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OPERATIONS AND ALGEBRAIC THINKING
OPERATIONS AND ALGEBRAIC THINKING
Lesson 1: Patterns and Functions
Identifying Pattern Attributes.pdfLesson 2: Factors and Multiples
GREATEST COMMON FACTOR (GCF), SIMPLIFYING FRACTIONS
LEAST COMMON MULTIPLE (LCM), ADDING & SUBTRACTING FRACTIONS
On your own:
FACTOR AND MULTIPLE PRACTICE
FACTORS AND MULTIPLES PRACTICE.pdfCount with your fingers! I do it all the time!
BEGINNING ALGEBRA
BEGINNING ALGEBRA
UNIT 1: Integers, Powers & Roots, PEMDAS
INTEGERS
Eventually you will need to know how to add, subtract, multiply, and divide with negative numbers. Although you can use your calculator very often, there are some math concepts that require an automatic sense for this.
Learning integers can feel difficult, but the good news is that there are many different ways to think about it. You should just find and stick with the framing that works for you. Framing the problem in terms of money can be helpful, or in terms of a contest or some kind of up/down analogy. Look at the different explanations below and find what works for you.
Integers are simply whole numbers. They can be positive or negative. When we don't see a sign, we should assume that number is positive. A negative number (worth less than zero) will have a negative sign.
Most of us know how to perform operations (adding, subtracting, multiplying, dividing) with just positive numbers. It's when we include some negative numbers that we find it harder.
Here are the rules:
Multiply & Divide: Perform the operation as you normally would, then attach a negative sign to the answer only if the two numbers you used have different (unlike) signs. For example, -2 x 3 = -6. If the two numbers you used have like signs (the same)--even if they are both negative--the answer is positive, so leave it alone. For example, 2 x 3 = 6. Or -2 x -3 = 6.
Add:
If the signs are the same, simply add the numbers and attach that same sign to the answer. For example, 2 + 5 = 7. Or -2 + -5 = -7.
If the signs are different, you will actually take the difference of the two numbers (subtract the smaller from the bigger) and give the answer the sign of the bigger number. It's helpful to think of the answer as having the "winning" sign--that is, taking the sign of the bigger number. For example, -8 + 5 = -3. Another example, -5 + 7 = 2. Or 9 + -6 = 3. To repeat: We take the difference between the numbers and give the answer the sign that the bigger number had. Remember that we don't actually see the positive sign, so in my last example when I say I give the answer the "winning" sign, I'm putting no sign in the answer because it's positive.
It's helpful to think of money and temperatures and contests when adding integers.
Subtract: Simply change the subtraction problem to an addition problem and follow the above rules.
To repeat: Turn the subtraction problem into an addition problem!
Here's how we do that: KEEP CHANGE CHANGE
We KEEP the first number exactly as is. We CHANGE the operation symbol from subtraction to addition (the - to a +). We CHANGE the sign of the second number.
[We are only allowed to change from subtraction to addition because we are also reversing that second sign. We can't just pretend we have an addition problem on our hands when we really have a subtraction problem! It's by reversing the second number's sign to its opposite that makes it ok to switch things up to addition.]
Now just follow the steps for adding integers, explained in Point 2.
For example, 8 - 12 becomes 8 + -12. To add 8 and -12 we take the difference (4) and give it the sign of the bigger, or winning, number (the 12). So our answer is -4.
Another example, -9 - -2: Think KEEP CHANGE CHANGE. KEEP the -9, CHANGE the subtraction to addition, and CHANGE the -2 to 2. So now we have -9 + 2, which = -7.
"Keep Change Change" is a mantra that may help you with the most difficult aspect: subtraction.
More integers, including word problems with negative numbers.
Adding & Subtracting Negative Numbers.pdfPOWERS & ROOTS
PEMDAS
Look at the colorful PEMDAS charts and let them sink in. They describe the order in which you perform operations in a number expression that has more than one operation. (The four operations are Addition, Subtraction, Multiplication, and Division.) Then watch the Khan Academy video that follows, and continue along with Khan, doing the schedule of events listed on the left. Please make sure you are logged in with Khan so the site can keep track of your work. Instructions are at the top of this page. If you have any trouble with that, I'd rather you did the work without signing in than not do the work. And if you can't do all of the work, I'd rather you did some than none.
UNIT 2: Simplify & Evaluate Expressions, Factors & Multiples
EVALUATE EXPRESSIONS
Evaluate expressions. (Plug in the given number for the variable and then simplify the expression to a single number). Remember that anything in fraction form can be framed as numerator divided by denominator. Fractions are division problems!
Follow along with the schedule of activities along the left side of the screen after you watch the video. Good luck and have fun!
SIMPLIFY EXPRESSIONS (combine like terms, distribute terms)
Today we will learn how to combine "like" terms. That just means collecting up the parts of an expression that can mix together. I like to think of the terms as different species, so it's like matching up the cats with other cats and the dogs with other dogs, etc. You don't want cats and dogs and snakes in the same cage!
In the expression 5x2 + 3x + 7, you have three entirely different species. They can't be combined.
In the expression 2x2 + 4x2 + 7x + 3 + 9, you have three species but five terms. Which can be combined?
Today we will also learn how to distribute a number across both terms in a parentheses in order to eliminate the parentheses. This just means to multiply the number sitting right in front of the parentheses by each term within the parenthesis. Then you can drop the parentheses.
For example, in the expression 2(4 + 3), you can multiply the 2 by the 4 and separately by the 3 and then add those two products together. So you'd have 8 + 6, which is 14. If instead, in the original, you added the 4 and 3 first and then multiplied that by 2, you'd also end up with 14. In math, we sometimes want to distribute the 2 across the two terms which are inside the parentheses.
Review all aspects of simplifying and evaluating algabraic expressions
BONUS: Polynomial Practice
UNIT 3: One- and Two-Step Equations, Multi-Step Equations
One-Step Equations
Two-Step Equations
solving two-step equations.pdfON YOUR OWN
Do these 10 practice questions & check your answers as you go. They are one-step equations to "solve for x." Remember to do the opposite (inverse) operation to get the x on one side of the equal sign. Then you know what x is! Some of you will find this exceedingly easy. Do it anyway. ;-)
2. Here's another round. Find the side with the x and do the opposite operation to undo (or cancel out) that relationship, and then do the same on the other side of the equation. You'll end up with x alone on one side.
3. OK, things are getting just a bit harder. This time the x is involved with two different numbers/ operations, so we need to undo (or cancel) each relationship, one at a time. Start with what I call "the trailer," which is the addition or subtraction relationship. Perform the opposite operation to cancel that out and make sure you do it on both sides of the equation. So, for example, if you are told 5x+2=17, you want to remove the 2 part first. You subtract 2 from both sides and you now have 5x=15. Now you just need to undo the relationship with the 5 to get the x alone. So you do division, which is the opposite of what's there (multiplication). Divide both sides by 5 and you end up with x = 3. Ta Da!
ON YOUR OWN REVIEW
Please do these pages for one- and two-step equations and scroll down to check your answers! You might find it easy, which is always fun. And if you struggle a bit, it means you needed to do this! Don't stop at the first sign of struggle. Persist, look at the answers, work backwards to figure out how they got that answer. This will all help build your understanding. If you'd like, go back to the Khan Academy videos from the past to refresh your skills.
One-Step Equations.pdfTwo-Step Equations.pdfMulti-Step Equations
Remember to consolidate your x's over to one side of the equal sign. You want to end up with just one x alone on ONE side. So if you see a variable on both sides, that's your first step. Move the smaller to the side of the bigger. THEN you start to work on combining like terms and canceling everything out 'til you get the x alone.
Remember to consolidate your x's over to one side of the equal sign. You want to end up with just one x alone on ONE side. So if you see a variable on both sides, that's your first step. Move the smaller to the side of the bigger. THEN you start to work on combining like terms and canceling everything out 'til you get the x alone.
For this "Algebra Topics" website, scroll down, looking at whichever videos and explanations will help you. Then do the problems at the bottom of the lesson.
For this "Algebra Topics" website, scroll down, looking at whichever videos and explanations will help you. Then do the problems at the bottom of the lesson.
UNIT 4: Algebraic Expressions, Write & Solve Equations, Apply Algebra to Formulas
Algebra_ Translating Words into Math.pdfTranslating Expressions- Gammache.pdfPlease do all of the activities for this Khan lesson. There are a few videos and a couple of practice pages.
(It's REALLY important that you try to do some math on your own. The struggle is the point.)
PRACTICE turning words into math symbols.
ON YOUR OWN
Scroll through the explanation on this website and then do the 10 questions at the bottom. You can check your answers. This is really good practice! The more you do on your own, the faster you will advance!
Agebra Practice: Putting it all together.
Algebra Word Problems.pdf
UNIT 5: Inequalities, Inequality word problems
BONUS: Polynomial Practice
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