In your own words, describe the concept of like terms. How would you identify a like-terms expression?
Identify the reason(s) that the expression x+y cannot be simplified any further.
Develop a word problem that incorporates a linear equation.
Solve your word problem, outlining each step in the process, as if you were teaching a room of students how to solve your word problem.
In your own words, identify the unique considerations that must be taken into account when solving linear equations.
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1.There are quite a few good analogies for like terms, but it's hard to come up with a perfect one. This is because math deals largely in absolutes; it is a "universal language" and it can be hard to come up with a more precise definition, or a variation on one, without losing some important clarity.
I think a good comparison might be cooking. Like terms would be like variations in a recipe, where the only thing that changes is the amount of one ingredient. Say you had made a bunch of cookies, and each one had a different amount of sugar. The sugar itself is the variable X, but you've changed the amount of that variable, and only that variable, so that each cookie is a little bit different. Each cookie is a term, and they are alike because they differ only in the amount of one variable.
2. X+Y cannot be simplified any further because, in analogous terms, the letters X and Y cannot be broken down any further. Sure, you could split the letter "X" up into a "/" and a "\" but these don't really have any intrinsic meaning that mean "X" when you combine them. Plus, if we ignore the fact that we're working with variables, you can't simplify a basic addition statement without losing the point of the statement; you would simply get "X", "Y" and "+". Think of it like the statement "I'm happy". You can't make this into a simpler statement without losing meaning.
3. A raiding party of orcs is planning to ambush a caravan in the mountain passes. Each orc needs at least 50 gold to cover the costs of his food and equipment. Additionally, the orc leader demands an extra 100 gold for himself. If the raiding party has 8 orcs, including the leader, how much gold do they need?
4. We need to identify the variables first. What elements in this problem can change? (This is where you'd ask the class to brainstorm a list of things). The total amount of gold can change, and the number of orcs can change. We'll make the orcs X and the gold Y. We know that each orc needs some amount of gold, so we're going to multiply or add numbers to X, until we reach Y.
How much gold does each orc need? 50. It's always 50 no matter how many orcs we have. So 50X is our first step.
How much EXTRA gold does the leader want? 100. So we add 100 as a bonus; 50x + 100. There's no X here because the total number of orcs doesn't matter; the leader always gets 100 gold to himself.
So 50x + 100 = Y, the total amount of gold they need.
Well, what's X? 8. So plug in 8 to replace X.
50(8) + 100 = 500. The orcs need at least 500 gold.
5. Linear equations can be deceptively simple; they are a relationship between two or more things. However, the nature of the relationship, and the identity of the things participating in the relationship is of utmost importance in actually articulating the problem. For example, in the problem above, if the student confuses the orc leader's bonus for a variable, the problem will be solved incorrectly. Thus language, abstract thinking and critical analysis are all vital to extracting the correct meaning from the information provided. One important consideration here would be fluency in the native language; a student who lacks fluency may have a difficult time understanding the word problem itself, let alone identifying the elements within it that are mathematically related.
1. In your own words, describe the concept of like terms. How would you identify a like-terms expression?
"Like terms" is referring to parts of a problem that have the same components or variables. In the equation 2x+3y+7x-2y, 2x and 7x can be combined because they both have a numeric component and an x variable. Same goes with the 3y and -2y. The expression can simplify to be 9x+y.
2. Identify the reason(s) that the expression x+y cannot be simplified any further.
x+y cannot be simplified because there are no like terms. Without knowing a numerical value for x and y, they have nothing in common. x represents a different value than y.
3. Develop a word problem that incorporates a linear equation.
68 less than 5 times a number is equal to the number. Find the number.
4. Solve your word problem, outlining each step in the process, as if you were teaching a room of students how to solve your word problem.
Your equation for this problem would be 5x-68=x. To solve this, you must isolate x/get it by itself on one side of the equals sign. If you subtract x from both sides you are left with 4x-68=0.
Your next step would be to add 68 to both sides. This gives you 4x=68. Now divide by 4 on both sides and you get x=17.
5. In your own words, identify the unique considerations that must be taken into account when solving linear equations.
You just have to be able to understand the math terms in the problem in order to set up the equations. Look for words like less than or times, making sure you are using the right operations. Other than that linear equations are simple. They have two variables and all you need to solve them is to get one variable by itself (it's usually the y variable).
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