c 1 unit 1: operations and rational numbers... chapter 1. unit 1: operations and rational numbers...

www.ck12.org Chapter 1. Unit 1: Operations and Rational Numbers

CHAPTER 1 Unit 1: Operations andRational Numbers

Chapter Outline1.1 ADDITION AND SUBTRACTION OF RATIONAL NUMBERS - 7.NS.1A,B,C,D

1.2 MULTIPLICATION AND DIVISION OF RATIONAL NUMBERS - 7.NS.2A,B,C,D

1.3 REAL WORLD AND MATHEMATICAL PROBLEMS WITH RATIONAL NUMBERS -7.NS.3

1.4 REFERENCES

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1.1. Addition and Subtraction of Rational Numbers - 7.NS.1a,b,c,d www.ck12.org

1.1 Addition and Subtraction of Rational Num-bers - 7.NS.1a,b,c,d

Students will be able to add and subtract fractions, decimals and whole numbers using both positive and negative

forms. Students will be able to model these concepts to demonstrate understanding as well as perform caluclations

using traditional algorithms.

Several students decided to play their classroom version of monopoly. Rebecca landed on chance and the card said

go back 3 spaces. The space she landed on said go to free parking. That was back another 8 spaces. How far back

did Rebecca travel on that single turn?

Modeling Addition and Subtraction of Integers

We can all count backwards, but did you realize that you were actually subtracting integers? Let’s imagine our

mathopoly board as a number line. Rebecca’s piece is on the ninth space on that side of the board. In black is the

back move of 3. In green is the back move of 8 to free parking.

FIGURE 1.1Mathopoly board as a number line

In mathopoly that would be 9 - 3 = 6 for the first move and then 6 - 8 = - 2 for the second move. Wouldn’t it be great

if all math were as easy as moving a mathopoly piece around a board. Well, adding and subtracting can be this easy.

First we need some sand.

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This method is called "heaps and holes". Did you ever dig a hole in the sand? The sand you dig out of the hole

and put in a pile is the positive. The hole is the negative. Fill in the hole and you have zero. Whatever is left after

filling all the holes you can, is the answer. That’s it!

Look at the model below. The number - 8 is represented by 8 holes. Each hole is -1 because it is missing sand. The

number 6 is represented by little piles or heaps of sand. There is just enough sand in each heap to fill one hole. Can

you tell how many holes will be filled?

Each time we fill a hole, we even things out; in other words we make 0. When we have all the holes filled, what we

have left is the answer.

Every time we "fill a hole" we are actually using a math property called the additive inverse. The additive inverse

is what allows us to make the zeros. The property is just what its name means - add the inverse (opposite) to make

(equal) zero. Here are some examples to help understand this very important property.

2 and - 2 are opposites (inverses) so 2 + - 2 = 0

-7 and 7 are opposites (inverses) so 7 + - 7 = 0

1.25 and - 1.25 are inverses so 1.25 + - 1.25 = 0

− 57

and 57

are inverses so −57+ 5

7= 0

We can even show this property using a number line with heaps and holes. - 4 and 4 are inverses so...

The additive inverse also says that subtracting a number is also the same as adding its inverse. Get it "additive inverse

→ adding the inverse". What does this mean in the language of math? Look at how we can change math equations...

2 - 2→ 2 + - 2

-9 - 8→ -9 + - 8

7.2 - .05→ 7.2 + - .05

−58− 1

6→− 5

8+− 1

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This property is not only used to make our heaps and holes method work, it can actually change the way a math

problem looks without changing the answer. It is mathmagic!

Why is this so important? We change the way the problem looks so we can find the answer using our heaps and

holes.

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If we start with 4 - 9, then by using the additive inverse we can write 4 + - 9. 4 heaps and 9 holes. Using a number

line and loops for heaps and holes we can find the answer. The number lines below show the answer.

We crossed out each pair of heap and hole to see how many holes were left. Therefore 4 - 9 = - 5. How simple is

that!

Absolute Value

Brian came in from outside and said that the temperature changed 8 degrees in the past hour. Can you tell whether

the temperature went up or down? When all we know is the amount of change, it is called the absolute value. The

absolute value is the distance away from a number, not the direction. It’s like someone telling you it is 25 miles

to the mall. You don’t have the directions, but you do have the distance. Absolute value symbols look like two

parallel lines around a math problem or just one number.

|−6|= 6 shows distance of 6

|7−2|= |5| complete the math problem

|5|= 5 and show the distance of 5

|−6+2|= |−4| complete the math problem

|−4|= 4 and show the distance of 4

Absolute value is like the "raw number" and the sign of the number tells the direction on the number line. The

distance between two numbers is the absolute value of their difference.

Example 1

Find the distance between 3 and -6. Explain how you got this distance using absolute value.

Mark where 3 and -6 are located on a number line. Connect the points to indicate the distance between them.

Distance is measured using absolute value. Using the symbol for absolute value, make an equation to showthe distance.

|3−−6|= |9| the absolute value of their difference

|9|= 9 absolute value to get distance

The distance on the number line and the answer to the math equation match. The distance is 9.

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Example 2

The following number line models a certain equation. Express this equation in two ways.

FIGURE 1.2heaps and holes as arrows on the numberline

This number line places heaps and holes on top of each other. Heaps are represented by an arrow going tothe positive(right). Holes are represented by an arrow going to the negative (left). It is a good idea to writeheaps and holes or even + and - next to the arrows to remember which one is which.

Count how many heaps. There are 7. So one part of our equation is +7.

Count how many holes. There are 3. Another part of our equation is -3.

Cross out all the heaps and holes that match. There are 4 heaps left because the arrow is going to thepositive. The answer is +4.

FIGURE 1.3-3 and 3 are additive inverses they be-come 0

Now two ways to write the equation: 7 - 3 = +4 or 7 + - 3 = +4

Watch more on integers and the number line at: http://www.allthink.com/v/oppositequantities

Example 3

An equation has been modeled using blocks. Identify the equation and the steps used to find the solution.

This model uses Algeblocks. The numbers above are the heaps(positive). The numbers below are theholes(negative). The positive is 5 and the negative is 8. The first part of the equation is 5 + -8. Remember, itcan also be written using the additive inverse; 5 - 8.

To find the solution (answer) we match blocks from the positive side with the negative side just like we matchheaps and holes. Thre are 3 blocks left over. They are in the negative. That means the answer is - 3.

The equation is 5 + - 8 = - 3.

The steps to find the solution are:

Count the blocks in the positive and negative. This makes the first part of the equation.

Use the additive inverse property to find out how many blocks make zero pairs.

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FIGURE 1.4Using blocks to model equations with in-tegers

FIGURE 1.5using the additive inverse (zero pairs) tofind the solution

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The blocks left become the answer.

http://nwsiders.com/Algeblocks/unit1-1.html is a video that teaches how to use algeblocks to model working withintegers.

http://sites.google.com/site/jillgergenselectronicportfolio/training-tools-for-algeblocks has print outs that you can

cut and use along with mats to practice.

Once you have mastered basic integers try your skill to complete the integer puzzle at:http://nlvm.usu.edu/en/nav/f

rames_asid_122_g_3_t_1.html?open=instructions&from=topic_t_1.html

Applying The Additive Inverse in Addition and Subtraction of Rational Numbers

Now that we understand how the additive inverse helps to find the answer to a problem with integers, we begin to

see if an answer to any problem will be positive or negative - just by looking at the problem. It is also possible to

say whether we will need to add or subtract to find the answer.

Any time we have all heaps, or all holes, we know just to count up how many we have all together. To count up all

together is to add.

Any time we have a mix of heaps and holes, we know that there will be some heaps or holes left. Any time we have

that mix, we always find how many are left over. Finding how many are left over means subtract.

Here are some model problems to help you understand this concept.

- 7 + -6 These are both "holes" so there are no heaps to fill up any holes. So we know that the answer will be

negative (all holes). Since they are all holes, we count them up which is adding.

4 + -10 We do have heaps and holes. How many holes can we fill? 4 holes. If we fill 4 holes, there are still 6 holes

left. The answer must be negative. When we have both heaps and holes, we will have something left over, so we

will subtract.

- 2.5 - 4.8 First we know to change the problem using the additive inverse. The problem becomes -2.5 + -4.8. Now

it is easy to see that these are all holes and the answer will be negative. Since these are all holes, we will just count

them up, which is adding.

Positive and negatives can be used with numbers other than integers. They can be used with all rational numbers. A

rational number is any number that can be written as a fraction. .625 is a rational number because it can be written

as 3/4 . 5 is a rational number because it can be written as 5/1 . Working with rational numbers means working with

numbers that can have decimals and fractions. It also means that these decimals and fractions can be either positive

or negative.

Example 4

Try to reason out the sign of the answer for the next four problems. Remember that these integer properties stay

true for all rational numbers.

1. -6.2 + 9.7

2. 14

- 5

3. 6.25 - (-7.3)

4. - 12

+ -1.5

To find the sign of each answer, we think of our heaps and holes as positives and negatives. If we have bothheaps and holes, we know to subtract.

1. We have more heaps than holes. Therefore the answer is positive. We will subtract to get the answer.

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2. We change the problem using the additive inverse. The problem becomes 14

+ -5. Now we see that there aremore holes than heaps. The answer is negative. We will subtract to get the answer.

3. Tricky! Any time there is subtraction we use the additive inverse. The problem becomes 6.25 + +7.3. Nowwe see that they are both heaps. All heaps is all positive and we add.

4. All we have is holes. Therefore the answer has to be negative. We add to get the answer.

(At this point we start to write the word negative for hole, and the word positive for heap.)

Mathematical Problems - Addition and Subtraction of Rational Numbers

Background Review

Before getting into working with all rational numbers with these properties of negatives, it is important to remember

how to add and subtract with fractions and decimals. Please view the following tutorials on decimals and fractions

before continuing this concept.

http://www.kidsmathgamesonline.com/videos/addandsubtractfractions.html - add and subtract fractions with word

problems.

http://www.mathplayground.com/howto_fractions_diffden.html - add and subtract fractions using modeling.

http://www.virtualnerd.com/middle-math/decimals/decimal-addition-method.php?id=15&subject=middle-math&type=T

opic&sid=8 - addition of decimals.

http://www.virtualnerd.com/middle-math/decimals/decimal-subtraction-method.php?id=15&subject=middle-math&typ

e=Topic&sid=8 - subtraction of decimals.

Addition and Subtraction

Did you ever write your name in a different order? Look at the three ways Amy wrote her name.

No matter what order she writes her name, it is still Amy. This is the same as the negative sign that goes with a

fraction. Look at the three ways a negative sign can be used with a fraction.

As you can see, no matter where the negative sign is the fraction is still -.625. This is important to know when

working with different ways of writing negative signs and fractions.

Adding and subtracting with negative fractions and decimals do not make the problem any harder. It just puts in a

thinking step before the problem is started.

Let’s say we want to solve the problem - 3.6 + -5.7. The thinking step is looking at the signs with the decimals. Are

they both negative (meaning both holes)? Yes they are, so the answer is going to stay negative. And since they are

both the same sign (both holes) we know to add.

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−3.5

+−5.7

−9.2

What about fractions? Same thing as decimals. Just put in the thinking step.

−34+ 5

12is a good model. Thinking step: do we have negatives (holes) AND positives (heaps) ? Yes we do. There-

fore we will have to subtract. Do we have more positives (heaps) or negatives (holes)? Not sure? Then let’s find

the common fractions for this problem and we will get our answer.

−3

4+

5

12⇒− 9

12+

5

12

By doing this, we can see that we have more negatives. The answer will be negative.

Now all we do is subtract the numerators and keep the denominator.

− 9

12+

5

12=− 4

12

− 4

12=−1

3reduce

How about some problems that mix different kinds of numbers up?

− 56+2.5 has a negative fraction and a positive decimal. Right away we know that the two numbers will be subtracted

( one is a positive and one is a negative). How can we tell if there are more positives than negatives? We make them

into a common form.

−56= .833333... This would not be a good form because the decimal repeats forever. So we keep the fraction form

and change 2.5 into fraction form.

2.5 = 2 12

or 52

We can use either the mixed number or the improper number. Always pick the form that is easier to

use. Now replace 2.5 with the fraction and solve.

−5

6+

5

2replace with fraction form

−5

6+

15

6change to common denominators

−5

6+

15

6=

10

6subtract the numerators

10

6= 1

4

6change to mixed number

14

6= 1

2

3reduce

Here is an problem that uses addition and subtraction with rational numbers.

17−33

4+1.8

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The first step must ALWAYS be to use the additive inverse. This problem now becomes

17+−33

4+1.8

. Even though the problem is longer, there are both negative and positive numbers. But here is a trick to make the

problem easier - we can flip around the problem to do whatever part we want first!

It is not really a trick. It is using another property of math - the Commutative Property. This property says that as

long as the problem is all addition, we can flip flop any two numbers we want. The answer will still come out the

same.

It is much easier to add two positive numbers first. The numbers added are in blue. Now we can "do the math" on

the final part of the problem.

18.8+−33

4= 18.8+−3.75 change to common form

18.8+−3.75 = 15.05 subtract; one is negative and one is positive

Example 5

Solve the following problem and justify each step in your solution.

4−91

3+4.4

1. Change to common form. Any fraction with a denominator of 3 repeats when it changes to a deci-mal. Therefore changing to fraction form is best.

4−9 13+4 4

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2. Use the additive inverse to change the problem to all addition.

4+−9 13+4 4

10

3. Use the commutative property to put like signs together.

4+4 410+−9 1

3

4. Add the positive numbers.

8 410+−9 1

3

5. Adding unlike signs means subtract. There are more negatives, so the answer is negative. Make commonfractions and solve the problem.

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84

10+−9

1

3

812

30+−9

10

30common denominator is 30

252

30+−280

30change to improper to subtract

−28

30subtract numerators

14

15reduce to lowest terms

http://www.sophia.org/subtracting-fractions-from-decimals/subtracting-fractions-from-decimals–11-tutorial?pathway=operations-with-fractions-and-decimals is a tutorial on subtracting fractions and decimals in one problem. After

the tutorial there is on line practice as well.

Real World Applications

FIGURE 1.6Scrolling stock prices

Stock prices - they go up and down almost every day. Stock prices are in decimal format. An upswing is a positive

and a downturn is a negative. People investing money need to know if they are making money or losing money. This

is working with positive and negative rational numbers.

Nike is a stock that people invest in. Below is a chart of the changes in Nike in a four week period. What would be

the price after 2 weeks (a)? What would be the price after 4 weeks (b)?

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The price of a share of stock of Nike started at $93.70. It dropped $14.41 in week one and dropped $2.19 in week

two. To find the price of the stock after two weeks the problem to solve is: 93.70−14.41−2.19

Using the additive inverse, the problem is written as 93.70+−14.41+−2.19

Add the two numbers that have the same sign. They are already next to each other. 93.70+−16.6

Now subtract because the signs are different. There are more positives than negatives, so the answer will be

positive. 93.70+−16.60 = 77.10 .

The price of the stock after two weeks is $77.10.

To find the price after 4 weeks we repeat the process, just use the new numbers.

77.10−4.98+0.28

77.10+−4.98+0.28 additive inverse

77.10+0.28+−4.98 commutative property

77.38+−4.98 add numbers with the same sign

77.38+−4.98 = 72.40 subtract, more positive means positive answer

After four weeks the price of Nike stock is $72.40 for one share.

Example 6

The remains of the Titanic are 12,500 feet below sea level. In comparison, Bronco Stadium, in Colorado, home of

the Denver Broncos football team, is 5,280 feet above sea level. What is the difference between the elevations of

these landmarks?

We need to find the distance between these two places. Distance between is absolute value. Difference meansto subtract. We can make an equation by using x as long as we say what x represents.

Let x be the distance between the landmarks. Distance below sea level is negative. Distance above sea levelis positive.

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|−12,500−5,280|= x set up the equation

|−12,500+−5,280|= x additive inverse

|−17,780|= x add; signs ar the same

|−17,780|= 17,780 take absolute value

The distance between the two landmarks is 17,780 feet.

Example 7

The temperature was 20 degrees when Carla left for school. At 12:00 school was dismissed early due to snow and

ice. By the time Carla’s mom got home, the temperature was 25 degrees colder. What was the temperature when

Carla’s mom got home? Using a number line, find the temperature at the time Carla’s mom got home.

A number line must have equal amounts between each break. The starting temperature is marked andlabeled. An arrow is drawn to show the decrease in temperature. The ending temperature is marked andlabeled.

FIGURE 1.7change in temperature

If the temperature is getting colder, then it is going toward the negative. It is expressed as 20 - 25.

Use the additive inverse: 20 - 25→ 20 + - 25.

Subtract because the signs are different and make the answer negative because there are more negatives. 20+ - 25 = -5. The temperature is -5 degrees.

Practice word problems with rational numbers at http://quizlet.com/9942524/comparing-rational-numbers-word-problems-flash-cards/. You may need to set up a free account, but it is worth it.

integer, whole number, rational number, additive inverse, absolute value, numerator, denominator

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1.2. Multiplication and Division of Rational Numbers - 7.NS.2a,b,c,d www.ck12.org

1.2 Multiplication and Division of RationalNumbers - 7.NS.2a,b,c,d

Students will know how to multiply and divide with rational numbers. Distributive property, division by zero,

multiplicative inverse are understood as they apply to rational numbers. Real world applications are presented and

worked through.

Modeling Multiplication of Integers

Did you ever hear of "skip counting"? Skip counting is skipping over equal groups of numbers again and again.

Skip count by 2? That is 2, 4, 6, ... which is really the even numbers.

Skip count by 4? That is 4, 8, 12, ...which is really the multiples of 4.

So multiplication is really skip counting. One factor tells how many times to skip count by the second factor. With

integers, the positive or negative of the first factor tells if we need to reverse the direction. We can model many

multiplication of integer problems using a number line.

Our first model is to simplify - 4 • 2. Notice that the skip always starts at 0. The first factor tells us that we need 4

groups and the groups are going in the reverse direction of the positive 2. Our second factor tells us that we are skip

counting by 2’s.

What happens if we reverse the problem? Now we model 2 • - 4. What do you notice about this number line and

it’s answer? You see two groups of - 4. The two is positive so we stay in the direction of the negative 4.

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It seems that the two answers are the same. It did not matter the order of the numbers. This in fact is the proof of

the Commutative Property in Multiplication. We can switch the order and the answer comes out the same. So if you

are multiplying any two numbers, switch the order if it makes the problem easier to solve!

Our next model has both factors as negatives. Simplify -2 • -3. Let’s break down the problem to see what it is really

asking us to do.

There are - 2 groups of - 3. The negative in front of the two (number of groups) tells us that we need to reverse

the direction of our move. So instead of making two groups of negative three, we make 2 groups of positive 3. The

number line shows this thinking.

Another model for multiplying with integers can be done with colored chips. View the lesson and try some problems

at the site http://www.learner.org/courses/learningmath/number/session4/part_c/multiplication.html

Patterns in Multiplication and the Rules for Multiplying Integers

Another way of understanding the signs in the problem and the sign of the answer is by looking at patterns in a

table. Chart 1 shows how answers become negative when going from two positives to one positive and one negative

numbers multiplied together. Chart 2 shows the pattern when we start with one positive and one negative to two

negative numbers multiplied together.

Looking at these two charts can you come up with some general rules for multiplying with negatives?

Try using these rules for multiplying 2, 3, even four integers at http://teachertech.rice.edu/Participants/mredi/lesso

ns/integermultiplication.html

Modeling Multiplication of Fractions

We can model multiplying fractions just by dividing squares into equal segments and overlapping the squares.

Our model is to simplify 13· 2

5.

15


We can read the above problem as one-third of two-fifths. First draw two congruent squares. Divide one vertically

into thirds and shade one section in. Divide the other into fiftths horizontally and shade two sections in.

Now overlap the two squares. Notice the number of blocks has increased and there are two blocks that are

purple. The purple blocks represent what one-third of two-fifths is.

The intersection of the two shaded represnts the answer. The whole has been divided into five pieces width-wise

and three pieces height-wise. We get two pieces that overlap. That is the numerator. The denominator is the total

number of pieces when overlapping the two grids. The denominator is 15.

Now we work our second model mathematically. We will multiply three fractions.

Example 1

1

4× 2

6× 4

5=

To start, let’s only look at the first two fractions.

1

4× 2

6

16


We start by simplifying. We can simplify these two fractions in two different ways. We can either cross simplifythe two and the four with the GCF of 2, or we can simplify two-sixths to one-third.

Let’s simplify two-sixths to one-third. Now rewrite the problem with all three fractions.

1

4× 1

3× 4

5=

Next, we can multiply and then simplify, or we can look and see if there is anything else to simplify. One-fourthand one-third are in simplest form, four-fifths is in simplest form. Our final check is to check the diagonals.

1

4× 1

3× 4

5=

The two fours can be simplified with the greatest common factor of 4. Each one simplifies to one.

1

1× 1

3× 1

5=

1

15

Our final answer is 115

.

This tutorial on multiplying fractions and mixed numbers can help you master this concept. http://www.sophia.or

g/multiplying-fractions-and-mixed-numbers/multiplying-fractions-and-mixed-numbers-tutorial?topic=multiplying-and

-dividing-fractions--2 View it and then try the problems that are attached. See how much better you have become.

Properties Used in Multiplying Rational Numbers

Rational numbers are any numbers that can be expressed as a fraction. Examples of rational numbers are:

27⇒ 27

1

4.35⇒ 435

100

.6666...⇒ 2

3

Rational numbers can be negative or positive.

There are 4 properties that are used in multiplying rational numbers. They are:

Commutative Property: The product of two numbers is the same whichever order the items to be multiplied are

written.

Example: 2 ·3 = 3 ·2

• Associative Property: When three or more numbers are multiplied, the product is the same regardless of how

they are grouped.

Example: 2 · (3 ·4) = (2 ·3) ·4

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• Multiplicative Identity Property: The product of any number and one is the original number.

Example: 2 ·1 = 2

• Distributive property: The multiplication of a number and the sum of two numbers is equal to the first number

times the second number plus the first number times the third number.

Example: 4(2+3) = 4(2)+4(3)

http://www.youtube.com/watch?v=8i-QQvroJdo has another explanation of the rules of multiplication.

Here are some models to help understand how these properties are used in actual math problems.

Example 2

Prove the rule for integer multiplication that a positive times a negative is a negative using the distributive prop-

erty. 5(7+−7) = 5(7)+5(−7)

We set up our model using the distributive property. We simplify everything BUT the 5(- 7) because that is what we are

5(7+−7) = 5(7)+5(−7)

5(0) = 35+5(−7)

0 = 35+5(−7)

Look at the problem left. 35 plus what number equals zero? It has to be -35 because it is the additiveinverse! So a positive number times a negative number is a negative number. 5( - 7) = -35.

0 = 35+−35

Example 3

Simplify:(

23

)(1.2)(−4)

We can multiply any two factors together first. It seems easier to multiply (1.2)(- 4) first.(23

)(−4.8)

Now we change either to both decimals or both fractions to multiply. If 2/3 is changed to a decimal it becomes

.6666... It repeats, so it is not a good choice to change to decimals. We change - 4.8 into a fraction. Then we

multiply the fractions.

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(2

3

)(−4.8)

(2

3

)(−4

8

10

)change decimal to fraction

(2

3

)(−48

10

)change to improper form

(2

3

)(−24

5

)simplify the fraction

(2

1

)(−8

5

)cross cancel the GCF of 24 and 3

(2

1

)(−8

5

)=−16

5

−16

5=−3

1

5

Example 4

What is 34

of the product of -12 and -3.5?

First we rewrite the problem using math symbols. Product means to multiply. So we multiply -12 and -3.5

first. Multiply that answer (remember that "of" means multiply) by 34.

3

4(−12 ·−3.5)

3

4(42) multiply in the parentheses

(.75)(42) change fraction to a decimal

(.75)(42) = 31.5

Sometimes multiplication of a decimal times a whole number can even be done using mental math. How? By

actually using the Distributive property. The Distributive property breaks down the math problem into simple small

pieces.

Let’s try to show mental math for the problem 7 x 4.3. Using the Distributive property we can break this down into

7(4 + .3). 7 x 4 = 28. 7 x .3 = 2.1. Then just add the two answers back together. 28 + 2.1 = 30.1. Math properties

can really make things easy.

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Example 5

Show how the Distributive property can be used to multiply 6 x 5.3

Break the problem down using the distributive property.

6×5.3 = 6(5+ .3)

Apply the distributive property by multiplying six by each number. This is called expanding the problem.

6(5+ .3) = 6(5)+6(.3)

Now simplify.

6(5)+6(.3) = 30+1.8

30+1.8 = 31.8

http://mathvids.com/lesson/mathhelp/85-multiplying-rational-numbers has a tutorial with several examples that aresolved step by step. A great review that is easy to understand.

Modeling Division of Integers

Division is the inverse operation of multiplication. That means it is the opposite math operation. Addition and

subtraction are also inverse operations. When solving for a variable, we always use inverse operations.

When modeling division of integers on the number line, it is always important to start at 0. Our first model has both

numbers as positive to understand the grouping.

Dividing does mean to split up evenly. So to split up 8 so there are two in a group, we have 4 groups. The inverse

operation to check this would be 4 x 2 = 8.

Negatives put a twist on this because the negative will mean direction.

If you were standing on the numberline and walked backward, you would be facing the negatives. Your answer is

a negative. Mathematically speaking we can look at the inverse. 10 ÷ - 2 is really asking what number times -2

equals +10. Eariler in this concept we proved that a negative times a negative is a positive. Either way you look at

it, the answer is - 5.

http://www.youtube.com/watch?v=Lh0tBKOTq8I is a live demonstration by students of how to use a number line in

division of negative numbers.

20


Example 6

Model the following equation on a number line and check using the inverse operation.

−16÷−4

Our number line will have to be made going by 2’s because the problem has bigger numbers in it. Set up thenumber line with a point on zero and a point on -16.

The negative next to the 4 tells us to "walk backward". Starting at the zero, make 4 equal groups or "hops"

back to the -16.

There are 4 in each group (remember that we are counting by 2’s). We also had to walk backward. When wewalk backward, we are actually facing the positive numbers. The answer is positive. Therefore -16 ÷ - 4 =+ 4.

To check using the inverse operation is to use multiplication. Does 4 x - 4 = -16? Yes, because 4 x 4 is 16 andwe know that a positive number times a negative number is a negative number.

Patterns in Division and the Rules for Dividing Integers

Just like multiplication, we can use a table to understand division of integers. Look at the patterns when the signs

change both in the problem and the answer. Chart 3 shows how answers become negative when going from two

positives to one positive number divided by one negative numbers . Chart 4 shows the pattern when we start with

one positive and one negative to a negative number divided by another negative number.

Look back at the two charts for patterns in multiplication. In charts 1 and 3, the pattern shows that if one number is

negative and the other is positive, the answer is negative. In charts 2 and 4, the pattern shows that if both numbers

are negative, the answer is positive. The rules for multiplication and division are generally the same.

21


We have to use two rules with one negative and one positive because division is not commutative. We also have to

use the words rational number instead of integer because 4÷ -8 = - 12. One half is not an integer.

http://www.youtube.com/watch?v=sEU5uaf-Tu4&feature=related is a quick review of the rules for multiplying and

dividing with negative numbers.

A Triangle Trick to Multiplying and Dividing Rational Numbers

Here is a neat trick if you ever get stuck trying to figure out what the sign of your answer is when multiplying or

dividing. All you need is to draw a triangle like the one above. Put two negative signs up top and one positive down

below. Then cover ups the two signs in the problem. The one uncovered sign is the sign of the answer!

Here are some models:

-32 ÷ 4 = ?8 This is a negative divided by a positive.

Cover up the negative and positive. There is a negative sign not covered. This is the sign of the answer. So -32 ÷4 = - 8.

- 17 x -1.2 = ?20.4 Cover up two negative signs.

The sign of the answer is positive.

Multiplicative Inverse

8 ÷ -2 can be written in two other ways and still mean the same (get the same answer). The first is to write the

division vertically.

8÷−2⇒ 8−2

. This may look like a fraction, but it is actually a division bar. Yes, this does mean that every fraction

is actually an unfinished division problem! The second way is to change it to a multiplication problem.

8÷−2⇒ 8×−12

. Dividing by -2 is the same thing as multiplying by - 12. Think about it - if you eat half of a pizza

that has 8 slices, you eat 4 slices. 12

of (and of means x in math) 8 is (is means = in math) 4.

One more thing to remember: when a division problem is expressed using the division bar, the negative sign can

appear in three different ways, yet mean the same thing.

22


−6

3⇒means take the negative of 6 divided by three =−2

−6

3⇒means -6 divided by 3 =−2

6

−3⇒means 6 divided by -3 =−2

Example 7

Express the following division problem in two ways and give the final quotient.

− 7224

Since the negative sign is before the entire division problem, we use parentheses when re-writing the problem.

For the first way, we re-write the problem using a ÷ sign.

− 7224⇒−(72÷24)

For the second way, we use multiplication of the fraction 124

.

− 7224⇒−(

72× 124

)Finally, we select any of these three expressions and solve.

− 7224⇒−(72÷24)

−(72÷24) =−(3)

−(3) =−3

Division with Zero

Using 0 in a division problem happens in two ways. One gives the answer of 0, and one has no answer. Can you

tell the difference between the two math expressions and give the answers to each?

0÷4 4÷0

The difference between the two is the zero being before the division sign or being after the division sign.

To give the answers to each, we look at these from the inverse operation, multiplication.

0÷4 is the same as asking what number times 4 equals zero. Of course the only number to multiply 4 by is zero.

This is called the multiplication property of zero. So 0÷4 = 0 because 0×4 = 0.

4÷ 0 is the same as asking what number times 0 equals 4. Wait! Any number times zero is always zero

(multiplication property of zero again). How can we ever get 4? We can’t. So there is no answer to this problem.

In math, when there is no answer it is called undefined.

http://www.youtube.com/watch?v=fcVnwjBgDmk is a mini lesson that reviews both multiplication and division by

zero.

23


Example 8

Simplify: 2434

This is called a complex fraction. It is a" fraction in a fraction". Remembering that a fraction bar is really adivision bar, the expression can be written in two different ways. Can you tell what was done?

24

1÷ 3

424÷ .75

The first way is using fractions. The second way is using decimals. Either way is fine.

8 ·31÷ 3

4factor 24

8 ·31× 4

3dividing by

3

4is multiplying by

4

38· �31× 4

�3 cancel out

32

1multiply numerators and denominators

The solution is 32.

http://www.sophia.org/dividing-rational-numbers--2/dividing-rational-numbers–6-tutorial?pathway=nc-skill-318 isa tutorial with concrete models and examples and solutions.

Dividing Rational Numbers using Long Division

Dividing rational numbers has only two things you need to do before it becomes the same thing as long division with

whole numbers. Set up the division, move the decimal, and you are ready to divide.

24


If you have difficulty in completing problems in long division, check out an another way to divide using the step by

step guide to using double division. http://www.doubledivision.org/

Example 9

Using long division, convert 78

into its equivalent decimal format.

Equivalent decimal format means to divide and make the fraction into a decimal.

Following the steps in the above model problem we set up the long division.

Now we divide. Two ways of division are shown, traditional division and partial quotient using only 10’s,5’s, and 2’s times tables. Both are good ways of dividing.

We know the decimal .875 is a rational number because it ended with 0 when we divided. A decimal that repeats is

also a rational number. All fractions with a denominator of 3, 7, and 9 repeat. All but one that has 6 as a denominator

(3/6 = 1/2) repeats as a decimal. And if you know what 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 are as decimals, all you do is

multiply to find out any of their fraction family decimals. Here is just one example of this "trick". First we change1/5 into a decimal by dividing. Then we look for the pattern in the multiplication to get the fraction → decimal

family.

25


So 3/5 would be 3 times .2, and so on. Just remember that ALL repeating decimals are rational numbers. That

means that all repeating decimals can be made into a fraction.

Mixed Multiplication and Division of Rational Numbers

Multiplication and division of rational numbers follows the rules of order of operations. All multiplication and

division in a math problem is solved from left to right, paying attention to all parentheses.

Example 10

Simplify:

(3

4× 2

7

)÷ .4

First multiply what is in the parentheses.

(3

4× 2

7

)÷ .4

(6

28

)÷ .4 multiply numerators and denominators

(6

28

)÷ 2

5change .4 into

4

10and reduce to

2

5

6

28× 5

2dividing by

2

5is the same as multiplying by

5

230

56

3056

is not in lowest terms. They are both even numbers. We can find the greatest common number that dividesinto both, or just keep reducing by simple numbers like 2, 3, etc.

30

56=

15 ·228 ·2

30

56=

15

28

Example 11

Simplify: − 38(6÷ .3)

Following order of operations, parentheses is simplified first. We divide 6 by .3.

26


Now we can change −38

into a decimal or change 20 into a fraction. Both numbers must be in commonform. Changing 20 into a fraction, 20

1, is much easier. Now multiply the two fractions. Remember that

a negative rational number times a positive rational number is a negative rational number (rules of integermultiplication).

−3

8· 20

1

− 3

2 ·2 ·2 ·5 ·2 ·2

1show numbers in factored form

− 3

�2 · �2 ·2 ·5· �2 · �2

1cancel out

−3

2· 5

1

−15

2multiply numerators, multiply denominators

−71

2simplify to a mixed number

http://www.youtube.com/watch?v=CMQrctqdz6g has a live example of a mixed multiplication that you can watch

as well.

integer, rational number, distributive property, division by zero, multiplicative inverse, commutative property, asso-

ciative property, product, quotient, factor, divisor, dividend

27

1.3. Real World and Mathematical Problems with Rational Numbers - 7.NS.3 www.ck12.org

1.3 Real World and Mathematical Problemswith Rational Numbers - 7.NS.3

Students will change between equivalent forms of rational numbers to perform addition, subtraction, multiplication,

and/or division with precision. Mathematical practice will lead to application in real world situations.

Did you know that there is only one thing that is universal? It is mathematics. Every country can talk about

mathematics in their own language, but pure mathematics, symbols and numbers, are used the same all over the

world.

Basic Words and Phrases into Symbols

The challenge is the changing of words into symbols and numbers. The chart below has shows common English

words and phrases with their math translation.

You may find more words and phrases to add to this list as more real world situations are modeled. Keeping a list

of your own is a great study aid.

Example 1

Jenna has 20 indian head nickels. She gives 14

of them to her sister . How many nickels does she give to her

sisiter?

28


The first sentence gives us our starting number. The second sentence gives us our math equation .

1

4of them to her sister

1

4×20 = number to her sister

1

4× 20

1make both into fraction form

1

�4 ·5· �41

factor and cancel

5


5nickels simplify

Example 2

The temperature at 2:30am was -100. At noon, the temperature had risen 160 . At 5:00pm the temperature had

dropped 80 . What was the temperature at 5:00pm?

The first sentence gives us our starting temperature. The next two sentences tell the amount of change and if it is a positive

−10+16−8 = the temperature at 5:00pm−10+16+−8 additive inverse−10+−8+16 commutative property−18+16

−10+16−8 = the temperature at 5:00pm

−10+16+−8 additive inverse

−10+−8+16 commutative property

−18+16 add like signs

−2 subtract unlike signs

The temperature at 5:00pm was -20.

Example 3

The difference in the cost of two skateboards is $13.50. If the first skateboard is priced at $98.45, what are the two

possible costs of the second skateboard? Explain your reasoning for each price.

The problem does not tell whether the first skateboard is the one that costs more or the one that costs less. This iswhere two possible prices can happen.

The price of $98.45 could be the higher price. Difference means to subtract. The first equation would be: 98.45−13.50 = price of the second skateboard

29


If the $98.45 skateboard is the lower price, we would want the second price to be more. This would change theequation to addition. The second equation would be: 98.45+13.50 = price of second skateboard

Now that we have explained the reason for each equation, the last step is to solve both equations.

98.45−13.50 = price of second skateboard 98.45+13.50 = price of second skateboard84.95 = price of second skateboard

98.45−13.50 = price of second skateboard 98.45+13.50 = price of second skateboard

84.95 = price of second skateboard 111.95 = price of second skateboard

The price of the second skateboard can be $84.95 or $111.95.

http://learnzillion.com/lessons/1150-use-addition-and-subtraction-to-solve-realworld-problems-involving-decimalsis a mini lesson on using addition and subtraction in real world problems.

http://www.youtube.com/watch?v=TxxKYne1C-M shows how to use decimals to find the perimeter of a unique

polygon.

Modeling Equations to Describe Situations

Math is not always purely equations. Often math comes from problems in real world situations. A problem happens

and math is the answer!

Example 4

A kindergarten class is making clay animals for a play. The teacher has 2 34

gallons of clay. Each child needs 1/16

of a gallon to make an animal. How many children can make a clay animal?

The problem gives a total amount, 2 34

gallons. The clay needs to be split up so each child gets a certainamount, 1/16 of a gallon. The units are the same (gallons). To split up means to divide.

Divide the total amount by how much each child gets.

23

4÷ 1

1611

4÷ 1

16make both fractions

11

4× 16

1dividing by

1

16is the same as multiplying by

16

111

�4 ×4· �41

cancel out common factors

44

1multiply across

children 44

30


Example 5

The stock market measures earnings per share of stock of companies. On Monday, Company ABC opened and

closed at $22.31 per share. On Tuesday, it rose $0.27 per share. On Wednesday it fell $1.12 per share. On Thursday

it fell another $1.16 per share. On Friday it gained $0.10 per share. What was the price of a share of stock for

Compay ABC at the close of the stock market on Friday?

The starting number is the price of the stock at the start of the week.To rise means to add. To fall means tosubtract. With this in mind the equation is made by adding and subtracting the numbers from the startingprice.

22.31+ .27−1.12−1.16+ .10 = ending price of stock

22.31+ .27+−1.12+−1.16+ .10 additive inverse

22.31+ .27+ .10+−1.12+−1.16 commutative property

22.68+−2.28 add like signs

20.40 subtract unlike signs

The final price of the stock is $20.40.

Example 6

Bob has saved $25.40. Rena saved 34

of the amount that Bob saved. Nick saved twice as much as Rena. How much

is the total saved of all three?

To find the answer to this problem, we must first find out how much each person saved.

The first sentence gives the starting number. The second sentence says 34

of. Of means to multiply. To findRena’s amount, multiply 3

4by 25.40.

3

4(25.40) = amount Rena saved

(.75)(25.40) make common forms

19.05 amount Rena saved

Rena saved $19.05.

The third sentence says twice as much as Rena. Twice means to multiply by 2. To find Nick’s amount, multiply2 by the amount Rena saved.

2(19.05) = amount nick saved

38.10 amount Nick saved

Total is to add. We add all three amounts together.

24.50+19.05+38.10 =total amount saved by all

81.65 =total amount saved by all

31


Bob, Rena, and Nick saved $81.65 together.

http://www.mathmaster.org/video/subtracting-mixed-numbers-word-problem/?id=867 is a step by step video that

helps understand what to look for in word problems and how to solve them.

http://www.mathmaster.org/video/dividing-real-numbers-with-different-signs/?id=928 gives a tutorial on mixing up

fractions, decimals, and integers in a problem.

Modeling Equations Using Diagrams and Charts

Often diagrams are used to help visualize word problems. Diagrams and charts help us see the relationships between

the numbers. Pictures are sometimes easier to understand than words.

Example 7

The following circle graph shows the five most popular snacks for people ages 20-24. There were120 people in this

survey.

What fractional part of people ages 20-24 like fruit?

How many people prefer chips based on the survey?

For the first question, the sum of the parts equals the whole. In other words, all the sections must add up to 1.To find the missing part (fruit) we subtract what we know (potato chips, tortilla chips, and pretzels) from thewhole (1).

1−(

1

3+

5

24+

5

12

)

We group the parts we know and subtract it from the total.

First we combine the three fractions. Find a common denominator for all three fractions. Make equivalentfractions to add.

32


1−(

1 ·83 ·8 +

5

24+

5 ·212 ·2

)common denominator of 24

1−(

8

24+

5

24+

10

24

)

1−(

23

24

)add numerators

Finally we subtract the two numbers. One can be expressed as any fraction when both the numerator and thedenominator are the same.

1−(

23

24

)

24

24−(

23

24

)change to common denominator

24

24− 23

24=

1

24

So never guess just by looking at a diagram what the answer might be. Do the math. Only 124

of the peoplesurveyed said fruit was their favorite snack.

The second question asks to find the actual number of people that like chips. Since it does not say what kindof chips, we combine both chips together. Combine is to add.

1

3+

5

12add both kinds of chips

1 ·43 ·4 +

5

12get a common denominator

4

12+

5

12=

9

12add numerators

Now to make fraction form into an actual number of people, we need to know what 912

of 120 is.

9

12of 120 = people who like chips

9

12×120 of means times

9

12× 120

1make equivalent forms

9

�1 2× �1 2 ·10

1factor to cancel

90


90

1⇒ 90 people

33


Example 8

George is making a picture frame. He wants the frame to be 1 12

inches wide. The picture is an 8in x 10in photo. How

long should he cut the length (side a) of the frame? Use the diagram as a model.

We will find each side separately. For the length (side a), the inside is 10 inches. There is also an extra 1 12

inches added on to each side for the frame.

1 12+10+1 1

2= length of side a

At this point, we can add as fractions or change to decimals and add. That is the great thing about rationalnumbers. We can use any form we want. We will show both ways here. You decide which you would prefer.

using fractions using decimals

11

2+10+1

1

21

1

2+10+1

1

2

11

2+1

1

2+10 1

1

2+1

1

2+10

use commutative property to put fractions together3

2+

3

2+

20

21.5+1.5+10

put numbers in common format to add26

213

add and reduce

13 13

The length is 13 inches.

34


Example 9

A scuba diver starts at 150km from the shore and descends to point B. Then he travels to point A. Realizing his

depth could not be held for a long time, he ascended to point C. Using the chart below, model an equation to find

his depth at point C.

Looking at the vertical axis, we see that 0 is our maximum number. This means that all numbers below it arenegative. The diver enters the water at the 0m,150km mark. Descend means to go down. So the distancefrom 0 to point B is -450m.

Is point A going down or up from point B? Up means add, down is subtract. Writing this part of the equationis: −450−450.

Now the diver goes up to point C. This is adding. The next part of the equation is now: −450−450+300

We know the diver is now at point C. Point C is the answer to complete our equation. Solve to find C. Usethe rules for adding and subtracting integers.

−450−450+300

−450+−450+300 additive inverse

−900+300 add negatives

−600 unlike signs subtract

still underwater so sign is negative

Now we write our model equation. −450−450+300 =−600

We can even check to see if our equation makes sense. Go back to the graph and see where point C islocated. It is about -600meters. We did it right!

Veiw a video lesson on mixing up operations with rational numbers in real world situations at: http://learnzillion.com/lessons/1152-use-addition-and-multiplication-to-solve-realworld-problems-with-rational-numbers

integer, rational number, distributive property, commutative property, multiplicative inverse, additive inverse, divisor,

dividend, quotient, factor, product, sum, difference, subtract from, order of operations,

35

1.4. References www.ck12.org

1.4 References

1. CK-12 Foundation. . CCSA






























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c 1 unit 1: operations and rational numbers... chapter 1. unit 1: operations and rational numbers...

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