Mistake of the week-8

Math Vocabulary:

Coefficient– Number that represents a constant value, usually multiplies the expression.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Like terms– Terms that have the same base and same exponent. Ex: x and x^2 are not like terms because they do not share the same exponent. xy^3 and 4xy^3 are like terms because their variables and exponents are the same (the coefficient is the only thing that can different in like terms)

Binomial– An algebraic expression consisting of two terms. Ex: 24x – 8

Polynomial–  An algebraic expression consisting of more that two terms. Ex:

Conjugate– Every binomial has a conjugate. This conjugate is the same binomial but you change the middle symbol. Ex: The conjugate of 2x + 8 is 2x – 8. As you can see you change the symbol between both terms.

Zero pairs– A set of two numbers that when added together equal zero.

Quadratic Equation– An equation in which the highest exponent of a function is 2. The equation has two solutions.

Best mistake of the week:

Chapter 3, Factoring with fraction and decimal coefficients:

My mistake:

Last week we learned the completing the square method. This is more difficult is you are doing it with fractions, I’ve tried some in the workbook and I think I can understand them now. I wanted to write a blog about it to refresh my memory and get better at it.

Why is this important? 

These different methods to factor will help you be more efficient.

What I did:

Rule #1: In a trinomial the x^2 has to have a coefficient of 1. To do this divide all the expression by the leading coefficient in x^2 (only if there is one.) It is also recomendable for it to be positive so if it is not try to divide by the negative coefficient.

Like before, for these problems you have to focus on the first two digits in the trinomial (x^2 and #x), put them into a square and compare the value on the low right side in the square and the last digit of the trinomial. Remember: If the value you get is bigger than the one you need then the symbol you writer down is negative, in the contrary it will be positive.

See example for a visual explanation.

Solution:

Another example:

I hope this helped 🙂

Mistake of the week-7

Math Vocabulary:

Coefficient– Number that represents a constant value, usually multiplies the expression.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Like terms– Terms that have the same base and same exponent. Ex: x and x^2 are not like terms because they do not share the same exponent. xy^3 and 4xy^3 are like terms because their variables and exponents are the same (the coefficient is the only thing that can different in like terms)

Binomial– An algebraic expression consisting of two terms. Ex: 24x – 8

Polynomial–  An algebraic expression consisting of more that two terms. Ex:

Conjugate– Every binomial has a conjugate. This conjugate is the same binomial but you change the middle symbol. Ex: The conjugate of 2x + 8 is 2x – 8. As you can see you change the symbol between both terms.

Zero pairs– A set of two numbers that when added together equal zero.

Quadratic Equation– An equation in which the highest exponent of a function is 2. The equation has two solutions.

Best mistake of the week:

Unit 2 Worksheet 13, number 3:

My mistake:

The new thing that we learned this week was using the square method. The confusing part is figuring out what to do with the third digit because it is a trinomial. However, after some of the problems I was able to figure out how the answer should be displayed. Another problem I encountered was that because it is a quadratic equation, the answer will have a square root, therefore I had to remember the rules that we need to follow with square roots.

Why is this important? 

If you do not have a clear idea on how to solve the problems using different strategies, you will be very lost on a test.

What I did:

For these problems you have to focus on the first two digits in the trinomial (x^2 and #x), you will then put them into a square and you will have to compare the value on the low right side in the square and the last digit of the trinomial. The thing that confused me was to see which symbol I was supposed to write down depending on the value of the trinomial digit. Something that really helped me was search for a pattern and it ended up showing me that if your value is bigger that the one you need (extra), it will be a plus sign; If it is a lower value then it will be a negative sign.

See example for a visual explanation.

Solution:

That is how i figured the answer out.

Extra example:

Mistake of the week-6

Math Vocabulary:

Coefficient– Number that represents a constant value, usually multiplies the expression.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Like terms– Terms that have the same base and same exponent. Ex: x and x^2 are not like terms because they do not share the same exponent. xy^3 and 4xy^3 are like terms because their variables and exponents are the same (the coefficient is the only thing that can different in like terms)

Binomial– An algebraic expression consisting of two terms. Ex: 24x – 8

Polynomial–  An algebraic expression consisting of more that two terms. Ex:

Conjugate– Every binomial has a conjugate. This conjugate is the same binomial but you change the middle symbol. Ex: The conjugate of 2x + 8 is 2x – 8. As you can see you change the symbol between both terms.

Zero pairs– A set of two numbers that when added together equal zero.

Best mistake of the week:

Chapter 3, Skills Check #1, number 6:

My mistake:

I was away for most of this week so I have not been able to catch up on much of this week’s content. Also I do not remember much from last year’s factoring. I couldn’t solve this problem at all and I thought it did not have any solution. By what I remember from last year, I needed to find a pair of numbers that multiplied together would give me the last digit (4) and when added together it would give me the middle digit (12); however, 4 is smaller than 12 so this rules did not work at all for me.

Why is this important? 

I was totally lost at what we were doing in class and couldn’t solve too many of the exercises. This could make me fall behind and drop my grades. If I hadn’t asked how we were supposed to solve them, I probably could not solve any of the problems on the test.

What I did:

As I said before I did not solve it at all, however, I tried to look for pairs of numbers that followed the previously stated rules. Because I did not find any pair of numbers that could help me, I just moved on to the next question.

Solution:

The answer is right now.

Remember- Always ask your teacher if you have any questions.

Mistake of the week-5

Math Vocabulary:

Mixed radical– Radicals expressed in the form ““, all mixed radicals can be converted into entire radicals. Mixed radicals are most of the time used to express large radicals in their simplest form. Ex: , if it was an entire radical it would be

Entire radical– Radicals expressed in the form ““. You can transform entire radicals to mixed radicals if there are perfect sections while factoring the radicand.

Radical– An expression containing the radical sign (square root symbol).

Radical sign– √.  The square root sign is called “radical sign.”

Coefficient– Number that represents a constant value, usually multiplies the expression.

Restrictions– Rules that you have to apply to variables. Ex: , the restriction would be 0. x should be a positive number bigger or equal to 0 because you cannot have square roots of a negative number (except if you are multiplying by a negative number that is also in the radicand and the variable has an odd exponent).

Equation– Mathematical statement with an equal to symbol.

Extraneous solution– A solution that comes from solving a problem, however, it is not a valid solution to it.

Isolate– Leave variable by itself.

Best mistake of the week:

Worksheet PC 11- Solving Radical Equations, number 2, g:

My mistake:

This was new content that we were just learning. It is usually really easy to solve equations (leave a variable by itself), however, when you are solving equations with radicals this becomes a little bit tricky.

You need to leave the variable by itself, therefore you start to move everything to the other side of the equal sign. When doing that with radicals, you have to think what is under and what is out of the radical sign. My mistake was that I thought a number was out of the radical sign so I could move it first but because it was under the radical sign it ended up giving me extraneous solutions.

Why is this important? 

When solving radical equations, moving a number before another one could affect the result greatly. This could make it harder for you to get to the right answers and most of the times will give you the wrong result.

What I did:

As I said I mistakenly moved a number even though it was under the radical sign, this ended up giving me an extraneous solution.

Solution:

Step 1. Recognize which numbers are under the radical sign. Move the ones that are not under the sign to the other side (when you move to the other side of the equal sign they become their opposite: numbers that multiply will divide now, positive numbers will be negative, etc.)

Step 2. Move the root to the other side. A root will become an exponent.

Step 3. Now it should look like a regular equation. Isolate variable.

The answer is right now.

Remember- Always analyze the question before solving it. If you want you can add a little downward stick when rewriting the equation to clearly see which numbers are under the radical sign.

Mistake of the week-4

Math Vocabulary:

Mixed radical– Radicals expressed in the form ““, all mixed radicals can be converted into entire radicals. Mixed radicals are most of the time used to express large radicals in their simplest form. Ex: , if it was an entire radical it would be

Entire radical– Radicals expressed in the form ““. You can transform entire radicals to mixed radicals if there are perfect sections while factoring the radicand.

Radical– An expression containing the radical sign (square root symbol).

Radical sign– √.  The square root sign is called “radical sign.”

Coefficient– Number that represents a constant value, usually multiplies the expression.

Index– A number located at the top left of the radical sign, it determines what type of root the expression is using. If there is no index we assume it has an index of 2 (square root).

Radicand– Number inside the radical sign.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Like terms– Terms that have the same base and same exponent. Ex: x and x^2 are not like terms because they do not share the same exponent. xy^3 and 4xy^3 are like terms because their variables and exponents are the same (the coefficient is the only thing that can different in like terms)

Binomial– An algebraic expression consisting of two terms. Ex: 24x – 8

Polynomial–  An algebraic expression consisting of more that two terms. Ex:

Rationalizing a denominator– Rewriting the fraction to make the denominator a rational number.

Conjugate– Every binomial has a conjugate. This conjugate is the same binomial but you change the middle symbol. Ex: The conjugate of 2x + 8 is 2x – 8. As you can see you change the symbol between both terms.

Zero pairs– A set of two numbers that when added together equal zero.

Best mistake of the week:

Workbook Page 83, number 4, b:

My mistake:

Dividing radicals that are binomials was the new thing we learned this week. While doing this problem, I made a mistake while rewriting the equation and its symbols.

Why is this important? 

If you are rewriting an equation or solving any type of math problem, it is really important to put attention at the symbols. If you do not have the right number/symbol, maybe because you are not focused in the question, the answer will be far from the right one. This could backfire on a test making you do tiny but dumb mistakes. This is why it is important to check your work after you are done, to make sure the numbers are right, and to make sure all the steps you made are correct.

What I did:

I understand the steps you have to do to divide radicals that are binomials, however, in this problem you have to multiply a binomial by a binomial. When doing this you will usually have a long polynomial and if you are not paying close attention to the problem it is really easy to make tiny mistakes.

In this case, I rewrote the positive symbol in the upper part as a negative symbol.

Solution:

Multiplying and Dividing Radicals:

Step 1. When dividing binomials that are radicals you need to get rid of any root that is in the denominator. To do this you need to multiply the denominator and numerator by the conjugate of the denominator, this action is called rationalizing the denominator. Coefficients should be multiplied with coefficients, roots with roots and variables with variables. Remember that there will also be adding and subtracting, so make sure apply the proper rules.

Step 2. Now that you multiplied the binomials, you should have zero pairs in the denominators and your irrational root should have disappeared.

Step 3. After getting rid of the irrational roots, you need to simplify the fraction (division). You will mostly end up with a binomial in the numerator and a integer as denominator. To simplify the fraction, you need to see if between your three terms have something in common. If there is nothing in common then your answer is simplified.

The answer is right now.

Remember- Only add and subtract like terms, and always check your answers.

Mistake of the week-3

Math Vocabulary:

Mixed radical– Radicals expressed in the form ““, all mixed radicals can be converted into entire radicals. Mixed radicals are most of the time used to express large radicals in their simplest form. Ex: , if it was an entire radical it would be

Entire radical– Radicals expressed in the form ““. You can transform entire radicals to mixed radicals if there are perfect sections while factoring the radicand.

Radical– An expression containing the radical sign (square root symbol).

Radical sign– √.  The square root sign is called “radical sign.”

Coefficient– Number that represents a constant value, usually multiplies the expression.

Index– A number located at the top left of the radical sign, it determines what type of root the expression is using. If there is no index we assume it has an index of 2 (square root).

Radicand– Number inside the radical sign.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Like terms– Terms that have the same base and same exponent. Ex: x and x^2 are not like terms because they do not share the same exponent. xy^3 and 4xy^3 are like terms because their variables and exponents are the same (the coefficient is the only thing that can different in like terms)

Best mistake of the week:

Workbook Page 62, number 7:

My mistake:

Adding and subtracting radicals was the new thing we learned this week, I did this math problem after that first lesson and it was challenging. My mistake is that I over complicated the problem.

Why is this important? 

In math you are looking to be as efficient as possible. The best thing to do when there is a difficult problem is to analyze the wording/ image/ equation/ etc. However, we usually overthink those math problems that look complicated, this can backfire during a test, making you lose time and get nervous.

What I did:

Instead of taking the time to analyze the picture I made up lines and tried solving it with my own logic.

Instead of looking at the figure as a rectangle I assumed that it was a square and that missing parts were the originals but divided by half. This clearly did not work because even though it looks like we are diving the full length by half, we do not know the measurements plus we still have not learned how to divide radicals.

Solution:

Adding and subtracting radicals:

Step 1. You can only add or subtract radicals that are like terms. To do this you have to simplify these radicals and/or convert them into mixed radicals.

Step 2. After you have simplified all your radicals, you can use them to add or subtract with your other radicals. When doing this remember that the only thing that changes are the coefficients (the roots will stay the same).

Remember- Only add and subtract like terms, do not worry if it is a two radical answer because they usually are.

The easy way to solve this problem was to take both values and subtract them-

x= +

y= –

This is the right answer.

Remember to always take the time to analyze the question.

Mistake of the week-2

Math Vocabulary:

Mixed radical– Radicals expressed in the form ““, all mixed radicals can be converted into entire radicals. Mixed radicals are most of the time used to express large radicals in their simplest form. Ex: , if it was an entire radical it would be

Entire radical– Radicals expressed in the form ““. You can transform entire radicals to mixed radicals if there are perfect sections while factoring the radicand.

Radical– An expression containing the radical sign (square root symbol).

Radical sign– √.  The square root sign is called “radical sign.”

Coefficient– Number that represents a constant value, usually multiplies the expression.

Index– A number located at the top left of the radical sign, it determines what type of root the expression is using. If there is no index we assume it has an index of 2 (square root).

Radicand– Number inside the radical sign.

Factorize/ factor/ factoring– In math factoring or to factor is to find the smallest numbers that multiplied together will give you the original number. This is used so you can work with simpler and smaller numbers.

Ex: 250= 25 x 10, you can still make those numbers smaller, 25= 5 x 5 and 10 = 2 x 5.

Therefore it will end up being 250 = 5 x 5 x 2 x 5.

Best mistake of the week:

Workbook Page 27, number 8:

My mistake:

The wording of this problem was not the best, looking back after I solved it, it was actually really easy but at the time I did not understand what they wanted me to do.

Why is this important? 

Word problems and instructions can be really challenging if you do not understand them or if the wording is difficult, it is really important to analyze these types of questions because they can make you lose a lot of time during a test.

What I thought I had to do:

Although the question clearly stated that it had to be converted into a mixed radical, it was weird because it was already a mixed radical.  So what I did instead was convert it to an entire radical, which was my error:

The real answer should be: 2.3, my answer ended up being 1.1.

Solution:

Radicals conversions and factoring:

Mixed to entire-

Entire to mixed-

The right way of solving this problem was to treat it as an entire radical and factor the radicand because it was not in its simplest form, exactly like the instructions said-

The answer is correct now.

Remember to always take the time to analyze the question, and if you are on a test try to solve the questions you understand the best first and leave questions like this to the end.