Book Reviews

Just wanted to leave a quick note to let you all know that the book reviews have moved to my new blog http://little-book-worms.blogspot.com/

Introduction to Computer Science 11.29.13 23:26

I restarted the class on computer science from Udacity this past week. So far this course discussed the difference between a toaster and a computer in that a toaster is designed to do one thing while a computer can do anything we program it to do. A program is a code that tells the computer what to do in a given situation. I also got to do a little bit of programming in Python which is the language they use for the course. We started out with arithmetic, typing in print 1 + 1 and running it in the interpreter produces the result 2 . The print function is a way to see what the code we built is doing. Then it moved on to assignment statements. Typing in any word (name for example) an equals symbol and an expression assigns that expression to that word so if we write name = 5 * 3  and then print name it will evaluate to 15.

So lets say we wanted to print out more than just numbers, this is where we need to use strings. To create a string we could just use print 'This is a test' and the output would be This is a test. To create a string it needs to have quotes or double quotes on each side so 'the', "isn't" and '"simple" she said.' would all be valid strings. As long as the string starts and ends with the same type of quote it will work. Simple enough, but what if we have a really long string and we don't want to type it out each time? Then we could assign the string to a name and just print the name. For example: einstein_quote = '"Education is what remains after one has forgotten what one has learned in school." Albert Einstein' we can then run print einstein_quote and the string "Education is what remains after one has forgotten what one has learned in school." Albert Einstein will be the result.  

This brings us to another question, what if we have an entire page as a string and we want to find where a specific word or link first appears? We can use the find function, so starting with the string we just used we could then write einstein_quote.find('one') and the output would be the location of the first instance of the word one in our string. Of course we would have to print that piece of code to actually see the location as a number. I'm going to explain this using a shorter string,  'one' for example, if I typed in print 'one'.find(n) I would get 1 as the result because the location is initialized by numbers starting at zero and n is the second letter in the word one. We could type print 'one'[1:] and this would result in ne, because it would print starting from position one through the rest of the string. If we typed print 'one'[1:2] we would get n because the end point (2 in this case) prints up to but not including that position. We could also write print 'one'[:2] and get on, because it would display from the start of the string up to position 2.

So why and how would we use this? Well, for lesson one the goal is to be able to extract a link from a webpage. Let's go back to our einstein_quote string and say we wanted to print just the part one has forgotten what one has learned in school. we could write some code to do this as follows.

first_one = einstein_quote.find('one')
end_point = einstein_quote.find('.') + 1
print einstein_quote[first_one:end_point]

This should produce the result we are looking for. The reason the second step includes a + 1 is because the end point of an index (the [:] used to show parts of a string  ) prints up to that position and we want to include that position.

Algebra 11.21.13

This past week I reviewed graphing basics and how to find the distance between two points. Let's start with some graphing basics. Points on a graph are denoted (x,y) where x is the distance a point is from the origin (0,0) on the x or horizontal axis and y is the distance from the origin on the y or vertical axis. A graph has 4 quadrants. Points in quadrant 1 will have both a positive x and y value so (x,y). Points in quadrant 2 will have a negative x and positive y (-x,y). In quadrant 3 both the x and y values are negative (-x,-y) and in quadrant 4 the y value is negative while the x value is positive (x,-y). It is important to note that while a graph will not show all the numbers the x and y axis extend indefinitely in both the negative and positive directions.

We can find the distance between two points using the Pythagorean Theorem a squared plus b squared = c squared where c is the hypotenuse of a right triangle. No matter which two points we look at they all can be looked at as being a part of a right triangle. So if we were going to find the distance between the point (1,2) and (2,4) we need to look at the big picture. The distance between the two x coordinates 1 and 2 is one and the distance between the two y coordinates 2 and 4 is 2. 1 squared is 1 and 2 squared is 4, one plus 4 is 5. Then we take the square root of 5 to get the distance between those two points. So the distance formula is      
                               (x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−−        Distance =      √

Mandarin Chinese 11.21.13

I started the Rosetta Stone Mandarin Chinese course today. So far I'm having trouble pronouncing some of the sounds but I'm starting to grasp word meaning. So here are the words I remember as close as I can get to what they sound like to me.

nu ren - woman
nan ren - man
nu hi zi - girl
nan hi zi - boy
you yong - swim
pao bu - run
hu - drink
ni hao - hello
si chen - goodbye
kan - read
shu - book
shuay - water
che - tea

As I learn more I will try to put in a few complete sentences.

Algebra 08.04.13

An equation is two expressions that are set equal to each other. Conditional equations, identities, and not equations. A Conditional equation is such that its truth is dependent on which number x stands for. For example 2x+1=5 is only true when x = 2. An Identity is when both sides are exactly the same so 3n+1=3n+1 because when we simplify we get 0=0. x-3 and 2/x +1= 2/x +2 are not equations simply because they do not equal anything or are false.

Twins 08.02.13

My girls are 7 months old today and very different. Rain is intense in her attention to things and small details, she is quiet and is always off exploring (even though she only scoots and rolls). Sometimes she squeaks when she is excited, she doesn't like sweet food as much as veggies. Her favorites are peas and plain cereal. Rain was the first to escape from the car seat when she's not strapped in and the first to figure out how to hold her own bottle. She's always examining objects and even her own hands just to see how they work. Today I watched her examine a Lego block for a good twenty minutes, just rolling it around putting her hand and fingers in all the holes.

River is more vocal and always wants to be the center of attention, where Rain examines new things River shakes them and tries to get them to make noise. She was the first to vocalize and giggle, her belly is ticklish too. River also growls at people and things, not an angry growl but just a very guttural sound, it's very comical, like a tiny dinosaur trying to be heard. She loves juice and the sweet fruity foods, when she doesn't want to eat something she blows raspberries to get it out of her mouth. Today while I was feeding her and Rain mixed veggies she blew most of it out and I got more on me then she ate. River is goofy, personable and loves to cuddle.

It's crazy to see how different they are. Rain is the little scientist trying to figure out how the world works and River is the social butterfly trying to mimic the big people.

Algebra 08.02.13

The last part of lesson two focuses on exponents. When a dealing with several variables multiplied together that have exponents the exponents can be added together to simplify the expression. For example if I have x^2*y^4*x^9*3 (where ^ is used to show there is an exponent) I can simplify to x^11*y^4*3.

Lesson three is about polynomials. The distributive property allows us to take the expression a(b+c) and simplify it to ab+ac. For example say we have the expression 5-(6-x) we can think of the negative sign as a -1 and distribute it to get 5-6+x or -1+x. When multiplying polynomials together make sure to multiply every term in the first polynomial by every term in the second polynomial. For example (x+2)(x+5)= x^2+5x+2x+10 which simplified is x^2+7x+10.

Algebra 08.01.13

Lesson 2 in Udacity's College Algebra course is about expressions. It starts by going back over the commutative property of both addition and multiplication. As a reminder the commutative property just means that the order of terms or factors doesn't matter. So a+b+c is the same thing as b+a+c and a*b*c is the same as c*a*b and so on. When combining like terms it is important to remember that they need to have the same variable and power (x cannot be added to x squared). So 2x+3x+4-2y can be simplified to 5x-2y+4 but no further.

A polynomial is an expression with constants and/or variables that are combined using addition, subtraction and multiplication, where all exponents are non-negative integers. The degree of a term is the sum of exponents on the variables in that term. So x squared times y cubed would have a degree of 5 because the exponents 2 and 3 add up to 5. The degree of a polynomial is equal to the highest degree of any of its terms. the standard form of a polynomial is to put the terms in order from highest degree to lowest degree.

Psychology 08.01.13

So over the last few days I have been working on Lesson 3 "The Biology of Behavior". Last post I talked about colorblindness which is a trait controlled by one gene. There are, however, many traits that are controlled by multiple genes such as eye color or height, these are called polygenic traits.  Traits are not just affected by genes but also by the environment. Identical twins can posses very different traits because of their life choices, if one overeats and smokes while the other doesn't  they are going to look very different. Epigenetics is the study of how our environment can turn different traits on and off. There is a study about two genetically identical mice whose mothers were fed different diets while pregnant, one high in methyl groups and one without. The end result was that one mouse came out obese and yellow while the other was normal and brown. These mice have the same genotype but different phenotypes. The same principals hold true for personality traits, there is a gene that is referred to as the warrior gene, this gene was thought to cause aggressive behavior but as it turns out many people have the gene without displaying aggressive behavior. In fact it takes an abusive childhood coupled with the Monoamine Oxidase A (warrior gene) to cause the more aggressive personality, making this an excellent example of how the environment can affect the gene expression.

The next part of Lesson 3 had to do with the nervous system. The nervous system can be broken down as into the Central and Peripheral nervous systems. The central nervous system consists of the brain and spinal cord while the peripheral accounts for all the other nerves. The peripheral system can be broken into the somatic and autonomic systems, the autonomic system controls functions like heart beat, breathing, digestion etc... the somatic is responsible for voluntary functions. Both of these systems have motor and sensory nerves. The autonomic nervous system has both a sympathetic and parasympathetic nervous system.   The sympathetic nervous system is responsible for the fight or flight response, when activated it does things like dilate the pupils, increase heart rate, inhibit digestion and relax the bronchi to make taking in more oxygen possible. When the parasympathetic nervous system is activated it has the opposite effect, the pupils and bronchi are restricted, heart rate is slowed and digestion is stimulated.

There are two types of cells that make up the nervous system, Glial cells and Neurons. Glial cells provide structural support and remove debris while Neurons process and transmit information. There are three different types of neurons; sensory, motor and interneurons, the interneurons connect neurons to one another. Below is a very rough drawing of a neuron. The Soma is the center of the cell and contains the nucleus. Branching off of the soma are the Dendrites, dendrites connect to other neurons and are the receivers for the cell. The axon transmits the electrical charge (when a neuron fires) to the terminal buttons, the mylin sheath (not present on all neurons) helps to speed up this process. The terminal buttons release neurotransmitters to neighboring neurons and the process repeats. Neurotransmitters can be either inhibitory or excitatory, excitatory neurotransmitters increase the chance that a neuron will fire while inhibitory ones decrease that chance. So to put it all together lets say an excitatory neurotransmitter is received by the dendrites of a neuron, if enough are received the neuron will fire off an electrical signal that runs down the axon and tells the terminal buttons to release more neurotransmitters once they do the cycle repeats with the next neuron.

Neuron

Algebra 07.28.13

Yesterday I finished up the Introductory Algebra Review on Udacity ( I know the certificate says 07.25.13 but that was for getting 80% of the questions right, I finished out the last unit and test on the 27th), I'm starting the College Algebra course today. Lesson one is about numbers, numbers are a construct in that they do not exist in the real world, we use the concept of numbers to do everything from balancing our checkbooks to launching satellites into orbit. Natural numbers are the numbers 1 to infinity, increasing by 1 at each step so 1, 34 and 375 are natural numbers but -3, 3.6, and 0 are not. Whole numbers are all natural numbers and the number 0. Integers are all whole numbers and negative numbers that are not fractions or decimals so they range from -infinity to +infinity. These are all sets of numbers, of course a set can be made up of any combination of numbers. Another way to express the relationship of of one set of numbers to another is to say x is a sub-set of y, for example whole numbers are a subset of integers because all whole numbers are integers. Rational numbers are any number that can be written as a/b where a and b are integers and b is not 0. Irrational numbers cannot be written as ratios of integers, so any square root that is not a perfect square is an irrational number, pi is also an irrational number.

There is also a review of variables, constants, expressions and equations. I am not going to go back over them because that information is in my previous algebra posts.

Psychology 07.28.13

Today in the Intro to Psychology class I went over good practices in experiments, specifically in reference to psychological experiments.  Randomly assigning participants to either the experimental or control group can help insure that both groups are similar. Not telling participants which group they are in eliminates bias on their part, having  both the participants and the experimenter in the dark is called a double blind study and this eliminates bias on both parts. An important thing to note about psychology is that a review board has to okay any experiments, the benefits must outweigh the risks and participants must sign informed consent. Participants must also be able to leave the study at anytime without negative consequences.

Lesson 3 starts out with an introduction into how long modern humans have been around as compared to other humans. Modern humans have existed for the last 200,000 years, before that were Neanderthals, homo-erectus and homo-habilus. Homo habilus, the first humans, lived from about 2.4 to 1.4 million years ago. Our closest relative would be the Neanderthals and we actually interbred with them before they died off around 28,000 years ago so many of us have Neanderthal DNA. Our closest living relative is the chimp, with whom we share about 98% of our DNA, the other 2% is what makes us uniquely human. These changes happened over a long period of time due to mutations and natural selection, in other words, we evolved.

The changes to our brains is probably what makes us most human. Our brains are bigger than our ancestors and the most differences between our brains and theirs is in the frontal lobe. Having a bigger brain did not come without consequences, childbirth is more painful because of larger skulls and we need more food to fuel our brains.

There is a section on DNA structure, from the nucleotides Thymine, Adenine, Guanine, Cytosine to the 23 chromosomes that make up our DNA. Also it is not the amount of DNA but the sequence that makes us more complex, in fact some amoeba's have more DNA than than anything else on earth. Only about 2% of our genome actually codes for proteins (these portions are called genes), most of the rest regulates which genes are activated, but some of this remaining 98% we're not really sure as to what its purpose is. Alleles are versions of genes, for example my dad has the allele for colorblindness where as my mom does not, so being that this allele only exists on the x chromosome (of which I have 2) I have one allele for colorblindness and one for normal vision. Luckily for me the colorblindness allele is recessive which means that I would need two in order to be colorblind. However this also means that there is a 50/50 chance my daughters will also carry the colorblindness allele and 50/50 chance that any sons I would have would be colorblind. Why? Because males have an x and a y chromosome so they only get one chance at a good allele, where as the x chromosome from my mom allows me to recognize all the colors males don't have a second x chromosome to cancel out or hide the colorblindness allele. This is why more men than women are colorblind, for a woman to be color blind her father would have to be colorblind and her mother would have to carry the colorblind allele.

Algebra 07.27.13

Yesterday we learned a bit about slope and how horizontal lines will always have a slope of zero because if you divide zero by any number it is still zero. This brings us to vertical lines, these lines have a change in x of zero, because we can't divide by zero the slope of any vertical line is undefined. Given two points (x₁, y₁) (x₂,y₂) we can then find the slope using this equation where "m" stands for slope, m=(y₁-y₂)/(x₁-x₂). So how do we find the graph or visual representation of a line from this information? We use the slope of a line and a point on the line, we can also represent a line using slope-intercept form. This uses the slope and the y intercept to represent a line or y=mx+b where again m is slope and b is the value of y at the y intercept. We can also use point-slope form y-y₁=m(x-x₁) where (x₁, y₁) is any point on the line. If we solve for y we can convert point-slope form to slope-intercept form. 

Psychology 07.26.13

I'm going through Udacity's Introduction to Psychology so here are my notes. Psychology is a field of science that explores the questions of why we do what we do. Psychology covers a wide area, from developmental psychologists who study why children acquire different skills at different times to industrial psychologists who review workplace dynamics to create a more productive environment, the classic image of a Freudian character asking a patient how that made them feel is only a small part of what psychologists do.

Lesson 2 is about research methods. The first part of the lesson is about correlations. Let's say that the price of pickles went up 2 dollars and the price of milk went up 1 dollar, so looking back we can see that as the price of pickles goes up so does the price of milk. So if we were to plot the data points on the price of pickles to milk we would likely see a correlation. However, that does not mean that one caused the other, the price of milk could be going up because of the economy or higher fuel prices, pickles could be in higher demand recently or a million other things could have affected the price. Correlation does not imply causation.

To determine whether one thing causes another is often more difficult than plotting points on a graph. One way is through an experiment. A good experiment starts with a testable hypothesis. I want to know what the best time to feed my girls vegetables is, I've noticed that they tend to eat more vegetables if they have them before meals or as snacks. So my hypothesis is, My girls will eat more of their vegetables if they are fed to them before their meals. Now I set up my experiment, I will feed my children vegetables before meals one week, during meals another week and after meals another week.. The independent variable (the one I will manipulate) in this experiment is the time they are fed vegetables. I will measure how much they eat by how much is left afterwards making quantity remaining my dependent variable. To ensure that my data actually measures what I want it to measure I will control the time of day they eat, what vegetables are served, and what the environment is while they are eating (tv on or off, living room or kitchen), these are my control variables. If in fact they do eat more when they are fed veggies before meals I will have proved my hypothesis. The difference between a hypothesis and a theory is that a theory encompasses many proven hypothesis to come up with an idea that ties them all together. Saying for example that evolution is a theory is really saying that it is the best explanation we have for all the data and proven hypothesis currently available to us.  As an aside, during my hypothetical experiment I recorded measurable data (how much of the veggies were left), if I wanted to know how much my kids like veggies I would have to find a way to measure "liking a food" liking something is a construct in that we know what someone means when they say they like something but it isn't a quality that we can measure. So we use operational definitions, in this case we could say kids will eat more of the foods they like so we will measure how much they eat.

Algebra 07.26.13

Unit 5 Section 1 starts out by introducing t-charts and ordered pairs for displaying data points. A t-chart looks like a t with the two variables (x and y) on the top and numbers on the bottom. This t-chart  shows some data points for the equation 2x = y.







This can also be shown with ordered pairs, for example (0,0) (1,2) (2,4). Both represent the same data and this data can then be plotted on a graph as shown above. An intercept is where the line crosses the x and y axis, in this case both the x and y intercepts are at (0,0). In any linear equation the x and y intercepts can be found by setting x or y to zero. If you set x to 0 and solve the result will be the y intercept, and if you set y to 0 and solve the result will be the x intercept of the line. Not all lines will have an x and y intercept, vertical lines will only cross the x axis and horizontal lines will only cross the y axis. Vertical lines will have equations that look like x=2 while horizontal lines will have equations that look like y=5.

Unit 5 Section 2 is about slope. Slope is how steep a line is and is calculated by rise/run, or the change in y divided by the change in x. Horizontal lines will always have a slope of 0 because the y value does not change and 0 divided by any number is still 0.

Algebra learning journal to date.

07.12.13 6:58

I am reviewing some basic algebra through the Udacity site. Some notes: the commutative property means that order of any two variables will not change the outcome. Addition and Multiplication both have and commutative property. Associative property means that the order of any three or more variables will not change the outcome, and again both addition and multiplication but not subtraction or division share this property. For example

5+2 = 7 and 2+5 = 7, also 2*5 = 10 and 5*2 = 10, this is an example of the commutative property. 5-2= 3 and 2-5 = -3, 3 is not = to -3 to subtraction does not have the commutative property. Just like 10/2 = 5 and 2/10 = 0.2 so division also does not share this property.

The next section is about powers, any thing to the first power is itself and anything to the zero power is one. So 3¹ is 3 and 3⁰ is 1.

Order of operations: ( ) and [ ] first, simplify exponents, multiply and divide from left to right, add and subtract from left to right.

Area of a square is l² or l*l (length²).

Area of a rectangle is l*w or Length times Width

Area of a triangle is 1/2 base times height.

07.14.13 3:08

This section is about fractions. An improper fraction is one in which the numerator is larger than the denominator. For example 4/3, this can be simplified to 1 1/3. 1 and 1/3 is a mixed number, which is a whole number and a fraction.

The denominator must be the same to add or subtract fractions, this is done by finding the lowest common denominator (the smallest number that both denominators go into). For example if I'm going to add 1/3 and 1/2 I have to find a number that both 3 and 2 go into. 12 is one but it's high and that would make the fraction more difficult so the lowest common denominator is 6. So to change both fractions I look first at what I have to multiply 3 by to get 6, that’s 2 so I multiply both the numerator and denominator by 2 to get 2/6. Going through the same process for 1/2 give me 3/6 then I just add the numerators, leaving me with 5/6.

Multiplying fractions is pretty straight forward just multiply the numerators together and the denominators together. Dividing fractions is the same as multiplying by the reciprocal. so if I want multiply 1/3 by 1/2 I get 1/6 (1*1 is 1 and 3*2 is 6) but if I divide 1/3 by 1/2 I get 2/3. This is because the reciprocal or 1/2 is 2/1 and then I multiply, so 2*1 is 2 and 1*3 is 3 so the answer is 2/3.

07.15.13 09:36

This section is on decimals and rounding. Starting with a decimal such as 3.462 and rounding up to the nearest whole number gives us 3, rounding to the first place gives us 3.5, this is because if a number to the right of the decimal is less than 5 it is rounded down and if it is greater than five it is rounded up.

The second part is about circles, Circumference is the distance around a circle or the perimeter of a circle. Diameter is the length of a line that passes through the center and whose end points lie on the circle. Radius is the distance of a line with one end point on the center of the circle or origin and the other on the circle.

Pi equals circumference/diameter. Circumference = 2*pi*r . Area equals pi*r².

07.19.13 07:52

This part of section 2 is about scientific notation. Scientific notation takes a number and simplifies is down to a decimal times 10^x . For example 535,000,000 can be written as 5.35 * 10⁸. The key to remember is that only one number should come before the decimal in scientific notation. Small numbers can also be written in scientific notation. For example .000632 can be written as 6.32 * 10^-4, and again remember only one number comes before the decimal.

Section 3 starts out with rates. Rates are things like miles per hour or diapers per package, they are used to compare two quantities with different units. So I can go 67 mph which would be the same as saying I went 67 miles in one hour. 16 diapers per 4 packages is the same as saying there are 4 diapers in each package or 4 diapers per package. Both of these are examples of a rate because miles, hours, diapers and packages are all different units.

A "Unit Rate" is any rate where the value of the denominator is equal to one.

A ratio is used when comparing two quantities with the the same units. For example 2 out 3 of my girls wear diapers, this is a ratio because there is only one unit, in this case it is "my girls". Ratios can also be written with colons so 2/3 of my girls can be written as 2:3.

07.19.13 04:56

This part of section 3 is about Conversion Factors, a conversion factor is a number that relates the quantity of one thing to the quantity of another thing and it always equals one. For example 12 eggs /1 carton is a conversion factor because 12 eggs is equal to 1 carton.

The last part of section three is about converting decimals and percents. To change a decimal to a percent move the decimal 2 places to the right, and to change a percent to a decimal move the decimal to places to the left. So if I want to say that 2 of my 3 children wear diapers that equals .667 of my children which is 66.7%. If my oldest daughter finishes 75% of her homework she has finished .75 of it and so on.

Unit 4 starts out with variables. A variable is a symbol or letter that can be used to represent an unknown value.

07.23.13

Unit 4 section one is about algebraic expressions, an algebraic expression has terms that include numbers and variables that are connected by addition, subtraction, multiplication and division. 3x-4 is an algebraic expression. To evaluate an expression the value of the variable is plugged in. For example if x=3 evaluate 3x-4 it would be 3(3)-4 or 9-4 which is 5. 4x+3(x-2) to simplify this I need to combine like terms, like terms are variables that are raised to the same power so x and 2x are like terms but not x^2 or 2y. So to simplify combine the coefficients of like terms (coefficients are numbers that variables are multiplied by) but leave the variable alone. Constants are numbers with out variables so in our example 4x+3(x-2), 4 is a coefficient and 2 is a constant. To simplify we follow the order of operations. So 4x+3x-6 turns into 7x-6, and our original expression is simplified.

Unit 4 Section two is about linear equations, linear equations have an equals sign, an expression on both sides of the equals sign and no variable has an exponent higher than one. So 5x+3=40 is a linear equation. These can be solved for the variable. There are some that don't have a real answer and others where x (the variable) can be all real numbers and still make the equation true. For example 2x-3=-3+2x, simplified down this is all real numbers because no matter what number is x is substituted for the equation remains true. Now 2x=2x +2 doesn't have a solution because if we simplify it we get 0=2 which is not true.

The same principles that apply to linear equations also apply to inequalities. Such as, if you do something to one side then you have to do the same thing to the other side. The only difference is that if you divide or multiply by a negative in an inequality the sign flips. So -3x<9 simplifies/ solves to x>-3.

Unit 4 Section three is about proportions and similar triangles. Proportions start with a rate and then can be used to find other data by cross multiplying and dividing. For example if I work out and lose 3 pounds a week how many days will it take me to lose 23 pounds. So 3 pounds/7 days = 23 pounds/x days. This problem can be made into a proportion. So now I multiply 23 pounds by 7 days to get 161 and then divide by 3 pounds, so because I have the variable pounds on the top and bottom of the equation I can cancel it out and I get 53 2/3 days.

07.25.13

Continuing with Unit 4 section three, Similar triangles are triangles that have the same angles but different sides. The angles will all have the same measure and the sides will be in proportion to each other. This is important because similar triangles can be used to find heights in the real world among other things. Lets say that we hold up a yard stick and it casts a 10 foot long shadow, we can then find the height of anything else by measuring its shadow because the sun is at the same angle to the yardstick as it is to every other vertical object. So lets say that we want to know how tall the tree outside is. We measure the shadow and it is 60 feet long, so now we can set up a proportion, 3/10 = x/60. Now we cross multiply and divide, 60 * 3 is 180, divided by 10 is 18. So the tree is 18 feet tall.

Unit 4 Section 4 is about the Pythagorean Theorem which states that in a right triangle a² + b² = c², where c is the length of the hypotenuse. The Hypotenuse is the longest side of a triangle and in a right triangle it is opposite the 90 degree angle.