This Is What We Have Learned 2

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

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Topic 4: Lesson 6 Solving Two-Step Equations

A 2 step equation is an equation where you have to get 2 variables instead of 1

What is prime factorization and prime numbers?

A composite number is a&nbsp; whole number that has more&nbsp; than two factors.

A prime number is a&nbsp;whole number with exactly two factors, 1 and the number itself. The number 1 is neither prime nor composite.

Writing a composite number as the product of its prime factors shows its prime factorization. You can use a factor tree to find prime factors.

The least common multiple (LCM) of two or more numbers is the least multiple that is common to all of the numbers.

The greatest common factor (GCF) of two or more numbers is the greatest number that is a factor of all the numbers.

Compound events are two or more events that happen together that is easy right? Let’s see how to solve one!

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By: Mateo Cano, Andrea Moreno, Alex Bruner

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The order of operations is Pemdas that states form parentheses, exponents, multiplication, division, addition, and subtraction.

In this example we have to do first the division then the addition we can’t and then divide we have follow the order of operations

In this examples we have to first do the operation in parentheses because if we follow pemdas that means that we have to answer the parentheses first and then multiply

1-9 Order of Operations and The Distributive Property

The Distributive Property says that multiplying a sum&nbsp;by a number is the same as multiplying each&nbsp;addend by the number and then adding the&nbsp;products.

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You can also change each unit capacity, length, and weight.

In this image you can see that if we do mL to L we need to divide, and when we do m to mm we need to multiply.

When you use measurements, the smaller the unit, the more precised

To estimate the population size we will have to make a proportion

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To multiply and divide integers you need to know this rules: 

By: Mauricio Garza, Julio C González , and Eduardo Bernal

Page 2- Exponents and Order of Operations Page 3- Prime Factorization Page 4- Simplifying Fractions Page 5- Comparing and Ordering Fractions Page 6- Mixed Numbers and Improper Fractions Page 7- Fractions and Decimals Page 8- Rational Numbers Page 9- Scientific Notation Page 10- Graphing Points in Four Quadrants Page 11- Graphing Linear Equations Page 12- Finding the Slope of a Line Page 13- Graphing Nonlinear Relationships Page 14- Translations Page 15- Line Symmetry and Reflections Page 16- Rotational Symmetry and Rotations

A sequence is a set of numbers that follow a pattern.

In an arithmetic sequence you find each term by adding the common difference to the previous term.

In a geometric sequence you find each term by multiplying the previous term by a fixed number called the common ratio.

To find the slope of a line we’re going to need this equation

We subtract Y from Y and then we divide by X minus X

Lesson 10-4: Graphing Nonlinear Relationships

A nonlinear graph is any graph in which the alined dots don’t form a straight line. A nonlinear graph has curved angles.

To graph nonlinear relationships, you’ll need to have a table or/and an equation. The table must only have 2 sides, for the X’s and the Y’s. If you follow correctly the pattern in the table you can get an equation. The table will tell you were to mark a dot in the graph and the equation will help you to verify.

If your equation has a number to any power, it’s nonlinear for sure.

Nonlinear graphs have always at least 1 curved part.

What are rational and irrational numbers?

A rational number is any number that is terminating or repeating. A terminating number is a number that you can see has ended, like numbers 1,2,3 or 3.97.Repeating numbers are numbers that actually never end, but you can actually predict which number is going to be in which order because this numbers are always repeating. To symbolize a repeating number, you can put a line on top of the number that is repeating. Example of repeating number: 3.3434343434…

An irrational number is a number that never ends, but it also never repeats its number. A good example of this is “Pie” it is a number that never ends. “Pie” starts with the digits 3.14.

Reminder: If you can’t tell the next digits of a number, it’s probably an irrational number.

To divide fractions you will need to multiply your first fraction times the reciprocal of the other fraction or mixed number. To divide mixed numbers you will need to first convert them into improper fractions and then multiply times the reciprocal of the other fraction or mixed number.

To complete a table you need to first know what the rule is in the sequence, it can be for example add 1 or multiply times -2.

To find the expression you need to find the rule and then apply it to the variable.

The scientific notation is a way to write really large or really small numbers ending in lots of 0’s. It uses the original number times 10 to the x power (representing the amounts of numbers there are at the beginning or end.) For example, let’s say you want to write 6,500,000, you’ll have to write 6.5 x 10 to the 6th power.

A scientific notation is just really a way to express large or small numbers using how many 0’s they have at the beginning or at the end.

Remember to count the 0’s and the amount of numbers before or after the 0’s so you can know by what number to multiply.

A scatter plot is a graph that uses dots to represent a set of data. Scatter plots can show a relationship, or trend, it can either be a positive trend or negative, or have no trend at all

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The first thing to know is what is an integer. An integer is a whole number (not a fractional number) that can be positive, negative , or zero

An example of integers could be:    -8,-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8

In this example -1 is greater than -3 because the more closer a negative number is to zero the greater it is

In this example we’re ordering the integers the ones to the left are the negatives and the ones to the right a positive

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To interpret a graph you have to know more less the speed distance or the measurements depending on the problem

Which one represents the height of a cricket as it jumps?

They start with a low speed not to get tired but as the race ends they sprint with all their speed.

If you sketch a graph of a runner in a 1mi race it would probably look like this:

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To estimate things you need to round first. Rounding means to replace the number to an approximate value that is shorter and simpler.

It rounds to 5 because it is the nearest whole number.

You first need to round both number to the nearest wholes number. Then add or subtract.

A fraction is a decimal, or a part of a whole represented in a special way, with the numerator on top of a line, and the denominator under it. Together, both numerators and denominators form a fraction.

A decimal is basically also a fraction, it’s used to represent a part of a whole. A decimal can also be showed as percent. Decimals are used to add, subtract, multiply and divide easier than with fractions. Careful, because you might miss the decimal point in any equation.

To graph a function you need to make a table substituting x and getting what y would be so you can graph all the points an make a line.

After you have your table to graph a function, with the x and y values you graph it.

Lesson 10-7: Rotational Symmetry and Rotations

A rotation is when you rotate a figure or shape by certain number of degrees. If you try to do a rotation in a figure, just grab it and rotate it by any degrees.

Rotational symmetry is when you rotate a figure or shape and it doesn’t change. For example, lets say you have a perfect square and you rotate it 90°, it will look exactly the same when you rotate it and when you don’t.

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Lesson 1: Evaluating and writing algebraic expressions

In this lesson you’ll have to do expressions depending on the substitutions of letters.

In the example above you would have to substitute m but you have no way of knowing what m is, so you need a substitution m=6

So now you can solve it you first need to multiply 5 x m(6) and then you can subtract by 6.

In transforming formulas you have to solve the equation for the indicated variable

To solve the equation you have to find the way to leave the variable alone 

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To add integers with the same sign you need to add the absolute values and then give the same sign.

Using the rules above you can know that the answers are 21 and -11

To add integers with different sign you need to find the difference of the absolute values and then give the sign of the integer with the greatest absolute value.

Subtracting an integer is the same as adding its opposite

You needed to subtract 26 so with the rules you need to add -26 and it gives you -17.

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There are two types of random samples and surveys, biased and fair.

Biased random samples and surveys say something biased toward something, example: ”Do you prefer the natural beauty of hardwood floors in your home?&nbsp;“ This is biased because it says “natural beauty”.

Fair random samples and surveys are fair to everyone, example: ”Do you exercise regularly?” This is fair because it doesn’t say biased words.

We can use data to create a double bar graph that can show us the information that was put there.

Sample space is basically the set of all possible outcomes from a probability experiment. In other words, it is all the possible outcomes you can get from doing something, like rolling a fair dice twice!

~sample space for rolling a fair dice twice~

As you can see here, you might get 4 in the first try and 2 in the second try. There are 36 possible outcomes on rolling 2 fair dice!

Operations with Fraction and Mixed Numbers

In these problems we have to find y for x = 1, 2, 3, and 4.

A function is a relation in which there is only one y-value for each x-value.

You can write a rule and an equation that tells how to find one value of the functions when the other value is known.

The vertical line test is a way to determine if a relation is a function.

Probability is the chance that something will happen.

For example, if you have 2 green shapes and 3 yellow shapes what is the probability you are randomly selecting green?

Well, how many green shapes are there? That’s right! 2 green shapes. Now, you are going to add the total of ALL shapes. In this case 2+3=5. 5 shapes in total. We can put probability as a fraction, decimal, or as a percent! I’m going to do the 3 of them. As a fraction: 2/5, as a decimal: 0.4, and as a percent: 40%.

And you are all done! That is how you find the probability of an event!

You may also do this with not only the color but the shape, size, and number, like what is the probability that you will randomly pick a triangle or a heart? Try to figure it out!

Lesson 2-4: Comparing and Ordering Fractions

Fractions are very easy to understand. First of all, to order fractions, you need to know how much their value is. The closer the numerator is to the denominator, the bigger it is. And if the denominator is a very big number, it means that the fraction is divided into plenty of pieces.

Another way of Ordering and Comparing is the LCD, or the Least Common Denominator.

To finding and LCD is fairly simple. You just need to find a number that is divisible by the 2 fractions’ denominators. If you can’t find a number that may be an LCD, just multiply the 2 denominator, but this also may not be the Least.

The LCD of this 2 fractions is 24, since it is both divisible by 12 and 8

Reminder— Even if 2 fractions don’t look the same, they might be equal, so simplify them!

Lesson 2-5: Mixed Numbers and Improper Fractions

A mixed number is a whole number and a part of a whole together. For example a mixed number will be 4 5/6. That mens that the original fraction was 4 x 6 + 5 (29)/6. So if you see 29/6, you can simplify it. You can see how many times 6 fits in 29, which is 4, and then the rest gies on the numerator, which will be 5. That’s how we get 4 5/6

An improper fraction is a fraction with the numerator larger than the denominator. Improper fractions are also mixed numbers written in as the original form. As we already saw, 4 5/6 is a mixed number, in which the original form was 29/6. Well 29/6 is the improper fraction. 

An improper fraction is NEVER in simplest form.

A spreadsheet and data displays are used to display data. These are some of them:

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Graphs can show trends. They can show positive or negative trends. Sometimes graphs show no trend.

On the example you can see it is a positive trend so it means that with more years of school the more income they get in their salary.

A stem-and-leaf plot is a graph that splits the number’s digits into a stem (the first digit or digits) and a leaf (the last digits on the left)

When adding or subtracting decimals, you need to always align the decimal point, no matter the digit difference.

In this lesson, there are 3 properties to keep in mind.

Permutations are ways a group of items can be arranged in ordered sets.

How many permutations are there in the word CHAIR?

The word CHAIR has 5 letters, so let’s solve 5 factorial which is expressed like this 5! So now, let’s solve 5x4x3x2x1. That gives you 120! 120 is your answer! 

When graphing a linear equation, you are going to start with making a table for the equation (Image to the right). First, you are going to put the axis that is alone, in one side of the equal sign, in the right, and the axis that is with other numbers or any addition/subtraction/multiplication/division to the left. Then, you are going to put the equation in the middle. In the left, your are going to put numbers to substitute the axis in the left side, preferably use the numbers -3,-2,-1, 0,1,2, and 3, or you could use the 3 middle ones or 0,1, and 2. Almost finally, you are going to solve the equation 1 time per number and write the solution to the right. Lastly, use the numbers in the left and in the right as coordinates depending if they represent the x or y axis to graph them in your graph Here is a video on how to do this.

Experimental probability is really easy! ☆

Even though it’s name sounds hard, experimental probability is basically the ratio of the number of times an event occurs. Here, let me show you an example!

Here is a video that will help you understand this topic a little bit better ☆!

Lesson 10-1: Graphing Points in Four Quadrants

To graph a point in the correct coordinates, you’ll have to mark it. But, how do we do that? Well, the first rule of graphing coordinates, is that you’ll first have to find X, and then Y. So if you find the coordinates put it in the order (X, Y) not (Y, X). To find the coordinates you’ll have numbers on the X-axis and on the Y-axis that will indicate in which coordinate you are at, and if it’s positive or negative.

In a graph with coordinates, there are four quadrants and to axis. As we already explained, there’s the x-axis and the y-axis, but for the quadrants there’s: quadrant I, quadrant II, quadrant III, and quadrant IV.

Point L = (-2, 2) Point B= (-7, 3) Point E= (-5, 0) Point H= (-2, -7) Point A= (2, 7) Point F= (8, 0) Point D= (4, -3) Point K= (8, -8)

There are rules for the quadrants. In quadrant I, the coordinates will always be (+, +). In quadrant II, the coordinates will always be (-, +). In quadrant III, coordinates will always be (-, -). And in the quadrant IV, the coordinates will always be (+, -).

To add or subtract fractions, you have to have the same denominator on both fractions. To do that you have to find the same multiples of both numbers

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The Median is that when you put the number in order from least to greatest it is the number in the middle. If there is no number in the middle you need to take the mean of the two middle numbers.

The mode is the number that appears the most in a set of numbers if none repeat there is no mode.

The range is subtracting the least value from the greatest value.

The mean is the sum of all the data divided by the number of items in the number set.

Using the metric units you can estimate the length of this examples.

This are all the conventions of all customary units

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Dividing decimals by 10 or by 100 is very easy. You just but the decimal’s decimal point 1 time to the left for every zero it has.

To divide decimals by decimals is more complicated

For dividing decimals by decimals you need to move the divisor’s decimals point until it is a hole number. You move the decimal point the same number of times for the dividend.

Lesson 2-1: Exponents and Order of Operations

An exponent is a little number standing on the top of another number. It symbolizes how many times you need to multiply the bigger number by itself.

As the name suggests, the order of operations tells you which operations in an equation you should do first or last

To remember the order, always use this acronym:

P: PARENTHESIS E: EXPONENTS M: MULTIPLICATION D: DIVISION A: ADDITION S: SUBTRACTION

Remember: An exponent is not multiplication, it is MORE than one multiplication.

Estimating with Fractions and Mixed Numbers

To estimate the sum, the difference, or the product you will need to round your numbers to the nearest half of a number. If you’re finding a quotient you will have to round the numbers to be compatible. Examples:

Now that you have your rounded numbers you can just add them, subtract them, divide them, or multiply them. Examples:

Lesson 4: Solving equations by multiplying and dividing

In this lesson we will be seeing equations but with only multiplying and dividing

The equations in this lesson will be harder since multiplying and dividing are harder than adding and subtracting

Remember all the equation steps which are: Identify the operation applied to the variable State the inverse operation Apply the inverse operation on both sides of the equal sign Inverse operations undo each other Solve Check by substituting.

In this example as shown below you have n divided by 7 = 8

You have to follow the equation steps from the top 

A translation is when a coordinate in a linear graph is moved left, right, up, or down. They can vary on the amount of units they move on and they also can combine different movements. Example to the right.

When you are translating a shape (a triangle, rectangle, etc.) Always write where a point ends up so you don’t get lost.

Interest is the amount of money that you pay when you borrow money, or the amount of money you earn on a deposit

Compound interest is the interest that is earned on both the principal and any interest that has been earned previously

This is the formula for compound interest:

Lesson 2: Using number sense to solve equations

In this lesson we will be solving equations

To solve an equation you need to first Identify the operation applied to the variable State the inverse operation Apply the inverse operation on both sides of the equal sign Inverse operations undo each other Solve Check by substituting.

When multiply decimals you can forget about the decimal and you just multiply as always then after you multiply you move the decimal point depending of how many numbers are after the decimal point if there a 2 numbers then you move the decimal point two times to the left

Topic 4 Lesson 5: solving 2 step problems

In this lesson you will be seeing problems that you can’t do in one go and they are called 2 step equations

In this lesson you need to define a variable and write an algebraic expression 

Lesson 3: solving equations by adding or subtracting

Remember this steps: Identify the operation applied to the variable State the inverse operation Apply the inverse operation on both sides of the equal sign Inverse operations undo each other Solve Check by substituting.

The way to solve the equations is the same as in lesson: 2 but now it’s only adding and subtracting 

To do this equation you’ll have to do the steps on top

Topic 4 Lesson 7 Graphing and Writing Inequalities

For this lesson you need an example to understand it

In this lesson you need to graph the solution of the inequalities.

To add or subtract a mixed number you first need to add or subtract the whole number and then the fraction.

When you multiply or divide fractions you can simplify both the numerator and denominator to make them smaller. If you have a mixed number, you have to make it into an improper fraction

A combination is a grouping of objects in which the&nbsp;order of the objects does not matter.

The formula for doing a combination is this one:

Let’s solve for you choosing 3 people from a group of four. Use the formula to guide you. Ready? The answer is 4! Great work!

A frequency table is used to show hoy many number of data value occurs in a set.

To simplify these, we would need to divide both numbers by the same number, for this to happen both, the numerator and the denominator should be divisible by that number you’re dividing by.

So, simplifying fractions is basically just dividing both the denominator and numerator by their greatest common factor. Simplifying fractions can make some jobs easier, it can also help to multiply and divide, etc.

Why is it 3/7 now? Well, it’s because both 15 and 35 can be divided by 5, so the fraction 15/35 isn’t on simplest form. When you divide both numbers by the greatest common factor, it is on simplest form.

Lesson 10-6: Line Symmetry and Reflections

A reflection acts like a mirror. When a set of points in a graph are reflected, they get reversed. There are 2 types of reflection in linear graphs, the vertical reflection and the horizontal. When a point gets reflected vertically, only the x axis is inverted, from negative to positive or viceversa. And when a point is reflected horizontally, only the y axis is inverted.

Line symmetry is when you can draw a line in a figure and both sides are congruent, meaning that they have the same size, dimensions, angles, etc. For example, if you draw a line at the middle of a perfect square, it’ll have line symmetry. But if you draw a line at the middle of any asymmetric shape, it won’t have.

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