This Is What We Have Learned...

Books created by Elementary students (2nd to 6th graders) in the Advanced Math Program at Latin American School in Monterrey, Mexico.

BOOKS BY AMP STUDENTS

How do you know when proportions are equal?

You know by doing cross products which is: when you multiply the up digit in the first ratio of the proportion by the lower digit in the second ration and then multiply the lower digit in the first ratio by the up digit in the second ratio, if the answer of both multiplications are the same then your proportion is true, but if the answers are different the proportion is not true.

Solving Two-Step problems

It’s the same steps but you need to do them 2 times

The most important thing measuring in metric units, is to do a proportion so if you want to do a proportion you need to convert the unit to the unit you want so here is a table

If you want to measure the first thing you need to do is to make a proportion to convert the number into a different unit, to make sure you did this correct you need to do cross products. So here is a table to do cross products if its from a greater to a smaller unit you multiply and viceversa.

E X P O N E N T S <a href="bookcreator://page/6CVdeOUGR6y60sjvSlJDiQ">4</a>

The exponent of a number says how many times to use that number in a multiplication.

It is written as a small number to the right and above the base number.

P E M D A S <a href="bookcreator://page/6CVdeOUGR6y60sjvSlJDiQ">4</a>

A R E N T H E S I S

X P O N E N T S

U L T I P L I C A T I O N

D D I T I O N

U B T R A C T I O N

A proportion is an equation that states that two ratios are equal.

exponents: a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression

PEMDAS: &nbsp;is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. Given two or more operations in a single expression, the order of the letters in&nbsp;PEMDAS&nbsp;tells you what to calculate first, second, third and so on, until the calculation is complete.

prime factorization: Prime factorization is all of the&nbsp;prime&nbsp;numbers that multiply to create the original number.&nbsp;Prime&nbsp;number is a number that can only be divided by one and itself.

simplifying fractions: The process is complete when the numerator and the denominator have no more factors in common. Reduced&nbsp;fractions&nbsp;are&nbsp;fractions&nbsp;that have been reduced to their simplest form. When the GCF of the numerator and denominator is equal to 1, the&nbsp;fraction&nbsp;cannot be reduced any further.

An inequality can be used to compare numbers or variables and you can graph them, for example: f&gt;7, you know by this sign “&gt;” that f is greater than 7 so on a number line you put it like this:

You can see that in number 7 there’s an open circle and an arrow pointing to the right, so if the operation would be f&lt;7, now 7 is greater than f so you put an open circle on 7 and an arrow pointing to the left.

But what if the operation is something like this:

The sign means that x is greater than or equal to 0 so x can be 0, and in a number line instead of putting an open circle you will put a closed circle on 0 and an arrow pointing to the right because x is greater than 0

Scientific Notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form.

P R I M E F A C T O R I Z A T I O N <a href="bookcreator://page/6CVdeOUGR6y60sjvSlJDiQ">4</a>

Prime Factorization is finding&nbsp;which prime numbers&nbsp;multiply together to make the original number.

An inequality is like an equation but the variable is not an exact number. It can be graphed on a number line.

Allison’s car can drive 450&nbsp;miles on just 12 gallons of gas.&nbsp;What’s the mileage per gallon? 

The copier company supplies 16 reams of paper for $55.84. What is the price per ream?&nbsp;

In his race, Speedy McGoo&nbsp;ran the 400 meter dash in&nbsp;&nbsp; 14.5 seconds.&nbsp;What is his&nbsp;speed in meters per second?&nbsp;

To convert an improper fraction into a mixed number you have to divide the numerator by the denominator. So you’ll have to divide 11 divided by 4 which is 2 remainder 3. And what you are going to do with the remainder is that you are going to put it as the numerator, and the denominator will stay the same. So the mixed number is 2 2/3. Now, to convert a mixed number into a fraction you have to multiply the whole number by the denominator and then add the numerator and the answer of that is going to be the numerator. And the denominator will stay the same.

In this example, 6 1/2, you have to multiply 6 x 2, which is 12, and then add 1. So you’ll end up with 13/2, since the denominator stays the same.

<a href="bookcreator://page/LwlluHhFSGKoku0QBsax6g">5</a>To compare fractions, you need to have the least common denominator, or the least common multiple, of the two fractions. To do that you need to know the time tables.

In this example, you need to compare the two fractions. The first thing to do is to find the least common denominator. In this case, the least common denominator is 8, because 8 is multiple of 8 (8 x 1) and 4 is multiple of 8 (8 x 2). We will end with 8 as a denominator for the two fractions. Then, you need to check by what number you multiplied the denominator to get to 8 (the new denominator), and by that same number you multiply the numerator. In the first fraction, you multiply by 1 (because 8 divided by 8 is 1) so it stays the same. While in the other fraction, you multiply by 2 because for 4 to get to 8, you need to multiply it by 2. So you multiply the numerator by 2, so 2 x 1 is 2. 

1.Make the mixed number an improper fraction by multiplying the denominator with the big number and then add the the numerator to the it. The denominator should stay the same and the result you get will be the numerator.

2.Multiply the denominator with the denominator of the other fraction, and multiply the numerator with the numerator of the other fraction.

3. If the result is an improper fraction divide the numerator by the denominator. The answer is the whole number, the remainder is the numerator, and the denominator stays the same

450 miles with 12 gallons = 37.5 miles per gallon

400 meters in 14.5 seconds is about 28 meters / second.

1. Identify the operation applied to the variable. 2. State the inverse operation. - Addition and subtraction are inverse operations. - Multiplication and division are inverse operations. 3. Apply the inverse operation on both sides of the equal sign. 4. Inverse operations undo each other. 5. Solve. 6. Check by substituting.

Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

Examples&nbsp;of&nbsp;rational numbers&nbsp;include , 0, 1, 1/2, 22/7, 12345/67, and so on.

Some examples of irrational numbers are:

&nbsp;Any repeating or terminating decimal represents a rational &nbsp;number.

Any number that can name a position on a number line is a real number.

A real number that is not rational is called irrational. Irrational numbers include √2, π, and e.

If we are going to divide fractions or mixed numbers first of all will have to make the mixed numbers improper fractions.

Next what we will is to get the reciprocal(3) of the divisor and multiply it to the dividend.

Whenever we are adding or subtracting 2 fractions, the first step will be to find the least common denominator. For this we will find the least common multiple(1) of both denominators.

Order of Operations and the Distributive Property

When you solve an equation, expression, etc. You have to follow order of operations. By not forgetting this, remember the word PEMDAS. PEMDAS means, Parentheses Exponents Multiplication Division Addition Subtraction It is written in a way that tells you the order you have to solve the operation. Always read from left to right!

The Distributive Property is a longer, simpler, or different way to solve multiplication. If you are multiplying by to numbers, it is the same as multiplying both of its addends, and then adding them.

<a href="bookcreator://page/LwlluHhFSGKoku0QBsax6g">5</a>Mixed Numbers &amp; Improper Fractions

Mixed numbers are made up of a whole number and a fraction, and it is more than 1. Improper fractions are fractions that are more than 1, and that the numerator is greater or equal to the denominator. You can convert an improper fraction into a mixed number and viceversa.

If you want to write an improper fraction for this model you just have to count in how many parts is the whole divided (this will be the denominator) and then count the total parts shaded (this will be the numerator). One whole is divided into 4 parts, so our denominator will be 4, and the total parts shaded are 11 so our fraction is 11/4.

Estimating with fractions and mixed numbers.

When we are adding or subtracting fractions and mixed numbers we will round the mixed number to the nearest whole. if it is a fraction we have to round it to 0, 1/2 , or 1.

For multiplication we will do the same as addition and subtraction.For division it is different, we will use compatible numbers.

To solve this operations you do the same process. The difference is that in the left one at the end of the process you just add the decimal point depending on the number of the decimal place.

There also are ways to express algebraic expressions there are a lot of ways to express each operation here are the expressions:

It’s the same steps but you need to do the process two times to get the variable

Evaluating and Writing Algebraic Expressions

A writing expression is an expression that uses words instead of symbols. You can recognize a writing expression with some of this words.

You can also simplify all you can by the number you are going to divide (in this example eight, but in more problems it will always be by the bottom number)

A unit rate is a simplification of a rate.

Precision means that state or quality of being precise. The precision is a value in a decimal after the whole number like 10.356.

Next we will find the equivalent fraction(2) with the least common denominator.

An inequality is like an equation but the variable is not exact and it can be graphed on a number line

Equations have 6 different steps, you need to follow them if you want to solve one; these are the steps:

1. Identify the operation applied to the variable; that means to see if the operation is addition, subtraction, multiplication, or division.

2. State the inverse operation; if the operation is addition, its opposite is subtraction, or if the operation is multiplication, its opposite is division.

3. Apply the inverse operation on both sides of the equal sign; imagine you have this equation: n+4=10 the inverse operation is subtraction so apply it like this: n+4-4=10-4.

4. Inverse operations undo each other; so you have to undo the plus 4 and minus 4 so you have n, and 10-4.

5. Solve; now you just subtract 10-4 so n=6.

6. Check by substituting; you need to check if your answer is correct, now put them correctly and you will have 6+4=10, and it is correct, so you finally put 10=10, and your answer will be n=6.

A ratio is a comparison of two quantities.

If you want to divide decimals in the first place is important is to multiply the decimal by 10’s until you get a whole number, but you must also multiply the other number by the 10’s you multiple the decimal.

You can also multiply decimal by decimal the only, you only multiply by 10’s until get a whole number in the first decimal and you multiply by the same quantity of 10’s in the second decimal.

scientific notation: Scientific notation&nbsp;is a mathematical expression used to represent a decimal number between 1 and 10 multiplied by ten, so you can write large numbers using less digits.

rational numbers: Rational numbers are simply numbers that can be written as fractions or ratios (this tells you where the term rational comes &nbsp;&nbsp;from).

irrational numbers: Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are &nbsp;irrational.

real number: Any number that can name a position on a number line is a real number. Every position on a number line can be named by a&nbsp;real number in some form.

Properties of operations are rules that you need to follow when doing an operation 

<a href="bookcreator://page/xwLFTcB4QieoAxZ5YoGQ6Q">Adding Mixed Numbers</a>

<a href="bookcreator://page/xwLFTcB4QieoAxZ5YoGQ6Q">Subtracting Mixed Numbers</a>

Denominator: The&nbsp;divisor&nbsp;of&nbsp;a&nbsp;fraction. The number under the line in a fraction 

Reciprocal: Exchange &nbsp;each&nbsp;of&nbsp;two&nbsp;or&nbsp;more&nbsp;things

<a href="bookcreator://page/GnRheW0jTCGPZNK0xxKE8g">Multiplying Mixed Numbers </a>

Numerator: top figure of a fraction. The&nbsp;dividend&nbsp;of&nbsp;a&nbsp;fraction.

To convert decimals into fractions you just have to see the last number and in which place value it is. For example, if you have 1.2, the 2 is in the tenths place so it is as if he had 1 2/10. To convert a fraction into a decimal you have to do the same. If you have 5 3/100, you just put is as 5.03. Now, there are many fractions that don’t have a multiple of 10 as a denominator. To those fractions, you will just have to divide the numerator by the denominator and that’s it. For example, if we had the fraction 5 1/6, we will have to do 1 divided by 6 which is 0.16 so we will just need to add the 5. So our answer is 5.16.

A proportion can be used to convert between different units

Use cross products to check your conversion

1Least common multiple of a and b is the least number that has a and b as factors.

Now you can do the same with multiplication or division like this:

It’s the same the only thing is that if you have multiplication the inverse operation is division, or if you have division you will have multiplication.

Solving Inequalities by adding, subtracting, multiplying, and dividing

Imagine you have s+4 &gt; 11, so then you do the same as in an equation so s+4-4&gt; 11-4 and the answer is s&gt;7

And this is how you do it by multiplying and dividing:

comparing fractions: If the denominators are the same, then the&nbsp;fraction&nbsp;with the greater numerator is the greater&nbsp;fraction. The&nbsp;fraction&nbsp;with the lesser numerator is the lesser&nbsp;fraction. And, as noted above, if the numerators are equal, the&nbsp;fractions&nbsp;are equivalent. Use &lt; or &gt; to&nbsp;compare&nbsp;the two&nbsp;fractions&nbsp;and .

ordering fractions: To&nbsp;order fractions&nbsp;from least to greatest, start by finding the lowest common denominator for all of the&nbsp;fractions. Next, convert each of the&nbsp;fractions&nbsp;by dividing the lowest common denominator by the denominator and then multiplying the top and bottom of the&nbsp;fraction&nbsp;by your answer.

mixed numbers: A&nbsp;mixed number&nbsp;is a combination of a whole&nbsp;number&nbsp;and a fraction

improper fractions: An&nbsp;improper fraction&nbsp;is a&nbsp;fraction&nbsp;in which the numerator (top number) is greater than or equal to the denominator (bottom number).

decimal: A&nbsp;decimal&nbsp;is a fraction written in a special form. Instead of writing 1/2, for example, you can express the fraction as the&nbsp;decimal&nbsp;0.5, where the zero is in the ones place and the five is in the tenths place.

Mean, Median, Mode, and Range

The Mean, Median, Mode, and Range are a set of numbers that describe a number line, graph, etc. The Mean is the average of all numbers in the graph. You get it by adding all the numbers and dividing the sum by all the addends. The Median is the number at the middle of the graph, but the numbers have to be written from least to greatest. You need to have something like this: 1, 2, 3, 4, 5, 6, 7, 8, 9. Not like: 3, 8, 5, 1, 2, 6, 4, 7, 9. In that case the Median would be four because the numbers are not even. If the numbers are even, you have to get the Mean of the two middle numbers. The Mode is the numbers that appears the most in a set of numbers. There can be no Mode, or there can be more than one Mode. For example: 11, 11, 8, 4, 6, 2, 3. The Mode should be 11. But here: 2, 4, 6, 8. There’s no Mode. Or: 2, 2, 3, 3, 5, 6, 7, 8. Now there’s 2 Modes, three and two. The Range is the difference between the largest number and the smallest number of a set of numbers, for example: 1, 2, 4, 6, 9, 12, 11, 16. Here the Range is 15.

The quantities in a ratio are called terms. The first term is a, and the second term is b.

Multiplying and Dividing Intengers

When you add integers with the same positive sign, you give the product the same sign. When you are multiplying with different signs, the answer will be negative. If you multiply by two negatives, the answer will be positive. It is the same with division.

From a greater to a smaller unit, multiply 

<a href="bookcreator://page/LwlluHhFSGKoku0QBsax6g">5</a>Comparing and Ordering Fractions

What is a fraction? Fractions are a part of a whole and they are less than 1. Fractions have a numerator (the number in the top of the fraction) and a denominator (the number in the bottom of the fraction). The numerator represents the parts we have, and the denominator represents the total parts in the whole.

Evaluating and Writing Algebraic Expressions

You can recognize expressions by these words

Steps to solve an equation

1. Identify the operation applied to the variable 2. State the inverse operation -Addition and subtraction are inverse operations -Multiplication and division are inverse operations 3. Apply the inverse operation on both sides of the equal sign 4. Inverse operations undo each other 5. Solve 6. Check by substituting

<a href="bookcreator://page/rawMSWKiTA2mwbhFpEclFw">Chapter 1</a>

<a href="bookcreator://page/djpc5af2T4yCjU6xxi7tfQ">Chapter 2</a>

<a href="bookcreator://page/FhrHqJyPSW6_PT6Oa74PJw">Chapter 3</a>

<a href="bookcreator://page/xGiI9E6RRS-0o_0ZgYG5yg">Chapter 4</a>

<a href="bookcreator://page/xGiI9E6RRS-0o_0ZgYG5yg">Chapter 5</a>

It is like a normal equation but you need to do the process two times to get the variable.

<a href="bookcreator://page/LwlluHhFSGKoku0QBsax6g">5</a>Fractions and Decimals

A decimal can be as if we had a fraction or a mixed number. Decimals (as all numbers) have place value. To the left side of the decimal point, we have the place value as we have with all numbers. But to the right side of the decimal point we have different place values like: tenths, hundredths, thousandths, ten thousandths, and so on.

When you read a decimal, you read the decimal point as and, and the rest you just say it as if it was a normal number, but you just need to add the word of the place value in which the last number is placed. The last two examples are the same because when you add a zero and there are no other counting numbers after the zero, then, you can take the zeros out.

When a number is missing in a proportion you need to multiply the up digit in the first ratio then the lower digit in the ratio that is missing one number and then divided it by the lower digit in the first ratio

Now, there are some exceptions of some fractions that will save you all this process. There are two examples: the first one consists if you have the same numerator. And the second one consists if you have the same denominator.

1/6 is less than 1/4 which is less than 1/2.

If you have the same numerator, you just need to see which denominator is the biggest, and the fraction with the biggest denominator will be the smallest fraction, and like that with the rest of the fractions.

Now, if you happen to have fractions with the same denominator, it will be like if you had just did all the process mentioned in page 3! So you will skip all this process and save time.

2/7 is less than 3/7 which is less than 5/7.

This is the process you should have done.

Estimating decimals is the same as adding, subtracting, multiplying, or dividing decimals. Just because you see decimals or fractions you don’t have to scare. They are the same.

What happens if in your exam you see this. Would you get scared? Would you stress? Or would you solve it without problem?

When you’re adding integers with the same sign, you add them normally and keep the sign they have. When you’re adding integers with opposite signs, you have to find the difference of the absolute value of both numbers, give it the sign of the integers with the greatest absolute value. When you subtract integers, it is the same as adding its opposite.

If you want to compare fractions you need to do this same process but with more fractions instead of just two. For example first, we will have to find the least common multiple for these 3 fractions. Which in these case seems to be 24. Now, by what number did you multiply 2 (the old denominator) to get to 24 (the new denominator)? By 12. Now you multiply the numerator (1) by 12, and we will end up with 12/24. The second fraction, by what number did you multiply 6 to get to 24? By 4. So now you multiply 5 (the numerator) by 4 and that’ll give us 20 as the numerator and 24 as the denominator. Finally, third fraction, by what number did you multiply 8 to get to 24? By 3. Now, multiply 7 by 3 and now the fraction is 21/24. With the same denominator, you compare just the numerators as if you were comparing just numbers.

1/2 is less than 5/6 and 5/6 is less than 7/8.

<a href="bookcreator://page/6CVdeOUGR6y60sjvSlJDiQ">4</a> S I M P L I F Y I N G F R A C T I O N S

A&nbsp;fraction&nbsp;is in simplest form when the top and bottom cannot be any smaller, while still being whole numbers.&nbsp;

To simplify a fraction, divide the top and bottom by the&nbsp;highest number&nbsp;that can divide into both numbers exactly.

1.Find the least common denominator of both fractions with mental math or using the normal process (shown in image below)

1.You aline the numbers as a normal subtraction.

2.Multiply the numerator by the same number you multiplied the denominator. For example in 2/3 + 1/6 you convert 2/3 into to 4/6, because you multiply both by two to make the denominator the same as the one of the other fraction.

2.Find the least common denominator of both fractions with mental math or using the normal process

3.Multiply the numerator by the same number you multiplied the denominator. 

3.Add the whole number of the mixed number.

4. Add only the numerators of both fractions. Leave the denominator the same. For example in 2/4 + 1/4 your answer would be 3/4.

4.Subtract the numerators as normal numbers and the denominator you leave it the same. If the numerator of the top mixed fraction is less than the one of the bottom, take one of the whole number.

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