# 4.6: Personal Referents - Mathematics

Think, for yourself, personally, how big areas or distances can be for you. So, if someone says, I think I am going to invite about 300 people to my wedding, she can picture the gym filled up to capacity and have an idea about how large the wedding will be.

## Partner Activity 1

Think about personal referents for yourself and then share with your partner:

1. 200 miles
2. 10 feet
3. 50 pounds

## Practice Problems

Write a personal referent for 1 meter, 1 inch, 1 foot, and 1 yard.

## MathJax basic tutorial and quick reference

To see how any formula was written in any question or answer, including this one, right-click on the expression and choose "Show Math As > TeX Commands". (When you do this, the '

## 4.6: Personal Referents - Mathematics

Everyday Mathematics is divided into Units, which are divided into Lessons. In the upper-left corner of the Study Link, you should see an icon like this:

The Unit number is the first number you see in the icon, and the Lesson number is the second number. In this case, the student is working in Unit 5, Lesson 4. To access the help resources, you would select "Unit 5" from the list above, and then look for the row in the table labeled "Lesson 5-4."

The University of Chicago School Mathematics Project

University of Chicago Press

## More importance of mathematics

When it comes to education, one of the biggest problems today is that high school students do not take mathematics seriously enough. They are simply not interested in this subject, despite the fact that this structural science can provide them well-paid jobs in engineering, statistics, education, and technology.

Teenagers see mathematics as something boring, difficult and irrelevant to their lives, and do not take into account all the benefits that math can provide them in the future, such as a bigger college choice, or as we have mentioned, a great paying job in the profession.

We bring you another 7 reasons why your child should be concerned with math and why is it important for his future:

Math makes your child smarter. Mathematics for learning is the same as the strength and durability for sport: basis that allows your child to surpass others and himself. Your child cannot become a big sports star if he is not strong and has problems with his health. Your child cannot become an authority in his work or prominent in his profession one day, if he doesn’t think smart and critically – and math, to a large extent, can help him with that.

Can make money with math. Let’s face it, not every child is predestined to be the winner of X-factor or a similar project. Even those who for a short time cannot afford to enjoy the glory and thus secure a bright future, usually, after a while, they return to school in order to finish up some education and to build their careers. Convince your child to skip several auditions and several sports games, and instead of that to do his math homework. Therefore, you will give him enough support to secure a job that will bring him a bright future and a stable income, more stable than singers and sports stars can earn for a living. Maybe this is not the case at the outset of his career, but it is certainly realistic to think so.

Mathematics is essential in order not to lose money. When a bunch of credulous people spends money on various pyramid schemes, thinking that they will make a fortune, they do so primarily because their math is not their strongest side. In particular, if you are the least bit familiar with statistics and calculations of interest, in a very easy way you will recognize the economic fraud and sellers of fog. With the help of science like mathematics, you will avoid a waste of money on various projects and tips that you believe can help you.

Math can provide your child with a ticket to the world. Global human consciousness is changing the world we live in. Clever children from Eastern Europe, India and China are considering mathematics and other “heavy” science as their ticket out of poverty and social degradation. Don’t you think that even your child can acquire knowledge that is payable everywhere in the world in a very wide range of occupations?

Mathematics is essential in a world of constant change. New technologies are changing the way we work and live. If you don’t want your child to use some instructions or constantly pay professionals in order not to be scared to press the wrong button, mathematics can be very useful in understanding how and why things work the way they work.

Math will be more represented in the future. Whether we like that or not, math is becoming an increasingly important factor in a variety of industries. Future journalists and politicians will speak less and analyze more. Future police and military personnel will use technology that is certainly an invention of scientists. Teachers and nurses will also rely on numbers and technology. Future mechanics and carpenters will use optimization electronics and analysis as much as they will use a hammer and a wrench.

Math makes up a large part of our everyday life. As a parent, you are bound to draw attention to your child all the advantages that this course provides. Of course, not everybody needs to become a mathematician or engineer, but this science can provide a bright future for your child can help him in the huge number of life situations to think critically, analyze and to bring the best possible decision.

## 2nd Grade Math Lesson Plans

### Even or Odd Nature Walk

Students will do a nature walk to find things in nature that are grouped in pairs that are odd or even.

### Exchanging Time

The lesson is used for students to practice basic time measurement, and understanding the basic units of time.

### Feed the Gator

Students will compare numbers with three or more digits using visual cues.

### Fun Fraction Pizza

Students will create a “pizza” from construction paper divided into 8 slices. They will decorate each slice and then exchange slices with classmates and then evaluate the fractions of slices that they have at the end. For example, 1/8 slices of my own pizza, 4/8 or ½ of pizza that was made by a female, 2/8 or ¼ that was made by my buddy. Note: Students should have already had some lessons about simplification of fractions.

### Graphing With Insects

This lesson is designed to teach students to draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Plus, solve simple put-together, take-apart, and compare problems.

### Odd or Even

This engaging lesson will help students determine whether a group of objects (up to 20) has an odd or even number of members.

### Scale It Up

This lesson will allow students to demonstrate knowledge use of scale.

### Shape Up

This lesson will allow students to demonstrate knowledge of various grade appropriate shapes.

### Skittles Graph

Students will learn to create a small bar graph using candy for information.

## 4.6: Personal Referents - Mathematics

Everyday Mathematics is divided into Units, which are divided into Lessons. In the upper-left corner of the Study Link, you should see an icon like this:

The Unit number is the first number you see in the icon, and the Lesson number is the second number. In this case, the student is working in Unit 5, Lesson 4. To access the help resources, you would select "Unit 5" from the list above, and then look for the row in the table labeled "Lesson 5-4."

The University of Chicago School Mathematics Project

University of Chicago Press

## Ken Ward's Mathematics Pages

The sum of the squares of the first n natural numbers is:

### Sum to n of the Squares of Natural Numbers Using Differences

We can use the same trick here that we used with the sum of the natural numbers, using differences.

n 0 1 2 3 4 5 6
n 2 0 1 4 9 16 25 36
Sn 0 1 5 14 30 55 91
Δ1
1 4 9 16 25 36
Δ2

3 5 7 9 11
Δ3

2 2 2 2

As usual, the first n in the table is zero, which isn't a natural number.
Because Δ3 is a constant, the sum is a cubic of the form
an 3 +bn 2 +cn+d, [1.0]

and we can find the coefficients using simultaneous equations, which we can make as we wish, as we know how to add squares to the table and to sum them, even if we don't know the formula.

In the table below, we create three equations, noting that d=0 from the first one (revealing the reason for the non-natural number zero)

n 0 1 2 3 0 1 5 14 d=0 a+b+c=1 8a+4b+2c=5 27a+9b+3c=14

Rewriting our equations:
a+b+c=1 [1.1]
8a+4b+2c=5 [1.2]
27a+9b+3c=14 [1.3]

Using 1.1 with 1.2 and 1.3 we can make two new equations:
6a+2b=3 [1.4]
24a+6b=11 [1.5]

By subtracting 3x Equation 1.4 from 1.5, we get:
6a=2
So a=1/3 [Also noting, by the way, that Δ3/3!=1/3]

Substituting a=1/3 in 1.4 gives
2b=1
So b=1/2

Finally, substituting these values into 1.1 we get
1/3+1/2+c=1
So c=1/6

[Actually, with the sum of the powers, the sum of the coefficients in the formula is always 1]

So, we can substitute our values into 1.0, to get the sum of the squares of the first n natural numbers (or first n positive integers):
n 3 /3+n 2 /2+n/6
Or, in various forms:

### Sum of the squares of the first n natural numbers using summation

We tried this with the sum of the natural numbers using summation, and fell flat on our faces, so this time we will go straight into setting up for the sum of the cubes, in the hope we will find our formula for the squares. (There is no reason stated here why this method should continue to work, however. Sometimes such approaches work only in specific cases. Fortunately, this approach does work for any sum of the powers of the natural numbers).

As before, we set up as follows:

Saying the sum to n is one term less than the sum to (n+1)

As expected, the cubic terms cancel, and we rearrange the formula to have the sum of the squares on the left:

Expanding the cube and summing the sums:

Dividing throughout by 3 gives us the formula for the sum of the squares:

### Sum of Natural Numbers Squared Using Errors

When we did this with the natural numbers, we found there was little work to do. With the squares, we have to go a little further. While the error on each term for the sum of the numbers is constant, the error on the squares depends on the term.

The graph below of y=x 2 , has the squares of the natural numbers represented by rectangles, and the area under the graph is approximately the sum of the squares (areas of the rectangles).

By integrating the x 2 , we find the area under the graph to be x 3 /3, so the area of the rectangles (sum of squares) is the area under the graph plus the error. If the number of squares is n, we can write n 3 /3 as the approximation:
[3.1]
Where is the sum of the squares and En is the error on approximating to n 3 /3.

The sum of the n-1 squares and the error En-1, gives us:
[3.2]

We note that the difference between 3.1 and 3.2 is
[3.3]
being the nth square

The difference between 3.1 and 3.2 is also:
[3.4]

Expanding (n-1) 3 in 3.4:
[3.5]
Rounding up similar terms:
[3.6]

Expressing the errors in terms of n:
[3.8]

This is a recursion formula. If we know one of the errors we can find the other. We do know what E0 is, because it is the error on the 0th term, which is zero. So, letting n=1, we have:
E1-E0=1-1/3

Letting n=2, and changing the relationship a bit:
E2=E1+2-1/3
E2=1-1/3+2-1/3

So we can generalise, the k-th term is:
[3.9]

We know the sum of the first n natural numbers,
()
so we can calculate (write down) the summation in 3.10 and put the result in
[3.11]

Substituting 3.12 in 3.13, we get the formula for the sum of the squares of the first n natural numbers:
[3.14]

### Using Infinite Calculus to find the Sum of the Squares of the First n Natural Numbers

We used this approach with the sum of the natural numbers.

The following graph is of y=x 2 , and the rectangles represent the sum of the squares.

y=x2 represents part of the sum of the squares, and the rest is the area between each rectangle and the function.

Looking at the graph, it seems the difference, or error En, varies at first it seems bigger than the term, and it gradually gets less.

The sum of the squares, where En is the error:
[4.1]

The error is the area of each square less the area under the graph:
[4.2]

Substituting 4.6 for En in 4.1:
[4.7]

### Summing the squares of the first n natural numbers by finding a general term (fails)

While this worked for the sum of the natural numbers, it does not work in the following example.

Set up for generalising:
[5.1]

It is now evident that the sum of the squares on both sides will cancel out. We could get a sum for the natural numbers out of this, but we can do that directly and much more easily.

After some thought, we can conclude that the last term must be negative for us to find an expression for the sum of the squares. It was negative for the sum of the natural numbers, and it will be negative again for the sum of the cubes, but will not work for the sum of the powers of four. That is, this method works for the squares of the odd numbers.

The expression corresponding to 5.4 for the sum of the cubes is:
[5.5]

Here the sum of the cubes on the right-hand-side is negative. I also note that in 5.4 and 5.5 the coefficients correspond to the binomial coefficients of the square and the cubic

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## How Is Math Related to Physical Therapy?

Physical therapists use math related to ratios, percents, statistics, graphing and problem-solving. Physical therapists need basic problem solving skills, group problem solving skills, inductive and deductive reasoning, and geometry skills.

Physical therapists treat people with injuries or abnormalities in their anatomy that limit their ability to move. They apply exercise and other rehabilitation techniques to reduce the pain and lack of mobility associated with the injury or abnormality. Because it is a medical field, math plays a large role in the knowledge physical therapists use.

Physical therapists need a good knowledge of physics to understand the mechanics of their patients' bodies. Geometry helps them understand how to manipulate angles to rehabilitate injured joints. In fact, basic math figures into the job from start to finish. Physical therapists have to administer assessment tests to determine a baseline of patients' abilities. They continue administering these tests to monitor improvement. Such tests also feature a threshold that marks when a patient has shown maximum improvement.

Solving mathematical equations teaches people both basic and group problem solving skills. This is necessary for physical therapists as they work around injuries and anatomical abnormalities to allow patients to live normal lives.

As physical therapists get promoted or go into business for themselves, they need to know business math. In this case, it is important to understand how to budget and staff a physical therapy clinic.

## Access the New York State Next Generation Learning Standards:

The revised learning standards for English language arts and mathematics are available at the links below.

If you have questions about the revised learning standards, please email us at [email protected]

The Department convened the New York State Early Learning Standards Task Force to discuss concerns around the P-2 grades, including standards, program decisions, social emotional needs, and how the content areas/domains work together in the early grades. To learn more read the Early Learning Task Force Update May 18, 2017.

## How to Combine Like Terms

In more technical speak, like terms have the same variables and the same exponents.

Like terms are two things that can added.

"Like Terms" means that you can add or subtract two terms. For instance, you know that you can add $2 + 3$ and get 5. You were able to add these two 'terms' ( the '2' and the '3') because they are both numbers! However, you might also know that you cannot 'combine' 2 and x. Since 2 is a number and 'x' is not, they are not like terms.

### Examples

Examples of Like Terms Examples that are NOT like Terms
$3 + 2$ $3 + x$
$x + 2x$ $x + 2$
$3x + 5x$ $3x^2 + 5x$
$x^2 + 3x^2$ $x^2 + 3x^3$
$2x^ <21>+ 3x^ <21>$ $2x^ <23>+ 3x^ <21>$
$2x^ + 3x^$ $2x^ + 3x^$ ($b e 1$)

### Video on how to combine Like Terms

• Exponents and Bases: You may have noticed that like terms always have the same base and exponent.
• Regarding Coefficients: Also, the coefficient in front of a variable does not change whether or not terms are alike. For instance 3x and 5x and 11x are all like terms. The coefficients ( the '3' in 3x, '5' in 5x and '11' in 11x) do not have anything at all to do with whether or not the terms are like. All that matters is that each of 'x' factors or 'bases' have the same exponent.

### Practice Combining Like Terms

##### Problem 1

Combine the like terms below. x + 2 +2x

##### Problem 2

Combine the like terms below. 5+ x + 2

##### Problem 3

Combine the like terms below. 3x + 2x + 6

##### Problem 4

Use your knowledge of like terms to simplify. 2x + 3 + x + 6

2x and x are like terms so you can combine (ie 'add') them to become 3x. Likewise, 3 and 6 are like terms and can be added to 9. The final answer is 3x + 9

### More Challenging problems

##### Problem 5

Use your knowledge of like terms to simplify. 2x 2 + 13 + x 2 + 6

2x 2 and x 2 are like terms so you can combine (ie 'add') them to become 3x 2 . Likewise, 13 and 6 are like terms and can be added to 19. The final answer is 3x 2 + 19

##### Problem 6

Use your knowledge of like terms to simplify. 2x 3 + 3x + x 2 + 4x 3

2x 3 and 4x 3 are the only like terms --combine (ie 'add') them to become 6x 3 . Likewise, 13 and 6 are like terms and can be added to 19. The final answer is 6x 3 + x 2 + 3x