Algebra Tiles and Variables

The thick purple line is Y and the purple square is Y squared because the height and length are made up of Y’s. The light blue thick line is X and the light blue square is X squared because it’s height and length are made up of X’s. The green square is XY because the height is Y and the length is X. The small dark blue square is a unit tile and at the bottom of that are 5 unit tiles. One unit tile is the height of a X and a Y. These tiles can also be negative by clicking on them and they would turn red.

Finding Perimeter and Combining Like Terms

The perimeter is counting what’s on the outside, not the inside. There’s a y2 and it’s made up of 4 y’s but on the left side it’s covered by an x, so it would be y – x. One unit covers the left side of x, so it would be x – 1 and the area of that unit would be 3, but 1 unit could go on the top and bottom of x. You lastly, add all the x’s, y’s, and 1’s. Since there is a negative and positive x, they are zero pairs and cancel each other out. There’s also, a negative and positive 1, and they’re zero pairs. You should be left with 4 positive y’s and 4 positive units. The equation should be 4y + 4.

Meanings of Minus

It’s a symbol and it has to do with math. It could mean subtracting a number and it’s also a part of PEMDAS, it’s the S which is subtract. It also has to do with fractions, it includes a numerator, a denominator, and a whole number at the side of the fraction. Negative numbers, numbers at the far right of the number line. It could also be a dash to separate words. We could also use it for repeating numbers when dividing.

Representing Expressions on an Expression Mat

For example, x – 1 – ( 2x – 3 ) could be shown in different ways. This expression on an expression mat would be like this :

Since there are 3x, you place them on the top side. Two of those x’s are negative, so you click on them to turn them negative and the other x is a positive, so you leave it as a light blue color. There’s three unit tiles, but one of them is negative and the other three are positive. Another way to show this would be like this :

The first portion of the expression, x – 1, is shown at the top and they’re all positives, but the other portion, – ( 2x – 3 ), is shown at the bottom because it’s the opposite of what you see. The regular expression of that portion would be negative 2x and positive 3. Then you place that on the ‘opposite’ side and turn them to their opposites, positive 2x and negative 3. These are two different ways to show x – 1 – ( 2x – 3 ).

Using Zeros to Simplify

The shape of the tile doesn’t matter, as long as it’s a negative and a positive to cancel each other out. This strategy is known as zero pairs. These tiles could be on the ‘what you see is what you get side,’ or on the ‘opposite’ side where the tiles that you see, their shape and color are actually the opposite of what they are.
For example if there were to be a positive x and a negative x on the opposite side, the positive would be a negative and the negative would be a positive because negatives and positives are opposites to each other. If there was a positive and a negative of the same shape tile, you could use zero pairs to cancel them out, this is one method to help you to simplify your equation.

Legal Moves for Simplifying and Comparing Expressions

Zero pairs, Balanced sets, and Opposite are the methods we learned in class. Zero pairs is two tiles are opposite and they cancel each other out. For example, negative x and positive x would be zero pairs. Balanced sets is when the same color and shape is on both sides of the mat. For example, if 2y + 1 is on the left side and 2y + 1 on the right side, you cancel them out because they’re the exactly the same tiles. Opposite is when there’s tiles on the opposite side, which is the ‘-‘ on the expression mat and you move them up to the what you see is what you get ‘+.’ For example if there was -2 on the ‘-‘ side, you move them up to the ‘+’ and they turn into positives. However if you have positive 2 on the ‘-‘ side and you move them up, they would turn into negatives because negatives and positives are opposites.

Solutions of an Equation

     One way is one solution is false because theres no solution. One solution is oneway because it means theres one answer for x. For it to be true, it would be like x = 5 or x = 9. When the value is uneven or not equal its because its false or theres no solution like 8 = 4, thats not equal because its not the same value. Infinite solutions is when theres more than one answer for x like x = 15 or x = 20.