Number Squares
Introduction
What is the total of the numbers inside the red square? How does the total relate to the position of the square?
Either investigate this on your own or look at the following prompts.
Support
You this worksheet to help you out.
How do you define the ‘position’ of the square? One way to do so: since the top left number of the square is 51, let's call this a "square 51".
What is the total of square 2? What about square 3?
Move the square to different positions and work out the total (do at least 6).
How would you move the square systematically? (You may overlap the squares).
How would you summarise your findings in a table? Take a look at the table on the right if you need a hint.
What patterns do you notice from the table? Can you find a rule linking the Square Code number and the total. Try to write this rule in words and algebra.
Now use your rule to work out what numbers would be in the square for each of these totals: 126, 190, 282, 550, 1246?
Why isn’t it possible to have an odd number total?
Further Questions and Challenges
What happens if you use another position in the Square as your Square Code? What would your rules now be?
What if you changed the size of the square, say to a 3 x 3? 4 x 4?
What if you used a different size grid? Use this to change the size of the grid.
What if you used a rectangle instead of a square? Or any other shape?
Further Practice
Some of the essential skills introduced in this lesson are "substitution", "collecting like terms", "forming expressions" and "solving equations". The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.
Try these pentomino puzzles on Desmos. This should consolidate your understanding of the lesson. Try to apply an algebraic approach.
Transum
Substitution:
Substitution Examples - not sure how substitution works, look at these examples.
Substitution - try this self-marked exercise. There are altogether 7 levels.
Connecting Rules - if you are given the values of x and y which of these equations is correct?
DiceGebra - a game for two players evaluating algebraic equations and inequalities.
Simplifying Algebra:
Algebra Matching - try this matching game with a friend become familiar with equivalent expressions.
Algebraic Notations - try this self-marked exercise. There are altogether 2 levels.
Collecting Like Terms - try this self-marked exercise. There are altogether 4 levels.
Algebragons - find the missing expressions in these partly completed algebraic arithmagon puzzles. There are altogether 5 levels.
Algebraic Perimeters - try this self-marked exercise. There are altogether 4 levels.
Brackets - try this self-marked exercise. There are altogether 10 levels, not all are appropriate.
Factorising - try this self-marked exercise. There are altogether 10 levels, only the first four are appropriate.
Changing the Subject - try this self-marked exercise. There are altogether 8 levels, not all are appropriate.
Writing Expressions - listen to the voice saying the algebraic expression then write it in its simplest form.
Superfluous -find a strategy to figure out the values of the letters used in these calculations.
For more goodies on algebra on transum, click on the hyperlink.
Extension
Take a look at the Multiplication Square on the right.
Pick any 2 by 2 square and add the numbers on each diagonal.
For example, if you take the green square on the right:
The numbers along one diagonal add up to 77 (32+45);
And the numbers along the other diagonal add up to 76 (36+40).
Try a few more examples, what do you notice?
Can you show (prove) that this will always be true?
Now pick any 3 by 3 square and add the numbers on each diagonal.
For example, if you take the purple square on the right:
The numbers along one diagonal add up to 275 (72+91+112);
And the numbers along the other diagonal add up to 271 (84+91+96).
Try a few more examples, what do you notice?
Can you show (prove) that this will always be true?
Now pick any 4 by 4 square and add the numbers on each diagonal.
For example, if you take the blue square on the right:
The numbers along one diagonal add up to 176 (24+36+50+66);
And the numbers along the other diagonal add up to 166 (33+40+45+48).
Try a few more examples, what do you notice?
Can you show (prove) that this will always be true?
Can you predict what will happen if you pick a 5 by 5 square, a 6 by 6 square ... an n by n square, and add the numbers on each diagonal?
Can you prove your prediction?
Desmos
"The Power of Expressions" Collection is a fantastic collection of lessons to investigate algebraic expression, below are some highlights:
Central Park - In this activity, you will write an algebraic expression that places the dividers for many different parking lots. This will help recap and deepen previous understanding with linear sequences you learnt in Year 7.
Picture Perfect - In this activity, you will use algebraic thinking to precisely (and efficiently) hang picture frames on the wall. This will help recap and deepen previous understanding with linear sequences you learnt in Year 7.
Pool Border Problem - In this activity, you will use those numerical expressions to help you write an expression with variables. Then you'll put the algebraic expression to the test and see if it helps you find the tiles for lots of pools very quickly. This again is a useful recap of the "Tiles" lesson in Year 7.
Lawnmower Maths - In this activity, you will learn how maths can give you the power to quickly mow dozens of lawns without breaking a sweat by creating an algebraic expression and see how it helps you mow lots of lawns very quickly.
Products and Sums - In this activity, you will use variable expressions to represent areas of rectangles in two different ways: as the sum of two areas and as the product of the two side lengths.
Equivalent Expressions and Expressions Mash-Up are great for consolidating understanding of multiplying expressions.
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.