Triangle Corners

Introduction

In today's lesson, you will learn how to solve different types of equations (including two-step equations, equations with unknowns on both sides, equations with brackets and equations with fractions.  You are highly recommended to take a look at the relevant skills on DrFrostMaths, CorbettMaths and MyiMaths.  Watch any video and/or go through any online lesson as you see fit.  

Take a look at the triangle corners puzzles above.  What numbers do you need to place on the lines to ensure that the boxes connected to the lines are equal in value?

Make your own triangle corners puzzles and solve them so that:

1. All the answers are integers

2. All the answers are positive integers

3. All the answers are the same

4. The answers are consecutive numbers, e.g. 1, 2, 3

5. Two of the answers add to make the third

When you think that you have the correct answers, get your partner to attempt your puzzles and check that they are correct.

Further Questions and Challenges

Now look at the "square corners" and the "square corners and diagonals" puzzles below.  How many equations are there for each to solve?  Can you solve them?

Can you make your own puzzles so that:

1. All the answers are 7

2. All the answers are distinct (different) integers

3. All the answers are all negative

4. The answers are consecutive even numbers, e.g. 2, 4, 6

5. The answers are consecutive square numbers

6. The answers are consecutive prime numbers

When you think that you have the correct answers, get your partner to attempt your puzzles and check that they are correct.

Here are some other questions to consider:

1. How many equations are there for a "pentagon corners and diagonals" puzzle?  What about a "hexagon corners and diagonals" puzzle?  What about an a "n-gon corners and diagonals" puzzle?  Why?

2. Try making your own "pentagon corners and diagonals"puzzles.  Come up with your own rules of what the answers have to be.

3. For a challenge, make a "cube corners" or a "cube corners and diagonals" puzzle.  Come up with your own rules of what the answers have to be.

4. Is it always possible to design an expression polygon that will produce any given set of numbers? Why or why not?

5. Very challenging and not in Year 8 - make puzzles that involve quadratic expressions.

Further Practice

As mentioned in the introduction, the essential skills in today's lesson are solving different types of equations (including two-step equations, equations with unknowns on both sides, equations with brackets and equations with fractions. Take a look at the relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.  Watch any video and/or go through any online lesson as you see fit.  

Transum

Equations:

• Stable Scales - ten balance puzzles to prepare you for solving equations.

• eQuation Generator - an unlimited supply of linear equations just waiting to be solved. Project for the whole class to see then insert the working in your own style.

• Equations - try this self-marked exercise.  There are altogether 5 levels.

• Missing Lengths - find the unknown lengths in the given diagrams and learn some algebra at the same time.

• Equations with Fractions - try this self-marked exercise.  There are altogether 5 levels.

• Old Equations - solve these linear equations that appeared in a book called A Graduated Series of Exercises in Elementary Algebra by Rev George Farncomb Wright published in 1857.

• Algebra In Action - real life problems adapted from an old Mathematics textbook which can be solved using algebra. There are altogether 7 levels.

• Nevertheless - a game for 2 players. You decide where to place the cards to make an equation with the largest possible solution.

• Mobiles - this is an excellent game to consolidate understanding on equations.