by collecting like terms.
• Use the concept and vocabulary of multiples.
• Use the 4 operations, including formal written methods, applied to integers.
• Use the concept and vocabulary of multiples.
• Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
• Solve problems.

### Teaching and Activities

Numeracy and Algebra
Write in expanded form (using digits or algebra); substitute into expressions; collect like terms and simplify; recognise multiples.

Numeracy and problem solving: Investigation: Same as… (involving revision of multiplying two-digit numbers by two-digit numbers and working with products).

Algebra and problem solving: Continuation of Same as… including generalisation, using algebra learnt in Lesson 1).

## Unit 2 Why is it so?

### Objectives

KS3 Programme of Study statements:

• Begin to model situations mathematically and express the results using a range of formal mathematical representations.
• Model situations by translating them into algebraic expressions.
• Use and interpret algebraic notation.
• Use the 4 operations, including formal written methods, applied to integers.
• Use the concept and vocabulary of multiples.Solve problems.
• Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
• Simplify and manipulate algebraic expressions to maintain equivalence by collecting like terms.

### Teaching and Activities

Numeracy and Algebra
Learn and practice problem solving strategies (e.g. simplifying a problem, being systematic, looking for patterns); express results using new mathematical representations; match algebraic expressions and numerical meanings.

Numeracy and problem solving
Investigation: Why is it so? (involving revision of adding two-digit numbers; adding three-digit numbers; and dividing large numbers)

Algebra and problem solving
Continuation of Why is it so? (including generalisation, using algebra learnt in B1)

## Unit 3 Minus Magic

### Objectives

KS3 Programme of Study statements:

• Use the four operations applied to integers and decimals.
• Model situations by translating them into algebraic expressions.
• Use and interpret algebraic notation.
• Use algebraic methods to solve linear equations in one variable.
• Make and test conjectures about patterns and relationships.
• Solve problems.
• Model situations or procedures by translating them into algebraic expressions or Formula.
• Substitute numerical values into formulae and expressions.

### Teaching and Activities

Numeracy and Algebra
Add positive and negative integers and decimals; multiply negative integers by a positive integer; model situations using algebraic expressions; solve simple equations involving decimal constants; and become familiar with the problem solving skill of ‘drawing a diagram’.

Numeracy and problem solving
Investigation: Minus Magic (including adding positive and negative integers; and multiplying negative integers by a positive integer).

Algebra and problem solving
Continuation of investigation: Minus Magic (including generalisation, using algebra learnt in C1).

## Unit 4 Factor Frenzy

### Objectives

KS3 Programme of Study statements:

• Use the concepts and vocabulary of factors, common factors, highest common factor.
• Understand and use the concepts and vocabulary of terms and factors.
• Solve problems by applying mathematics to non-routine problems with increasing sophistication.
• Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.

### Teaching and Activities

Numeracy and Algebra:
Find factors and highest common factors of positive and negative integers, and algebraic terms; Substitute into algebraic terms; become familiar with the problem solving skill of ‘working systematically’.

Algebra and problem solving:
Investigation: Factor Frenzy (including finding factors and highest common factors of algebraic terms; using the problem solving skill of ‘working systematically’).

Algebra and problem solving:
Continuation of investigation: Factor Frenzy (including substitution of positive and negative integers into algebraic terms; using the problem solving skill of ‘working logically’).

### Objectives

KS3 Programme of Study statements

• Use conventional notation for the priority of operations, including brackets.
• Use and interpret algebraic notation.
• Use conventional notation for the priority of operations, including brackets.
• Substitute values in expressions.
• Manipulate algebraic expressions to maintain equivalence by taking out common factors and multiplying a single term over a bracket.
• Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships.
• Simplify and manipulate algebraic expressions to maintain equivalence by taking out common factors and multiplying a single term over a bracket.

### Teaching and Activities

Numeracy and Algebra
Understand the action of expanding brackets (involving number only calculations, and then algebraic expressions), practice adding, subtracting and multiplying positive and negative integers.

Algebra and problem solving
Investigation: Blinking Brackets (including using brackets correctly, working with consecutive integers; adding, subtracting and multiplying positive and negative integers; substitution into expressions; using trial and improvement to find solutions, solving problems using logical reasoning; generalising).

Algebra and problem solving
Continuation of investigation: Blinking Brackets (including using brackets correctly; working with consecutive integers; adding, subtracting and multiplying positive and negative integers, substitution into expressions, using trial and improvement to find solutions, simplifying expressions, solving problems using logical reasoning; generalizing).

## Unit 6 Prime Suspects

### Objectives

KS3 Programme of Study statements

• Use the concept and vocabulary of prime numbers.
• Understand and use the concepts and vocabulary of expressions and equations.
• Use algebraic methods to solve linear equations in one variable.
• Model situations or procedures by translating them into algebra.
• Use the concepts and vocabulary of prime numbers, factors and multiples.
• Substitute numerical values into expressions.
• Use algebra to generalise.
• Model situations or procedures by translating them into algebra.
• Solve problems.

### Teaching and Activities

Numeracy and Algebra
Know about prime numbers. Understand the language of algebra (including variable, substitution, equations and expressions). Form and solve equations from given situations. Substitute into an equation and then solve.

Numeracy, Algebra and problem solving
Investigation: Prime Suspects (including multiplication and division facts, multiples, factors and prime number, recognising patterns and using mathematical reasoning to explain these, generalising, spotting exceptions to the general rule and then explaining these, substituting into expressions and solving equations).

Algebra and problem solving
Continuation of investigation: Prime Suspects (including multiplication and division facts, multiples, factors and prime number, recognising patterns and using mathematical reasoning to explain these, generalising, spotting exceptions to the general rule and then explaining these, substituting into expressions and solving equations).

