Solving equations (beginning algebra)
Section 1: One strategy is that you can find an input from an output if you can follow a function rule backwards. For example, the function table:
| x | y |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
has the function rule y=2x+1, or “double the input andadd 1). If I want to know what inputgoes with an output of 23, I can work backwards:
| the opposite of adding 1 is subtracting 1: 23 – 1 = 22 |
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| The opposite of doubling, is dividing by 2: 22 / 2 = 11 |
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A trickier one is what input goeswith an output of 22:
| the opposite of adding 1 is subtracting 1: 22 – 1 = 21 |
|
| The opposite of doubling, is dividing by 2: 21 / 2 = 10.5 |
|
The first homework problem isto do this same thing with the function rule: y=2x+2; “double the inputand add 2”. I can do similar things to find the input for the output 25:
| the opposite of adding 2 is subtracting 1: 25 – 2 = 23 |
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| The opposite of doubling, is dividing by 2: 23 / 2 = 11.5 |
|
When you are doing thehomework problems, show your steps as you go along, and think about how youwould explain your steps to someone else.
Section 2: Another technique is using algebra tiles:
One of the key ideas was thatyou can always add (or subtract) the same amount to (or from) both sides of anequation and keep things balanced. We did two examples using algebra tiles:
Example 1
In the first example, wesimplify by taking off equal amounts from both sides, which leaves us with avalue for x.
Example 2
In the first steps, we add 4to each side (keeping the equation still balanced), and note that +4 and–4 cancel out to give 0.
Finally, we take twox’s away from each side, and divide each side into two equal groups tofind that x has the value 4.
If you have more than one variable, you can use substitution to help you solve the problem. I asked us all to think about these problems as though we were doing them with physical objects of a given (but mostly unknown weight) on a 2-pan balance scale. This set of problems (in which we are looking for the values of the square, star, and triangle) comes from the teacher resource book Get It Together.
The equations are:
The main idea here is toreplace shapes by other shapes that you know they are equal to (have equalvalue/weight as), until you are able to compare a single shape to a number.
Example 3:
Another example showingsubtraction and division:
What did I do to show thesubtraction?
What did I do to show thedivision?
Section 3: Elementary algebra readiness includes mental math. I often share with this class my exprience of visiting a class doing second grade algebra. A second grade algebra problem looks something like:
3 + ____=14 – 2
Not only is this goodaddition and subtraction practice, but these problems teach students that themissing number can be anywhere in a problem, and that “=” meansthat each side has an equal amount, not that “the answer is comingnext”.