Focus:

Subtract dissimilar fractions

Multiply mixed form by a fraction

Solve 1-step word problem involving addition of fractions

Identify congruent polygons

Solve word problems involving body temperature

Interpret data presented in a line graph

Read and interpret reading from electric meter/ water meter Solve word problems involving measurement of surface area

triangles Solve word problems involving measurement of surface area

trapezoid Solve word problems involving measurement of solids

prisms Add dissimilar fractions in mixed forms with regrouping

Subtract dissimilar fractions in mixed forms with regrouping

Solve 1 to 3-step word problems involving addition and subtraction of decimals including money

Solve word problems with proportions

Solve word problems involving finding the percentage

Solve word problems involving finding the rate

Read and interpret data presented in a circle graph

Prepared by:

Prof. Gerry C. Areta

SUBTRACT DISSIMILAR FRACTION

Change the dissimilar or unlike fraction into similar fractions

1. Find the least common multiple (LCM) of the two denominators.

2. Get the equivalent fractions using the LCD.

3. Subtract the similar fractions and simplify the answer.

MULTIPLY MIXED FORM BY A FRACTION

1. Change the mixed form into improper fraction

2. Multiply as usual

Example: 4

8 -

2

6

Step 1: LCD of 2 and 6. 8=2.2.2 6=2.3 LCD= 2.2.2.3=24 Step 2: making the fractions into equivalent

fractions 4

8 =

12

24

2

6 =

8

24

Step 3: subtact

12

24 -

8

24 =

4

24 or

1

6

Example: 2 1

3 x

2

6

Step 1: changing mixed form into improper

fractions 2 1

3= 2 +

1

3 =

6+1

3 =

7

3

Step 2: multiply

7

3x

2

6 =

14

18 or

7

9

SOLVE 1-STEP WORD PROBLEM INVOLVING ADDITION OF

FRACTIONS

Problem: Sharmaine baked 200 pcs. of cookies. She gave 1

4 of it to her younger sister

Shina and 3

10 to her younger brother Jay-ar. What part of the cookies where given to

her siblings?

To answer what word problem, these are the things to remember:

1. understand the problem by knowing what is/are given and what is asked

2. solve and always label your answer by indicating the unit of measure or things the

problem is asking.

(Apply AGONA)

Step 1. Analysing

Asked- What part of the cookies where given to her siblings?

Given- 200 pcs. of cookies, 1

4 given to Shina,

3

10 given to Jay-ar

Operation- Addition

Step 2. Solving and labelling answer

Number sentence - 1

4 +

3

10 = N

5

20 +

6

20 =

11

20

Answer- 11

20 of the cookies was given to her two siblings

IDENTIFY CONGRUENT POLYGONS

Definition of terms:

Polygon- a simple closed figure in a plane formed by three or more line segments

Congruent- having the same shape and size

Congruent polygons- polygons that have exactly the same shape and size

Two polygons are if they have the same shape but not necessarily the same size. The symbol is used to indicate that two polygons are similar are similar polygons that have the same shape and the same size.

Example 1:

Finding Measures of Congruent Polygons

Given that RST XYZ, name the corresponding sides and corresponding angles.

Then find XY.

Corresponding sides: RS and XY, RT and XZ, ST and YZ

Corresponding angles: R and X, S and Y, T and Z

Because XY and RS are corresponding sides, they are equal in length.

XY = RS = 6 inches

Example 2:

Find

C = (5/9) 66.6 C = 37

98.6 Fahrenheit is equal to 37 Celsius.

The boiling point of water is 100 Celsius. What is this value on the Fahrenheit scale?

(F = degrees Fahrenheit and C = degrees Celsius)

To solve this problem, replace the value of temperature given in the problem with the matching value in the conversion equation that solves for the units of temperature required in the problem.

F = ((9/5) C) + 32 F = ((9/5) 100) + 32 F = 180 + 32 F = 212

100 Celsius is equal to 212 on the Fahrenheit scale.

LINE GRAPH

Graphs are used to present a set of data in a simple and clear way. That is why

we can immediately visualize a lot of information in a graph.

Line graphs are used to sow the data that change over time

Example of line graph and how to construct it as follows:

C = (5/9) (F - 32)

F = ((9/5) C) + 32

The title of the line graph indicates that it is about the daily temperature in Manila

during the first week of August.

The labels of the graph include the y-axis category and the x-axis category. The

y-axis category is the number of degrees centigrade (Celsius) and the x-axis category is

composed of the names of the week.

Before we construct a graph, we need to have a set of data or information. In our

example, the data used in a line graph is a record of the daily temperatures in Manila

during the first week of August.

Monday- 34C Thursday- 31C Saturday- 33C

Tuesday- 33C Friday- 32C Sunday- 32C

Wednesday- 30C

The first thing to do is to find the range and determine the scale to use. The

range is the difference between the highest number and the lowest number in data.

34C-30C= 4C, the range is 4. Use the range to determine the number of units

the graph will have in the y-axis by dividing the highest value by the range. 34/4= 8, so

there will be 8 units in the y-axis. The number of units in the x-axis must correspond to

the number of days that will be 7.

The scale must be in the intervals of 5s or 10s, because multiples of 5 or 10 are easy

numbers to deal with. The range of our data is 4, which is close to 5, so we will make

the scale, 1 unit for every 5C (1:5). There are 8 units in the y-axis and the ratio is 1:5,

so multiply 8x5= 40. The y-axis value will range from 0 to 40 as shown in the graph.

The scale in the x-axis is 1 unit : 1 day. There are 7 days so the range will reach

7 units.

ELECTRIC METER

The unit for measuring electric energy used in our household is kilowatt-hour

(kWh). One kilowatt-hour is equal to 1000 watts and a kilowatt-hour is 1000 watts flow

of electricity in an hour.

Every house or building that uses electricity has an electric meter that measures

the amount of electric energy consumption.

The following are MERALCO instructions on how to read your electric meter.

Step 1:

Stand directly in front of your meter and look at the four dials (A, B, C and D).

Note that adjacent dials move in opposite directions. Dials A and C move counter-

clockwise while Dials B and D move in a clockwise direction.

Step 2:

Read the meter starting from the leftmost dial. First, read Dial A followed by Dials

B, C and D. Write down the number that the pointers have passed in each of the four

dials. In this example, the reading on the meter is 5949.

Remember:

When a pointer is directly pointing to a number in any of the dials, always

consult the dials on its right. Look at the example on Figure 1. The pointer on Dial C

points directly at 5, that is why we should look at dial D to determine the reading of Dial

C.

Figure 1

If the pointer of Dial D points directly at 0 or between 0 and 1, then Dial C will

read as 5

Figure 2

But if the pointer of Dial D

point between 9 and 0, the Dial C will

read as 4.

Figure 3

Step 3:

Write down your own meters new reading and this will be the present reading.

Look for the previous reading which can be found on your bill. Then, subtract your

meters previous reading from the present reading. The difference is the number of

kilowatt-hours that your household has consumed since your last billing period.

Example:

Present Reading 6 649

Less: Previous Reading 6 532

KWh Electricity used 117

Source: www. Meralco.com.ph

TRAPEZOID

A trapezoid is a quadrilateral that has one pair of sides which are parallel. These two sides are called the bases of the trapezoid. The height of a trapezoid is a segment that connects the one base of the trapezoid and the other base of the trapezoid and is perpendicular to both of the bases. Example 4: Find the area of the figure

Solution: For this trapezoid, the bases are shown as the top and the bottom of the figure.

The lengths of these sides are 45 and 121 units. It does not matter which of these we

say is b1 and which is b2. The height of the trapezoid is 20 units. When we plug all this

into the formula, we get square

units.

TRIANGLE

The height of a

triangle is the

perpendicular distance from any vertex of a triangle to the side opposite that vertex. In

other words the height of triangle is a segment that goes from the vertex of the triangle

opposite the base to the base (or an extension of the base) that is perpendicular to the

base (or an extension of the base). Notice that in this description of the height of a

triangle, we had to include the words or an extension of the base. This is required

because the height of a triangle does not always fall within the sides of the triangle.

Another thing to note is that any side of the triangle can be a base. You want to pick the

base so that you will have the length of the base and also the length of the height to that

base. The base does not need to be the bottom of the triangle.

You will notice that we can still find the area of a triangle if we dont have its

height. This can be done in the case where we have the lengths of all the sides of the

triangle. In this case, we would use Herons formula.

Example 5:

Find the area of the figure.

Solution:

Notice that in this figure has a dashed line that is shown to be perpendicular to

the side that is 8.2 units in length. This is how we indicate the height of the triangle (the

dashed line) and the base of the triangle (the side that the dashed line is perpendicular

to). That means we have both the height and the base of this triangle, so we can just

plug these numbers into the formula to get

square units.

Notice that the number 6 is given as the length of one of the sides of the triangle.

This side is not a height of the triangle since it is not perpendicular to another side of the

triangle. It is also not a base of the triangle, since there is no indication of the

perpendicular distance between that side and the opposite vertex. This means that it is

not used in the calculation of the area of the triangle.

SPACE FIGURES

Space figures are figures whose points do not all lie in the same plane. There are different types of space figures. Included to the different types of space figures are the Polyhedrons which are space figures with flat surfaces, called faces, which are made of polygons.

Prisms and pyramids are examples of polyhedrons. Prism is a polyhedron with at least a pair of parallel faces.

Face the plane surface

Edge - the side(line segment)

of each plane figure

Vertex the common end point

where the edges meet

The parallel faces of the prism are the bases which are the top and bottom faces, and the other faces are the lateral faces. A prism is named according to its bases.

Pyramid is a polyhedron with a polygonal base which is the bottom face and triangular lateral faces with a common vertex.

Pyramids are named according to the kind of base they have.

Square Pyramid

Rectangular Pyramid

Cylinders, cones, and spheres are not polyhedrons, because they have curved, not flat, surfaces. A cylinder has two parallel, congruent bases that are circles. A cone has one circular base and a vertex that is not on the base. A sphere is a space figure having all its points an equal distance from the center point.

SURFACE AREA

Surface Area (SA) is the measure in square units of the regions covering all the

faces of the space figure.

In getting the surface area of a space figure, we have to find the area of each face

and then add all the areas. Here are the formulas for prisms:

o Surface Area of a Cube = 6 a 2

a = length of the side of each edge of the cube

Therefore, the surface area of a cube is the area of the six squares that cover it.

o Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

a, b, and c are the lengths of the 3 sides

Therefore, the surface area of a rectangular prism is the area of the six rectangles that cover it. To solve its surface area easily, always remember that the top and bottom of the rectangular prism are the same, the front and back are the same, and the left and right sides are the same.

o Surface Area for any Prism

b = shape of the ends

Surface Area = Lateral area + Area of two ends

Lateral area = (perimeter of shape b) x L

Surface Area = (perimeter of shape b) x L+ 2* x (Area of shape b)

VOLUME

Volume is the number of cubic units (unit3) contained in a space figure.

The volume is measured in "cubic" units, and the volume of a figure is the number of cubes required to fill it completely, like blocks in a box. Here are the formulas for volume of prisms and other space figures:

o cube = a 3

o rectangular prism = a b c

o irregular prism = b h

o cylinder = b h = pi r 2 h

o pyramid = (1/3) b h

Remember that the general formula in getting the volume of space figures is

V = B x H

That is the formula where B is the area of the base of the figure and H is the measure from the top to the bottom of the base.

ADD AND SUBTRACT DISSIMILAR FRACTIONS IN MIXED FORMS WITH REGROUPING

In adding or subtracting mixed numbers, one must add or subtract the fraction parts and the whole number parts separately. Then, the answers will be placed together. Still, the way of adding or subtracting similar fractions in mixed forms is different from the way of adding or subtracting dissimilar fractions in mixed forms.

ADDING DISSIMILAR FRACTIONS IN MIXED FORMS

To add unlike fractions in mixed forms, change the fractions to similar fractions by

finding their LCD. After that, well add the numbers in the usual way.

Example:

5 3

4 + 8

5

6

53

4 = 5

9

12

+ 85

6 = 8

10

12

_______

1319

12

1319

12 = 13 + (1

7

12)

= 147

12

SUBTRACTING DISSIMILAR FRACTIONS IN MIXED FORMS

To subtract unlike fractions in mixed forms, change them first to similar fractions

before subtracting the numbers. Then, put the difference of the whole numbers

and fractions together.

Example:

257

8 - 14

3

6

257

8 = 25

21

24

- 143

6 = 14

12

24

__________

119

24 or 11

3

8

First, find the LCD.

LCM of 4 and 6. 4 = 22

6 = 23

LCM = 2 2 3 = 12

The LCD is 12.

Then, get the equivalent fraction

and add the similar fraction.

After that, add the whole

numbers.

If you got an improper fraction,

change it to mixed number.

Lastly, get the sum of the mixed

number and whole number.

Subtract the fractions, and change

the unlike fractions into similar

fractions.

Then, get the LCD.

LCM of 8 and 6 = 24

The LCD is 24.

Then, get the equivalent fraction

and subtract the similar fraction.

After that, subtract the whole

numbers.

Put together the difference of the

fractions and the whole numbers.

RENAMING AND REGROUPING GROUP NUMBERS

Renaming and regrouping are done if the fraction part of the minuend is less than

the fraction part of the subtrahend.

Example:

53

4 = 5

6

8 = (4 + 1) +

6

8 = 4 +

8

8 +

6

8 = 4 +

14

8 = 4

14

8

- 37

8 = 3

7

8 = - 3

7

8

17

8

The shortcut for renaming a mixed number like 56

8 is to subtract 1 from the whole

number (5-1=4). Next, add the numerator and denominator of the fractions in the minuend

(6+8=14) and write the sum (14) over the denominator.

ADDITION AND SUBTRACTION OF DECIMALS

A number line shows that there are decimal numbers in between whole numbers.

The number line above presents decimal numbers between 0 to 1.

Knowing how to add and subtract decimals is something that is very important in

life. That is because this will help you in many ways. For example, when you handle

money at a shop, the bank or post office, you should use your knowledge of decimals to

calculate the money and any change exchanged.

ADDING DECIMALS

Here are the things that you have to remember in adding decimal numbers:

1. Put the numbers in a vertical column, aligning the decimal points 2. Add each column of digits, starting on the right and working left. If the sum of a

column is more than ten, "carry" digits to the next column on the left. 3. Place the decimal point in the answer directly below the decimal points in the

terms.

SUBTRACTING DECIMALS

Here are the things that you have to remember in subtracting decimal numbers:

1. Put the numbers in a vertical column, aligning the decimal points. 2. Subtract each column, starting on the right and working left. If the digit being

subtracted in a column is larger than the digit above it, "borrow" a digit from the next column to the left.

3. Place the decimal point in the answer directly below the decimal points in the terms.

4. Check your answer by adding the result to the number subtracted. The sum should equal the first number.

Examples for addition and subtraction of Decimal numbers:

PROPORTION

A proportion is a name we give to a statement that two ratios are equal. It can be written

into two ways:

two equal fractions:

=

c

d

using colons: a:b= c:d

It is read as a is to b as c is to d. In problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. To find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

Example:

20:4 = 25:5

20x5=25x4

=100

Here, 20 and 5 are the extremes, and 25 and 4 are the means. Since the cross products are both equal to one hundred, we know that these ratios are equal and that this is a true proportion.

KINDS OF PROPORTION

There are two kinds of proportions, the indirect and direct.

The term direct proportion means that two (or more) quantities increase or decrease in

the same ratio. In short, as the other one increase, the other one also increases and

vice versa. If y is directly proportional to x, this can be written as y=kx

Example #1:

Fe works as a tutor of a grade 6 pupil. She earns Php 100 per hour. Everyday she

works for 4 hours. How much did she earn everyday?

Steps in Solving Problem:

Write the proportion

1:100 = 4:x

Multiply to find the cross products.

1 x x = 100 x 4

1

1 =

400

1

X=400 pesos

Example #2:

If y is directly proportional to x.

When x = 12 then y = 3

Find the constant of proportionality and the value of x when y = 8.

We know that y is proportional to x so y = kx

We also know that when x = 12 then y = 3

To find the value of k substitute the values y = 3 and x = 12 into y = kx

3 = k 12

So k = 3/12 = 1/4

To find the value of x , when y = 8 substitute y = 8 and k = 1/4 into y = kx

8 = (1/4) x

So x = 32 when y = 8

The term indirect proportion also known as inverse proportion means that one quantity

increases while the other decreases. If y is inversely proportional to x, this can be

written as y=k/x

Example #1:

If 6 students can finish a project in 2 days, how many days can 8 students finish the

same project?

Steps in Solving Indirect Proportion:

Write the proportion

(Note: instead of using the ration students:days, use the ratio students:students)

6:8 = 2:x

Multiply the extremes.

6 x x= 6x

Multiply the means.

8 x 2= 16

Multiply to find the cross products.

6

6 x

16

6

X=2.66666

Round off your answers to the nearest hundredths.

X=2.67

Example #2:

y is inversely proportional to x. When y = 3, x = 12 .

Find the constant of proportionality, and the value of x when y = 8.

y 1/x

y = k/x

So xy = k

Substitute the values x = 12 and y = 3 into xy = k

3 12 = 36

So k = 36

To find the value of x when y = 8, substitute k = 36 and y = 8 into xy = k

8x = 36

So x = 4.5

PERCENTAGE

Percentage (P) represents a number that is a part of a whole quantity. The whole

quantity is the base (B), and the rate (R) is the number expresses with % symbol. The

formula is: P= RxB

Example #1:

10% of 50 is 5.

5 is the percentage (P)

10% is the rate (R)

50 is the base (B)

Steps in Solving the problem:

What is asked?

What are the given?

Formula you will used.

Solve the problem by substituting using the formula.

Example #2:

Last Thursday was costume day at Donavan's school. 70% of the 200 students wore a

costume. How many students wore a costume?

Solution:

What is asked?

Number of students wore a costume.

What are the given?

70%- Rate

200 students wore costume- Base

Formula you will used

P= BxR

Solving.

P= 200 x 0.70

P= 140 students wore costume

RATE

The amount of discount corresponds to percentage in the formula: P= B x R or Amount

of Discount= Percent of discount x Original price . The rate is the percent of discount.

The original price corresponds to the base.

How to Solve Problem involving rate:

What is asked?

What are the given?

Formula.

Solution.

Example #1:

Nikka wants to buy her favorite book which is cost P400, but luckily the bookstore sells

it at 20% discount. How much did Nikka pay for the book?

Answer:

What is asked?

Amount of money did Nikka pay for the book.

What are the given?

P400- Original price

20%- discount

Formula

Amount of Discount= Percent of discount x Original price

Sale price= Original price Amount of discount

Answer

P= 20% x P400.00

P= 0.20 x 400.00

=P80.00

SP= P400.00 P80.00

=P320.00

Example #2:

In a video store, a DVD that sells for P155 is marked, "10% off". What is the sale price

of the DVD?

Answer:

What is asked

The sale price of the DVD

What are the given?

P155.00- original price

10%- discount

Formula

Amount of Discount= Percent of discount x Original price

Sale price= Original price Amount of discount

Answer

P= 10% x P155.00

P= 0.10 x P155.00

=P15.50

SP= P155.00 P15.50

=P139.50

INTERPRETING DATA IN A CIRCLE GRAPH

Circle graph, or also known as, Pie graph. It is a circular graph which is divided into

sectors with each sector representing a part of a set of data.

Example #1:

Grade 6 Amethyst makes a list of their favorite sports as follows: Gymnastics 11%,

Swimming 27%, Soccer 30%, Track 20% and Tennis 12%.

Grade 6 Amethyst Favorite Sports

Question:

1. What is the title of the graph?

Grade 6 Amethyst Favorite sports.

2. What are the favorite sports of Grade 6 Amethyst?

Gymnastics, Swimming, Soccer, Track and Tennis

3. Which sport gets the highest percentage?

Soccer

4. Which sport gets the lowest percentage?

Gymnastics