MATH

**Math Chapter Eight Notes**

## MATHEMATICAL PROPERTIES

## MULTI-STEP EQUATIONS

**The most important things to think about when solving multi-step equations are:**

1. Read the story as a story and determine what they are talking about, who the characters are, and what is being asked.

2. Determine if it is a multiplication, division, addition, or subtraction problem.

3. Solve for each part of the equation.

4. Remember to label the parts.

1. Read the story as a story and determine what they are talking about, who the characters are, and what is being asked.

2. Determine if it is a multiplication, division, addition, or subtraction problem.

3. Solve for each part of the equation.

4. Remember to label the parts.

## MIXED NUMBERS AND RECIPROCALS

## DIVIDE BY A FRACTION

**Dividing a whole number by a fraction is finding the number of fractional units in the whole. So 4 divided by 2/3 is finding how many units of 2/3 are found in the number 3. To actually divide the whole number by a fraction, you need to RENAME THE WHOLE NUMBER into a fraction form.**

THE BEST THING ABOUT DIVIDING FRACTIONS

Is that you can EASILY find the quotient by multiplying a fraction by its reciprocal! Multiplying by a reciprocal is the same as dividing by the original! SO COOL!!!

THE BEST THING ABOUT DIVIDING FRACTIONS

Is that you can EASILY find the quotient by multiplying a fraction by its reciprocal! Multiplying by a reciprocal is the same as dividing by the original! SO COOL!!!

**Math Chapter Seven Notes**

## BETWEEN NUMBERS

**RATIONAL NUMBERS: Any two numbers have an infinite number of numbers between them! These "in-betwen" numbers are found by partitioning the space between two numbers into fractions or decimals. Think of a number line. There could be an un-limited number of dashes between the numbers 3 and 4. There are 3.25, 3.5, 3.75 just in quarters alone! Remember what you have learned about place value. It is a lot like alphabetizing numbers! You need to look past the first letter sometimes just like you have to look past the first number. Be sure to take your time and think about the ways that numbers can be partitioned (separated) ... tenths ... eighths? Look closely and determine the place values and the places numbers need to be in.**

## MULTIPLY DECIMALS

ESTIMATE THE PRODUCT: of a decimal by rounding each decimal to the greatest place value and multiplying the factors

EX: 3.46 x 0.5 = 3 x 1

MULTIPLY BY A DECIMAL: by a whole number using repeated a edition, or the DISTRIBUTIVE PROPERTY. The number of decimal places in the product is the same as the number of decimals represented in both factors.

EX:

3.46

x 0.5

_______

1730 *SINCE THERE ARE THREE NUMBERS BEHIND DECIMAL POINTS IN THE EQUATION, THERE ARE THREE NUMBERS

+ 6920 BEHIND THE DECIMAL POINT IN THE PRODUCT.

_______

8.650

USE MENTAL MATH TO MULTIPLY A DECIMAL BY A POWER OF TEN: Write the digits of the decimal factor in the answer. Move the decimal point from its placement in the decimal factor one place to the right for each 0 in the POWER OF 10.

EX: 5.68 x 100 = 568, because the decimal point moves two places to the right.

ESTIMATE THE PRODUCT: of a decimal by rounding each decimal to the greatest place value and multiplying the factors

EX: 3.46 x 0.5 = 3 x 1

MULTIPLY BY A DECIMAL: by a whole number using repeated a edition, or the DISTRIBUTIVE PROPERTY. The number of decimal places in the product is the same as the number of decimals represented in both factors.

EX:

3.46

x 0.5

_______

1730 *SINCE THERE ARE THREE NUMBERS BEHIND DECIMAL POINTS IN THE EQUATION, THERE ARE THREE NUMBERS

+ 6920 BEHIND THE DECIMAL POINT IN THE PRODUCT.

_______

8.650

USE MENTAL MATH TO MULTIPLY A DECIMAL BY A POWER OF TEN: Write the digits of the decimal factor in the answer. Move the decimal point from its placement in the decimal factor one place to the right for each 0 in the POWER OF 10.

EX: 5.68 x 100 = 568, because the decimal point moves two places to the right.

## MULTIPLY MIXED NUMBERS

**When multiplying a MIXED NUMBER you can solve the equation one of two ways:**

1. RENAME THE MIXED NUMBER AS AN IMPROPER FRACTION

2. USE THE DISTRIBUTIVE PROPERTY TO MULTIPLY EACH PART OF THE EQUATION

* Remember that you can round a fraction to 1 or 1/2 depending on what it is closer too to estimate the product of an equation.

1. RENAME THE MIXED NUMBER AS AN IMPROPER FRACTION

2. USE THE DISTRIBUTIVE PROPERTY TO MULTIPLY EACH PART OF THE EQUATION

* Remember that you can round a fraction to 1 or 1/2 depending on what it is closer too to estimate the product of an equation.

## SIMPLIFY

**Before you can multiply fractions. You need to be able to simplify them. You can use one of two methods:**

1. USE PRIME FACTORIZATION:

- list the prime factors of each term

- cancel fractions until you can get no lower

- multiply the simplest form

2. FIND A COMMON FACTOR (try to find the GREATEST COMMON FACTOR)

- divide numerator and denominator by a common factor

- multiply the simplified numerators and denominators

To get a fraction to turn into 1, multiply the fraction by its RECIPROCAL (opposite) 3/4 = 4/3

Any whole number actually has a denominator of 1, so the reciprocal of 8 is 1/8.

1. USE PRIME FACTORIZATION:

- list the prime factors of each term

- cancel fractions until you can get no lower

- multiply the simplest form

2. FIND A COMMON FACTOR (try to find the GREATEST COMMON FACTOR)

- divide numerator and denominator by a common factor

- multiply the simplified numerators and denominators

To get a fraction to turn into 1, multiply the fraction by its RECIPROCAL (opposite) 3/4 = 4/3

Any whole number actually has a denominator of 1, so the reciprocal of 8 is 1/8.

## MULTIPLYING FRACTIONS

**Remember that when you multiply a whole number by a fraction, your answer will always be SMALLER than the original number. You are multiplying by a "piece" of something ... by something LESS than one, so your solution will be less than it would be if it were multiplied by a whole number.**

**Math Chapter Six Notes**

## 3-D FIGURES

**The points of 3-D FIGURES do not lie in the same plane. A three dimensional figure has length, width, and height. POLYHEDRONS are 3-D FIGURES enclosed by a flat surface that is a polygon. Pyramids and Prisms are named by the shape of their bases (bottoms). POLYHEDRONS have no curved surfaces. A FACE is a flat surface. An EDGE is a segment where two faces meet. A VERTEX is the point where three edges meet and also the point on a cone, opposite of its base. Look below at the quick guide to 3-D shapes.**

## CIRCLES

**A CIRCLE contains all of the points in a plane that are the same distance from a given point, the CENTER. A circle is named by its center point. A RADIUS is aline segment that extends from the center of a circle to a point on the circle and is written like a line segment. A CHORD is a line segment with endpoints on the circle. A DIAMETER is a chord that passes through the center of a circle and extends all the way to the other side. A CENTRAL ANGLE is an angle with its vertex at the center of the circle.**

**A CIRCLE can be broken up into fractions using central angles. There are 360 degrees in a full circle, by renaming fractions into lowest terms, you can see what part of a circle you are looking at ... for example, 90/360 is 1/4 in lowest terms. 180/360 in lowest terms is 1/2. 270/360 in lowest terms is 3/4.**

## TRANSFORMATIONS & SYMMETRY

**Translation, rotation, and reflection are transformations. They change the location, the position, or both the location and position of a polygon. The original figure moves from one place to another place. The transformed figure or image is congruent to the original.**

TRANSLATION: changes the location only

ROTATION: turns the position of a figure and changes its location by making it turn clockwise or counter clockwise

REFLECTION: changes the location and position by flipping over a "LINE OF REFLECTION" like a mirror image

TRANSLATION: changes the location only

ROTATION: turns the position of a figure and changes its location by making it turn clockwise or counter clockwise

REFLECTION: changes the location and position by flipping over a "LINE OF REFLECTION" like a mirror image

## FUN INTERRUPTION OF IRISH VACATION BUDGETING PROJECT

## TRIANGLES

**TRIANGLES are polygons with three sides, three vertices, and three angles. A triangle can be measured by angles or sides in the following ways ...**

**All angles of a triangle add up to 180 degrees. If you know the measure of two of the angles, you can easily figure out the final angle's measurement.**

## QUADRILATERALS

**QUADRILATERALS are polygons with four sides, four vertices, and four angles. A quadrilateral can be measured by angles or sides in the following ways ...**

**The angles of a QUADRILATERAL add up to 360 degrees. This is because when you cut a quadrilateral on a DIAGONAL (line segment connecting two nonadjacent vertices of a polygon) you create two triangles. Since the sum of the angles of a triangle are 180 degrees, adding two of them together creates 360 degrees.**

If you know the measures of three of the angles in a quadrilateral, you can figure out the fourth easily.

If you know the measures of three of the angles in a quadrilateral, you can figure out the fourth easily.

## CONGRUENT & SIMILAR FIGURES

**CONGRUENT FIGURES: two geometric figures with the same size and shape**

CONGRUENT POLYGONS: have corresponding angles of equal measure and corresponding sides of equal length

SIMILAR FIGURES: two geometric figures with the same shape but not necessarily the same size

SIMILAR POLYGONS: have corresponding angles of equal measure. The lengths of corresponding sides must be proportional

CONGRUENT POLYGONS: have corresponding angles of equal measure and corresponding sides of equal length

SIMILAR FIGURES: two geometric figures with the same shape but not necessarily the same size

SIMILAR POLYGONS: have corresponding angles of equal measure. The lengths of corresponding sides must be proportional

## POLYGONS

Remember that each POLYGON, whether regular or irregular, still forms INTERIOR ANGLES, and those angles can be classified as acute, right, or obtuse!

## ANGLE RELATIONSHIPS

## TYPES OF LINES & CLASSIFYING AND MEASURING ANGLES

PARALELL LINES: lines that are perfectly across one another in the same plane and never intersect

INTERSECTING LINES: lines that share a common point or will obviously cross when extended

PERPENDICULAR LINES: lines that intersect at a perfect 90 degree angle (perfect cross)

*** Pairs of coordinates can be drawn on a coordinate plane to create lines. There is even sometimes a pattern between the x and y-axis points. For example:

x - 2 = y

X Y

2 0

4 2

6 4

INTERSECTING LINES: lines that share a common point or will obviously cross when extended

PERPENDICULAR LINES: lines that intersect at a perfect 90 degree angle (perfect cross)

*** Pairs of coordinates can be drawn on a coordinate plane to create lines. There is even sometimes a pattern between the x and y-axis points. For example:

x - 2 = y

X Y

2 0

4 2

6 4

RAY: a part of a line that goes endlessly in one direction, the ray is named by the point side first

ANGLE: two rays sharing an endpoint create an angle, the VERTEX (shared endpoint) is how you name an angle

RIGHT ANGLE: an angle that measures perfectly at 90 degrees

ACUTE ANGLE: an angle that measures less than 90 degrees

OBTUSE ANGLE: measures greater than 90 degrees and less than 180 degrees

STRAIGHT ANGLE: a straight line that measures at 180 degrees

ANGLE: two rays sharing an endpoint create an angle, the VERTEX (shared endpoint) is how you name an angle

RIGHT ANGLE: an angle that measures perfectly at 90 degrees

ACUTE ANGLE: an angle that measures less than 90 degrees

OBTUSE ANGLE: measures greater than 90 degrees and less than 180 degrees

STRAIGHT ANGLE: a straight line that measures at 180 degrees

## BASIC GEOMETRIC FIGURES

GEOMETRY: is the study of shapes formed by points in a plane or space. The word geometry comes from Greek GEO which means earth, and METIRA meaning measure.

POINT: an exact location in space. It has no length, width, or thickness. The location is described using a point and a letter.

LINE: a line is a straight path connecting two points and extending endlessly in both directions. It is described by putting a line with arrows over the two letters that are points on either side.

PLANE: a plane is a flat surface that extends endlessly in all directions. It is named by three non-collinear points on the plane and is written by saying the word plane and then three letters of the points.

COLLINEAR:this term means "together on a line." A set of points is collinear when one line can be drawn through the points.

NONCOLLINEAR: this means "not together on a line." A line cannot be drawn straight through all of the points.

COORDINATE PLANE: a coordinate plane is formed by two number lines that intersect at right angles (90 degrees).

X-AXIS COORDINATE: the horizontal number line in a coordinate plane running left and right. Positive numbers are to the right. Negative numbers are to the left.

Y-AXIS COORDINATE: the vertical number line in a coordinate plane running up and down. Positive numbers are up. Negative numbers are down.

QUADRANTS: there are four sections on a coordinate plane and they are numbered 1-4. The top right is 1. The top left is 2. The bottom left is 3. The bottom right is 4.

ORDERED PAIR: this is the description of any point on a line designated by the X point and the Y point. It is labeled (X,Y).

*LOOK AT THE IMAGES BELOW OF THE VOCABULARY TERMS. CLICK ON EACH IMAGE TO GET IT TO ITS FULL SIZE.

POINT: an exact location in space. It has no length, width, or thickness. The location is described using a point and a letter.

LINE: a line is a straight path connecting two points and extending endlessly in both directions. It is described by putting a line with arrows over the two letters that are points on either side.

PLANE: a plane is a flat surface that extends endlessly in all directions. It is named by three non-collinear points on the plane and is written by saying the word plane and then three letters of the points.

COLLINEAR:this term means "together on a line." A set of points is collinear when one line can be drawn through the points.

NONCOLLINEAR: this means "not together on a line." A line cannot be drawn straight through all of the points.

COORDINATE PLANE: a coordinate plane is formed by two number lines that intersect at right angles (90 degrees).

X-AXIS COORDINATE: the horizontal number line in a coordinate plane running left and right. Positive numbers are to the right. Negative numbers are to the left.

Y-AXIS COORDINATE: the vertical number line in a coordinate plane running up and down. Positive numbers are up. Negative numbers are down.

QUADRANTS: there are four sections on a coordinate plane and they are numbered 1-4. The top right is 1. The top left is 2. The bottom left is 3. The bottom right is 4.

ORDERED PAIR: this is the description of any point on a line designated by the X point and the Y point. It is labeled (X,Y).

*LOOK AT THE IMAGES BELOW OF THE VOCABULARY TERMS. CLICK ON EACH IMAGE TO GET IT TO ITS FULL SIZE.

*Be sure to take time to notice where the positive and negative numbers are. QUADRANT ONE (+,+) QUADRANT TWO (-,+) QUADRANT THREE (-,-)

QUADRANT FOUR (+,-).

QUADRANT FOUR (+,-).

**Math Chapter Five Notes**

## ADD & SUBTRACT UNLIKE FRACTIONS

**Adding and subtracting UNLIKE FRACTIONS means that both of the denominators are different and unrelated. In order to solve any equation, you need to get the denominators to be the same! To do this, you simply RENAME the fraction by finding a COMMON DENOMINATOR. It is important that your sum or difference is always put into lowest terms.**

There are three ways that you can get denominators to be the same:

1. MULTIPLY THE DENOMINATORS:

If I have the fraction 2/5 and I want to add 2/3, I need to get their denominators the same by multiplying the numerators and denominators by the same number, in this case, the denominator over itself 3/3 and 5/5 (it doesn't change the relationship of the fraction when I multiply both by the same number because any number over itself equals 1).

So ... 2/5 x 3/3 is 6/15 and 2/3 x 5/5 is 10/15.

6/15 + 10/15 = 16/15 This is an improper fraction, so the correct answer is to make it into a mixed number which is 1 and 1/15.

2. LIST MULTIPLES:

If you find the LCM of the two denominators, you multiply the top and bottom of the fraction by the number that will get your denominators to be that same LCM. 5/6 minus 1/4.

4: 4, 8, 12

6: 6, 12

Since 12 is the LCM, we need to multiply 2/2 by 5/6 to equal 10/12, and 3/3 times 1/4 to equal 3/12.

10/12 - 3/12 is 7/12.

3. USE PRIME FACTORIZATION:

If the numbers are too far apart to list multiples, remember you can find the LCM by using PRIME FACTORIZATION of the two numbers.

If my problem is to add 5/8 and 7/18 ... I need to do the following ...

5/8 I need to find the prime factorization of 8 which is 2 x 2 x 2

7/18 I need to find the prime factorization of 18 which is 2 x 3 x 3

The LCM of 8 and 18 would be 2 x 2 x 2 x 3 x 3 or 72

Since I know this, I multiply 5/8 by 9/9 to get 45/72

And 7/18 I multiply by 4/4 to get 28/72

Now I can add 45/72 + 28/72 to get 73/72

Since this is improper, I need to turn it into a mixed number, so it would be 1 and 1/72.

*RANDOM BUT IMPORTANT INFORMATION: If you have mixed numbers that you need to subtract, and your fraction is too small, you can rename the number by taking a whole number away and adding to the fraction like this ...

4 and 3/5 can be renamed to 3 and 8/5

I did this by adding 5/5 (1) to 3/5 and taking that whole number away from 4, turning it into 3.

There are three ways that you can get denominators to be the same:

1. MULTIPLY THE DENOMINATORS:

If I have the fraction 2/5 and I want to add 2/3, I need to get their denominators the same by multiplying the numerators and denominators by the same number, in this case, the denominator over itself 3/3 and 5/5 (it doesn't change the relationship of the fraction when I multiply both by the same number because any number over itself equals 1).

So ... 2/5 x 3/3 is 6/15 and 2/3 x 5/5 is 10/15.

6/15 + 10/15 = 16/15 This is an improper fraction, so the correct answer is to make it into a mixed number which is 1 and 1/15.

2. LIST MULTIPLES:

If you find the LCM of the two denominators, you multiply the top and bottom of the fraction by the number that will get your denominators to be that same LCM. 5/6 minus 1/4.

4: 4, 8, 12

6: 6, 12

Since 12 is the LCM, we need to multiply 2/2 by 5/6 to equal 10/12, and 3/3 times 1/4 to equal 3/12.

10/12 - 3/12 is 7/12.

3. USE PRIME FACTORIZATION:

If the numbers are too far apart to list multiples, remember you can find the LCM by using PRIME FACTORIZATION of the two numbers.

If my problem is to add 5/8 and 7/18 ... I need to do the following ...

5/8 I need to find the prime factorization of 8 which is 2 x 2 x 2

7/18 I need to find the prime factorization of 18 which is 2 x 3 x 3

The LCM of 8 and 18 would be 2 x 2 x 2 x 3 x 3 or 72

Since I know this, I multiply 5/8 by 9/9 to get 45/72

And 7/18 I multiply by 4/4 to get 28/72

Now I can add 45/72 + 28/72 to get 73/72

Since this is improper, I need to turn it into a mixed number, so it would be 1 and 1/72.

*RANDOM BUT IMPORTANT INFORMATION: If you have mixed numbers that you need to subtract, and your fraction is too small, you can rename the number by taking a whole number away and adding to the fraction like this ...

4 and 3/5 can be renamed to 3 and 8/5

I did this by adding 5/5 (1) to 3/5 and taking that whole number away from 4, turning it into 3.

## ADD & SUBTRACT RELATED FRACTIONS

**Adding and subtracting RELATED FRACTIONS is a lot like adding and subtracting LIKE FRACTIONS. The difference is, that related fractions have different denominators. They are related because one is a multiple of the other. Before you can solve the problem, you need to be able to get the denominators to be the same. This is easy when you multiply the numerator and denominator by the same number to create a fraction that is LIKE!**

## ADDING & SUBTRACTING LIKE FRACTIONS

**Adding and subtracting LIKE FRACTIONS is as easy as adding or subtracting regular numbers. As long as your denominator is the same, you can add or subtract just like any normal problem.**

## ESTIMATE SUMS AND DIFFERENCES

**The best way to estimate the sums and differences of fractions is to simplify the math by comparing each fraction to either 0, 1/2, or 1. You can estimate each fraction to one of these three, and then proceed with the problem. Read through the image below, and practice by watching the video attached below it.**

**Math Chapter Four Notes**

## COMPARE AND ORDER FRACTIONS

**There are multiple ways to compare fractions:**

1.

2.

3.

EX: 7/8 vs. 2/6 When we look at these, we know that 7/8 is more than 1/2, and 2/6 is less than one half, so 7/8 is greater than 2/6.

4.

5.

6.

EX: 2 and 2/4 vs. 2 and 3/4 Since they both have 2 as a whole number, you'll compare the fractions 2/4 vs. 3/4 and since 3/4 is larger, 2 and 3/4 is the larger number.

1.

__SAME SIZE PARTS:__Fractions with the same denominator are called LIKE FRACTIONS and have the same size parts. Whatever numerator is larger (and closer to the denominator) is the larger fraction. EX: 5/8 vs. 3/8 Five is closer to the whole eight, so 5/8 is bigger than 3/8.2.

__SAME NUMBER OF PARTS:__When fractions have the same numerator, but different denominators, you compare by the size of the denominator. EX: 2/6 vs. 2/4 Fourths are bigger pieces than sixths are, so 2/4 is larger than 2/6.3.

__GREATER OR LESS THAN 1/2:__Analyze each fraction's relationship to 1/2. See if one fraction is greater or less than 1/2.EX: 7/8 vs. 2/6 When we look at these, we know that 7/8 is more than 1/2, and 2/6 is less than one half, so 7/8 is greater than 2/6.

4.

__CLOSENESS TO 1/2 or 1:__Analyze each fractions closeness to 1/2 or 1 whole. When both fractions are only one part away from 1/2 or 1, compare the number of parts to see which is bigger or smaller. EX: 7/8 vs. 6/7 Eighths are smaller than sevenths, so since they are both only one apart, 6/7 is a smaller fraction than 7/8 because only needing to add 1/8 to get to a whole is a smaller amount left to add.5.

__RENAME FRACTIONS TO COMPARE:__Sometimes you need to find a common denominator and rename fractions to see which is bigger by making them LIKE FRACTIONS because you really cannot tell any other way. EX: 3/4 vs. 7/8 You can rename 3/4 to 6/8 by multiplying the top and bottom by two. When you do this, you can compare 6/8 vs. 7/8 and you'll know that since 6/8 is smaller, 3/4 is smaller because they are equal fractions.6.

__COMPARE MIXED NUMBERS:__When you have a mixed number, you can first compare the whole numbers, then if they have the same whole number, start to compare the fraction.EX: 2 and 2/4 vs. 2 and 3/4 Since they both have 2 as a whole number, you'll compare the fractions 2/4 vs. 3/4 and since 3/4 is larger, 2 and 3/4 is the larger number.

## EQUIVALENT FRACTIONS

**EQUIVALENT FRACTIONS name the same part of a whole or the same part of a set and are equal to one another in value. When a fraction is renamed to HIGHER TERMS, the numerator and denominator of the new fraction are greater (having been multiplied by the same number on the top and bottom to keep them equal). When a fraction is renamed to LOWER TERMS, the numerator and denominator of the new fraction are less (having been divided by the same number on top and bottom to keep them equal).**

**A fraction is in LOWEST TERMS when it is in its SIMPLEST FORM, and when 1 is the only common factor of the numerator and denominator (neither can be divided by any other same number to equally reduce them).**

Sometimes the quickest way to simplify bigger fraction numbers is to use the strategy of CANCELLATION. This process cancels out fractional names and replaces them with 1, by using the prime factorization of the numerator and denominator. STEPS: 1. Use mental math to list the prime factors of the numerator and denominator from least to greatest. 2. Identify and cancel all the fractional names for 1. 3. The canceled numbers removed the GCF. Multiply the remaining factors in the numerator and in the denominator. The result is the simplified fraction. |

## IMPROPER FRACTIONS

**An improper is a fraction with a value LARGER than 1. The numerator is larger than the denominator, meaning that there are more pieces than the whole. To balance improper fraction, we can change it to a mixed number, which is basically found the same way we do division.**

**Below you can see how to convert an IMPROPER FRACTION to a MIXED NUMBER and back again. Using simple multiplication and division, both are easily accomplished.**

## PROPER FRACTIONS

**Fractions are used to name a part of a whole. The NUMERATOR is the top number and tells the number of parts selected and the DENOMINATOR is the bottom number and tells the equal parts of a whole. A fraction is a PROPER FRACTION if its value is less than one.**

## LEAST COMMON MULTIPLE

**There are an infinite number of multiples for any given number. A MULTIPLE is the product of two whole numbers. A COMMON MULTIPLE of two numbers can always be found by multiplying the two numbers together.**

**The LEAST COMMON MULTIPLE (LCM) is the smallest multiple that they have in common. The LCM of any two numbers is equal to or greater than the greater number.**

When the greater number is a multiple of the lesser number, the LCM is equal to the greater number.

EX:

12: 12, 24, 36, 48, 60

24: 24, 48, 72, 96

When the greater number is NOT a multiple of the lesser number and the two numbers share a common factor, the LCM is greater than the greater number and less than the product of the two numbers.

EX:

12: 12, 24, 36, 48, 60, 72

18: 18, 36, 54, 72

18 is not a multiple of 12, but 12 and 18 share the common factor 3; the LCM is greater than 18 and less than the product of 12 x 18 (216); the LCM of 12 and 18 is 36.

THERE ARE TWO MAIN METHODS FOR FINDING THE LEAST COMMON MULTIPLE.

For smaller numbers or numbers that are easy to count by, list the multiples and see what they have in common.

For bigger numbers, write out the PRIME FACTORIZATION WITH EXPONENTS and then multiply the highest power of each factor listed.

When the greater number is a multiple of the lesser number, the LCM is equal to the greater number.

EX:

12: 12, 24, 36, 48, 60

24: 24, 48, 72, 96

When the greater number is NOT a multiple of the lesser number and the two numbers share a common factor, the LCM is greater than the greater number and less than the product of the two numbers.

EX:

12: 12, 24, 36, 48, 60, 72

18: 18, 36, 54, 72

18 is not a multiple of 12, but 12 and 18 share the common factor 3; the LCM is greater than 18 and less than the product of 12 x 18 (216); the LCM of 12 and 18 is 36.

THERE ARE TWO MAIN METHODS FOR FINDING THE LEAST COMMON MULTIPLE.

For smaller numbers or numbers that are easy to count by, list the multiples and see what they have in common.

For bigger numbers, write out the PRIME FACTORIZATION WITH EXPONENTS and then multiply the highest power of each factor listed.

## GREATEST COMMON FACTOR

**Remember that a FACTOR is a number that you multiply by to get to a product. For example, the factors of the number 4 are 1, 2, and 4 because you can multiply 1 x 4 to get to the product of 4, or 2 x 2 to get to the product of 4.**

Also, when a number only has factor of 1 and itself, it is called a PRIME number. When numbers have more than that (such as the number 4) it is called a COMPOSITE number.

When two or more composite numbers have a factor that is the same, each shared factor is called a COMMON FACTOR. The GREATEST COMMON FACTOR (GCF) is the largest number out of the factors that they have in common. The GCF can be found by LISTING FACTORS of each number.

Also, when a number only has factor of 1 and itself, it is called a PRIME number. When numbers have more than that (such as the number 4) it is called a COMPOSITE number.

When two or more composite numbers have a factor that is the same, each shared factor is called a COMMON FACTOR. The GREATEST COMMON FACTOR (GCF) is the largest number out of the factors that they have in common. The GCF can be found by LISTING FACTORS of each number.

**A composite number can be**

**expressed as the product of a set of PRIME FACTORS. A FACTOR TREE (breaking down the product into multiplication sets) can be used to find the PRIME FACTORIZATION (what prime numbers are multiplied together to get to the product) of a composite number.**

**PRIME FACTORIZATION can be used to find the GCF of two numbers.**

1. List the prime factors of each number after doing a factor tree

2. Select the factors they have in common on both lists

3. Multiply the common factors and you'll get the GCF

* Remember that factors are all the numbers that multiply up to the original number, PRIME factors are the numbers that are only divisible by one and themselves. All numbers can be broken down into factors and prime factors.

1. List the prime factors of each number after doing a factor tree

2. Select the factors they have in common on both lists

3. Multiply the common factors and you'll get the GCF

* Remember that factors are all the numbers that multiply up to the original number, PRIME factors are the numbers that are only divisible by one and themselves. All numbers can be broken down into factors and prime factors.

**Math Chapter Three Notes**

## MULTI-STEP PROBLEMS

One thing that you want to pay attention to is that multi-step problems can have several "correct" equations that get you to the right answer. The ORDER OF OPERATIONS RULES can help you figure out the way to put a story problem into an equation.

Remember our Order of Operation Rules

1. Do operations in PARENTHESES first.

2. Solve all EXPONENTS next.

3. MULTIPLY or DIVIDE left to right.

4. ADD or SUBTRACT left to right.

If you are feeling "stuck" on a word problem, ask yourself if you understand the story problem as a story, then evaluate it as a problem. Use the CUBES strategy to help you slow down and evaluate all of the information.

One thing that you want to pay attention to is that multi-step problems can have several "correct" equations that get you to the right answer. The ORDER OF OPERATIONS RULES can help you figure out the way to put a story problem into an equation.

Remember our Order of Operation Rules

1. Do operations in PARENTHESES first.

2. Solve all EXPONENTS next.

3. MULTIPLY or DIVIDE left to right.

4. ADD or SUBTRACT left to right.

If you are feeling "stuck" on a word problem, ask yourself if you understand the story problem as a story, then evaluate it as a problem. Use the CUBES strategy to help you slow down and evaluate all of the information.

**Example:**

Miss Robbins needs a border for a bulletin board that has a length of 90 centimeters and a width of 75 centimeters. How many centimeters of border does she need?

First, remember that she has four sides. A bulletin board is either a square or rectangle.

So know that she has two sides that are 90 centimeters (2 x 90).

Then, think about that she has another two sides that are 75 centimeters (2 x 75).

Finally, realize they are asking you for the total ... so the equation is (2 x 90) + (2 x 75) = 330 centimeters.

Miss Robbins needs a border for a bulletin board that has a length of 90 centimeters and a width of 75 centimeters. How many centimeters of border does she need?

First, remember that she has four sides. A bulletin board is either a square or rectangle.

So know that she has two sides that are 90 centimeters (2 x 90).

Then, think about that she has another two sides that are 75 centimeters (2 x 75).

Finally, realize they are asking you for the total ... so the equation is (2 x 90) + (2 x 75) = 330 centimeters.

## ORDER OF OPERATIONS

**The ORDER OF OPERATIONS is a very specific set of rules so that when we have multi-step problems, we know which part of the problem to solve first. If we do not solve these problems in order ... we get them wrong!!! Be sure to ALWAYS solve problems in the correct order!**

1. DO ANYTHING IN PARENTHESES FIRST (It's like a limo driving up a V.I.P. they always get let in to the place first because they are very important customers)!

2. FIND THE VALUE OF ANY EXPONENTS

3. MULTIPLY AND DIVIDE FROM LEFT TO RIGHT (Just like when we read a book. Start on one side and move to the other.)

4. ADD AND SUBTRACT FROM LEFT TO RIGHT

There is a silly ACRONYM (word that is an abbreviation for a set of words) that can help you remember the order of operations ... it is ... "PLEASE EXCUSE MY DEAR AUNT SALLY."

P = PARENTHESES

E = EXPONENTS

M = MULTIPLICATION *But remember, multiplication and division and addition and

D = DIVISION subtraction are equal operations, meaning you just do

A = ADDITION them left to right in order M & D first, then A & D.

S = SUBTRACTION

1. DO ANYTHING IN PARENTHESES FIRST (It's like a limo driving up a V.I.P. they always get let in to the place first because they are very important customers)!

2. FIND THE VALUE OF ANY EXPONENTS

3. MULTIPLY AND DIVIDE FROM LEFT TO RIGHT (Just like when we read a book. Start on one side and move to the other.)

4. ADD AND SUBTRACT FROM LEFT TO RIGHT

There is a silly ACRONYM (word that is an abbreviation for a set of words) that can help you remember the order of operations ... it is ... "PLEASE EXCUSE MY DEAR AUNT SALLY."

P = PARENTHESES

E = EXPONENTS

M = MULTIPLICATION *But remember, multiplication and division and addition and

D = DIVISION subtraction are equal operations, meaning you just do

A = ADDITION them left to right in order M & D first, then A & D.

S = SUBTRACTION

## DIVIDE BY A POWER OF TEN

**When dividing a whole number or a decimal by 10, the movement of the decimal point one place to the left renames each digit to the next lesser place, thereby making the quotient 1/10 (one tenth) of the value of the dividend!**

**When dividing by a power of ten, move the decimal point one place to the left for each zero in the divisor. Annex zeros as needed.**

**NOTICE A PATTERN!!!**

**Remember dividing makes the quotient SMALLER. If your quotient is not smaller than your dividend ... something went wrong.**

## DIVIDE A DECIMAL BY 2-DIGIT DIVISORS

**Look over the image of division problems below. Make sure that they make sense to you.**

**An AVERAGE is found by adding two of more quantities in a set and then dividing that total by the number of addends. (Sometimes the total is given and only the division step is necessary.)**

When a number is NOT divisible by the divisor, you can continue dividing by annexing (adding) zeros in the dividend and ROUNDING THE QUOTIENT to any given place. There is a symbol that means "IS APPROXIMATELY EQUAL TO" and is used to show that the quotient was rounded.

When a number is NOT divisible by the divisor, you can continue dividing by annexing (adding) zeros in the dividend and ROUNDING THE QUOTIENT to any given place. There is a symbol that means "IS APPROXIMATELY EQUAL TO" and is used to show that the quotient was rounded.

## DIVIDING DECMALS BY WHOLE NUMBERS

**Remember that just like multiplying a whole number by a decimal, your answer is going to have THE SAME NUMBER OF NUMERALS BEHIND A DECIMAL POINT as your equation. Before dividing decimals, however, s**

**ometimes numbers are not easily divisible so instead of putting a remainder, you can continue dividing the number by ANNEXING ZEROS (adding zeros) after a decimal point. Then, you will have a DECIMAL QUOTIENT instead of the fraction remainder we are currently used to.**

Be aware that sometimes it is more accurate to have a reminder in the form of a fraction and sometimes it is more accurate to have a decimal quotient.

EX: Jack and John are sharing 5 packs of gum. How many packs can each boy have?

5 / 2 = 2.5 packs of gum

EX: Chef Watson has 5 cups of flour. If he divides the flour evenly between 2 bowls, how much will be in each bowl?

5 / 2 = 2 1/2

For this lesson, ESTIMATE RANGE works the same way.

Remember that fractions ARE ALSO decimals. You just divide the numerator by the denominator, annexing the zeros as needed. the QUOTIENT of a fraction will ALWAYS be less than one.

Take some time to watch these three videos BEFORE you try a few problems on your own.

Be aware that sometimes it is more accurate to have a reminder in the form of a fraction and sometimes it is more accurate to have a decimal quotient.

EX: Jack and John are sharing 5 packs of gum. How many packs can each boy have?

5 / 2 = 2.5 packs of gum

EX: Chef Watson has 5 cups of flour. If he divides the flour evenly between 2 bowls, how much will be in each bowl?

5 / 2 = 2 1/2

For this lesson, ESTIMATE RANGE works the same way.

Remember that fractions ARE ALSO decimals. You just divide the numerator by the denominator, annexing the zeros as needed. the QUOTIENT of a fraction will ALWAYS be less than one.

Take some time to watch these three videos BEFORE you try a few problems on your own.

## 2-DIGIT DIVISORS

**Before solving a problem, you can estimate the quotient by rounding the divisor to a multiple of ten. Then, use a number that is compatible with the rounded divisor (a multiple of the rounded divisor). By using two multiples of the divisor as dividends, you can find an**

__ESTIMATE RANGE__for the quotient.**When solving division problems, you may need to ADJUST THE QUOTIENT up or down. Adjust the quotient UP when the remainder is the same as or greater than the divisor. Adjust the quotient DOWN when the product is greater than the dividend.**

**Remember that dividing by 2-digits is no different than dividing by one. You still need to take one step at a time. Whether using traditional or partial-quotient method, you still need to figure out what place value to begin your quotient at.**

**Be careful when doing word problems. If there is a remainder, think closely about what is being asked. Sometimes, the quotient needs to be increased by one. Sometimes, the remainder can be dropped for the answer.**

EX: Amanda has 22 baby carrots to give to 7 friends. How many carrots can she give to each friend?

22 / 7 = 3 with one left over,

So she can only give each friend 3 carrots. The remainder does not matter in this case.

EX: Timothy and 9 of his friends planned a camping trip. They will travel with 4 friends in each car. How many cars are needed?

10 / 4 = 2 with two left over

You cannot leave two kids behind, so you will need to round up to 3 cars, so that the whole group can go.

EX: Amanda has 22 baby carrots to give to 7 friends. How many carrots can she give to each friend?

22 / 7 = 3 with one left over,

So she can only give each friend 3 carrots. The remainder does not matter in this case.

EX: Timothy and 9 of his friends planned a camping trip. They will travel with 4 friends in each car. How many cars are needed?

10 / 4 = 2 with two left over

You cannot leave two kids behind, so you will need to round up to 3 cars, so that the whole group can go.

## LESSON TWO: MULTIPLES OF 10

**Let's take a moment to review the rules of divisibility. These rules are helpful in any combination of facts to get you to realize whether or not you will have a remainder.**

____Remember that a whole number is DIVISIBLE by another whole number if there is no remainder. You can use DIVISIBILITY RULES to determine whether a number is divisible by another whole number.

Look for COMPATIBLE NUMBERS (numbers that share basic multiplication or division facts).

4,500/50

Think about the fact that 45 is divisible by 5, the quotient is 9.

Realize then, that 4,500 is nothing more than 50 X 90.

If that is true, then 4,500/50 is simply 90.

In the equation 1,651/40, think about compatible numbers.

4 x 4 = 16, so 40 x 40 would be 1,600.

Now you know that you will need MORE than 40, but it gave you a suitable start.

When you continue with the equation, you'll have a remainder in the form of a fraction, but the starting point is what is most important to realize.

____When you are dividing by multiples of 10 (any number that ends in zero or a set of zeros) use mental math to find the quotient if your DIVISOR and DIVIDEND both end in zeros.Look for COMPATIBLE NUMBERS (numbers that share basic multiplication or division facts).

4,500/50

Think about the fact that 45 is divisible by 5, the quotient is 9.

Realize then, that 4,500 is nothing more than 50 X 90.

If that is true, then 4,500/50 is simply 90.

In the equation 1,651/40, think about compatible numbers.

4 x 4 = 16, so 40 x 40 would be 1,600.

Now you know that you will need MORE than 40, but it gave you a suitable start.

When you continue with the equation, you'll have a remainder in the form of a fraction, but the starting point is what is most important to realize.

**This second video is FANTASTIC for reminding you about decimal places and that a decimal is ALWAYS there, even when you cannot see it. Watch it and try the problems along with him.**

## LESSON ONE: DIVISION

**Division is the INVERSE (opposite) of multiplication. A DIVIDEND is the number being divided. A DIVISOR is the number that is dividing the dividend into equal parts. The QUOTIENT is how many times the divisor equally splits the dividend. The REMAINDER is the part over the whole fraction that remains from NOT BEING DIVIDED equally.**

**The Part-to-Whole Model of Division is where you can create an equation by taking the whole (quotient) and dividing it into the equal pieces.**

**When you divide traditionally, there are multiple steps to follow. It is a piece by piece, number by number process that you can see above. Read through the steps carefully. Watch the video below to help you remember these steps. There is also a song to help you.**

**If the above description of long division is too confusing to you, take a peek at the PARTIAL QUOTIENT method below. It is another way to use smaller multiplication equations to make sense of larger division equations. There is also a song to help you remember.**

**Math Chapter Two Notes**

## LESSON ONE: MULTIPLICATION

**Factor x Factor = Product**

MULTIPLICATION is a form of addition! It is used to find the PRODUCT (total) when you multiply. The first FACTOR of a

multiplication equation tells the number of sets; the second factor tells the size of each set. When drawing the equation

to help you make sense of it, determine the number of sets, and how many are in each set.

MULTIPLICATION is a form of addition! It is used to find the PRODUCT (total) when you multiply. The first FACTOR of a

multiplication equation tells the number of sets; the second factor tells the size of each set. When drawing the equation

to help you make sense of it, determine the number of sets, and how many are in each set.

**This is an illustration of 5 x 6**

**A FACTOR is a number that can be multiplied together to make a product (For example: the factors of the number 12 are: 1, 2, 3, 4, 6, and 12)**

A MULTIPLE of a number, is the product of any whole number and a given number (For example: some multiples of 2 are: 2, 4, 6, 8, 10 ... ect.)

A MULTIPLE of a number, is the product of any whole number and a given number (For example: some multiples of 2 are: 2, 4, 6, 8, 10 ... ect.)

**There are many PROPERTIES, or rules that are true about multiplication.**

**LESSON TWO: MULTIPLES OF 10**

*** Knowing a MULTIPLE OF 10 will help you do MENTAL MATH, or, math in your head. You simply need to remember that any number ending in 0 is a multiple of 10! For example:**

10 x 10 = 100

10 x 100 = 1,000

10 x 1,000 = 10,000

10 x 5 = 50

10 x 24 = 240

* When you are multiplying by 10, 100, or 1,000, first multiply the basic fact, then write the same number of zeros in the product as the number of zeros in the equation. For example:

2,400 x 20 =

24 x 2 = 48

Now I need to add my three zeros from the equation ... so 2,400 x 20 = 48,000

* When both factors are multiples of 10, 100, or 1,000, multiply the basic fact, and write the number of zeros of both combined after the answer in your product. For example:

20 x 30 =

2 x 3 = 6

Now I need to add my two zeros from the equation ... so 20 x 30 = 600

* You can use the DISTRIBUTIVE PROPERTY to find the product of an equation mentally when one factor is a multiple of 10. For example:

20 x 17 =

20 x (10 + 7) =

(20 x 10) + (20 x 7) =

200 + 140 = 340

10 x 10 = 100

10 x 100 = 1,000

10 x 1,000 = 10,000

10 x 5 = 50

10 x 24 = 240

* When you are multiplying by 10, 100, or 1,000, first multiply the basic fact, then write the same number of zeros in the product as the number of zeros in the equation. For example:

2,400 x 20 =

24 x 2 = 48

Now I need to add my three zeros from the equation ... so 2,400 x 20 = 48,000

* When both factors are multiples of 10, 100, or 1,000, multiply the basic fact, and write the number of zeros of both combined after the answer in your product. For example:

20 x 30 =

2 x 3 = 6

Now I need to add my two zeros from the equation ... so 20 x 30 = 600

* You can use the DISTRIBUTIVE PROPERTY to find the product of an equation mentally when one factor is a multiple of 10. For example:

20 x 17 =

20 x (10 + 7) =

(20 x 10) + (20 x 7) =

200 + 140 = 340

## LESSON THREE: EXPONENTS

**Multiplication is a short way to write a repeated addition problem. When a factor is repeated in a multiplication equation, it can be written in exponent form.**

BASE: the number that is repeated as a factor

EXPONENT: an exponent is the number of times the base is repeated as a factor

EXAMPLE: 5 to the 4th power is 5 x 5 x 5 x 5

6 to the 3rd power is 6 x 6 x 6

*Exponents can be used to express numbers to the POWER OF TEN also. If, for example, I wanted to say 1,000,000 as an exponent, I would say ten to the power of 6. You can also use this when you are writing numbers in expanded form.

EXAMPLE: 5,429 is (5 x 10 to the power of 3) + (4 x 10 to the power of 2) + (2 x 10 to the power of 1) + (9 x 10 to the power of 0)

BASE: the number that is repeated as a factor

EXPONENT: an exponent is the number of times the base is repeated as a factor

EXAMPLE: 5 to the 4th power is 5 x 5 x 5 x 5

6 to the 3rd power is 6 x 6 x 6

*Exponents can be used to express numbers to the POWER OF TEN also. If, for example, I wanted to say 1,000,000 as an exponent, I would say ten to the power of 6. You can also use this when you are writing numbers in expanded form.

EXAMPLE: 5,429 is (5 x 10 to the power of 3) + (4 x 10 to the power of 2) + (2 x 10 to the power of 1) + (9 x 10 to the power of 0)

**Look at these images and make sense of bases and exponents, then watch the video on EXPONENTS below.**

## LESSON FOUR: 1 & 2 DIGIT MULTIPLIERS

**Remember that when you are multiplying by a one digit factor, every place in the**

**multiplicand (what is being multiplied) needs to be multiplied by the multiplier (how many times).**

2,345 So you need to multiply 6 x 5, 6 x 4, 6 x 3, and 6 x 2, of course renaming place values

x 6 if need be.

____

2,345 So you need to multiply 6 x 5, 6 x 4, 6 x 3, and 6 x 2, of course renaming place values

x 6 if need be.

____

**Sometimes a problem will ask you to ESTIMATE, and when it does, remember there are two main types of estimation:**

1. ROUNDING: Here, you would round to the greatest place value

2. FRONT-END ESTIMATING: Here, you would multiply the two greatest place values (remember it is like the front two seats in a car). Front-End Estimation is more accurate.

EXAMPLE:

4 x 5,280

ROUNDING: 4 x 5,000 = 20,000

FRONT-END ESTIMATING: 4 x 5,200 = 20,800

1. ROUNDING: Here, you would round to the greatest place value

2. FRONT-END ESTIMATING: Here, you would multiply the two greatest place values (remember it is like the front two seats in a car). Front-End Estimation is more accurate.

EXAMPLE:

4 x 5,280

ROUNDING: 4 x 5,000 = 20,000

FRONT-END ESTIMATING: 4 x 5,200 = 20,800

**Remember that when you multiply, you can use the DISTRIBUTIVE PROPERTY to make your numbers more manageable and easier to work with. You need to pull the multiplier apart and make individual equations. The answers to these small equations are called PARTIAL PRODUCTS, and you will add them together to give you the ultimate product.**

**There is another way to multiply called SHORT FORM, which is the traditional way of multiplying. Study the picture below, notice how you need to put a place value holding 0 as the first digit of the second round of products when multiplying by a multiplier in the 10's place. Watch the video below to help you remember, then do a few practice problems.**

**Watch the video on Multiplying by 2-Digit Numbers, and then do a few practice problems of your own. You may use a calculator to CHECK your answers.**

## CHAPTER TWO: LATTICE MATH

**Click on the linked image below to remind you how to do lattice math! Exciting and fun!**

**CHAPTER TWO: MULTIPLY DECIMALS BY A WHOLE NUMBER**

**When multiplying decimal by a whole number, the process is the same as multiplying by two whole numbers. You just need to be sure to put the decimal in the answer in the same place as it was in the equation. This type of multiplication is so important, because this is how we would multiply money. FOR EXAMPLE:**

$ 2.56 * To estimate your answer, you can just

x 2 multiply the two whole numbers.

____

$ 5.12

$ 2.56 * To estimate your answer, you can just

x 2 multiply the two whole numbers.

____

$ 5.12

**Watch the video below to review estimating, and multiplying decimals by whole numbers. Watch the whole video and maybe even do the problem with her to help you pay attention and not lose focus on the lesson.**

**Here is another example.**

**Below, there is an example of multiplying two decimal numbers by one another. This is a bit advanced, but you notice that the process is the same. However many digits are behind the decimal point in the equation is STILL the same number of digits behind the decimal point in the product.**

**Math Chapter One Notes**

## LESSON ONE: WHOLE NUMBERS PLACE VALUE

**Click on the picture of "Place Value" after you really look at it. The link will take you to a reminder article of how to find place value of whole numbers and what the digits mean in each place.**

**Look below at the different examples of how to write numbers. STANDARD FORM is writing the number as a number form (2,345 is Standard Form). WORD FORM is when you write the number as an actual set of words (two-thousand, three-hundred, forty-five). EXPANDED FORM is when you write a number by stretching it out and showing the place value of each number and adding them together (2,000 + 300 + 40 + 5). Take a second to write one number each way as practice.**

**Below you will see an example of EXPANDED NOTATION also known as EXPANDED WITH MULTIPLICATION FORM. This form (like Expanded Form, it becomes an addition problem, but this form includes multiplication as a way to mark the place value. For example, the number 5,000 would be written as 5 X 1,000. Writing the number 2,345 would be notated as 2 X 1,000 + 3 X 100 + 4 X 10 + 5. It is pretty logical, but it takes a long time, and a lot of room, to write it.**

**Look at the image below and think about what you remember when comparing numbers. Click on the picture and watch the video below to remember how how to compare and order whole numbers. It is all about remembering place value!**

**Remember when you are Rounding Whole Numbers, you need to check the digit to the right. If the digit is LESS than 5, the rounding number stays the same. If it is MORE than five, the rounding number increases by one. If you do round UP, replace all the numbers to the right of the rounding number with zeros! Example: 5,678 ... if I round to the nearest thousand, I round up to 6,000 because the 6 in the hundreds place is higher than five, so it makes me round the 5 in the thousands place up to 6! Look at the image gallery below closely to help you.**

## LESSON TWO: ADD WHOLE NUMBERS

**Addition is used to find the value of two or more number sets. The**

**numbers being added together are called ADDENDS and the answer is called the SUM. Be sure to begin adding at the place of least value (farthest to the right) and then go toward the place of greatest value (all the way to the left). Remember to regroup when you need to depending on adding numbers of greater place value. WATCH THE VIDEO BY CLICKING ON THE IMAGE BELOW TO REVIEW REGROUPING ADDITION.**

**Estimation is to guess an approximate answer. Estimation is used to get near to an answer or to check your answer mentally. ROUNDING is a practice of where you estimate a number to a certain place value. For example, rounding 4,178 to the nearest thousand would be 4,000. FRONT-END ESTIMATION is where you add the digits of the two greatest place values to get a more accurate guess of the answer. For example, 15,678 + 26,311, I would add 15,000 + 26,000 = 41,000. That would give me a great estimate of my answer. CLICK ON THE IMAGES THAT SAYS ROUNDING TO PRACTICE.**

## LESSON THREE: SUBTRACT WHOLE NUMBERS

**Subtraction is used to find the difference between two numbers, or, the amount left when something is taken away. The MINUEND is the number something is taken away from. The SUBTRAHEND is the number being subtracted. The DIFFERENCE is the answer. Subtraction begins at the lowest place value (far right) and works up to the greatest place value (far left). Click on the image below to WATCH THE VIDEO on subtraction!**

**Addition and Subtraction are INVERSE OPERATIONS, or, opposites of one another. Because they are opposites, we can write related equations to one another using addition and subtraction. WATCH the video link below to help you remember this property.**

**Estimation is used in subtraction to get an approximate answer. To find this approximation, we can use rounding to the greatest place value, or front-end estimation. Do the rest of these sample problems and have a parent check your work. The numbers with a space should have a comma between them because they are in the thousands. Insert a comma into your work.**

## LESSON FOUR: DECIMAL PLACE VALUE

**A DECIMAL is a number that contains one or more digits to the**

*right*of the ones place. Those digits are a decimal or a FRACTION of a whole number! Decimals have a value of LESS THAN ONE! The decimal portion of a number is separated from the whole number by a DECIMAL POINT.**The DECIMAL PLACE VALUE is like any place value, and the farther you go in one direction or the other, the larger or smaller a number becomes. For example, 45.67 is forty-five and sixty-seven**

**hundredths, and that is smaller than 45.736 forty-five and two hundred and thirty-six thousandths, because the number in the tenths place is larger! Look at the chart of place value below, and then WATCH the video to help you.**

**Decimals can be written in various forms, just like regular whole numbers. There is still a STANDARD FORM, (written like a regular number: 4.56) WORD FORM, (written in words: four AND fifty-six**

**hundredths) EXPANDED FORM (stretched out: 4 + 0.5 + 0.06) and EXPANDED FORM WITH MULTIPLICATION ( (4x1) + (5 x O.1) + (6 x 0.01) = 4.56). But there is also a**

*new* form called FRACTION FORM, which is where the number is written as it is said! For example, 4 56/100 (four and fifty-six hundredths). Look closely and make sense of each of the pictures below.**Remember that decimals can be rounded just like any other number. Look to the largest digit, and then look behind it for a "high five" round up, for a "low five" round down. For example, the number .362 rounds UP to the nearest tenth .4, or the number .362 rounds DOWN to the nearest hundredth of .36 because the 2 was not large enough to round up. Be careful that you know what decimal place the question is asking you to round to. Study the picture below to make sense of what you just read in these notes.**

**Something you might not know is that you can "annex" or get rid of zeros that you don't need to help you line up decimals. You can also add zeros without changing the number if they are AFTER the last number of a decimal. You cannot add numbers in between digits. Zeros can make an EQUIVALENT DECIMAL (or an equal decimal) to help you order them. For example:**

3.15 3.5 3.01 and 0.3 can be written like this to help you make sense of the numbers and make them "seem" the same size.

3.15

3.50

3.01

0.30

Adding the red zeros did NOT change the numbers, but it made it easier to line them up. Now I can see easily that the order from least to greatest should be:

0.3 3.01 3.15 3.5

3.15 3.5 3.01 and 0.3 can be written like this to help you make sense of the numbers and make them "seem" the same size.

3.15

3.50

3.01

0.30

Adding the red zeros did NOT change the numbers, but it made it easier to line them up. Now I can see easily that the order from least to greatest should be:

0.3 3.01 3.15 3.5

## LESSON FIVE: ADD & SUBTRACT DECIMALS

**Adding and subtracting decimals is much the same as adding and subtracting whole numbers. You still need to align the numbers according to place value. Remember that you can add or annex zeros to help you as long as it does not change the amount of the number. Read the picture below and CLICK TO WATCH THE VIDEO on adding and subtracting decimals.**

**Remember that there are four ADDITION PROPERTIES. The COMMUTATIVE PROPERTY says we can change the order of numbers in an equation and still get the same sum. In the ASSOCIATIVE PROPERTY, you can change the grouping of addends and still get the same sum. The IDENTITY PROPERTY says that adding zero to a number doesn't change it at all. The DISTRIBUTIVE PROPERTY combines addition and multiplication, and it says that multiplying the sum of an addition equation will be the same as adding the independent addends multiplied by that number. These are useful rules that you will need to remember when working with decimal equations. The video linked below reviews two of these properties that you will use most in this lesson.**

**Remember that you should estimate the answer of adding or subtracting a decimal by ROUNDING each decimal to the greatest place value. That means, look at the number that is in the largest place, then look behind that number. If the digit is five or over, round up; if it is less than five, round down. Look at the picture below to reming you.**

## LESSON SIX: SOLVING PROBLEMS

*** Being a good problem solver is an essential skill to every part of life! Before you are able to solve math story problems, it is important that you understand the story before you'll be able to understand the problem. Ask yourself the following questions:**

- Is there enough information to solve it?

- Is information from a previous problem required?

- Does solving the problem require more than one step?

- Is there more than one way to solve it?

* Sometimes deciding on a good STRATEGY is a smart way to solve a problem. This could include: making a list, drawing a picture, creating a graph, solving the problem backwards, guessing and checking, or solving simpler problems within the question.

* A PART-WHOLE MODEL can help you visualize the problem when you don't quite know how to solve it. It shows you what part of the equation you know, and what part is missing, especially if it is an addition, subtraction, multiplication, or division problem. Remember that a VARIABLE is a letter that symbolizes the part we do not yet know. Look below and click on the image to understand part-whole models.

- Is there enough information to solve it?

- Is information from a previous problem required?

- Does solving the problem require more than one step?

- Is there more than one way to solve it?

* Sometimes deciding on a good STRATEGY is a smart way to solve a problem. This could include: making a list, drawing a picture, creating a graph, solving the problem backwards, guessing and checking, or solving simpler problems within the question.

* A PART-WHOLE MODEL can help you visualize the problem when you don't quite know how to solve it. It shows you what part of the equation you know, and what part is missing, especially if it is an addition, subtraction, multiplication, or division problem. Remember that a VARIABLE is a letter that symbolizes the part we do not yet know. Look below and click on the image to understand part-whole models.

## LESSON SEVEN: POSITIVE AND NEGATIVE NUMBERS

**POSITIVE NUMBERS are numbers that are to the right of zero on the number line. NEGATIVE NUMBERS are numbers to the left of zero on a number line. Negative numbers are always worth less than zero. ZERO is neither a positive nor a negative number.**

**You can use a number line to help you solve equations that involve positive and negative numbers being added or subtracted. Adding a negative number will always make the original addend smaller, so your sum will be less than your original number.**

EXAMPLE: 3 + -2 = 1 Because I am starting at three, and essentially going backward on the number line, which make it act like a subtraction problem.

Just like in regular whole number addition or subtraction, the COMMUTATIVE PROPERTY works in negative numbers too. It doesn't matter whether I say 3 + -2 or -2 + 3 either way, the answer will be 1 either way. Look at the number line below, and create equations that make you go up or down the number line. Then, watch the video attached to the picture.

EXAMPLE: 3 + -2 = 1 Because I am starting at three, and essentially going backward on the number line, which make it act like a subtraction problem.

Just like in regular whole number addition or subtraction, the COMMUTATIVE PROPERTY works in negative numbers too. It doesn't matter whether I say 3 + -2 or -2 + 3 either way, the answer will be 1 either way. Look at the number line below, and create equations that make you go up or down the number line. Then, watch the video attached to the picture.

**There are multiple different combinations of addition and subtraction using positive and negative numbers, but the important rules to remember are these:**

1. Adding a positive to another positive will give you a positive answer. (3 + 6 = 9)

2. Adding a negative to another negative will give you a negative answer. (-3 + -6 = -9)

3. Adding a negative and a positive will give you either a positive or negative answer depending on which addend was bigger. If the negative number was bigger, it will be a negative answer. If the positive addend was bigger, it will be a positive answer.

(-4 + 7 = 3 versus 4 + -7 = -3)

4. Subtracting positive numbers will give you a positive answer, unless the first addend is smaller than the second.

(9 - 2 = 7 versus 2 - 9 = -7)

5. Subtracting two negative numbers will act like adding. If you subtract a negative, you are "taking away" negative numbers, or essentially, adding. The answer could be positive or negative depending on how big your initial addend is.

(-5 - -6 = 1) another example is (-4 - -10 = 6)

Click on the image below to see some practice with these ideas.

1. Adding a positive to another positive will give you a positive answer. (3 + 6 = 9)

2. Adding a negative to another negative will give you a negative answer. (-3 + -6 = -9)

3. Adding a negative and a positive will give you either a positive or negative answer depending on which addend was bigger. If the negative number was bigger, it will be a negative answer. If the positive addend was bigger, it will be a positive answer.

(-4 + 7 = 3 versus 4 + -7 = -3)

4. Subtracting positive numbers will give you a positive answer, unless the first addend is smaller than the second.

(9 - 2 = 7 versus 2 - 9 = -7)

5. Subtracting two negative numbers will act like adding. If you subtract a negative, you are "taking away" negative numbers, or essentially, adding. The answer could be positive or negative depending on how big your initial addend is.

(-5 - -6 = 1) another example is (-4 - -10 = 6)

Click on the image below to see some practice with these ideas.

## LESSON EIGHT: PATTERNS

* Remember that you can often notice patterns in numbers. It is usually an operation of (+, -, x, or /) that creates this pattern.

EXAMPLE:

1, 5, 9, 13

Here, you are adding four between every number, so the operation is +4.

The next number would be 17.

*Sometimes more than one operation is used to create a pattern.

EXAMPLE:

3, 4, 6, 10, 18

The pattern began by adding 1, then 2, then 4, then 8 ... so you can see that there was addition and multiplication used in this number pattern.

*So how do we determine patterns? It is as easy as three steps.

1. Examine each pair of numbers in the sequence to determine a repeating pattern.

2. Apply the repeating operation(s) to the pattern in the sequence.

3. If the operation(s) works for the sequence, use the repeating pattern to extend the sequence.

* Click on the video below to review determining patterns!

* Remember that you can often notice patterns in numbers. It is usually an operation of (+, -, x, or /) that creates this pattern.

EXAMPLE:

1, 5, 9, 13

Here, you are adding four between every number, so the operation is +4.

The next number would be 17.

*Sometimes more than one operation is used to create a pattern.

EXAMPLE:

3, 4, 6, 10, 18

The pattern began by adding 1, then 2, then 4, then 8 ... so you can see that there was addition and multiplication used in this number pattern.

*So how do we determine patterns? It is as easy as three steps.

1. Examine each pair of numbers in the sequence to determine a repeating pattern.

2. Apply the repeating operation(s) to the pattern in the sequence.

3. If the operation(s) works for the sequence, use the repeating pattern to extend the sequence.

* Click on the video below to review determining patterns!