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.

Mistake of the week-1

Math Vocabulary:

Radical– An expression containing the square root symbol.

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).

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

Radicand– Number inside the radical sign.

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

BEDMAS– Math acronym that helps to remember the order of operations in a math problem.

Best mistake of the week:

Workbook Page 14, number 13:

My mistake:

I did this problem using my calculator because we are working with numbers with many decimals. However, when I was entering the numbers, my calculator got the expression differently; giving me the wrong answer to the problem.

Why is this important? 

If you do not know how to fix your calculator errors, it might be very stressful to figure out what to do if it happened during a test.

What I did:

I wrote the expression like this in the calculator- 

The result to that expression ended up being: -0.56700304 which rounded up to the hundredth should be -0.57.

The real answer should be: 0.960689239 which rounded up to the hundreth should be 0.97, my original answer is negative and is not even near 0.97.

The problem is that instead of the organizing the expression like this:

The calculator organized it like this:

Even though they might look similar, when we do the math they give really different answers. This is because of BEDMAS, the calculator assumes that we want to calculate the expression a certain way if you do not put brackets, however because I wrote down the expression exactly as I saw it, it automatically added its own bracket pairs.

If you were doing an easy expression like 4 (1/2) it does not really affect the expression, but because we are working with radicals our brackets should be in the radicans like this √(4). It is usually not shown but we all assume it is like that, we assume that the calculator will think the same as us but it does not, that is why we have to be really careful when using it.

Solution:

If you are writing down everything in the calculator:

Make sure that you put brackets around the numbers that have to be always together-

Results:

Now we got the right answer.

If you are using the calculator only for the numbers but wanting to show your progress:

You also need to make sure the brackets are around the numbers that should stay together-

This is equal to -0.97364718.

This is equal to 2 x 0.96716821 = 1.93433642.

-0.97364718 + 1.93433642= 0.96068924 = 0.97.

The answer is correct now.

I hope this helped figuring out the importance of BEDMAS and how it is a common error in calculators.

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