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MATH 7 Q1 - This learning activity sheet focuses on the basic
terms and concepts of measurement
College Algebra (Don Honorio Ventura Technological State University)
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7
MATHEMATICS
MODULE
Quarter 2: Week 1-2
Module 1: Measurement
Module 2: Converting measurements and solving
problems involving conversion of units
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MATHEMATICS 7
Quarter 2 – Module 1:
Approximates the Measures of Quantities Particularly Length, Weight/Mass,
Volume, Time, Angle and Temperature and Rate
Name
Week 1
Section
Date
Background Information
What’s New
You use measurements to describe things: the size of your apartment or house, the
distance you travel to school, the weight you gain after a vacation, the temperature of your
room, the number of paint gallons you need for your bedroom, and the cost of carpeting your
living room. Our country uses both the English and Metric Measurement. In this chapter, we
will explore ways of measuring things in the English System as well as in the Metric System.
When one knows how units of measure are used to describe his world, he will make decisions
based on understanding the numbers and units he encounters every day.
Warm-Up Activity: Measuring Ourselves
Using your own body, determine:
•
•
•
How many hands tall are you?
How many finger-widths tall is your head?
How many finger-widths is it from your elbow to the tip of your finger?
Compare your results to your friends or siblings.
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What is It
A. Historical Development of Measurement
Measurement is the process by which human beings obtain useful quantitative
information about the different physical aspects of objects. In ancient times, part of
the human body and nature were used which is known as primitive unit of
measurement.
The metric system and English system, also called the imperial system of
measurements, are both common systems of measurement used today. The main
difference between imperial and metric units is that metric units are easier to convert
because those conversions require only multiplying or dividing by powers of 10.
There are 10 millimeters in a centimeter, 100 centimeters in a meter, and 1,000
meters in a kilometer. Converting imperial units is much less straightforward. Take
imperial length units, for example. There are 12 inches in a foot, 3 feet in a yard and
1,760 yards in a mile.
The units of measurement used in ancient times were chosen for convenience rather
than accuracy. Historical records indicate that the first units of length were based on
people’s hands, feet, and arms. The hand, span, foot and cubit appear in the early
records of Babylonians and Egyptians. Our Filipino ancestors, on the other hand used
dangkal, dipa, hakbang, talampakan, and dakot
In 1790, the French Academy of Sciences devised a new system of measurement.
The new unit was called a meter, which was taken from the Greek word metron, “to
measure.” The metric unit used for determining mass (weight) is called gram. The
liter is the metric unit used for determining volume.
B. Four Units of Measurement used by Ancient Civilization
a. Span – the length from one end of the middle finger to the other end of the other
middle finger when the arms are spread out.
b. Palm – The distance across the base of the four fingers that form the palm.
c. Digit – The thickness or width of the index finger.
d. Foot – The length of a foot.
e. Cubit – The distance from the tip of the middle finger of the outstretched hand
to the front of the elbow.
f. Pace – The distance of one full step.
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C. Physical Quantities and Their Units
Length – describes how long something is. Distance, height, thickness, and
depth also use the same units.
Units of Length
Metric System
Millimeter
mm
Centimeter cm
Meter
m
Kilometer
km
English System
Inch
in
Foot
ft
Yard
yd
mile
mi
Mass – describes how heavy something is.
Units of Mass
Metric System
Milligram mg
Gram
G
Kilogram kg
English System
Ounce oz
Pound lb
Ton
T
Time – describe how long it takes to do something.
Units of Time
Second sec
Minute min
Hour
Hr
Day
D
Week
Wk
Month
Mo
year
Yr
Area – describe how much surface is occupied by something.
Units of Area
Metric System
Square
𝑐𝑚2
centimeter
𝑚2
Square meter
Hectare
English System
Square foot
𝑓𝑡2
Acre
ac.
ha
Volume – describes how much space (or liquid) is occupied (or contained) by
something. Similar to area, volume also uses the units of length but the only
difference is that the unit is “cubed”. Cubed units are typically used for
volumes of space while there are also units of volume dedicated for liquid
measurements
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Units of Volume
Metric System
cubic centimeter
𝑐𝑚3
Millimeter
ml
Liter
l
English System
Fluid ounce
fl oz
Cup
c
Pint
pt
Quart
qt
Gallon
gal
Temperature – describes how hot or cold something is.
Metric System
Celsius scale oC
Units of Temperature
English System
Fahrenheit scale oF
SI
Kelvin scale K
D. Measuring Instruments
LENGTH
Tape Measure
Ruler
Steel Measuring Tape
MASS/VOLUME
WEIGHING SCALE
TIME
Clock
Calendar
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VOLUME
Measuring cup
Graduated Cylinder
Measuring Spoon
TEMPERATURE
Forehead Thermometer
Digital Thermometer
Mercury Thermometer
ANGLE
Protractor
What I need to know
Approximates the measures of quantities particularly length, weight/mass, volume, time,
angle and temperature and rate (M7ME-IIa-3).
General Instruction
Write your answer on a clean sheet of paper.
What I Know
Identify the following. Write your answer on a separate sheet of paper.
1. It describe how long it takes to do something.
2. A chart or series of pages showing the days, weeks, and months of a particular year,
or giving particular seasonal information.
3. What is a primitive unit of length equivalent to a span of the forearm
4. What is based on the decimal system and allows easy conversion from one unit to
another?
5. What is the process of comparing an unknown quantity to a standard known quantity
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What’s In
Identify the measuring instrument needed in each of the situations below. Write your
answer on a separate sheet of paper.
1. Mia’s sister is not feeling well. What instrument does she need to find out if her sister
has a fever?
2. Victor wants to make sure that he has drawn a right angle. What instrument does he
need?
3. Yaya Dub wants to sew a shirt for her Lola. What instrument will she use to measure
her Lola’s skirt?
4. “A kilo of iron is heavier than a kilo of cotton,” said Cardo. “You are wrong” said Alyana.
What instrument does Alyana need to prove that she is right?
5. Pirena wants to find out how tall she is now. What instrument does she need?
What’s More
Choose the best estimate of each of the following. Write the letter of the correct
answer on a separate sheet of paper.
4.
1. A sack of rice
a. 5,000 g
b. 50 kg
c. 5 tons
2. Temperature in a cool day
a. 3oC
b. 10oC
c. 24oC
3. A glass of water hold about
a. 250 mL b. 500 mL c. 1 L
The time it takes to sing the Philippine national anthem
a. 56 sec
b. 1 min
c. 2 min
5. Maximum speed of a car in the NLEX
a. 5 kph
b. 80 kph
c. 120 kph
6. A normal body temperature
a. 37oC
b. 39oC
c. 40oC
7. A pack of 3 in 1 coffee
a. 50 L
b. 48 g
c. 29 mm
8. A bottle of drinking water
a. 350 mL
b. 50 mL
c. 10 mL
9. A cracker sandwich
a. 50 kg
b. 10 g
c. 32 mg
10. A pack of sugar
a. 3 mL
b. ¼ kg
c. 10 cm
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What I Can Do
Identify the appropriate Metric Unit and Measuring Device used for each item. Write your
answer on a separate sheet of paper.
ITEM
MEASURING UNIT
MEASURING DEVICE
1. length of your pen
meter, centimeter, kilometer
meter stick, ruler
2. a person’s waistline
centimeter, meter, kilometer
millimeter, milliliter,
dekameter
protractor, tape measure
3. thickness of Math book
4. normal body
temperature
5. weight of your
classmate
6. time it takes you to
reach school
degree, degree Celsius
meter, kilogram, centimeter
minutes, weeks, months
ruler, meter stick
thermometer, tape
measure, clock
meter stick, weighing
scale, graduated cylinder
clock, thermometer
Assessment ACTIVITY 1
Read each statement carefully. Write the letter of the correct answer on a separate
sheet of paper.
1. The average mass of an adult person is about:
a. 6 kg
b. 60 kg
c. 600 kg
2. What primitive unit of length equivalent to the distance of the one full step?
a. cubit
b. digit
c. pace
3. What is the process by which human beings obtain useful quantitative information about
different physical aspects of objects?
a. algebra
b. measurement
c. probability
4. The time it takes to sing the Philippine national anthem
a. 56 sec
b. 1 min
c. 2 min
5. A kitchen utensil used to measure the volume of liquid or ingredients like flour and sugar.
a. clock
b. graduated cylinder
c. measuring cup
6. A normal body temperature
a. 37oC
b. 39oC
c. 40oC
7. The metric unit used for determining length
b. gram
b. liter
c. meter
8. It refers to the degree of hotness and coldness of a body.
a. Mass
b. temperature
c. volume
9. What is based on the decimal system and allows easy conversion from one unit to
another?
a. English system b. metric system
c. primitive unit
10. An instrument commonly used for measuring the mass of fruits, vegetables, and meat
a. Meter stick
b. protractor
c. weighing scale
Additional activities ACTIVITY 2
Using a meter stick, tape measure, ruler and your fingers, measure the items listed
below.
ITEM
SPAN
RULER
METER STICK
Height of the door
Height of the chair back
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TAPE MEASURE
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Distance from window to floor
Length of your pencil
Length of your foot
Answer Key
What I know
1.Time
What’s In
1.Thermometer
2. Calendar
3. Cubit
4. Metric System
2. Protractor
3.Tape Measure
4. Weighing scale
What’s More
1. B
3.
2. C
4.
What Can I Do
1. centimeter, ruler
2. centimeter, tape measure
3. millimeter, ruler
Assessment
1. B
3.
2. C
4.
Additional Activities
A
A
5. B
6. A
5.Measurement
7. B
8. A
5. Tape
Measure
9. C
10. B
4. degree Celsius, thermometer
5. kilogram, weighing scale
6. minutes, clock
B
A
5. C
6. A
Students’ answer may vary.
7. C
8. B
9. B
10. C
References
Nivera, Gladys C,Ph.D. Grade 7 Mathematics Patterns and Practicalities(Makati City:
SalesianaBOOKS by Don Bosco Press, Inc.,2013) pp.126-161
Gamboa, Job D., Elementary Algebra (Lipa City, Batangas: United Eferza Academic Publications
Co.,2010) pp.86-93
“Measurement Units” Accessed on September 10, 2020
https://www.ipracticemath.com/learn/measurement/measurement-units
“Difference Between English and Metric System” Accessed on September 11, 2020
https://sciencing.com/difference-between-english-metric-system-12742341.html
Prepared by:
VANESSA A. VILLANUEVA
SST – III, FGNMHS
MA REVA G. CASTRO
SST – I, FGNMHS
Illustrated by:
ROMMEL G. SALEM
SST- III, FGNMHS
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MATHEMATICS 7
Quarter 2 – Module 1:
Conversion of Measurements and Its Application
Name of learner:
Section:
_
_
Week 2
Date:
Background Information
What’s New
FIND ME!
Fifteen different units of measures are hidden in this puzzle. How many can you find? Look
up, down, across, backward, and diagonally.
H
J
A
B
G
R
A
M
E
R
A
S
J
P
P
O
U
N
D
S
T
U
Z
F
Y
A
R
E
T
E
M
I
L
L
I
M
Z
Y
A
R
D
U
M
P
A
Y
M
J
T
R
E
T
I
L
I
T
N
E
C
A
E
M
A
T
M
E
T
E
R
N
O
O
E
U
E
L
K
I
L
O
G
R
A
M
F
X
Y
I
P
P
I
N
T
A
N
I
H
E
C
T
O
G
R
A
M
L
N
E
G
Z
T
E
E
L
I
M
J
C
T
W
K
T
J
R
X
E
H
U
H
Z
O
Q
W
C
E
L
S
I
U
S
O
T
N
S
Answer the following questions:
1. How many words in the puzzle are familiar to you?
2. What word(s) is/are unfamiliar to you?
3. Which are the units for the Metric System?
4. Which are the units for the English System?
What Is It
Length, mass, capacity, temperature and time are some of the things that we measure in
everyday living. To measure quantities like this, units of measurements are required. Two
standard system of measurement were developed. The English system and the metric system.
A. MEASURE OF LENGTH
Length is a measure of distance. The inch is the smallest commonly used in the English
system. For longer distances; foot, yard and mile are used.
Meter is the basic unit of length in metric system. The shortest length can be millimeter. For
longer distances; the decameter, hectometer, and kilometer are used. The table shows the
equivalent measurement of length.
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CONVERSION UNITS OF LENGTH
ENGLISH SYSTEM
METRIC SYSTEM
1 foot (ft) = 12 inches(in)
1 yard (yd) = 3 feet (ft)
1 yard (yd) = 36 inches
1 mile(mi) = 5280 feet (ft)
1 mile(mi) = 1760 yards
1 meter (m) = 10 decimeters (dm)
1 meter (m) = 100 centimeters (cm)
1 meter (m) = 1000 millimeters (mm)
1 decameter (dam) = 10 meters
1 hectometer (hm ) = 100 meters
1 kilometer(km) = 1 000 meters
Converting from one unit to another might be tricky at first, so an organized way of doing it
would be a good starting point. As the identity property of multiplication states, the product of
any value and 1 is the value itself. As a result, dividing a value by the same value would be
equal to one. Thus, dividing a unit by its equivalent in another unit is equal to 1. For example:
1 foot / 12 inches = 1
3 feet / 1 yard = 1
These conversion factors may be used to convert from one unit to another. Just remember
that you’re converting from one unit to another so cancelling same units would guide you in
how to use your conversion factors.
Example
1. Convert 5 feet into inches
5𝑓𝑡 𝑥
1 𝑓𝑡
2. Convert 72 inches into yard
=60 inches
72
𝑥
36 𝑖𝑛
= 2 yds
3.There are 60,000 miles of blood vessels in a human body. How long is this in feet?
60, 000 𝑚𝑖𝑙𝑒𝑠 𝑥
5280 𝑓𝑡
=316, 800, 000 ft
The metric system is based on powers of 10. Each metric unit is ten times as great as the unit
to its right and one-tenth as great as the units to its left.
There are rules to apply when converting the metric system.
1. To convert from a larger unit to a smaller unit, multiply by a power of ten.
2. To convert a smaller unit to a larger unit, just divide by a power of ten.
It is as simple as moving the decimal point to the right (for smaller prefixes) or to the left (for
larger prefixes). By lining up these prefixes in a chart, you can easily figure out how many
centimeters are in a kilometer, and how many millimeters are in a meter.
METRIC CONVERSION
cm
m
m
dm
To help you set up a chart always remember the mnemonic device “King Henry Died
Unexpectedly Drinking Chocolate Milk”. Each word in the King Henry phrase represents a
metric unit.
km
hm
da
King
Henry
Died
Unexpectedly Drinking
Chocolate Milk
Kilo
Hecto
Deca
UNIT
Centi
Deci
Milli
The number of moves to the left or right from one unit to another indicates the number of
decimal places to move either to the left or to the right.
Example:
Solution:
1. Convert 5.5 km to meters
Since 1 km = 1000 m
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Another Solution: Start from kilometer and move three places to the right to obtain meter.
km
5.5 km = 5 5 0 0
hm
dam
m
5.5 km = 5 500 m
cm
dm
mm
2. Convert 67.9 cm to hm
Solution:
Since 1 hm = 10000 cm
67.9 ÷ 10000 = 0.00679 We divide because we are converting a smaller unit
to a larger unit. Therefore, 67.9 cm =0.00679 hm
Another Solution : Start from centimeter and move four places to the left to obtain
hectometer.
km
hm
dam
67.9 cm = 0.0 0 6 79
m
cm
dm
mm
67.9 cm = 0.00679 hm
3. A typical full-sized guitar is around 96.5 cm long. Express the length in meters
Solution:
km
96.5 cm = 0.96 5
hm
dam
m
cm
dm
mm
Answer: 0.965 meters
B. MEASURE OF MASS
Mass is the measure of the amount in an object. The SI standard unit of mass is kilogram.
The table below shows the measurement of mass.
ENGLISH SYSTEM
16 ounces (oz) = 1 pound(lb)
2 000 pounds (lbs) = 1 ton (t)
CONVERSION UNITS OF MASS
METRIC SYSTEM
1 gram (g) = 10 decigrams (dg)1 decagram (dag) = 10 g
1 gram (g) = 100 centigrams (cg)1 hectogram (hg ) = 100 g
1 gram (g) = 1000 milligrams (mg)1 kilogram(kg) = 1 000 g
Example: Convert the following:
Solution: 1. 2 lbs to oz
Since 16 oz = 1 lb
16 𝑜𝑧
2 lb x
= 32 oz
1 𝑙𝑏
2 lbs = 32 oz
2. 10 000 lbs to t
Since 2 000 pounds (lbs) = 1 ton (t)
10000 𝑙𝑏𝑠 𝑥
1𝑡
2000 𝑙𝑏𝑠
= 5
10 000 lbs = 5 t
3. 5600 g =
kg
Solution: Start from gram and move three places to the left to obtain kilogram.
kg
hg
dag
g
5600 g = 5. 6 0 0
5600g = 5.6 kg
4. 98.45 dag =
mg
cg
dg
or 5600 𝑔 𝑥
mg
= 5.6 kg
Solution: To change decagrams to milligrams, move four units to the right.
10000 𝑚𝑔
= 984 500 mg
98.45 dag = 984500 mg
or 98.45 𝑑𝑎𝑔 𝑥
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5. A slice of a pizza is about 4 ounces. How heavy is it in pounds?
Solution:
4 ounce x 1 𝑙𝑏
= 0.25 lbs
16 𝑜𝑧
C. MEASURE OF CAPACITY
Capacity refers to how much a container can hold. In the metric system, liter is the basic unit
used to measure capacity. The table below shows the measurement of capacity.
CONVERSION UNITS OF CAPACITY
ENGLISH SYSTEM
METRIC SYSTEM
1 cup (c ) = 8 fluid ounces (oz)
1 liter (L) = 10 deciliters (dL)
1 pint (pt) = 2 cups (c )
1 liter (L) = 100 centiliters (cL)
1 quart (qt) = 2 pints (pt)
1 liter (L) = 1000 milliliters (mL)
1 gallon (gal) = 4 quarts (qt)
1 decaliter (daL) = 10 liters (L)
1 cup (c ) = 16 tablespoons (tbsp)
1 hectoliter (hL ) = 100 liters (L)
1 tablespoon (tbsp) = 3 teaspoons(tsp)
1 kiloliter(kL) = 1 000 liters (L)
Example:
1. How many pints is 2.75 gallons?
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛:
2.75 𝑔𝑎𝑙 𝑥
4 𝑞𝑡
1 𝑔𝑎𝑙
𝑥
2 𝑝𝑡
1 𝑞𝑡
= 22 pints
2. Convert 6.76 mL to liters
Solution: Start from milliliter and move three places to the left to obtain liter.
kL
6.75 mL = 0.00675
hL
daL
L
dL
cL
mL
6.75 mL = 0.00675L
3. The average volume of blood in an adult is about 4.5 L. What is the volume in millimeters?
Solution: To change liters to milliliters, move three units to the right.
4.5 L = 4 5 0 0
4.5L = 4500 mL
D. MEASURE OF TIME
Time is the interval between two events. It measures the passing of years. The unit of
measurement of time are second, minute, hour, day, week, month, year, century and
millennium. The table shows the equivalent measurement of time.
CONVERSION UNITS OF TIME
1 minute (min) = 60 seconds (sec)
1 leap year = 366 days
1 hour (hr) = 60 minutes (min)
1 decade = 10 years
1 day = 24 hours
1 score = 20 years
1 week = 7 days
1 century = 100 years
1 year = 12 months
1 millennium = 1000 years
1 year = 365 days
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Example
1. How many weeks are there in 42 days?
Solution: Since, 1 week = 7days, then 42 days ÷ 7 = 6 weeks
42 days =6 weeks
2. How many minutes are there in 1 day?
Solution:
1 𝑑𝑎𝑦 𝑥
24 ℎ𝑟
𝑥
60 𝑚𝑖𝑛
1 ℎ𝑟
= 1440 minutes
D. MEASURE OF TEMPERATURE
Temperature refers to the degree of hotness or coldness of something. The commonly used
units of measure of temperature are degree Fahrenheit (⁰F) and degree Celsius (⁰C). In the
Celsius scale, 0⁰C is the freezing point of the water and 100⁰C is the boiling point.
Fahrenheit to Celsius
Celsius to Fahrenheit
⁰C = 5 (℉ − 32)
⁰F = 9 ℃ + 32
1. The normal body temperature is 37⁰C. What is the equivalent in Fahrenheit?
Solution: To convert Celsius to Fahrenheit, we use the formula:
⁰F = 9 ℃ + 32
= 9 (370) + 32
= 66.6 + 32
= 98.6
37⁰C =98.6 ⁰F
5
5
2. An oven must be set at 400 ⁰F to bake a muffin. What is the equivalent temperature in C⁰?
Solution: To convert Fahrenheit to Celsius, we use the formula:
⁰C = 5 (℉ − 32)
= 5 (400 − 32)
= 5 (368)
= 204.44 400 ⁰F = 204.44⁰C
9
9
9
What I need to know
Converts measurements from one unit to another unit in both Metric and English systems.
(M7ME-IIb – 1)
Solves problems involving conversion of units of measurement. (M7ME – IIb- 2)
General Instruction
Write your solutions and answer in a clean sheet of paper.
What I Know
Write how many places the decimal point should be moved to the left or right. The first one is
done for you.
3 places to the right 1. kilometer to meter
2. millimeter to decameter
3. hectometer to centimeter
4. milligram to gram
5. decigram to hectogram
6. gram to decigram
7. liter to milliliter
8. kiloliter to centiliter
9. decimeter to meter
10. centigram to gram
What’s In
Convert the following to the indicated unit:
A. Metric Units of Measurement
1. 225 613 cm
=
2. 786 mm
=
3. 0.75 hm
=
4. 45 g
=
km
m
cm
cg
6 . 6 hg
7. 3 500 dag
8. 160 L
9. 4.723 mL
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=
=
=
=
g
kg
mL
hL
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5. 393 000 mg
=
B. English Units of Measurement
1. 10 ft
=
2. 42 ft
=
3. 144 in
=
4. 8 lbs
=
5. 128 oz
=
kg
10. 953 L
=
dL
in
yd
yd
oz
lb
6. 3.5 t
7. 5.5 lb
8. 8 pt
9. 5 cups
10. 16 pint
=
=
=
=
=
lb
oz
qt
tbsp
gal
What’s More
Complete the MAZE. Make your way from start to end by coloring the stations that are correctly
converted from one unit to another. Use different color per station.
4800 mg
3 lb
4.8
40
dg
48 oz
70
cm
24
in
5 century
50 min
24 feet
500yea
rs
yd
300
min
END
7 000 m
7 km
10⁰C
65⁰ C
50 ⁰F
yd
5 500
kg
years
5 hours
2750 g
200
min
What I Can Do
CROSS NUMBER PUZZLE
Complete the puzzle by converting units of measurement.
e
32
qt
32 pt
27.5
kg
3 000 lb
32
cups
4 gal
1.6
START HERE
CONGRATULATIONS
a
1 ton
b
c
f
i
J
k
o
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2.75
kg
1600 mL
lOMoARcPSD|23567711
ACROSS
=
a. 2.54kL
c. 9.4825 dam =
f. 0. 5 L
=
g. 135⁰ C
=
i. 1 year
=
j . 5 decades =
k. 4 century =
l. 8cups
=
m. 432.2 dL =
o. 4 km
=
DOWN
b. 48. 15 g
c. 9 century
d. 85. 294 hm
e. 0. 63841 kg
f.585. 253 dam
h. 58. 618 kg
m. 2 ton
n. 2yrs
dL
mm
mL
⁰F
days
years
years
tbsp
cL
dam
=
=
=
=
=
=
=
=
cg
yrs
dm
cg
cm
dg
lbs
mos
Assessment ACTIVITY 1
Multiple Choice. Choose the letter of the correct answer. Write the letter of the correct
answer.
1. How many decimeters are there in a decameter?
a. 10
b. 100
c. 1 000
d. 10 000
2. The weight of a powdered milk is 400 grams. What is the equivalent weight in
kilogram?
a. 4.00 kg
b. 40.0 kg
c. 0.4 kg
d. 0.04 kg
3. An OFW experienced 14 days self quarantine. What is the equivalent of 14 days
in hours?
a. 336 hours
b. 600 hours
c. 720 hours
d. 7200 hours
4. The distance between Angeles City and City of San Fernando is 23 km. What is
the equivalent distance in meter?
a. 0.23 m
b. 23 m
c. 230 m
d. 23 000 m
5. How many hours are there in one week?
a. 120 hours
b. 168 hours
c. 200 hours
d. 218 hours
0
6. A thermal scanner reads at 95⁰F. What is its equivalent in C?
a. 37 0C
b. 36 0C
c. 35 0C
d. 34 0C
7. How many years are there in 2 decades?
a. 2
b. 20
c. 200
d. 2 000
8. A face shield weighs 1 ounce. How many pounds is it?
a. 16 lbs
b. 6 lbs
c. 1 lb
d. 0.0625 lb
9. The size of a standard milk tea is 473 mL. Convert 473 mL in liters.
a. 0.473 L
b. 4.73
c. 47.3 L
d. 4730 L
10. A bag of sugar weighs 35 kilograms. What is its weight in gram?
a. 3.5 g
b. 350 g
c. 350 g
d. 35000 g
Additional activities ACTIVITY 2
Read the following problems and write your answer on the blank.
1. A bottle of vitamin C contains 90 tablets. If each tablet contains 500 mg of vitamins, how
many grams of vitamins are there in all?
2. Mayor distributed 300 sacks of rice. A sack of rice weighs 50 kg. How many grams of
rice did the mayor distribute?
3. Mitz took one hour and 25 minutes to finish her test. How many minutes in all did she
spend for the quiz?
4. Jasmin drinks 8 glasses of water everyday. Each glass holds 375mLof water. How
many liters of water does Jasmin drink each day?
5. Mill had a flu and his temperature was 104⁰ F. What is his temperature in Celsius
scale?
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Rubric
Excellent (5)
The answer is correct
and complete.
The solution is
complete and correct.
Very Good (4)
The answer is correct
but incomplete.
The solution lacks
clarity and incomplete.
Good (3)
The answer is
incorrect.
The solution lack
clarity and incomplete.
Fair (2)
The answer is
incorrect.
There is no solution.
Answer Key
What’s New
1.
2.
3.
4.
Answer may vary
Answer may vary
Gram, millimeter, centiliter, meter, kilogram, Celsius, hectogram, liter
pounds, yard, feet, pint, mile, inch, ton
What I Know
2.
3.
4.
5.
6. 1 place to the right
7. 3 places to the right
8. 5 places to the right
9.1 place to the left
10. 2 place to left
4 places to the left
4 places to the right
3 places to the left
3 places to the left
What’s In
A.
1. 2.25613 km
6. 600 g
B.
1. 120 inches
6. 7000 lb
2. 0.786 m
7. 35 kg
2. 14 yd
7. . 88 oz
3. 7500 cm
8. . 160 000 mL
3. 4 yd
8. 4 qt
4. 4500 cg
9. 0.00004723 hL
4. 128 oz
9. 80 tbs
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5. 0.393 kg
10. 9530 dL
5. 8 lb
10. 2 gal
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What’s more
What I Can Do
Assessment
1. b
2. c
6. c
7. b
Additional Activities
3. a
8. d
4. d
9. a
5. b
10. d
1. 45 kg
2. 15 000 000 g
3. 85 min
4. 3 liters
5. 40⁰C
References
Villano, Ma. Luisa V., et. al.,Phoenix Math for the 21st Century Learners ( 2nd edition, Phoenix
Publishing House, Inc., 2016) pp. 173 – 183
Nivera, Gladys C., Grade 7 Mathematics Patterns and Practicalities( SalesianaBook by Don Bosco
Press, Inc., 2012) pp 131 – 152
Oronce, Orlando A., e-Math I (Elementary Algebra).(Rex Book Store, Inc., 2007).. pp 26- 83
Licardo, Edna D, and Jisela Naz Ulpina, Math Builders ( Jo-Es Publishing House, Inc., 2007).pp 4- 42
Latonio, Rhett Anthony C., Department of Education Learning Guide Lesson 15 – Measurement and
Measuring Length) pp 1-7
Department of Education Module 1, Be Precise and Accurate. pp. 1 – 10
Prepared by:
RUCHELYN T. PASION
SST-III, FGNMHS
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7
DRRM INTEGRATION
in Mathematics
MODULE
Quarter 2: Week 3- 4
Module 3: Translating English Phrase/Sentence into
Mathematical Phrase/Sentence and Vice Versa
Module 4: Evaluating Algebraic Expressions and Adding and
Subtracting Polynomials
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Mathematics 7
Supplementary Learning Resource
Quarter 2 – Module 3 – Weeks 3:
Translates English phrases to mathematical phrases and English sentences to mathematical
sentences, and vice versa.
First Edition, 2021
Republic Act 8293, section 176 states that no copyright shall subsist in any work of
the Government of the Philippines. However, prior approval of the government agency or
office wherein the work is created shall be necessary for exploitation of such work for profit.
Such agency or office may, among other things, impose as a condition the payment of
royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,
trademarks, etc.) included in this book are owned by their respective copyright holders. Every
effort has been exerted to locate and seek permission to use these materials from their
respective copyright owners.
The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education - Region III
May B. Eclar, PhD, CESO III
Rhoda T. Razon, EdD, CESO V
Librada M. Rubio, PhD
Development Team of the Module
Writer: KAREN KRISTINE V. HENSON
Editor: ZCHARINN GENNE V. CUNANAN
Reviewer: ELIZA M. ROQUE
Illustrator: ROMMEL G. SALEM
Layout Artist: JOHN PAUL E. PRING
Management Team:
EMILY F. SARMIENTO, PhD, Division EPS In-charge of LRMS
MA. ESPERANZA MALANG, PhD, Division EPS-Mathematics
Printed in the Philippines by ________________________
Department of Education – Region III
Office Address:
Telefax:
Email Address:
Diosdado Macapagal Government Center, Brgy. Maimpis,
City of San Fernando, 2000 Pampanga
(045) 598-8580 to 89; (045) 402–7003 to 05
region3@deped.gov.ph
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DRRM INTEGRATION
in Mathematics
MODULE
Quarter 2: Week 3
Translating English Phrase/Sentence into
Mathematical Phrase/Sentence and Vice Versa
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Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our
dear learners, can continue your studies and learn while at home.
Activities, questions, directions, exercises, and discussions are
carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide
you step-by-step as you discover and understand the lesson
prepared for you.
Pre-tests are provided to measure your prior knowledge on
lessons in each SLM. This will tell you if you need to proceed on
completing this module or if you need to ask your facilitator or your
teacher’s assistance for better understanding of the lesson. At the
end of each module, you need to answer the post-test to selfcheck your learning. Answer keys are provided for each activity
and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the
Teacher are also provided to our facilitators and parents for
strategies and reminders on how they can best help you on your
home-based learning.
Please use this module with care. Do not put unnecessary
marks on any part of this SLM. Use a separate sheet of paper in
answering the exercises and tests. And read the instructions
carefully before performing each task.
If you have any questions in using this SLM or any difficulty in
answering the tasks in this module, do not hesitate to consult your
teacher or facilitator.
Thank you!
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Translating English Phrase/Sentence
Lesson into Mathematical Phrase/Sentence
3
and Vice Versa
What I need to know
In this module, you are about to learn to translate algebraic expressions. It will help
you to understand more about the Mathematics language.
The activities in this module are utilized to allow you to understand the competency in
Math 7. They consist of several experiences and tasks to help you improve and enhance the
essential skills needed.
The learning module is aligned on the learning competency:
Translates English phrases to mathematical phrases and English sentences to mathematical
sentences, and vice versa.
At the end of this module, you are expected to:
1. Differentiate verbal phrases/sentences from mathematical expressions;
2. Translates verbal phrases/phrases into mathematical expressions and vice versa; and
3. Appreciate the importance of symbols.
What I Know
Identify if the given expression is a phrase or a sentence. Write your answer on a separate
sheet of paper.
1. at the school clinic
6. behind
2. We saw the 3Rs sign yesterday.
the
3. with the DRRM officers
firewall
4. under the sturdy table
7. Where is the fire exit?
5. Learning to administer CPR is fun.
8. Take the first aid kit.
9. Refrain from throwing trash.
10. Be alert during heavy rainfall.
What’s In
Identify each expression as a sum, difference, product, quotient, power, or root. Write your
answer on a separate sheet of paper.
6. 𝑥 2 − 2𝑦
1. x + 17
4
7. 4 +
2. 𝑥 2
𝑥
3.
2+𝑥
5
4. 5(x – 4)
8.
𝑥−6
𝑥+2
9. (3𝑥 − 4)2
5
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10. 9x – 2y
5. √5𝑥
What’s New
Typhoon In-fa (local name Fabian) brought heavy rainfall and hit several parts of the
Philippines from July 28 to August 1, 2021. The typhoon triggered landslides and floods,
affecting most areas of Luzon. Many volunteers went to Angeles City Relief Center to donate
relief goods for the victims. The volunteers brought 100 boxes filled with different relief goods.
If each box has 2 plastic bags of relief goods, the total number of relief goods = 100 x 2 = 200 .
If each box has 3 plastic bags of relief goods, the total number of relief goods = 100 x 3 = 300 .
If each box has 4 plastic bags of relief goods, the total number of relief goods = 100 x 4 = 400 .
If each box has 5 plastic bags of relief goods, the total number of relief goods = 100 x 5 = 500 .
Complete the table below. Write your answer on a separate sheet of paper.
Numbers of plastic bags of relief goods per box
2
3
4
5
n
Total number of plastic bags of relief goods
What is It
Algebra is a branch of mathematics that involves expressions with variables. The goal
of algebra is to use it as a tool in applying mathematical logic to real-life situations and problem
solving. The result of combining numbers, variables, and symbols is called Algebraic
Expression. In What’s New, 100n is an algebraic expression, where n is called a variable.
Other algebraic expressions are:
x + 6,
2x - y,
2x,
and
𝑥−4
5
Expressions are usually named by the last operation to be performed in the expression.
For example, the expression x + 6 is a sum because the only operation in the expression is
addition. The expression 2x – y is referred to as difference because subtraction is the last
operation to be performed. The expression 2x is a product while
𝑥−4
5
is a quotient.
In translating an English to mathematical phrase, assign a variable to one unknown
quantity. Then, write an expression for any other unknown quantities involved in terms of that
variable.
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Below are some English expressions leading to algebraic expressions with addition and
subtraction:
Operation
Key
Symbol
Addition
+
Subtraction
-
Key Word/Phrase
plus
the sum of
added to
more than
increased by
greater than
the total of
exceeds – by
minus
the difference of
subtracted from
less than
less
decreased by
take away
reduced by
Example
Translation
x plus three
the sum of a and two
seven added to m
four more than s
m increased by one
nine greater than v
the total of five and c
exceeds d by ten
x minus three
the difference of a and two
seven subtracted from m
four less than s
eleven less g
m decreased by 1
eight take away j
ten reduced by d
x+3
a+2
m+7
s+4
m+1
v+9
5+c
d + 10
x–3
a–2
m–7
s–4
11 – g
m–1
8–j
10 – d
Note:
You must be careful about the order of the number. For example, eleven less g is 11 – g and
g less eleven is g – 11.
Example 1
Write an English phrase for each expression.
a. 2 + 5
b. 7 – 6
c. x + 2
Solution:
Expression
a. 2 + 5
Key
Symbol
plus
c. x + 2
plus
English Phrase
d. 6 – n
Expression
the sum of two and
five
x increased by two
English Phrase
b. 7 – 6
Key
Symbol
minus
d. 6 – n
minus
n less than six
seven les six
Example 2
Translate each phrase into an algebraic expression.
a. ten more than m
c. u less than twenty-nine
b. six decreased by x
d. d greater than two
Solution:
Phrase
a. ten more than
m
c. u less than
twenty-nine
Key word/
Phrase
Expression
more than
m + 10
less than
29 – u
Phrase
b. six decreased
by x
d. d greater
than two
Key word/
Phrase
decreased
by
greater than
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Expression
6–x
2+d
lOMoARcPSD|23567711
Below are some English expressions leading to algebraic expressions with multiplication and
division:
Operation
Key Symbol
Multiplication
•,()
Division
÷
Key Word/Phrase
times
the product of
multiplied by
twice, thrice
Example
four times x
the product of six and m
b multiplied by nine
twice k, thrice m
of
half of p
the quotient of
the quotient of six and m
divided by
ten divided by h
the ratio of
the ratio of x and seven
splits into - equal
parts
d splits into three equal
parts
divided into
f divided into eight
Translation
4x
6m
9b
2k, 3m
1
2
𝑝
𝑝 or 2
6
𝑚
10
ℎ
𝑥
7
𝑑
3
8
𝑓
Note:
The multiplication sign (x) is seldom used in algebra since it could be mistaken for the letter x.
For two numbers, parentheses are often preferred over a raised dot, which may be confused
with a decimal.
The division symbol (÷) is rarely used in algebra. More often, we use the fraction bar.
Example 3
Write an English phrase for each expression.
a.
20
𝑥
b. 9x
c.
Solution:
Expression
a.
c.
20
𝑥
3+𝑥
5
Key
Symbol
English Phrase
divide
twenty divided by x
divide
the quotient of three
plus x and five
3+𝑥
5
Expression
b. 9x
Key
Symbol
English Phrase
multiply
nine of x
Example 4
Translate each phrase into an algebraic expression.
a. two more than seven times x
c. the product of ten and f, increased by one
b. r divided by eleven
Solution:
a.
Phrase
two more than seven times x
Key Word/Phrase
more than/times
b.
r divided by eleven
divided by
Expression
7x + 2
𝑟
11
10f + 1
the product of ten and f, increased by one
the product of/increased by
c.
Note:
In Example 4c, if there was no comma after the word number, the expression “The product of
ten and f increased by one” would be ambiguous. Ambiguous means the expression has more
than one possible meaning.
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Below are some English expressions leading to algebraic expressions with power and root:
Operation
Key
Symbol
𝑥𝑛
Power
𝑛
√𝑥
Root
Key Word/Phrase
Example
the square of
the cube of
the square root of
the square of x
the cube of y
the square root of x
the cube root of
the cube root of y
Translation
𝑥2
𝑦3
√𝑥
3
√𝑦
When we put together numbers, variables, and symbols, we may form either a
mathematical phrase or sentence. A mathematical sentence can be transformed into an
equation where the equal signed is used.
Operation
Equal
Key
Symbol
=
Key
Word/Phrase
equals
is equal to
is
is the same as
yields
amounts to
Example
b.
x – 1 = 10
x+y=7
2p = 11
9=1–m
k+4=6
5 – y = 10
One less than x equals ten.
x plus y is equal to seven
Twice p is eleven.
Nine is the same as one minus m.
Four added to k yields six.
Five less y amounts to ten.
Example 5
Write an English sentence for each expression.
a. 5x3 = -10
b. √𝑦 = 𝑥 + 4
Solution:
a.
Translation
Expression
Key Symbol
5x3 = -10
multiply/cube/equal
√𝑦 = 𝑥 + 4
root/equal/plus
English Phrase
Five times the cube of x is equal to
negative ten.
The square root of y is x plus four.
Example 6
Translate each into an algebraic expression.
a. The sum of y and 3 multiplied by x is 15.
b. Half the square of x is nine more than y
Solution:
Phrase
Key Word/Phrase
a. The sum of y and 3 multiplied by x is 15.
b.
the sum of/multiplied by/is
Half the square of x is nine less than y.
half/the square of/less than
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Expression
(y+3)x = 15 or
x(y + 3) = 15
1
2
𝑥 2 = 𝑦 − 9 or
𝑥2
= 𝑦−9
2
lOMoARcPSD|23567711
What’s More
Translate each of the following Mathematical phrases or sentences into equivalent English
phrases or sentences. Write your answer on a separate sheet of paper.
6. 4𝑥 + 6𝑦 − 3𝑧
1. a – b
2. x – 12 = 15
7. n + 6 = 9
3. V = s3
8. √𝑘
4. 6w
9. 5 – 7x
𝑡
10. 9e + 20 = 2
5. = 15
7
What I Have Leaned ACTIVITY 1
Choose the letter of the phrase or sentence that best matches each expression. Write your
answer on a separate sheet of paper.
1.
2.
3.
4.
5.
6.
2–x
4x – 1
2(x + 7)
3(x + 5) = 10
2x
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
7
𝑥
7. 8 – y
8. b – 20
9. x + 12 = 5
10. n2 + 2n
Three times the sum of a number and five is equal to ten.
seven divide by x
twenty less than a number b
eight diminished by y
one subtracted from four times a number
twice the sum of a number and seven
the square of a number increased by twice the number
the difference of two and a number
twice a number
Twelve added to a number yields five.
What I Can Do
Write your answer on a separate sheet of paper.
Performance Based Assessment: Translating and Evaluating Cell Phone Plans
1. Calamities are happening within the Philippines almost every day. These
calamities may be naturally occurring or man-made. Hence, it is a must to have a
cell phone plan which includes unlimited data, texting and calls for emergencies.
To purchase a cell phone, go to the nearest cell phone service provider and find a
cell phone you would want to buy. You need to record the price and type to help
you decide which phone you will buy. All of the plans at cell phone service providers
have a base service charge of Php 499 per month plus the cost of the phone. Write
an expression to represent the service charge.
2. There are three unlimited plans being offered in any cell phone service provider.
Each plan incorporates the service charge. Translate each plan into an
algebraic expression using the service charge expression from the above
question.
Plan 1: Php 10 less than twice the amount of the service charge.
Plan 2: The quotient of the service charge and 2 increased by a fee of Php 599 per
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month.
Plan 3: The sum of half the number of months cubed and a third of the service charge.
3. Which plan would be the most cost-effective for a 2-year contract? Show all
calculations that led to your results. Explain your results comparing the three
(3) plans.
4. One cell phone service provider is having a special promotion by challenging
customers to create their plans. Customers must write a verbal expression that
must include the service charge and at least two operations. The operations to
choose from are listed below. If the plan is within Php 10 per month of one of the
original three (3) plans, then the customer could choose that plan. Create a 4th
plan and show all calculations that led to your results.
Student Assessment List
Assessment points
Points
Possible
Element
1. Cell phone type, price and expression to represent
the service charge.
5
2. Setup an expression for the 3 plans.
5
3a. Calculations for the most cost effective plan
5
3b. Explanation with complete sentences.
5
4a. Setup an expression with at least two of the
given operations representing a 4th plan.
5
4b. Calculations that shows that the 4th plan is within
Php 10 per month of one of the original plans.
5
Total points
30
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Earned
Assessment
Self
Teacher
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Teacher Rubric for Performance Based Assessment
#
5 points
4 points
3 point
2 points
1.
Stated the cell
phone name, price
and correctly
wrote an algebraic
expression to
represent the
service charge.
Correctly wrote an
algebraic
expression for the 3
plans
Stated the cell
phone name,
price and wrote
an algebraic
expression to
represent the
service charge.
Correctly wrote an
algebraic
expression for 2 out
of the 3 plans
Stated only the cell
phone name and
price.
Only stated the cell
phone name or
price.
Correctly wrote an
algebraic
expression for 1 out
of the 3 plans
Did not set up
either of the
expressions
correctly
3a.
If the students
correctly
demonstrated
and calculated
the cost of all 3
plans.
If the students
correctly
demonstrated and
calculated
the cost of 2 out of
the 3 plans. OR
Correctly
demonstrated and
calculated
the cost of all 3
plans using the
incorrect number
for the months.
If the students did
not correctly
calculate any of the
plans.
3b.
Uses mathematical
language
to thoroughly
explain which plan
is most cost
effective
Correctly setup an
algebraic
expression, which
includes two of the
given operations.
Uses mathematical
language to
partially explain
which plan is most
cost effective
If the students
correctly
demonstrated and
calculated
the cost of 1 out of
the 3 plans. OR
Correctly
demonstrated
and calculated
the cost of 2 out of
the 3 plans using
the incorrect
number for the
months.
Does not use
correct
mathematical
language to
explain.
Correctly setup an
algebraic
expression, which
includes one of the
given operations.
Setup an
expression but did
not use the given
operations.
Did not setup an
expression
Correctly evaluated
the expression for
a 2-year contract,
correctly converted
to a monthly cost
and demonstrated
that the plan was
within Php 10 of
one of the original
plans.
Evaluated the
expression for a 2year contract using
the incorrect
number for months,
correctly converted
to a monthly cost
and demonstrated
that the plan was
within Php 10 of
one of the original
plans.
Correctly evaluated Evaluated the
the expression for a expression for a 22-year contract and year contract
correctly converted
to a monthly cost
but the plan was
not within Php 10 of
one of the original
plans.
2.
4a.
4b.
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Uses no
explanation.
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Assessment
Choose the letter of the correct answer. Write your answer on a separate sheet of paper.
1. Which equation represents the phrase, “the sum of x and eight”?
a. 8 – x
b. x + 8
c. x8
d. 8x
2. What is y – 5 = 12 in English sentence?
a. y minus five and twelve
c. y decreased by five is twelve
b. y divided by five is twelve
d. five added to y yields twelve
3. The following phrases belong to division, EXCEPT.
a. the quotient of
c. divided into
b. the ratio of
d. the square of
4. What is ab in English phrase?
a. a times b
b. a plus b
c. a minus b
d. a divided by b
5. What is “the sum of five and x divided by fifteen is y” in mathematical sentence?
a. 5x + 15 = y
b.
5+𝑥
15
c. 5 + x – 15 = y
=𝑦
d. (5 + x)15 = y
Additional Activities
Directions: Look at the puzzle board below. Match the Mathematical phrases with their corresponding
English phrases to form some measures to remember if an earthquake happens.
Disaster Preparedness during Earthquake
Mathematical Translations Matching
twice a number
diminished by five
n+5
five more than
a number
(6n – 3)2
the square of three
less than six times
a number
2n - 5
Answer Key
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What I Know
sentence
9.
sentence
8.
sentence
7.
phrase
6.
sentence
5.
phrase
4.
phrase
3.
sentence
2.
phrase
1.
10. sentence
Downloaded by Jenilyn Espejo (jenilyn.espejo@deped.gov.ph)
14
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2
What's In
sum
power
quotient
product
root
difference
sum
quotient
power
difference
What's New
3
4
200
What I Have Learned
1. h
300
400
100n
500
n
5
c
8.
5. b
d
7.
4. a
b
6.
3. d
i
5.
2. c
a
4.
Assessment
1. b
f
3.
e
2.
What’s More
Activity 1
1. a minus b
2. The difference of x and 12 is
fifteen.
3. V is equal to the cube of s.
4. six times w
5. The quotient of t and seven is
fifteen.
6. Four of x increased by six of y
minus three of z
7. Six more than n is nine.
8. the square root of k
9. five minus seven of x
10. Nine times e plus twenty
yields twenty-five.
*Student’s answer may vary.
Additional Activity
What I Can Do
*Student’s answer may vary.
j
9.
twice a number diminished by five
2n – 5
five more than a number
n+5
the square of three less than six
times a number (6n – 3)2
10. g
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References
Department of Education. Mathematics MELCs. p. 304.
Guren, Paulino T. Ph.D. Realistic MATH Scaling Greater Heights 6. Sibs Publishing House, Inc., 2016. pp. 267 –
274.
Orines, Fernando B. Next Century Mathematics 7. Phoenix Publishing House, Inc., 2012. pp 242 – 246.
Oronce, Orlando A. and Mendoza, Marilyn O. E-Math 7. REX Book Store, Inc., 2012. Pp. 150 – 157.
http://cneacs.weebly.com/uploads/2/4/8/5/24853122/2015_lesson_1_translating_words_into_mathematical_symbols
.pdf
https://www.radford.edu/rumath-smpdc/Performance/src/Abbie%20Brewer%20%20Translating%20and%20Evaluating%20Cell%20Phone%20Plans.pdf
https://www.palmbeachstate.edu/prepmathlw/Documents/translatingkeywords.pdf
https://reliefweb.int/disaster/fl-2021-000091-phl?fbclid=IwAR0bHeD8bZ-_JblXXKXnk2tnDTxZZqrNk_o-VFNXf8lVhRvDu5H0T3gBX8
https://www.liveworksheets.com/worksheets/en/Grammar/Sentences/Sentence_or_Phrase_Worksheet_sy430741rq
?fbclid=IwAR1nGKhnOrw2hBncphbNgNpnRqwkOH2n03PxB91SzesfBrYPMx_762EqrAI
https://www.google.com/search?q=ndrrmc+clipart+earthquake&tbm=isch&ved=2ahUKEwjtzNbXjfPyAhVtEqYKHZm
gDnIQ2cCegQIABAA&oq=ndrrmc+clipart+earthquake&gs_lcp=CgNpbWcQAzoECAAQHlDbkSRYsqkkYJOrJGgAcA
B4AIABkAGIAd0JkgEEMC4xMZgBAKABAaoBC2d3cy13aXotaW1nwAEB&sclient=img&ei=_KA6Ye3eIe2kmAWZwb
qQBw&bih=657&biw=1366#imgrc=UwIbmYEv5s8ueM&imgdii=HxOHSa7976_YCM
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MATHEMATICS 7
Quarter 2 – Module 3:
Related Terms in Algebra
Week 3 LC 2
Date
Name
Section
Background Information
What’s New
Let’s help Anna find the new words that she will encounter. Find and circle the words
listed below. Words appear straight across, back word straight across, up and down, down
and up, and diagonally.
A
B
C
C
O
N
S
T
A
N
T
D
E
F
T
G
H
I
Q
W
EXPONENT
MONOMIAL
VARIABLE
S
D
F
G
H
H
J
K
L
P
O
I
U
Y
R
T
R
E
W
P
S
A
V
Z
X
C
V
B
N
M
M
N
B
V
I
C
X
Z
A
O
X
C
M
A
N
B
V
H
G
T
Y
R
S
E
N
Q
E
D
S
L
M
N
B
V
R
C
X
X
Z
A
S
D
E
F
O
G
H
J
K
Y
F
D
S
A
Q
I
W
E
R
T
Y
U
L
I
M
O
P
L
K
N
G
H
J
K
L
M
A
N
B
V
C
X
I
Z
I
A
Q
W
D
O
Y
T
Y
L
K
J
M
B
N
V
B
H
T
G
A
T
R
F
E
M
L
U
I
O
L
K
O
P
L
T
Y
E
R
Q
L
E
W
E
D
I
A
M
S
A
Z
X
C
V
B
E
N
M
L
K
J
H
X
T
G
A
I
U
D
F
G
H
J
K
L
P
O
I
U
Y
T
P
R
E
W
L
POLYNOMIAL
CONSTANT
TRINOMIAL
M
L
E
S
A
B
H
G
F
E
D
E
D
C
O
S
X
Z
A
Q
O
T
Q
A
W
S
D
E
R
F
T
G
Y
N
H
U
J
K
I
O
N
I
T
R
E
Q
W
Q
S
A
Z
X
E
C
V
B
N
M
P
L
O
N
A
S
E
W
T
G
H
J
B
N
D
K
L
M
H
N
U
T
BASE
BINOMIAL
TILES
M
O
L
K
H
F
V
D
R
E
T
Q
L
K
F
G
H
I
K
J
T
M
A
X
Q
I
O
K
Z
Q
Q
X
Y
U
I
D
F
G
J
K
Q
I
S
B
W
U
P
H
X
W
A
G
M
N
V
E
R
T
J
F
A
A
D
V
E
Y
A
D
C
T
D
H
J
K
L
W
J
K
L
X
S
L
F
G
R
T
S
F
L
A
I
M
O
N
I
B
S
X
C
V
MULTINOMIAL
What is It
You have learned in the previous lessons that we use expressions to write products of
repeated factors:
32 = 3 • 3 = 9
53 = 5 • 5 • 5 = 125
24 = 2 • 2 • 2 • 2 = 16
The expression 32 , the number 3 is the base and 2 is the exponent. The expression is
called an exponential expression and it is the exponential form of 3 • 3.
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Exponent is a small number written to the right of slightly above another number or
letter to indicate how many times the letter or number is used as a factor.
Example 1
Find the power.
a. 33
Solution:
b. (−4)2
c. 34
a. 33 = 3 • 3 • 3 = 27
b. (−4)2 = −4 • −4 = 16
d. 53
c. 34 = 3 • 3 • 3 • 3 = 81
d. 53 = 5 • 5 • 5 = 125
In the expression 4𝑥3 + 6𝑥2 + 2𝑥, the quantities 4𝑥3, 6𝑥2 and 2𝑥 are called terms. Term is a
number, a variable or a product or quotient of numbers and variables. The terms of an
expression are separated by the symbols + and -.
Variable (literal coefficient) is a symbol, usually a letter, such as x, y or z, used to represent
any unknown number.
Constants are symbols that have fixed values.
Numerical Coefficient is the number in an algebraic term.
Example 2
Identify the numerical coefficient, variable and constant in each expression.
a. 3n + 2
b. x
c. -5m + 9
d. 15y
Solution:
a. The number 3 is the numerical coefficient of the variable n and 2 is the constant.
b. The variable x is assumed to have a numerical coefficient of 1.
c. The number -5 is the numerical coefficient of the variable m and 9 is the constant.
d. The number 15 is the numerical coefficient and y is the variable.
Polynomial is an algebraic expression that represents a sum of one or more terms containing
whole number exponents on the variables.
Example 3
Tell whether each expression is a polynomial or not.
a. 5𝑥2 + 2𝑥 + 8
b. 3𝑥−4 + 7𝑥 − 1
c. 1 𝑥 3 − 8𝑥 + 6
2
Solution:
a. Polynomial
b. Not a polynomial because the first term has a negative exponent on a variable base
c. Polynomial
Classifying Polynomials based on the number of terms of:
- A polynomial with one term is called monomial.
- A polynomial with two terms is called binomial.
- A polynomial with three terms is called trinomial.
- A polynomial with four or more terms is called multinomial.
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Procedure
To identify an expression as a monomial, binomial, trinomial or multinomial, follow this
procedure:
a. Look for the plus or minus signs separating the terms. (Although we defined a
polynomial as a sum of one or more terms, any one of those could be a negative
term.)
b. Count the number of terms and name the expressions accordingly.
Example 4
Classify the following polynomials
a. 3𝑥 2𝑦
b. 𝑦 2 − 5
c. 10
Solution:
a. Monomial – consist of one term
b. Binomial – consist of two terms
c. Monomial – consist of one term
d. 3𝑥3 − 2𝑥 +7
e. 2𝑥3 − 3𝑥2 + 5𝑥 + 3
d. Trinomial – consist of three terms
e. Multinomial – consist of four terms
We can also classify polynomials using their degree. The degree of a monomial is the total
number of times its variable occurs as factor. The degree of nonzero constant monomial is 0.
The degree of a polynomial is the greatest of the degrees of its terms.
Example 5
State the degree of each polynomial.
a. 4𝑥2𝑦 b. −3𝑎𝑏2𝑐4
c. 2𝑥2 + 5𝑥− 1
Solution:
a. The degree of 4𝑥2𝑦 is 3.
4𝑥2𝑦 = 0 + 2 + 1 = 3
b. The degree of −3𝑎𝑏2𝑐4 is 7.
−3𝑎𝑏2𝑐4 = 0 + 1 + 2 + 4 = 7
d. 𝑎2𝑏 + 𝑎2 𝑏2 − 𝑎𝑏
We also use algebra tiles to represent algebraic expressions.
Let:
c. The degree of 2𝑥2 + 5𝑥 −1 is 2.
2𝑥2 + 5𝑥 − 1
2𝑥2 = 0 + 2 = 2
5𝑥 = 0 + 1 = 1
−1 = 0
d. The degree of 𝑎2𝑏 + 𝑎2𝑏2 − 𝑎𝑏 is 4.
𝑎2𝑏 = 2 + 1 = 3
𝑎2𝑏2 = 2 + 2 = 4
𝑎𝑏 = 1 + 1 = 2
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Example 5
Create an algebra tile model on each polynomial.
a. 4x + 3
b. 5x – 10
c. 4x2 + 3x + 5
d. -3x2 – 2x + 3
Solution:
a. 4x + 3 =
b. 5x – 10 =
c. 2x2 + 3x + 1 =
d. -3x2 – 2x + 3 =
What I need to know
Illustrates and differentiates related terms in algebra: (a) a n where n is a positive integer, (b)
constants and vari
ables, (c) literal coefficients and numerical coefficients, (d) algebraic expressions, terms and
polynomials, and (e) number of terms, degree of the term and degree of the polynomial.
General Instruction
Write your answer on a clean sheet of paper.
What I Know
Match each term in Column A with its meaning in Column B. Write your answer on a separate
sheet of paper.
A
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Constant
Polynomial
Monomial
Binomial
Trinomial
Multinomial
Exponent
Base
Variable
Term
B
a. a polynomial with three terms
b. an algebraic expression that represents a sum of one or more terms
containing whole number exponents on the variables
c. a number or letter with an exponent
d. a polynomial with four or more terms
e. symbols that have fixed values
f. a number, a variable or a product or quotient of numbers and variables
g. a polynomial with one term
h. a small number written to the right of slightly above another number or
letter to indicate how many times the letter or number is used as a
factor
i. a polynomial with two terms
j. a symbol, usually a letter, such as x, y or z, used to represent any
unknown number.
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What’s In
Identify the exponent, base, constant and variables by completing the table. Write your
answer on a separate sheet of paper.
Exponent
Base
Constant
Variable
1.
2. −2𝑥2 + 7
3. 9a + b
4. 8𝑎7 + 3
5. 15𝑚2 − 6
13𝑥5
What’s More
Activity 1
Fill in the
or ( ) with the correct integer. Write your answer on a separate sheet of paper.
1. 52 =
2. (−2)3
3. (3𝑥𝑦)2
4. 6( ) = 216
5. 7( ) = 2401
Activity 2
Classify each polynomial in the box. Write your answer on a separate sheet of paper.
3a + 2
𝑎4 + 3𝑎3 − 2𝑎2 − 𝑎 − 9
3(a)(b)(c)(d)
6𝑎7 + 3𝑎 −5
9𝑏3 + 4𝑏2 − 𝑏 + 8
5x + y
-10
2
𝑥 + 𝑥𝑦 + 𝑦3
-4x
3x - 4y
MONOMIAL
BINOMIAL
TRINOMIAL
y
xyz
𝑎𝑏 + 4𝑎2𝑏2 − 8𝑎𝑏2 + 4𝑎4𝑏3
x+y
12𝑎2𝑏2𝑐2𝑑2
MULTINOMIAL
What I Can Do
Give the degree of each polynomial. Write your answer on a separate sheet of paper.
4. 6𝑎3𝑏6
5. 3𝑥 3 𝑦 3 + 5𝑥2𝑦 − 2𝑥𝑦
1. 3𝑥5
2. −10
3. 15𝑚9 + 11𝑚12 − 7𝑚4
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Assessment
A. Tell whether each polynomial is a monomial, binomial, trinomial or
multinomial.
1. 𝑥 2 𝑦 4
4. 2𝑎2𝑏 + 6𝑎𝑏 − 3𝑏
2. 2𝑥3 − 2𝑥2 + 𝑥 + 9
5. 5𝑥 + 9
4
3
2
3. 8𝑎 𝑏 𝑐 𝑑
B. Determine the degree of each polynomial.
1. −9𝑥
4. −4𝑥 3𝑦4
2. 8
5. 2𝑥 2𝑦 3 − 5𝑥𝑦2
3. 9𝑥5 − 3𝑥9 + 4𝑥12
C. Create an algebra tile model in each polynomial
4. 2𝑥2 + 3𝑥 −4
5. 4𝑥 − 10
1. 3𝑥 + 5
2. 2𝑥2 − 3𝑥 + 5
3. 5𝑥 − 3
Additional Activities
Identify whether the following algebraic expressions are polynomials or not. If they are not
polynomials, state the reason why. Write your answer on a separate sheet of paper.
1. 7𝑥
4. 3𝑏−4
1
2. 4𝑛2 + 25
3.
5. 8𝑥2
3𝑥+𝑦
𝑥𝑦
Answer Key
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References
Department of Education. Mathematics MELCs. p. 304.
Oronce, Orlando A. and Mendoza, Marilyn O. E-Math 7. REX Book Store, Inc., 2012. Pp. 154 – 161.
Herrera, Lucia D. Skillbook in Math I, St. Bernadette Publishing House Corporation, 2009. Pp. 126 146
Prepared by:
Daizeyrell N. Pedraza
SST – I, FGNMHS
Illustrated by:
Rommel G. Salem
SST – III, FGNMHS
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MATHEMATICS 7
Quarter 2 – Module 4
Evaluating Algebraic Expressions
Week 4
Date:
Name:
Section:
Background Information
What’s New
Are You Ready to Evaluate Algebraic Expressions?
In the past, you've evaluated numerical expressions by using the order of operations.
We are going to use these same rules to evaluate algebraic expressions.
An algebraic expression is a combination of numbers, letters and symbols. It
composes of one or more terms. To evaluate algebraic expression, we represent any value
of variables until we determine the number of values it represents. You have to substitute a
number for each variable and perform the arithmetic operations. If we know the value of our
variables, we can replace the variables with their values and then evaluate the expression.
An algebraic expression consists of numbers, variables, and operations. Here are a
few examples:
Algebraic Expression
4n
3x + 5
8y -7
𝑛
+2
3
Meaning
4 times n
Three times x plus 5
Eight times y minus 7
𝑛 divided by 3 plus 2
What is It
In order to evaluate an algebraic expression, you must know the exact values for
each variable. Then, you will simply substitute and evaluate using the order of operations.
Suppose numbers are assigned to letters of the Alphabets as follows:
A= 1, B= 2, C=3, D= 4, E= 5 and so on, up to 26.
Using this, find the value of a name by adding values of its digits.
For example:
LUCAS
L = 12
U=21
C=3
A= 1
S = 19
LUCAS = 12 + 21 + 3 + 1 + 19 = 56
Find the numerical value of each name:
1. JOHN
2. HARVEY
3. MATTHEW
4. ELEANA
5. Find the value of your name. Compare it with the values of the names of others in
your class. Are there two names with the same value.
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Replacing a variable with a number demonstrates the Substitution Property of Equality
Subtraction Property of Equality
If two quantities are equal, then the two quantity can be replaced by the other.
In symbols:
For all numbers a and b, if a = b , then a may be replaced by b
There are at least two steps involved in evaluating algebraic expression:
1. Replacing the variable by the given value (substitution);
2. Performing the indicated arithmetic following the order of operations.
Example 1: Evaluate the expression: 5x2 + 1 when x = 2
5x2 + 1 when x = 2
5(2)2 + 1
-Replacing the variable x by the given value 2 in the expression (substitution);
5(4) + 1
-Using the Order of operations, we must simplify the expressions within the
grouping symbols
2
(2) = 4 -simplify the powers
20 + 1
- Simplify the product. (5)(4) = 20
21
- Simplify the sum. Evaluate the final addition. (20 + 1) = 21
Now, let’s evaluate algebraic expressions with more than one variable. Don't forget to
always use the order of operations when evaluating the expression after substituting.
Example 2: Find the value of the polynomial expression 2x3 – 3y + 5, if x = -2 and y = 4
2x3 – 3y + 5
2(-2)3 – 3(4) + 5
- substitute the given values for each variable
2(-8) – 3(4) + 5
- evaluate the powers (-2)3= (-2)(-2)(-2) = -8
-16 – 12 + 5
-28 + 5
- simplify sum and differences in order from left to right
- perform the indicated operation
-23
- the final answer
Example 3: Evaluate the polynomial 3x2 + 2x – 4 when:
a. x = -1
b. x = -2
c. x = 0
d. x = 1
X
3x2 + 2x – 4
-1
-3
3(-1)2 + 2(-1) – 4 = -3
-2
4
3(-2)2 + 2(-2) – 4 = 4
0
-4
3(0)2 + 2(0) – 4 = -4
1
1
3(1)2 + 2(1) – 4 = 1
2
12
3(2)2 + 2(2) – 4 = 12
e. x = 2
Solution
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Example 4: Evaluate the expression 2x2y
x–y
2x2y
x–y
when x = 5, y = 1
2(5)2 (1)
5–1
- substitute the given values for each variable
2(25)(1)
5-1
- evaluate the numerator first starting with the power (exponent) (52 = 25)
50
5-1
- simplify numerator. (2)(25)(1) = 50
50
4
- evaluate the denominator (5-1 = 4)
25 or 12.5
4
-simplify the fraction. This is the final answer
What I need to know
Evaluates algebraic expressions for given values of the variables. (M7ALIIc-4)
General Instruction
Write your solutions and answers on a clean sheet of paper.
What I Know
Tell whether each statement is True or False
1. If x = 3, then the value of 2x is 6.
2. If b = -4, then 3b + 1 is 11.
3. Evaluating 3x2 with x = 2 is 12.
4. If m=-2, simplifying 2m2 – 3m resulted to 14.
5. In the expression 3w3 + 4w – 7, if w= 1 by evaluating it will give you 0
What’s In
Find the value of the following algebraic expressions using the indicated constant
1. a + b; a = 2 and b = 3
6. 4a +5; a= -3
2. m – n; m= 5 and n = 4
7. 2p -3r; p = 2 and r = 1 3.
x + y – z; x = 1, y = 2 and z = 3
8. 3m – 2n; m=1 and n=4 4.
9x; x= 2
9. 2x2 + 1; x= -1
5. 5m; m= 1
10. 3w3 + 2w2 - 5w + 4; w= 1
What’s More
Evaluate each expression using substitution
1.
2.
3.
4.
5.
4s2
s=5
LWH L = 10, W= 8 and H= 9
πr2
r=3
2
2
a + b + c2 a=1, b = 2 and c = 3
5 (oF
9
6. am+n ; a = 2,m=3, n=2
7. b – 4ac; a=1, b=8, c=2
8. 180o(n – 2); n=8
9. oC + 273.15; oC = 20
– 32) ; F=36o
10. 9 oC + 32; oC = 25
5
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What I Can Do ACTIVITY 1
Evaluate the polynomial. Complete the table
1.
x
x2 + 5
Solution
3x2 – 5x + 2
Solution
-1
-2
0
3
4
2.
x
-1
-2
0
1
2
Assessment
Find the value of each expression and write it in the square. Let m = 5
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Additional activities
A magic square is a puzzle in which the sum of its numbers in any row, column and along
the diagonal are the same.
When 5, 3 7 are substituted in place of a, b, and c respectively for the expressions in the left
squares the result is the square at the right.
a-b
a+b-c
a+c
a+b+c
a
a–b-c
a-c
a+c-b
a+b
Guide questions:
1. Examine the sum of the numbers in any row, column and along the diagonals, what
did you notice?
2. Assign any value for, a, b, and c.; then substitute your numbers for the variables in
each expression.
Answer Key
What’s In
What is It
1. 47
2. 79
3. 90
4. 38
5. Answer may vary
What I Know
1. True
2. False
3. True
4. True
5. True
1. 5
2. 1
3. 0
4. 18
5. 5
6. -7
7. 1
8. -5
9. 3
10. 4
What’s More
1. 100
2. 720
3. 28.26
4. 14
5. 2.22o
6. 32
7. 0
8. 1 080o
9. 293.15
10. 77o
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What I Can Do
x
-1
-2
0
3
4
1.
x2 + 5
6
9
5
14
21
Solution
(-1)2 + 5 = 6
(-2)2 + 5 = 9
(0)2 + 5 = 5
(3)2 + 5 = 14
(4)2 + 5 = 21
x
-1
-2
0
1
2
2.
Assessment
3x2 – 5x + 2
10
24
2
0
4
Solution
3(-1)2- 5(-1) + 2 = 10
3(-2)2- 5(-2) + 2 = 24
3(0)2- 5(0) + 2 = 2
3(1)2- 5(1) + 2 = 0
3(2)2- 5(2) + 2 = 4
Additional activities
Guide questions:
1. the sum of its numbers in any row, column and along the
diagonal are the same.
2. Answer may vary
References
Karin, Hutchinson, Copyright 2019-2020, algebra-class.com, Making Algebra Easier for you,
accessed September 08, 2020. https://www.algebra-class.com/evaluate-algebraic-expressions.html
mathplanet.com, Evaluate Algebraic Expressions, accessed, September 08, 2020
https://www.mathplanet.com/education/pre-algebra/introducing-algebra/evaluate-expressions
Oronce Orlando A, and Mendoza Marilyn O., E-Math Worktext in Mathematics 7 (Rex Book Store :
856 Nicanor Reyes St., Sampaloc, Manila, Third Edition 2012), 164 – 168
Prepared by:
JENNY L. CAPITULO
Master Teacher I, FGNMHS
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MATHEMATICS 7
Quarter 2 – Module 4
Adding and Subtracting Polynomials
Week 4
Date
Name
Section
Background Information
What’s New
Familiarize yourself with the tiles below:
Stands for (+1)
+
Stands for (-1)
+
+
-
Stands for (+x)
+
Stands for (+x2)
Stands for (-x)
Stands for (-x2)
You can use algebra tiles to add or subtract polynomials
Consider the polynomials 3x2 – 2x + 4 and -x2 + 4x -1
This is how to model the addition of the
given polynomials.
+ +
-
-
+ +
3x – 2x + 4
2
+
+
-
+
-
+
+
+
+
-x2 + 4x -1
When a positive tile is paired with
a negative tile that is of the same shape, the
result is called a zero pair
Combine the tiles that have the
same shape.
Remove any zero pairs
0
0
0
+
+
+
+
+
+
+
+ +
+ +
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+ +
+
+
+
+
The tiles that remain give the
answer.
+
2x2 + 2x + 3
(3x2 – 2x + 4) + (-x2 + 4x -1) = 2x2 + 2x + 3
Subtractions can also be modeled with tiles
+
+
A tile pair having opposite signs is referred to as a neutral pair.
The value of the neutral pair is zero (0)
Subtract 3
tiles
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
(4x + 3) – (2x + 1) =
This leaves 2
Example 4: (3x + 2) – (5x + 1) =
zero pair
Example 2: (4x + 3) – (2x + 1) =
+
-
zero pair
4x – 6x = -2x
This leaves 1
+
+
+
+
+
+ +
+ +
4x – 3x = x
-
To subtract 4
tiles, add 2 zero pairs
of x tiles. Then take
away 4
tiles
+
zero pair
+
Start with 2
tiles
Start with 4
tiles
+
-
Example 3: 4x – 6x =
+
Example 1: 4x – 3x =
+
zero pair
(3x + 2) – (5x + 1) = -2x + 1
2x + 2
What is It
Just as you add and subtract real numbers, you can perform each of these basic
operations with polynomials.
To add two polynomials, write the sum and simplify by combining like terms. The
following properties can be used to find the sum of polynomials.
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Associative Property for Addition
Commutative Property for Addition
For any number a and b,
For any number a, b and c
a + b = b + a.
(a + b) + c = a + (b + c).
Addition Rule
To add polynomials, simply combine similar terms. To combine similar terms, get
the sum of the numerical coefficients and annex the same literal coefficients. If there is
more than one term, for convenience, write similar terms in the same column.
Do you think you can add polynomials now without the tiles?
Perform the operation.
1. Add 4a - 3b + 2c, 5a + 8b - 10c
and -12a + c.
1.
Add 13x4 - 20x3 + 5x - 10
and -10x2 - 8x4 - 15x + 10.
13x4 - 20x3
+ 5x - 10
+ -8x4
- 10x2 - 15x + 10
5x4 - 20x3 - 10x2 - 10x
4a - 3b + 2c
5a + 8b - 10c
+ -12a
+c
-3a + 5b - 7c
You can subtract a number from another by using opposites. This is the given rule below
Subtraction Rule
To subtract polynomials, change the sign of the subtrahend then proceed to
the addition rules. Also, remember what subtraction means. It is adding the negative
of the quantity.
Perform the operation.
1.
2x2 - 15x + 25
-3x2 + 12x – 18
2x2 - 15x + 25
+ -3x2 - 12x + 18
-x2 - 17x + 43
2. (30x3 - 50x2 + 20x - 80) - (17x3 + 26x + 19)
30x3 - 50x2 + 20x - 80
+ -17x3
- 26x - 19
13x3 – 50x2 - 6x - 99
What I need to know
Adds and subtracts polynomials. (M7ALIId-2)
General Instruction
Write your solutions and answers on a clean sheet of paper.
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What I Know ACTIVITY 2
Perform the indicated operations using the algebra tiles
1.
2.
3.
4.
5.
6. 3x – 2x
7. (5x -3) - (3x + 2)
8. (4x + 1) - (-x +2)
9. (x2 – 4x + 1) - (x - 3)
10. (2x2 – 3x + 3) - (-x2 + 2x -1)
3x + 2x
(-5x -3) + (3x -2)
(4x + 1) + (-x +2)
(x2 – 4x +1) + (x + 3)
(3x2 – 2x + 4) + (-x2 + 4x -1)
What’s In
Find the sum/difference of the following polynomials expressions
1. 3x + 10x
2. 12y - 18y
3. 14x3 + (-16x3)
4. -5x3 -4x3
5. 2x – 3x
=
=
=
=
=
6. 10xy - 8xy
=
2 2
2 2
7. 20x y + 30x y
=
8. -9x2y + 9x2y
=
9. -10x2y3 - 10x2y3
=
10. 5x - 3x - 8x + 6x =
What’s More
Write the resulting polynomial.
1.
2.
3.
4.
(4y + 2) + (-5y -4)
(-x2 + 8x) – (7x + 3x2)
(4m2 + 6m +3) – (3m + 5)
(8n3 + 5n – 7) + (6n3 – n +3)
5. (7b2 -3b – 5) – (-2b -4b2 +3)
=
=
=
=
=
What I Can Do
Answer the following questions. Show your solutions.
1. What is the sum of 3x2 - 11x + 12 and 18x2 + 20x - 100?
2. What is 12x3 - 5x2 + 3x + 4 less than 15x3 + 10x + 4x2 - 10?
3. What is the perimeter of the triangle shown below?
(2x2+7) cm
(3x2 - 2x) cm
(x2 + 12x - 5 ) cm
4. If you have (100x3 - 5x + 3) pesos in your wallet and you spent (80x3 - 2x2 + 9)
pesos in buying foods, how much money is left in your pocket?
5. What must be added to 3x + 10 to get a result of 5x - 3?
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Assessment
Error Analysis. Study how Lucas works in adding or subtracting polynomials. Check
whether his answer is correct or not. If he is correct just write Correct. Write incorrect if it is
wrong then write the correct answer
Correct or Incorrect?
If incorrect, explain and find the correct answer
Lucas Works
1. (3x + 2) + (4x +3)
=(3x + 4x) +(2 + 3)
=7x + 5
2. (-5x + 3y) + (3x +2y)
=(-5x + 3x) + (3y + 2y)
= 8x + 5y
3. (4x -3) – (2x + 1)
=(4x -2x) + (-3 – 1)
=2x + (-4)
=2x - 4
4. (3a + 2b) – (a -b)
= (3a -a) + (2b -b)
=2a + b
5. (5x2 + 3x +3) + (3x2 -2x -7)
=(5x2 +3x2) + (3x -2x) + (3 -7)
= 8x2 + x + 4
Additional activities
RIDDLE TIME!
Why did the school cook become a history teacher?
W
8
11
5
4
6
3
11
7
T
X
4
2
11
9
11
10
11
1
12
2
G
T
5
11
2
12
11
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3
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Perform the indicated operations
1. (5x4 + 7x3 ) + (8x4 + 8x3)
2. (4x4 + x3) + (– 7x4 + 5x3)
3. (2x3 + 5) + (-6x3 + 1)
4. (6x4 + 2x2 + 2) + (8x4 + x2)
5. (6x4 + 7x + 6) + (7x4 + 3x)
6. (8x4 – 5x2 + 5) + (3x2 -1)
7. (7x + 6) – (2x - 3)
8. (4x – 7) - (7x + 7)
9. (3 + 2x2) – (– 3+ 4x2)
10. (6x4 + x3 + x) – ( 5x4 + 6x3)
11. (8x4 – 6x3 + 7x) – (-x4 + 6x)
12. (-6x2 – 5x + 3) – (-x2 + 6x)
A. 14x4 + 3x2 + 2
B. -3x -14
C. 13x4 + 10x +6
E. 9x4 – 6x3 + x
H. 5x + 9
I. x4 - 5x3 + x
N. -3x4 + 6x3
O. 13X4 + 15X3
P. 6 – 2x2
R. -5x2 + 11x + 3
S. -4x3 + 6
U. 8x4 – 2x2 + 4
Answer Key
What I Know
Perform the indicated operations using the algebra tiles
6. 3x – 2x
1. 3x + 2x
+
+
+
+
Answer 5x
+
+
+
=x
+
zero pair
2. (-5x -3) + (3x -2)
-
-
-
-
-
-
7. (5x -3) - (3x + 2)
+
+
+
+
+
+
3. (4x + 1) + (-x +2)
+
+
+
4. (x – 4x +1) + (x + 3)
-
+
-
-
+
+
Answer = x – 3x + 4
+
+
-
-
-
-
Answer = 2x2 + 2x + 3
-
10. (2x2 – 3x + 3) - (-x2 + 2x -1)
+
+
+
+
+
+ +
+
+
Answer = x – 5x + 4
+
+
+
+
2
5. (3x2 – 2x + 4) + (-x2 + 4x -1)
+
+
+
9. (x2 – 4x + 1) - (x - 3)
2
+
+
Answer = 5x -1
2
-
+
+
Answer = 3x + 3
+
+
8. (4x + 1) - (-x +2)
+
+ +
+
Answer: 2x - 5
Answer -2x - 5
+
-
+
7
+
+
Answer = 3x2 -5x + 4
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+
+
+
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What’s More
What’s In
1. 13x
2. -6y
3. -2x3
4. -9x3
5. -x
6. 2xy
7. 50x2y2
8. 0
9. -20x2y3
10. 0
1. -y -2
2. -4x2 + x
3. 4m2 +3m -2
4. 14n3 + 4n - 4
5. 11b2 – b - 8
What I Can Do
1. 21x2 + 9x – 88
2. 3x3 + 9x2 + 7x – 14
3. (6x2 + 10x + 2) cm
4. ₱ (20x3 + 2x2 - 5x - 6)
5. 2x – 13
Assessment
Error Analysis. Study how Lucas works in adding or subtracting polynomials. Check
whether his answer is correct or not. If he is correct just write Correct. Write incorrect if it is
wrong then write the correct answer.
Lucas Works
1. (3x + 2) + (4x +3)
=(3x + 4x) +(2 + 3)
=7x + 5
2. (-5x + 3y) + (3x +2y)
=(-5x + 3x) + (3y + 2y)
= 8x + 5y
3. (4x -3) – (2x + 1)
=(4x -2x) + (-3 – 1)
=2x + (-4)
=2x – 4
4. (3a + 2b) – (a -b)
= (3a -a) + (2b -b)
=2a + b
5. (5x2 + 3x +3) + (3x2 -2x -7)
=(5x2 +3x2) + (3x -2x) + (3 -7)
= 8x2 + x + 4
Correct or Incorrect?
If incorrect, explain and find the correct answer
Correct
Incorrect.
-5x + 3x should be -2x. Expressions are
unlike signs so subtract them and copy the
sign of the bigger number. So the answer
should be -2x + 5y
Correct.
Incorrect.
The statement should = (3a -a) + (2b + b) on
the second mathematical statement. Adding
them will result to 2a + 3b.
Incorrect.
3 - 7 should be -4, so the final answer will be
8x2 + x - 4.
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Additional activities
RIDDLE TIME!
Why did the school cook become a history teacher?
C
E
B
8
11
5
A
S
U
4
6
3
E
4
4
S
11
3
X
11
2
E
P
9
N
C
I
E
N
2
5
10
11
2
11
E
H
7
R
11
O
T
12
1
G
T
W
E
A
S
4
3
N
2
R
E
A
S
E
12
11
4
3
11
References
Lambert G. Quesada and Catherine P. Vistro-Yu, Ed.D. Mathematics – Grade 7 Learner’s
Material First Edition, 2014, Lesson 22: Addition and Subtraction of Polynomials, 130-133
Oronce Orlando A, and Mendoza Marilyn O., E-Math Worktext in Mathematics 7 (Rex Book
Store : 856 Nicanor Reyes St., Sampaloc, Manila, Third Edition 2012), 169 – 173
pinterest.ca, Riddle Time, accessed September 11, 2020
https://www.pinterest.ca/pin/477592735468922727/
pngwave.com, Animated film Storyboard Television, accessed September 08, 2020
https://www.pngwave.com/png-clip-art-fxkez
Prepared by:
JENNY L. CAPITULO
Master Teacher I, FGNMHS
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7
MATHEMATICS
MODULE
Quarter 2: Week 5-6
Module 5: Deriving the Laws of Exponents,
Multiplying and Dividing Polynomials
Module 6: Special Products
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MATHEMATICS 7
Quarter 2 – Module 5
Deriving the Laws of Exponents
Name
Week 5
Section
Date
Background Information
What’s New
I was the first to discuss zero, negative and fractional exponents in depth.
I am
!
12
19
27
56
120
8
25
25
38
Evaluate the following algebraic expressions to decode the name of the Mathematician.
Write your answer and solution on a clean sheet of paper.
GIVEN
SOLUTION
I
What is 47 – n if n = 9?
N
What is u + 48 if u = 8?
W
If s=12, what is 10s?
S
92 – c; when c=6
J
If b is 5, what is 60 ?
A
If q = 2, what is 4q?
Z
If c = 6, what is 60 ?
O
What is 26 – y if y= 7?
L
If t = 4, what is 100 ?
H
If z = 3, what is 30 – z?
𝑏
𝑐
𝑡
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What is It
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
The exponent says how many times a number is to
be multiplied.
A negative exponent is equal to its reciprocal.
A fractional exponent like 1 to take the nth root: 𝑥 𝑛=
𝑛
𝑛
√𝑥 .
And all the laws below are based on these ideas.
Laws of Exponents
Law
Explanation
𝑚
𝑛
To
multiply
when
the
bases are the same, write the
𝑥 ∙𝑥 =
base
and
add
the
exponents.
(Product Law)
𝑚+𝑛
𝑥
𝑥𝑚
To divide when the bases are the same, write the
base and subtract the exponents. (Quotient Law)
=
𝑥𝑛
𝑥𝑚−𝑛
(𝑥𝑚)n =
𝑥𝑚𝑛
(xy)n =
𝑥𝑛𝑦𝑛
𝑛
To raise a power to another power, write the base
and multiply the exponents. (Power Law)
𝑦
For any nonzero number x and y, and any number
n, just distribute their exponents to x and y.
(Product of Fraction Law)
For any nonzero number x raised to 1 is equal to
its number x.
𝑥0 = 1
1
For any nonzero number x raised to 0 is equal 1.
(Zero Exponent)
For any nonzero number x raised to -1 is equal to
1
. (Negative One Exponent)
𝑥1 = X
𝑥−1 =
𝑥−𝑛 =
𝑥
1
𝑥𝑛
𝑥 ∙ 𝑥 = 𝑥4+3 = 𝑥7
𝑋
𝑥8−2 = 𝑥6
(x4)3 = x4 ∙ 3 = x12
(xy)4 = 𝑥4𝑦4
( 𝑥 )3 = 𝑥
𝑦
81 = 8
3
𝑦3
60 = 1
5 -1
1
=5
1
𝑥−3 = 𝑥3
𝑥
For any nonzero number x raised to a negative
number, flip it and change the exponent to positive.
(Negative Exponent)
CAUTION: -x ≠ 1
3
𝑥8
=
𝑥2
For all numbers x and y and any integer n, just
distribute the exponents to x and y. (Power of
Product Law)
(𝑥)n = 𝑥 𝑛
𝑦
Example
4
-3 ≠
1
3
(-3)2 = (-3)(-3) =9
-32 = - 9
REMEMBER: An exponent applies only to the
factor directly next to it, unless parentheses
enclose other factors.
And the Law about fractional exponents will be discussed to next grade level.
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To simplify exponential expressions using the laws will mean writing the expression with
positive exponents only. Therefore, when algebraic exponents have zero or negative
exponents, they have to be rewritten in a way that all exponents must be positive.
Exponential expressions are said to be in simplest form if:
the exponents are positive;
there are no powers of powers;
each base appears only once; and
all fractions are in simplest form.
a-6 ∙ a-3 ∙ a5
Example 1:
= a-4
By the Product Law of Exponents
=
By the Negative Exponent Law
(a4b3c-5) (a-4b2c3)
= a0b5c-2
Example 2:
By the Zero and Negative Exponents Law
𝑐2
Example 3:
(a6b-3)-2
Example 4:
By the Product Law of Exponents
= a-4
By the Quotient Law of Exponents
=4
By the Negative Exponent Law
= a-12b6
By the Power of a Product Law
By the Negative Exponent Law
Example 5:
Example 6:
-2
(
(
)-3
=
By the Power of a Quotient Law
= 24 a18
= 16a18
By the Quotient Law of Exponents
By the definition of exponent
=(
=
=
)3
5
23
125
By the Negative Exponent Law
By the Product of Fraction Law
By the definition of exponent
What I need to know
Derives the laws of exponent. (M7ALIId-e-1).
General Instruction
Write your solutions and answers on a clean sheet of paper.
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What I Know
Write the following expression using exponents.
1. 8 ∙ 8 ∙ 8 ∙ 8
2. a ∙ a ∙ a ∙ a ∙ a ∙ a
3. (4c)(4c)(4c)(4c)
4. 7 ∙ m ∙ m ∙ m ∙ m
5. (2ef)(2ef)(2ef)(2ef)
=
=
=
=
=
6. (6xy)(6xy)(6xy)
=
7. 8 (ab)(ab)(ab)
=
8. (y + z)(y + z)(y + z)
=
9. 11 (2d + e)(2d + e)(2d + e) =
10. (4h - 5i)(4h - 5i)
=
What’s In
Simplify the following algebraic expressions.
A. Zero and Negative Exponents
1. (a2b3)0
=
2 3 -1
2. (a b )
=
3 2 4 0
3. (2p q r )
=
4. (x2y4)-2
=
5. 8a0
=
B. Product Law
1. 52 ∙ 52
=
8
2. a ∙ a
=
7
8
3. c ∙ c
=
4. t3 ∙ t10
=
5. s7∙ s8∙ s9
=
C. Quotient Law
D. Power Law
1. 75 ÷ 73
2. 106 ÷ 103
3. a10 ÷ a4
=
=
=
1. (w2)5
2. (e6)3
3. (a20)4
=
=
=
=
4. (3m5)2
=
=
5. (5a2)3
=
4.
5.
𝑠5𝑡 4
𝑠𝑡
24 𝑑4
8𝑑2
E. Power Product Law
F. Fraction Law
1. (yz)3
=
1. ( 2 )2
2. (pqr)5
=
3. (abcd)8
2. ( 𝑎)3
=
4. (mnop)7
=
=
3
=
𝑛
3. (𝑎𝑏 )5
𝑐𝑑
=
)3
=
5. ( ℎ )3
=
4. (
2
𝑐4
4
5. (2xyz)3
=
𝑖5
What’s More
Insert >, < or = to make each statement true. Write your answer on a clean sheet of paper.
5 -1
)
6
1. 50
-50
6. (
2. -42
3. 7-2
(-4)2
(-7)2
7. (8 + 2)2
8. 112
4. (b + 10)0
(b + 9)0
9. 92
5. 70 + 80 + 90
(7 + 8 + 9)0
10. 6(7)0
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(
6
)
5
82 + 22
11-2
1
9−2
(6 ∙ 7)0
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What I Can Do
A. Puzzle: Evaluate the exponential expressions. Write the word corresponding to the answer in the boxes
below. What message is formed?
b4 ∙ b3
AND
BLESSINGS
(b5)4
COUNT
SHARE
YOUR
(
b20
b7
)
b8
b14
b9
B. Real World Connections (with Values Integration)
It is good to invest your money. A simple way of investing is saving. Explain the quotation
“Save for rainy days.”
Assessment ACTIVITY 1
Perform the indicated operations by applying the laws of exponents, then simplify using positive
exponents. Assume that no denominator is zero. Write your answer on a clean sheet of paper.
15𝑎2𝑏3
5
-10
6.
1. x ∙ x
=
=
5𝑎5 𝑏7
2. a5 ∙ a3 ∙ a-6
3. (4x3) (5x-6)
7. (a2)-4 ∙ (a3)4 =
8. (-5x3) (3x-4) =
𝑥−4 . 𝑥−2 0
9. (
) =
𝑥6
−2𝑎3𝑏2𝑐0
=
=
4. (-7x2) (-5x-3) (-6x-6) =
5.
24𝑥4𝑦5𝑧6
10. (
=
8𝑥 6𝑦 5𝑧 4
3𝑎2 𝑏3𝑐 7
)-2
Additional activities
A. Show the difference among the following expressions:
1. k4 ∙ k4
2. k4 + k4
3. (k4)4
B. Show that each statement is true.
1.
f6 f5 f4 f3
+ + +
+
f5 f4 f3 f2
2. (c4)2 + c3 ∙ c5 +
𝑐12
f2
+ f = 6f
f
= 3c8
7
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=
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Answer Key
What’s New
J
I am
12
O
_H
19
27
N
56
W
A
L
120
8
25
L
25
I
S
!
38
86
What I know
1. 84
6. (6xy)3
2. a6
7. 8(ab)3
3. (4c)4
8. (y + z)3
4. 7m4
9. 11(2d + e)3
5. (2ef)4
10. (4h - 5i)2
What’s In
A. Zero and Negative Exponents
1.) 1
2.)
3.)1
4.)
2.) a9
3.) c15
4.) t13
5.) s24
2.) 103 or 1000
3.) a6
4.) s4t3
5.) 3d2
2.) e18
3. a80
4.) 32m10 or 9 m10
2.) p5q5r5
3.) a8b8c8d8
4.) m7n7o7p7
1
𝑎2𝑏3
1
5.) 8
𝑥4 𝑦8
B. Product Law
1.) 54 or 625
C. Quotient Law
1.) 72 or 49
D. Power Law
1.) w10
5.) 53a6 or
125 a6
E. Power Product Law
1.) y3z3
5.) 23x3y3z3
or 8 x3y3z3
F. Fraction Law
2
1.) 2 or 4
32
1. >
6. =
9
3
2.)
𝑎
3.)
𝑛3
What’s More
2. <
7. >
𝑎5𝑏5
4.) 2 or 8
𝑐5𝑑5
𝑐12
3. <
8. >
3
4. =
9. =
12
5.)
𝑐12
ℎ
𝑖15
5. >
10. >
What Can I Do
A. Puzzle
b20
COUNT
b7
AND
b8
SHARE
b14
YOUR
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b9
BLESSINGS
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B. Real World Connections (with Values Integration)
Answers may vary
Assessment
1.
2. a2
1
𝑥5
6.
3. 203
7. a4
3
𝑎3𝑏4
8.
Additional Activities
1. k4 ∙ k4 = k8
A.
4.
𝑥
−15
−210
𝑥7
9. 1
𝑥
2. k4 + k4 = 2k4
5.
3𝑧2
𝑥2
10.
9𝑏 2 𝑐14
4𝑎2
3. (k4)4 = k16
B.
6
5
4
3
2
1. f + f + f + f + f + f = 6f
f5
f4
f3
f2
f
Solution: f6 + f6 + f6 + f6 + f6 + f6 =
= 6f
f5
12
2. (c ) + c ∙ c + 𝑐 4 = 3c8
4 2
3
5
𝑐
Solution: c8 + c8 + c8 = 3c8
References:
De Leon Cecille, Bernabe Julieta. Elementary Algebra Textbook for First Year. (Quezon City: JTW
Corporation, 2002) pp. 85-86
Soledad Jose-Dilao, Bernabe Julieta. Intermediate Algebra Textbook for Second Year. (Quezon City: SD
Publications, Inc., 2009) pp. 120-133
Sundiam,Lutgarda S. Elementary Algebra A Simplified Approach. (Quezon City: New Horizon
Publications, 2010) pp.110-114
Sundiam, Lutgarda S. Intermediate Algebra Intermediate Algebra. (Quezon City: New Horizon
Publications, 2002) pp.27-40
Zara Ivy, Sanchez Priscilla. Worktext in Intermediate Algebra 2 nd Year RBEC Based. (Lipa City:
EFERZA Academic Publications, 2007) pp.59-62
“John Wallis” Accessed on Sept 1, 2020.
https://www.google.com/search?q=john+wallis&tbm=isch&source=iu&ictx=1&fir=rLa8QQdBtyihnM%2
52CIzsdfzXTGhcQgM%252C%252Fm%252F01jp_7&vet=1&usg=AI4_-kSa6mkTgnSce64oOtaMfLjWrrGqQ&sa=X&ved=2ahUKEwiK6p3V8t3rAhWhIaYKHdLnCGsQ_B16BAgMEAM
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MATHEMATICS 7
Quarter 2 – Module 5:
Multiplication and Division of Polynomials
Name
Section
Week 5
Date
_
Background Information
What’s New
You can visualize 10 ∙ 12 by thinking of the surface area of a 10 ft. by 12 ft room. You can use
algebra tiles to visualize a product involving a variable.
1- tile
x2 – tile
x - tile
-
-
+
+
+
Dimensions
1 by 1
1 by x
x by x
Area
1∙1
1∙x=x
x ∙ x = x2
To write an expression modeled by the algebra tiles.
+
+
+
+
+
+
3x
find the lengths of the sides of the rectangle;
+
one side is 3x and the other side is 2x;
+ +
2x
this model shows the product (3x)(2x);
+
there are 6 x2- tiles; and
+
+
the tiles model: (3x)(2x) = 6x2
Model each product using algebra tiles. Write your answer on a clean sheet of paper.
a. (3x)(4x)
b. (4x)(2x)
c. (2x)(5x)
What is It
Now, find the following products and use the tiles whenever applicable:
1) (3x) (x)
2) (-x)(1+ x)
3) (3 - x)(x + 2)
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Examples
Illustrations
Show the tiles to know the product.
Example 1
Recall the
Laws of
Exponents.
By the Laws
of Exponents,
(3x) (x) =
3x2.
x
x
x2
So, 3x2 is represented by three of the big shaded squares.
x2
x2
x2
Example 2
(-x)(1+ x)
The product (-x)(1+ x) can be represented by the following.
-x
x
1
-x2
-x
The picture shows that the product is (-x2) + (-x).
Can you explain what happened?
Recall the sign rules for multiplying.
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Example 3
(3 - x)(x + 2)
x
1
1
-x2
-x
-x
x
1
1
x
1
1
x
1
1
-x
(-x2)+ (-2x) +3x + 6 = (-x2) + x + 6
Rules in Multiplying Polynomials
A. To multiply a monomial by another monomial, simply multiply the numerical coefficients then multiply the
literal coefficients by applying the basic laws of exponent.
Examples:
1) (a3)(a5) = a3+5 = a8
2) (3a2)(-5a10) = -15a12
3) (-8a2b3)(-9ab8) = 72a3b11
B. To multiply monomial by a polynomial, simply apply the distributive property and follow the rule in
multiplying monomial by a monomial.
Examples:
1) 3a
(a2 – 5a + 7) = 3a3 – 15a2 + 21a
2) -5a2b3 ( 2a2b – 3a + 4b5) = -10a4b4 + 15a3b3 – 20a2b8
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C. To multiply binomial by another binomial, simply distribute the first term of the first binomial to each term
of the other binomial then distribute the second term to each term of the other binomial and simplify the
results by combining similar terms. This procedure is also known as the F-O-I-L method or Smile method.
Another way is the vertical way of multiplying which is the conventional one.
Examples:
1) (a + 3)(a + 5) = a2 + 8a + 15
First terms Last terms
F – first terms
( a + 3)( a + 5)
(a)(a) = a2
O – outer terms (5)(a) = 5a
Since 5a and 3a are
I – inner terms (3)(a) = 3a
similar terms, we can
L – last terms (3)(5) = 15
combine them. 5a + 3a=8
So, the final answer is a2 + 8a + 15.
Inner terms
Outer terms
2) (x - 5)(x + 5) = x2 + 5x – 5x – 25 = x2 – 25
3) (x + 6)2 = (x + 6)(x + 6) = x2 + 6x + 6x + 36 = x2 + 12x + 36
4) (2x + 3y)(3x – 2y) = 6x2 – 4xy + 9xy – 6y2 = 6x2 + 5xy – 6y2
5) (3a – 5b)(4a + 7) = 12a2 + 21a – 20ab – 35b (There are no similar terms so it is in simplest
form)
Another Way of Multiplying Polynomials
1. Consider this example.
x
86
75
430
602
6450
2) Now, consider this.
This procedure also
applies the distributive
property
2a + 3
a-7
14a - 21
2a2 + 3a
2a2 + 17a – 21
This one looks the same
as the first one.
(Align the same terms)
Consider the example below.
3a – 5b
12a2 – 20ab
4a + 7
21a – 35b
12a2 – 20ab + 21a – 35b
D. To multiply a polynomial with more than one term by a polynomial with three or more terms,
simply distribute the first term of the first polynomial to each term of the other polynomial. Repeat
the procedure up to the last term and simplify the results by combining similar terms.
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Examples:
1) (a + 3)(a2 – 2a + 3) = a(a2 – 2a + 3) + 3(a2 – 2a + 3)
= a3 – 2a2 + 3a + 3a2 - 6a + 9
= a3 + a2 - 3a + 9
2) (a2 + 3a – 4)(4a3 + 5a – 1) = a2(4a3 + 5a – 1) + 3a(4a3 + 5a – 1) - 4(4a3 + 5a – 1)
= 4a5 + 5a3 – a2 + 12a4 + 15a2 – 3a – 16a3 – 20a + 4
= 4a5 + 12a4 – 11a3 + 14a2 – 23a + 4
3) (2a – 3)(3a + 2)(a2 – 2a – 1) = (6a2 – 5a – 6)(a2 – 2a – 1)
= 6a4 – 17a3 – 2a2 + 7a + 6
*Do the distribution one by one
In dividing polynomials, we can also use tiles. Recall also that division is the reverse
operation of multiplication. Let’s see if you can work out this problem using Tiles:
(x 2 + 7x + 6)  (x +1)
x
1
1
1
1
1
x
x2
x
x
x
x
x
x
1
x
1
1
1
1
1
1
1
The answer is x + 6.
Rules in Dividing Polynomials
A. To divide polynomial by a monomial, simply divide each term of the polynomial by the given
divisor.
Examples:
1.) Divide 12x4 – 16x3 + 8x2 by 4x2
a.
=
12𝑥4− 16𝑥3+ 8𝑥2
4𝑥 2
12𝑥4
16𝑥3
2
4𝑥
4𝑥 2
= 3x2 - 4x + 2
8𝑥2
+ 4𝑥2
b.
-16x3
(-)-16x3
8x2
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2. Divide 15x4y3 + 25x3y3 – 20x2y4 by -5x2y3
=
= -3x2 – 5x + 4y
2. To divide polynomial by a polynomial with more than one term (by long division), simply follow
the procedure in dividing numbers by long division.
These are some suggested steps to follow:
a. Check the dividend and the divisor if it is in standard form.
b. Set-up the long division by writing the division symbol where the divisor
is outside the division symbol and the dividend inside it.
c. You may now start the Division, Multiplication, Subtraction and Bring
Down cycle.
d. You can stop the cycle when:
i. the quotient (answer) has reached the constant term.
ii. the exponent of the divisor is greater than the exponent of the dividend
Examples:
1. Divide x2 – 3x – 10 by x + 2
-5x - 10
-5x - 10
0
1) divide x2 by x and put the result on top
2) multiply that result to x + 2
3) subtract the product to the dividend
4) bring down the remaining term/s
5) repeat the procedure from 1.
2. Divide x3 - 6x2 + 11x - 6 by x – 3
-3x2 + 11x
-3x2 + 9x
2x – 6
2x – 6
0
3. Divide 2x3 – 3x2 – 10x – 6 by 2x + 1
-4x2 – 10x
-4x2 – 2x
- 8x - 6
- 8x - 4
-2
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remainder over the
divisor
lOMoARcPSD|23567711
What I Need to Know
Multiplies and divides polynomial (M7ALIIe-2)
General Instruction
Write your solutions and answers on a clean sheet of paper.
What I Know
Write your answer on a clean sheet of paper.
NUMBER PUZZZLE
1
2
3
4
5
6
7
8
9
10
11
12
ACROSS
DOWN
1. 2 ∙ 42 – 75 ÷ 5
2. 2 – 15 ÷ 5 + 23 ∙ 2
3. (32 + 11) ∙ 2 – 28
5. -32 ∙ (18 – 15) + 51
6. 39 – 3 ∙ 122 ÷ 24
7. 82 ∙ (10 – 11) + 79
8. 330 – 2 ∙ (53 + 25)
9. (90 – 5) ÷ 5 + 22 ∙ 2
11. 2 ∙ 131 + 35 ÷ 5
12. 1 – 15 ÷ 5 + 23 ∙ 2
1. 10 - 25 ÷ 5 + 10
2. 42 – 3 ∙ (23 + 2)
3. 72 ∙ 2 – 588 ÷ 7
4. 41 – 2 ∙ 52 + 20
5. 150 ÷ (53 – 75) + 22
6. 16 + 64 ÷ 25 ∙ 2
7. (22 + 16) ∙ 2 – 21
8. 47 – 3 ∙ 43 ÷ 16
9. 22 ∙ (47 – 33) – 57
10. 4 ∙ (22 + 26) ÷ 5
What’s In
What Did the Girl Mushroom Say About the Boy Mushroom After Their First Date?
Direction: For each exercise below, multiply the polynomial by monomial. Find your answer in
the set of answers under the exercise and notice the letter next to it. Write this letter in the box
that contains the number of that exercise. (Write your answer on a sheet of paper.)
1. 5(2a2 + a)
6. 4a(a2 – 2a + 3)
11. a2b(2a2 – 4ab + b2)
2. 3a(8a2 – 2a)
7. -2a2(9 – a – 4a2
12. -2ab2(2a4 – 5a2b2 – 3b4 )
3. a2(4a – 3)
8. a2b(a2 – b2)
13. 4a3b(-a2b + 2ab – 5ab2)
4. -2a(4 + 5a3)
9. -3ab2(a3b2 - 2a2b)
14. -a2b3(7ab3 – a2b2 + 3a3b)
5. -6a2(4a2 – 9)
10. 2ab(a2 + 4ab – 3b2)
15. 3a2b2(2a4b2 – 3a2b – 1)
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Answers
Answers
Answers
B -24a4 – 54a
M 4a3 – 8a2 + 10
N -4a5b2 + 10a3b4 + 6ab6
T 24a3 – 4a
H -18a2 + 2a3 + 8a4
S 2a4b – 4a2b3 + a2b4
R -24a4 + 54a2
E 2a3b + 8a2b2 - 6ab3
E -4a5b2 + 8a4b2 – 20a4b3
U 4a3 – 3n2
I 2a3b + 8ab2 – 4ab
U -4a5b2 + 10a2b4 – 20a2b3
S 10a2 + 5a
A a4b – a2b3
Y 2a4b – 4a3b2 + a2b3
L 24a3 – 6a2
G 4a3 – 8a2 + 12a
F 6a6b4 – 9a4b3 – 3a2b2
O -8a – 6a3
W -18a2 + 2a3 + 6a5
T -7a3b6 + a5b4 – 3a3b4
A -8a – 10a4
L -3a4b4 + 6a3b3
I -7a3b6 + a4b5 – 3a5b4
7
10
1
5
13
4
9
2
11
8
15
3
12
6
14
What’s More
Why Is a Stick of Gum Like a Sneeze?
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12a3 + 14a2 – 4a + 3
4a3 – 30a2 +21a+10
a3 + 7a2 + 7a - 6
12a3 – 10a2 + a - 12
12a3 – 9a2 – 2a - 12
a3 + 6a2 + 9a – 6
6a3 + 10a2 + 8a - 4
4a3 – 33a2 + 27a+10
(a + 2)(a2 + 5a – 3)
(3a – 1)(2a2 + 4a + 4)
(2a + 3)(6a2 – 2a + 1)
(4a – 5)(a2 – 7a – 2)
(3a – 4)(4a2 + 2a + 3)
(a + 8)(6a2 – a – 4)
C R I H E A N W D
6a3 + 44a2 - 9a – 32
13
14
15
16
17
18
T
6a3+ 47a2 – 12a - 32
12a2 – 29a + 14
5a2 – 17ab – 12b2
4a2 + 4ab + 3b2
4a2 - 25
12a2 + 22a - 4
5a2 – 11ab – 12b2
(4a – 7)(3a – 2)
(2a + 5)(2a – 5)
(6a – 1)(2a + 4)
(a + 2b)(4a + b)
(5a + 3b)(a – 4b)
(3a – 8b)(2a – b)
I S E R A N O
6a2 – 19ab + 8b2
7
8
9
10
11
12
T
4a2 + 9ab + 2b2
2a2 + 5a - 18
a2 – 13a + 18
a2 + 11a + 18
a2 – 9a + 18
a2 + 8a + 15
6a2 + 7a + 4
(a + 3)(a + 5)
(a + 2)(a + 9)
(a – 8)(a + 1)
(a – 3)(a – 6)
(2a + 9)(a – 2)
(3a + 1)(2a + 4)
E S I A U T N
6a2 + 14a + 4
1
2
3
4
5
6
B
a2 – 7a – 8
Direction: For each exercise, multiply the two polynomials. Find your answer in the set of
answers under the exercise. Cross out the letter above your answer. When you finish, the
answer to the title question will remain!
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What I Can Do
What Did the Carpenters Call Their Bass Quartet?
Direction: Simplify the expression. Assume that no divisor equals zero. Find your answer in
the set of answers under the exercise and cross out the box above it. When you finish, the
answer to the title question will remain.
Assessment ACTIVITY 2
A. Multiply each expression. Using FOIL Method or vertical form.
1. 5x( x + 3)
2. -4x(x2 – 5x + 10)
3. 6x2(2x + 8)
4. (a + 8)(a + 4)
5. (2a – 7)(a – 8)
=
=
=
=
=
B. Divide and write your answer as a polynomial or mixed expression.
6.
7.
8.
9.
4𝑥−8
2
−10𝑥+ 5𝑥2
−5𝑥
𝑥2 +8𝑥 +15
𝑥+5
𝑥2−3𝑥−54
𝑥+6
=
=
=
=
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-a2 + 3ab – 8b2
5a3b + a2b3 -4ab2
5a3b + a2b2 -2ab3
U
3b3 – 3ab – 8b2
15
2a2 + a - 6
14
-a2 - 4ab + 1
13
3b3 + 2ab +5a3
12
8𝑎3 + 4𝑎2 − 24𝑎
4𝑎
21𝑎𝑏3 + 14𝑎2𝑏 + 35𝑎4
7𝑎
3
2𝑎 𝑏 − 6𝑎2𝑏2 + 16𝑎𝑏3
−2𝑎𝑏
45𝑎2𝑏4 − 60𝑎3𝑏2 − 15𝑎2𝑏
15𝑎2𝑏
5
15𝑎 + 3𝑎4𝑏5 − 6𝑎3𝑏6
3𝑎2𝑏3
R O B E A S 4 N
3b3 –4ab - 1
N
11
2a2 – a - 1
2
-5b3 + 1
H
4c2 – 7bc2
S
9a2 – 25
𝑎2
3a - 25
O
10
4a2 + 3a
T
2𝑎3 − 7𝑎2
2a + 3
4
4a2 + 5
2a - 7
A
3a + 5
5
2bc – c4
9
𝑥2
4c3 + 2c
4
8
b+c
3
7
12𝑐2 − 27𝑐4
3𝑐2
30𝑏4 − 6𝑏
−6𝑏
2
𝑏 𝑐 + 𝑏𝑐2
𝑏𝑐
8𝑏𝑐4 − 14𝑏2𝑐3
2𝑎𝑏
3
−10𝑏 𝑐2 + 5𝑏2𝑐5
−5𝑏2𝑐
E B 8 T A L
4c3 – 9c2
2
6
2bc – 5c
6𝑎 + 9
3
18𝑎2 − 50
2
2
12𝑎 + 20𝑎
4𝑎
20𝑎3 + 5𝑎2
1
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10.
2𝑏 4− 𝑏3−2𝑏+1
𝑏3−1
=
Additional Activities
Supply the missing term in Column B that will make the division procedure in Column A
correct. Write the letter of your answer on a clean sheet of paper.
Column
A
5x2 + ( 3 )
)(1 ) – 14x2
x– (2)
-(6)
– 26x
5x – 20x
6x2 – 26x
6x2 - ( 4 )
- 2x
(7)
3
Column B
1.
2.
3.
4.
5.
6.
7.
8.
+ 12
2
+ (5)
+ (8)
4
a. 24x
b. -2x
c. 5x3
d. 8
e. 4
f. 12
g. 6x
h. 2
Answer Key
What’s New
a. 12x2
b. 8x2
What I know
c. 10x2
NUMBER PUZZLE
1
2
7
1
1
5
3
5
4
2
1
5
1
6
4
2
1
2
7
8
5
1
0
3
9
9
10
5
2
11
2
12
3
3
1
4
What’s In
7
10
1
5
13
4
9
2
11
8
15
3
12
6
14
H
E
S
R
E
A
L
L
Y
A
F
U
N
G
I
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-5b3 + 1
2a2 – a - 1
A
12a3 + 14a2 – 4a + 3
2bc – c4
E
4a3 – 30a2 +21a+10
4c3 + 2c
B
a3 + 7a2 + 7a - 6
b+c
12a3 – 10a2 + n - 12
4c3 – 9c2
12a3 – 9a2 – 2a - 12
U
3b3 – 3ab – 8b2
L
a3 + 6a2 + 9a – 6
A
2a2 + a - 6
T
H E A N W D
6a3 + 10a2 + 8a - 4
4a2 - 25
8
4a3 – 33a2 + 27a+10
12a2 + 22a - 4
B
6a3 + 44a2 - 9a – 32
5a2 – 11ab – 12b2
6a3+ 47a2 – 12a - 32
6a2 – 19ab + 8b2
T C R I
12a2 – 29a + 14
E R A N O
5a2 – 17ab – 12b2
S
4a2 + 4ab + 3b2
I
4a2 + 9ab + 2b2
a2 – 13a + 18
T
S
H
2
N
E
4a2 + 5
2a + 3
4a2 + 3a
3a - 25
9a – 25
4c2 – 7bc2
2bc – 5c
R O
S
2 3
3
5a b + a b -2ab
2 2
3
-a2 - 4ab + 1
3b3 + 2ab +5a3
3b –4ab - 1
3
2
4
N
-a2 + 3ab – 8b2
O
5a b + a b -4ab
T
2
4
3
A
3a + 5
THE TUBA 4 (two-by-four)
2a - 7
What I Can Do
N
2a2 + 5a - 18
A U T
a2 + 11a + 18
I
a2 – 9a + 18
6a2 + 14a + 4
a2 + 8a + 15
a2 – 7a – 8
B E S
6a2 + 7a + 4
What’s More
Assessment
A.
1.) 5x2 + 15x
2.) -4x3 + 20x2 – 40x
B.
1.) 2x – 4
2.) 2 – x
Additional Activities
1. c
2. e
3. g
4. a
5. f
3.) 12x3 + 48x2 4.) a2 + 12a + 32
3.) x + 3
4.) x – 9
6. h
8. d
7. b
5.) 2a2 -23a + 56
10.) 2b – 1
References
Department of Education. Grade 7 Mathematics Teaching Guide, Lessons 23-24
De Leon Cecille, Bernabe Julieta. Elementary Algebra Textbook for First Year. (Quezon City: JTW
Corporation, 2002) pp. 104 – 115
Oronce Orlando, Mendoza Marilyn. E-math Work text in Mathematics 7. (Manila: Rex Book
Store, Inc., 2012) pp. 178 – 218
Sundiam,Lutgarda S. Elementary Algebra A Simplified Approach. (Quezon City: New Horizon
Publications, 2010) pp.115 – 124
Marcy Steve, Marcy Janis. Algebra with Pizzazz!. (Chicago: Creative Publications, 1996) pp. 68 – 77
Prepared by:
AILEEN O. LAXAMANA
Master Teacher I, FGNMHS
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MATHEMATICS 7
Quarter 2 – Module 6
Special Products
Name
Week 6
Section
Date
Background Information
What’s New
From the pool of answers below, find the product of the given polynomials:
1. (𝑥 + 2)(𝑥 + 5)
2. (𝑥 − 9)(𝑥 + 9)
3. (𝑥 − 4)2
4. (𝑥 + 3)3
5. (𝑥 + 2)(𝑥2 − 2𝑥 + 4)
What is It
What method did you use to answer the previous activity? Did you know that there are
other ways to get their products? In this lesson you will learn how to use models and other
algebraic methods in getting the product of certain polynomials, particularly, binomials and
trinomials.
Method 1: Models
1.A. Algebra Tiles
Algebra tiles are mathematical manipulatives that are useful in evaluating
algebraic expressions. These tiles are usually square and rectangle to represent
numbers or variables. To understand the discussion in the next part of this module, we
must remember what each tile represent.
- denotes −𝒙𝟐
-
- denotes −𝒙
- denotes 𝟏
- denotes −𝟏
1.B. Generic Rectangle
Generic rectangles allow you to multiply any kind of terms. They are generally
called generic because the size of the rectangles is not fixed. One can associate it as
Double Distributive Property or in some cases, similar to the well-known FOIL Method.
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Method 2: Algebraic Method
We can solve the product of certain binomials and trinomials by applying what you
have learned on the lesson about the Multiplication of Polynomials. However, in this
module, you will learn more about the FOIL Method and Special Products.
FOIL Method and Special Products
FOIL Method is the general form for ac + ad + bc + bd which is the product of two
binomials (a + b) (c + d). FOIL stands for First, Outer (or outside), Inner (or inside) and Last.
Special products, on the other hand, are distinctive cases of multiplying polynomials.
They are very useful in finding the products of certain polynomial factors. This module will
introduce the different kinds of special products and their examples.
A. Product of Two Binomials
Find the product of the following using (a) algebra tiles, if applicable, (b) generic
rectangle and (c) algebraic method:
1. (𝑥 + 3)(𝑥 + 4)
2. (2𝑥 + 1)(𝑥 − 5)
Solution
1. (𝑥 + 3)(𝑥 + 4)
a) using Algebra Tiles
𝒙
𝟏
𝟏
𝟏
(𝑥 + 4)
𝟏
= 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟐
b) using Generic Rectangles
Distribute each term in (𝒙 + 𝟑) to each term in (𝒙 + 𝟒). For instance,
𝒙 𝒕𝒊𝒎𝒆𝒔 𝒙 𝒊𝒔 𝒙𝟐, 𝒙 𝒕𝒊𝒎𝒆𝒔 𝟒 𝒊𝒔 𝟒𝒙, 𝟑 𝒕𝒊𝒎𝒆𝒔 𝒙 𝒊𝒔 𝟑𝒙 𝒂𝒏𝒅 𝟑 𝒕𝒊𝒎𝒆𝒔 𝟒 𝒊𝒔 𝟏𝟐.
𝒙
𝒙
𝟒
Therefore, the answer is 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟐.
Note that 𝟕𝒙 is the sum of the terms in
the encircled part of the rectangle.
𝟑
c) by Algebraic Method
To solve the product of (𝑥 + 3)(𝑥 + 4) algebraically, we can use the FOIL
Method.
F:
O:
I:
(𝑥)(𝑥) = 𝒙𝟐
(𝑥)(4) = 𝟒𝒙
(3)(𝑥) = 𝟑𝒙
L
:
(3)(4) = 𝟏𝟐
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2. (2𝑥 + 1)(𝑥 − 5)
a) using Algebra Tiles
(𝑥 − 5)
= 𝟐𝒙𝟐 − 𝟗𝒙 − 𝟓
b) using Generic Rectangles
Distribute each term in (𝟐𝒙 + 𝟏) to each term in (𝒙 − 𝟓).
𝟐𝒙
𝒙
−𝟓
Therefore, the answer is 𝟐𝒙𝟐 − 𝟗𝒙 − 𝟓.
Note that −𝟗𝒙 is the sum of the terms in
the encircled part of the rectangle.
𝟏
c) by Algebraic Method
Again, we will use the FOIL Method in solving the product of (2𝑥 + 1)(𝑥 − 5).
F:
O:
I:
L:
(2𝑥)(𝑥) = 𝟐𝒙𝟐
Thus, the answer is
(2𝑥)(−5) = −𝟏𝟎𝒙
(1)(𝑥) = 𝒙
−𝟗𝒙
(1)(−5) = −𝟓
𝟐𝒙𝟐 − 𝟗𝒙 − 𝟓
B. Product of The Sum and Difference of Two Terms
Find the product of the following using (a) algebra tiles, if applicable, (b) generic
rectangle and (c) algebraic method:
1. (𝑥 − 2)(𝑥 + 2)
2. (2𝑥 + 3)(2𝑥 − 3)
Solution
1. (𝑥 − 2)(𝑥 + 2)
a) using Algebra Tiles
(𝑥 + 2)
= 𝒙𝟐 − 𝟒
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b) using Generic Rectangles
Distribute each term in (𝒙 − 𝟐) to each term in (𝒙 + 𝟐).
𝒙
𝒙
𝟐
Therefore, the answer is 𝒙𝟐 − 𝟒.
Note that there is no linear term because
𝟐𝒙 + (−𝟐𝒙) = 𝟎.
−𝟐
c) by Algebraic Method
The product of the sum and difference of two terms is given by the
general formula: (𝒂 + 𝒃)(𝒂 − 𝒃) = 𝒂𝟐 − 𝒃𝟐
Applying the above concept, the product of (𝑥 − 2)(𝑥 + 2) is 𝒙𝟐 − 𝟐𝟐
which can be written as 𝒙𝟐 − 𝟒.
2. (2𝑥 + 3)(2𝑥 − 3)
a) using Algebra Tiles
𝒙𝟐
𝒙𝟐
𝒙𝟐
𝒙𝟐
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
−𝒙 −𝒙 −𝒙
−𝒙 −𝒙 −𝒙
−𝟏 −𝟏 −𝟏
−𝟏 −𝟏 −𝟏
(2𝑥 − 3)
= 𝟒𝒙𝟐 − 𝟗
−𝟏 −𝟏 −𝟏
b) using Generic Rectangles
Distribute each term in (𝟐𝒙 + 𝟑) to each term in (𝟐𝒙 − 𝟑).
𝟐𝒙
−𝟑
Therefore, the answer is 𝟒𝒙𝟐 − 𝟗.
𝟐𝒙
Note that there is no linear term because
(−𝟔𝒙) + 𝟔𝒙 = 𝟎.
3
c) by Algebraic Method
Applying the general formula: (𝒂 + 𝒃)(𝒂 − 𝒃) = 𝒂𝟐 − 𝒃𝟐 , the product of
(2𝑥 + 3)(2𝑥 − 3) is (𝟐𝒙)𝟐 − (𝟑)𝟐 which can be written as 4𝒙𝟐 − 𝟗.
C. Square of a Binomial
Find the product of the following using (a) algebra tiles, if applicable, (b) generic
rectangle and (c) algebraic method:
1. (𝑥 − 4)2
2. (3𝑥 + 5)2
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Solution
1. (𝑥 − 4)2
a) Using Algebra Tiles
(𝑥 − 4)
= 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟔
b) Using Generic Rectangles
Distribute each term in (𝒙 − 𝟒) to each term in (𝒙 − 𝟒).
𝒙
𝒙
−𝟒
Therefore, the answer is 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟔.
Note that −𝟖𝒙 is the sum of the terms in
the encircled part of the rectangle.
−𝟒
c) By Algebraic Method
The square of a binomial (𝒂 ± 𝒃)𝟐 is the product of a binomial when
multiplied to itself. It has a general formula, (𝒂 ± 𝒃)𝟐 = 𝒂𝟐 ± 𝟐𝒂𝒃 + 𝒃𝟐.
Applying the above concept, the product of (𝒙 − 𝟒)𝟐 is
𝒙𝟐 − 𝟐(𝒙)(𝟒) + 𝟒𝟐 which can further be simplified as 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟔.
2. (3𝑥 + 5)2
a) Using Algebra Tiles
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
𝒙𝟐
𝒙
𝒙
𝒙
𝒙
𝒙
𝟏
𝟏
𝒙𝟐
𝒙𝟐
𝒙𝟐
𝒙
𝟏
𝟏
𝒙𝟐
𝟏
𝟏
𝟏
𝒙𝟐
𝒙𝟐
𝟏
𝒙
𝒙𝟐
𝒙
𝒙
𝒙
𝒙𝟐
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
𝟏
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
𝟏
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
(3𝑥 + 5)
𝒙
𝒙
𝟏
𝟏
𝟏
𝟏
𝟏
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b) Using Generic Rectangles
Distribute each term in (𝟑𝒙 + 𝟓) to each term in (𝟑𝒙 + 𝟓).
𝟑𝒙
𝟑𝒙
𝟓
Therefore, the answer is 𝟗𝒙𝟐 + 𝟑𝟎𝒙 + 𝟐𝟓.
Note that 𝟑𝟎𝒙 is the sum of the terms in
the encircled part of the rectangle.
𝟓
c) By Algebraic Method
Applying the general formula: (𝒂 ± 𝒃)𝟐 = 𝒂𝟐 ± 𝟐𝒂𝒃 + 𝒃𝟐 , the product of
(3𝑥 + 5)2 is (𝟑𝒙)𝟐 + 𝟐(𝟑𝒙)(𝟓) + (𝟓)𝟐 which can be simplified as
𝟗𝒙𝟐 + 𝟑𝟎𝒙 + 𝟐𝟓.
D. Cube of a Binomial
Find the product of the following using (a) algebra tiles, if applicable, (b) generic
rectangle and (c) algebraic method:
1. (𝑥 + 6)3
2. (3𝑥 − 2)3
Solution
1. (𝑥 + 6)3
a) Using Algebra Tiles
Algebra tiles are seldom used in getting the product of the Cube of a Binomial
because it would be a tedious job. Thus, for practical reasons, we shall skip
this part.
b) Using Generic Rectangles
For this type of special products, generic rectangles will be used twice. To start
solving, let us consider this: (𝒙 + 𝟔)𝟑 = (𝒙 + 𝟔)(𝒙 + 𝟔)(𝒙 + 𝟔). Then, perform the
first generic rectangle using the first two factors, (𝒙 + 𝟔)(𝒙+ 𝟔).
𝒙
𝒙
𝟔
Thus, the answer is 𝒙𝟐 + 𝟏𝟐𝒙 +𝟑𝟔.
Note that 𝟏𝟐𝒙 is the sum of the terms in
the encircled part of the rectangle.
𝟔
Next, perform the second generic rectangle using the third factor of (𝒙 + 𝟔)𝟑
and the answer from the first rectangle, (𝒙 + 𝟔)(𝒙𝟐 + 𝟏𝟐𝒙 + 𝟑𝟔).
𝒙
𝒙𝟐
𝟏𝟐𝒙
𝟑𝟔
𝟔
Therefore, the answer is
𝒙𝟑 + 𝟏𝟖𝒙𝟐 + 𝟏𝟎𝟖𝒙 + 𝟐𝟏𝟔.
Note that 𝟏𝟖𝒙𝟐 and 𝟏𝟎𝟖𝒙 are the sum of
the terms in the encircled parts of the
rectangle, respectively.
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c) by Algebraic Method
The cube of a binomial has the general form,
(𝒂 ± 𝒃)𝟑 = 𝒂𝟑 ± 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 ± 𝒃𝟑
Applying the above concept,
(𝒙 + 𝟔)𝟑
=
𝒙𝟑 + 𝟑(𝒙)𝟐(𝟔) + 𝟑(𝒙)(𝟔)𝟐 + (𝟔)𝟑
=
𝒙𝟑 + 𝟑(𝒙)𝟐(𝟔) + 𝟑(𝒙)(𝟑𝟔) + 𝟐𝟏𝟔
=
𝒙𝟑 + 𝟏𝟖𝒙𝟐 + 𝟏𝟎𝟖𝒙 + 𝟐𝟏𝟔
Thus, the product of (𝑥 + 6)3 is 𝒙𝟑 + 𝟏𝟖 + 𝟏𝟎𝟖𝒙 + 𝟐𝟏𝟔.
2. (3𝑥 − 2)3
a) Using Algebra Tiles
We will skip this part for the same reason as in number 1.
b) using Generic Rectangles
Let us consider this: (𝟑𝒙 − 𝟐)𝟑 = (𝟑𝒙 − 𝟐)(𝟑𝒙 − 𝟐)(𝟑𝒙 − 𝟐). Then, perform the
first generic rectangle using the first two factors, (𝟑𝒙 − 𝟐)(𝟑𝒙 − 𝟐).
𝟑𝒙
𝟑𝒙
−𝟐
Thus, the answer is 𝟗𝒙𝟐 − 𝟏𝟐𝒙 + 𝟒.
Note that −𝟏𝟐𝒙 is the sum of the terms in
the encircled part of the rectangle.
−𝟐
Next, perform the second generic rectangle using the third factor of (𝟑𝒙 − 𝟐)𝟑
and the answer from the first rectangle, (𝟑𝒙 − 𝟐)(𝟗𝒙𝟐 − 𝟏𝟐𝒙 + 𝟒).
𝟑𝒙
𝟗𝒙𝟐
−𝟏𝟐𝒙
𝟒
−𝟐
Therefore, the answer is
𝟐𝟕𝒙𝟑 − 𝟓𝟒𝒙𝟐 + 𝟑𝟔𝒙 − 𝟖.
Note that −𝟓𝟒𝒙𝟐 and 𝟑𝟔𝒙 are the sum of
the terms in the encircled parts of the
rectangle, respectively.
c) by Algebraic Method
Applying the general form of the cube of a binomial,
(𝟑𝒙 − 𝟐)𝟑
=
(𝟑𝒙)𝟑 − 𝟑(𝟑𝒙)𝟐(𝟐) + 𝟑(𝟑𝒙)(𝟐)𝟐 − (𝟐)𝟑
=
𝟐𝟕𝒙𝟑 − 𝟑(𝟗𝒙𝟐)(𝟐) + 𝟑(𝟑𝒙)(𝟒) − 𝟖
=
𝟐𝟕𝒙𝟑 − 𝟓𝟒𝒙𝟐 + 𝟑𝟔𝒙 − 𝟖
Thus, the product of (3𝑥 − 2)3 is 𝟐𝟕𝒙𝟑 − 𝟓𝟒𝒙𝟐 + 𝟑𝟔𝒙 − 𝟖.
E. Product of a Binomial and a Trinomial
Find the product of the following using (a) algebra tiles, if applicable, (b) generic
rectangle and (c) algebraic method:
1. (𝑥 − 10)(𝑥2 + 10𝑥 + 100)
2. (2𝑥 + 5)(4𝑥2 − 10𝑥 + 25)
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Solution
1. (𝑥 − 10)(𝑥2 + 10𝑥 + 100)
a) Using Algebra Tiles
We will skip this part for the same reason as in the Cube of a Binomial.
b) Using Generic Rectangles
Solve the given directly into the generic rectangle.
𝒙
𝒙𝟐
𝟏𝟎𝒙
𝟏𝟎
𝟎
−𝟏𝟎
Therefore, the answer is 𝒙𝟑 − 𝟏𝟎𝟎𝟎 which
is equivalent to 𝒙𝟑 − 𝟏𝟎𝟑.
Note that there are no quadratic and linear
terms because 𝟏𝟎𝒙𝟐 + (−𝟏𝟎𝒙𝟐) = 𝟎 and
𝟏𝟎𝟎𝒙 + (−𝟏𝟎𝟎𝒙) = 𝟎.
c) By Algebraic Method
The product of a binomial and a trinomial can be expressed as the sum or
difference of two cubes if they are in the following form.
(𝒂 + 𝒃)(𝒂𝟐 − 𝒂𝒃 + 𝒃𝟐) = 𝒂𝟑 + 𝒃𝟑 (𝒂
− 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐) = 𝒂𝟑− 𝒃𝟑
Applying the above concept,
(𝑥 − 10)(𝑥 2 + 10𝑥 + 100) = 𝒙 𝟑 − 𝟏𝟎𝟑
= 𝒙𝟑 − 𝟏𝟎𝟎𝟎
You might be wondering how do we know that the answer to the product of a
binomial and a trinomial is either the sum or the difference of two cubes. What
you need to do is to observe the trinomial and check the following:
(1) its first and third terms are the squares of the two terms in the binomial;
(2) its second term is the product of the two terms in the binomial;
(3) the sign of its second term is opposite the sign of the second term in the
binomial.
2. (2𝑥 + 5)(4𝑥2 − 10𝑥 + 25)
a) Using Algebra Tiles
We will skip this part for the same reason as in the previous item.
b) Using Generic Rectangles
Solve the given directly into the generic rectangle.
𝟐𝒙
𝟒𝒙𝟐
−𝟏𝟎𝒙
𝟓
𝟐
𝟓
c) by Algebraic Method
Therefore, the answer is 𝟖𝒙𝟑 + 𝟏𝟐𝟓 which
is equivalent to (𝟐𝒙)𝟑 + 𝟓𝟑.
Note that there are no quadratic and linear
terms because (−𝟐𝟎𝒙𝟐) + 𝟐𝟎𝒙𝟐 = 𝟎 and
𝟓𝟎𝒙 + (−𝟓𝟎𝒙) = 𝟎.
Checking the validity of the given and applying the general form for the
product of a binomial and a trinomial,
(2𝑥 + 5)(4𝑥2 − 10𝑥 + 25)
27
= 𝟖𝒙𝟑 + 𝟏𝟐𝟓
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What I need to know
Uses models and algebraic methods to find the: (a) product of two binomials; (b) product of
the sum and difference of two terms; (c) square of a binomial; (d) cube of a binomial; (e)
product of a binomial and a trinomial.
(M7AL-IIe-g-1)
General Instruction
Write your solutions and answers on a clean sheet of paper.
What I Know
NOTE: Write all your answers on a separate sheet of paper.
Match the given factors in Column A with their products in Column B. Write the letter of the
correct answer .
B
A
_1.
_2.
_3.
_4.
_5.
(3𝑥 − 7)(𝑥 + 2)
(𝑥 + 9)(𝑥 − 9)
(2𝑥 + 1)2
(𝑥 − 3)3
(5𝑥 − 4)(25𝑥2 + 20𝑥 + 16)
a. 4𝑥2 + 4𝑥 + 1
b. 𝑥2 − 81
c. 125𝑥3 −64
d. 3𝑥2 − 𝑥 − 14
e. 𝑥3 − 9𝑥2 + 9
f. 𝑥 3 − 9𝑥2 + 27𝑥 − 27
What’s In
1. Given the Algebra Tiles below, identify the polynomial factors and their product.
−𝒙
−𝒙
𝒙
𝒙
−𝒙𝟐
−𝒙𝟐
−𝒙𝟐
−𝒙𝟐
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
𝒙
−𝟏 −𝟏 −𝟏
2. Given the Generic Rectangle below, identify the polynomial factors and their product.
−𝟐𝒙
𝟑
𝟓𝒙
−𝟏𝟎𝒙𝟐
𝟏𝟓𝒙
−𝟖
𝟏𝟔𝒙
−𝟐𝟒
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What’s More
Write TRUE if the statement is correct and write FALSE if it’s incorrect.
1. When the difference of two terms is multiplied by the sum of their squares
plus the product of these terms, the result is the difference of their cubes.
2. When the sum and the difference of a binomial are multiplied to one
another, the product is the sum of the square of the two terms.
3. The square of a binomial is the product of a binomial when multiplied to
itself.
4. The product of a trinomial and a binomial does not always give a product
of four terms.
5. The cube of the binomial (𝑥 + 1) is equivalent to (𝑥 + 1)(𝑥 +1)(𝑥 − 1).
What I Can Do ACTIVITY 1
Using any method (Models or Algebraic), find the products of the following.
1. (4𝑥 + 7)(4𝑥 − 7)
2. (5𝑥 − 4)2
3. (3𝑦 + 4)(9𝑦2 – 12𝑦 + 16)
4. (3𝑥 − 2)3
5. (– 4𝑧 + 1)(5𝑧 – 2)
Assessment ACTIVITY 2
Apply the concept of special products in solving the following.
1. (8𝑧 − 5)(8𝑧 + 5)
2. (1 + 9𝑥)2
3. (2𝑦 + 7)(4𝑦2 − 14𝑦 + 49)
4. (𝑥 + 3𝑦)3
5. (𝑥2 + 4)(2𝑥 − 1)
Additional activities ACTIVITY 3
Find the products of the following.
1. (4𝑥 + 3𝑦)(2𝑥 + 𝑦)
2. (5𝑥𝑦 − 2𝑧)(5𝑥𝑦 + 2𝑧)
3. (𝑥2 + 3𝑦)2
4. (2𝑥𝑦 + 3)3
5. (7𝑦 − 5)(49𝑦2 + 35𝑦 + 25)
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Answer Key
What’s New
What I Can Do
(Solutions may vary)
1. 𝑥2 + 7𝑥 + 10
2. 𝑥2 − 27
3. 𝑥2 − 8𝑥 + 16
4. 𝑥3 + 9𝑥2 + 27𝑥 + 27
5. 𝑥3 + 8
1. 16𝑥2 − 49
2. 25𝑥2 − 40𝑥 + 16
3. 27𝑦3 + 64
4. 27𝑥3 − 54𝑥2 + 36𝑥 − 8
5. −20𝑧2 + 13𝑧 − 2
What I Know
1.
2.
3.
4.
5.
Assessment
d
b
a
f
c
1. 64𝑧2 − 25
2. 1 + 18𝑥 + 81𝑥2
3. 8𝑦3 + 343
4. 𝑥3 + 9𝑥2𝑦 + 27𝑥𝑦2 + 27𝑦3
5. 2𝑥3 − 𝑥2 + 8𝑥 − 4
What’s In
1. Factors: (−2𝑥 + 1)(2𝑥 − 3)
Product: −4𝑥2 + 8𝑥 − 3
2. Factors: (−2𝑥 + 3)(5𝑥 −8)
Product: −10𝑥2 + 31𝑥 − 24
Additional activities
1. 8𝑥2 + 10𝑥𝑦 + 3𝑦2
2. 25𝑥2𝑦2 − 4𝑧2
3. 𝑥4 + 6𝑥2𝑦 + 9𝑦2
4. 8𝑥 3𝑦 3 + 36𝑥2𝑦2 + 54𝑥𝑦 + 27
5. 343𝑦3 − 125
What’s More
1.
2.
3.
4.
5.
TRUE
FALSE
TRUE
TRUE
FALSE
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References:
Grade 7 Math Learning Guide. Accessed on September 11,
2020.https://cejerl.files.wordpress.com/2012/06/special-products-final-lg.pdf
Special Products on Factoring. Accessed on September 11, 2020.
http://lrmds.depedldn.com/DOWNLOAD/SPECIAL_PRODUCTS_ON_FACTORING.PDF
Multiplying Binomial and Factoring Trinomials Using Algebra Tiles and Generic Rectangles.
Accessed on September 11, 220.
https://www.wccusd.net/cms/lib/CA01001466/Centricity/domain/60/lessons/algebra%20i%20le
ssons/MultiplyingBinomialsFactoringTrinomialsV3.pdf
Prepared by:
MARIA LOURDES T. REYES
SST I, FRANCISCO G. NEPOMUCENO MEMORIAL HIGH SCHOOL
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7
MATHEMATICS
MODULE
Quarter 2: Week 7-8
Module 7: Solves problems involving algebraic expressions,
Differentiates and illustrates linear equation and
inequality in one variable
Module 8: Finds solution and solves linear equation or
inequality in one variable
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MATHEMATICS 7
Quarter 2 – Module 7:
Problems Involving Algebraic Expressions
Week 7 LC 1
Date
Name
Section
Background Information
What’s New
Barangay Captain Manalo divided some kilos of rice equally among the 72 families during the
Enhanced Community Quarantine in barangay San Pablo. Each of the families got 5 kilos of
rice. How many kilos of rice did Barangay Captain Manalo divide among the families in the
barangay?
Let x = number of kilos of rice Barangay Captain Manalo divided.
𝑥 = 5
Multiply both sides by 72.
72
𝑥
)
Perform the operation
(
72 ( ) = 72 5
72
x = 360
Therefore, there are 360 kilos of rice Barangay Captain Manalo divided among the
families in the barangay.
1. What x represent?
2. Why it is important to use letters in solving problems in Algebra?
What is It
You are familiar with the use of letters or variables, to stand for unknown numbers in
equations or formulas. Variables are also used to represent quantities that change over time
or in different situations. For example, N represents the number of people infected with Covid19 and d days after the start of a pandemic.
An algebraic expression is any meaningful combination of numbers, variables, and
symbols of operation. Algebraic expressions are used to express relationships between
variable quantities.
Example 1
Represent the following Mathematical phrases in terms of x.
1. a number increased by two
2. the age of Mario decreased by six
4. an amount increased by 75 pesos
5. John’s weight diminished by 8 kg
2
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3. one hundred ten less than a number
Solution:
1. a number increased by two
2. the age of Mario decreased by six
3. one hundred less than a number
4. an amount increased by 75 pesos
5. John’s weight diminished by 8 kg.
Key Phrase
increased by
decreased by
less than
increased by
diminished by
Expression
x+2
x–6
x – 100
x + 75
x–8
Example 2
Give and represent the following algebraic expressions.
a. Sarah has pens (p) and notebooks (n). Christine has five fewer pens than Sarah, but twice
as many notebooks. Write an expression for Christine’s school materials.
Solution:
Christine has five fewer pens than Sarah
→
twice as many notebooks
→
expression for Christine’s school materials, (pen and notebook)
Answer
→
p–5
2n
p – 5 + 2n
b. A waiter is paid Php 2,500 a week (w) plus Php 80 for each hour (h) of overtime worked.
Write an expression for the waiter’s salary.
Solution:
Php 2,500 a week (w)
→
2500w
Php 80 for each hour (h)
→
80h
expression for the waiter’s salary (week plus hour)
Answer
→
2500w + 80h
c. The denominator of a fraction is three less than the numerator. Write an expression for the
fraction.
Solution:
numerator
→
x
three less than the numerator
→
x–3
an expression for the fraction (denominator is three less than the numerator)
𝑥
Answer
𝑥− 3
Problem solving often involves translating a real-life problem into algebraic
expressions. We can then use algebra to solve the mathematical problem and interpret the
solution in the context of the original problem.
Example 3
Solve the following problems.
1. If the sum of 3x + 7 and 6x – 5 are doubled, find the value of the resulting expression.
Solution:
the sum of 3x + 7 and 6x – 5 →
doubled
→
Answer
→
(3x + 7) + (6x – 5) = 9x + 2
2(9x + 2) = 18x + 4
18x + 4
2. The length of a rectangle is 2x + 4 cm and the width is 3x – 5 cm. Find the area of the
rectangle.
Given:
l = 2x + 4
w = 3x – 5
→
→
length of the rectangle
width of the rectangle
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A = lw
→
area of the rectangle
Solution:
(2x + 4) (3x – 5)
6x2 – 10x + 12x – 20
6x2 + 2x – 20
Answer
→
→
→
→
multiply the length to the width
combine the like terms
area of the rectangle
6x2 + 2x – 20 cm ----- area of the rectangle
What I need to know
Solves problems involving algebraic expressions. (M7AL-IIg-2)
General Instruction
Write your solutions and answers on a clean sheet of paper.
What I Know
Write an expression in terms of x.
1. the product of a number and eleven
2. six increased by twice a number
3. seven times a week
4. a liter of water divided by two
5. Carlo’s age is subtracted by five
What’s In
Write an expression in terms of x for each problem.
1. A number is twenty more than another number. Write an expression for another number.
2. The numerator of a fraction is four more than twice the denominator. Write an expression
for the fraction.
3. The width of a rectangle is five less than twice the length. Write an expression for the
perimeter of the rectangle.
4. Carl is three years younger than his brother. Write an expression for the age of Carl.
5. The tens digit of a number is seven more than the ones digit. Write an expression for the
number.
What’s More
Solve the following problems. Show your solution.
1. Ryan is 10x + 8 years old, his father is 45x + 2 years old. How many years older is the
father than Ryan?
2. Find the perimeter of the rectangle whose length is 5x + 10y and width is 4x + 6y.
3. If 2x – 4y – 7 and 3x + 5y + 6 are doubled, then it is equal to
.
4. A box is x + 3 cm by x – 5 cm by x + 3 cm. Find the volume of the box.
5. Find the area of the rectangle if its length is 6x – 4cm and its width is 9x + 2cm.
What I Can Do
Match each problem to the correct algebraic expression. Choose your answer inside the box
below.
20(5) – x
5+4+7+x
2x – 8
𝑥
4
7(2x)
5x
(x + 6) – 1
6 + 4x
15 + x
(x – 8) – 4
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1. If Robert feed his cat the same amount of food twice a day, how much food will he feed
the cat in a week?
2. Lance takes 6 more cookies than her friend Miya. If Lance puts 1 cookie back, how many
cookies does he have?
3. James will drive a car each day, he is at the beach. If he drives the same distance each of
the 4 days he is there, how far will he drive each day?
4. Bantay fetched 15 sticks and some rocks when he played outside. How many objects did
he fetch in all?
5. Shiela answered 8 problems incorrectly on her math test. If she didn’t finish the test the
last 4 problems, how many problems did she answer correctly?
6. Tony is 8 years younger than twice her brother’s age. How old is Tony?
7. Joy ran 5 miles on Saturday, 4 miles yesterday, and 7 miles today. If she runs an unknown
number of miles tomorrow, how many miles will she have run in all?
8. Gino earns an hourly wage for delivering pizza. How much will he earn if he delivers pizza
for 5 hours?
9. Micah’s mother baked 6 cookies for Micah and 4 cookies for each of her friends. How
many cookies did she bake in all?
10. Farmer Dante has 20 hens. Each hen laid 5 eggs, but some eggs cracked. How many
eggs not crack?
11.
Assessment ACTIV ITY 1
Choose the letter of the correct answer.
1. Roman had 7 books. Then he went to a book sale and bought b more books. Choose the
expression that shows the number of books Roman has now.
a. b
b. 7 + b
c. b – 7
d. 7b
2. Write “Gina’s age is twice as her sister’s age” in expression.
a. 2x
b. x + 2
c. 2 – x
d. x2
3. Suppose the length of one side of a square is 7x + 3 cm. Find the area of the square.
a. 14x + 6
b. 14x2 + 21x + 9
c. 49x2 + 42x + 9
d. 49x2 + 9
4. A calamity fund is represented by 10x2 + 13x + 9. Represent the amount after an increase
of 7x + 5 in calamity fund.
a. 10x2 + 20x + 14
c. 10x2 + 6x + 4
2
b. 17x + 18x + 9
d. 3x2 + 4x + 5
5. A typist earns c2d3 + c2d2 + c2d for working cd hours. How much does he earn per hour?
b. c2d + cd
c. cd + cd2
d. cd2 + cd + c
a. cd2
Additional Activities
Use the following clues to find the secret number.
I am a 3-digit number.
All of my digits are odd.
I am greater than 600.
I am less than 800.
The sum of the digits is 15.
My last digit is 5.
What am I?
a. Make a table showing starting and ending numbers.
b. What pattern do you notice?
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Answer Key
References
Department of Education. Mathematics MELCs. p. 305.
Guren, Paulino T. Ph.D. Realistic MATH Scaling Greater Heights 6. (Quezon City: Sibs Publishing
House, Inc., 2016). pp. 267 – 299.
Orines, Fernando B. Next Century Mathematics 7. (Quezon City: Phoenix Publishing House, Inc.,
2012). pp 242 – 278.
Oronce, Orlando A. and Mendoza, Marilyn O. E-Math 7. (Manila: REX Book Store, Inc., 2012). pp.
150 – 211.
Taay, Sheryly F., Gamboa, Marivic V., and Gumangan, Amado R. Mathematical Skills and Talent
Enhancer and Reviewer I. (Makati City: Eureka Scholastic Publishing, Inc. 2008). pp. 32 – 36.
“Algebraic Expressions and Problem Solving,” Google, last modified September 7, 2020,
https://yoshiwarabooks.org/mfg/appendix-Algebraic-Expressions-and-Problem-Solving.html
Prepared by:
Ma. Reva G. Castro
SST – I, FGNMHS
Rommel G. Salem
SST – III, FGNMHS
Illustrated by:
Rommel G.
Salem
SST – III, FGNMHS
MATHEMATICS 7
6
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Quarter 2 – Module 7:
Algebraic Expressions, Equations and Inequalities
Week 7 LC 2
Date
Name
Section
Background Information
What’s New
Unscramble the letters to find words related to our topic for today.
SCRAMBBLED WORDS
1. BARIVALSE
2. IQETUALIYN
3. OXSPNESIER
4. TNNSOTAC
5. ENUATQIO
`
ANSWER
What is It
A. By Definition
a. Algebraic Expression – a statement containing one or more terms connected by
plus or minus signs.
b. Constants – a number, has a fixed value
c. Variables –refers to letter
d. Equation – a mathematical statement that shows two numbers or two
expressions are equal
e. Inequality - a mathematical statement that one algebraic expression is not
equal to another algebraic expression. It contains the symbols;
>, <, ≥ , ≤, or ≠
f. Linear equation in one variable –an equation which has one variable in
the first degree. It is of the form ax + b = 0, where x is the variable. This equation
has only one solution
g. Linear Inequality in one variable –an inequality that can be written in one
of the following forms: ax + b < c ; ax + b > c
ax + b ≤ c ; ax + b ≥ c
- have either infinitely many solutions or no solution.
The chart below shows the phrases that indicate inequalities with the corresponding
symbols.
<
Is less than
Is fewer than
>
Is greater than
Is more than

At most
No more than
Less than or
equal to
≥
At least
No less than
Greater than or equal to
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B. Examples
ALGEBRAIC EXPRESSION
EQUATION
INEQUALITY
9x + 27
2 + 3y
-3y + 6
9x = 27
2 + 3y = 6
-3y + 6 = 18
9x ≥ 27
2 + 3y < 6
-3y + 6 ≠ 18
LINEAR EQUATION IN ONE VARIABLE
C.
LINEAR INEQUALITY IN ONE VARIABLE
x - 7 ≥ 10
-4x < 25
y ≠ -18
3x = 1
22x-1=0
4x+9=-11
Graph of Inequality
To effectively interpret inequality, it is recommended that we first illustrate its
graph. Drawing and interpreting its graph is illustrated below. The graph of a linear
inequality in one variable is a number line. Use a non-shaded circle for < and > and a
shaded circle for ≤ and ≥.
Graph the following inequality, where x is a real number
1. n<2
The solution of the inequality n < 2 is all real numbers less than 2. To graph n<2, shade
the numbers to the left of 2 since they are less than 2, then put an open circle on 2 to
indicate that 2 is not a solution.
2. n ≤ 2
This graph shows all real number values of 𝑛 are less than or equal to 2, including 2.
3. n ≥ -2
This graph shows all real number values of 𝑛 are less than or equal to -2, including -2.
4. x ≠ 2
This graph shows all the real number values of x except 2. The solutions of x ≠ 2 are
all real numbers not equal to 2.
D. By Properties
Properties
Addition Property
Subtraction Property
Linear Equations
If there exists a=b, then
𝑎+ 𝑐 = 𝑏+ 𝑐
If there exists a=b, then
𝑎− 𝑐 = 𝑏− 𝑐
8
Linear Inequalities
If there exists a<b, then
𝑎 + 𝑐 < 𝑏 + 𝑐.
If there exists a<b, then
𝑎 − 𝑐 < 𝑏 − 𝑐.
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Multiplication Property
If there exists a=b, then
𝑎𝑐 = 𝑏𝑐
Same as when multiplying
negative a number on both
sides; 𝑎(−𝑐) = 𝑏(−𝑐)
Division Property
If there exists a=b, then
𝑎
𝑐
=
If there exists a<b, then
𝑎𝑐 < 𝑏𝑐.
However, when multiplying a
negative number on both sides,
the sense of inequality sign will
be reversed;
𝑎(−𝑐) > 𝑏(−𝑐)
If there exists a<b, then
𝑎 𝑏
<.
𝑐 𝑐
However, when diving a
negative number on both sides,
the sense of inequality sign will
be reversed;
𝑎 𝑏
>
−𝑐 −𝑐
𝑏
.
𝑐
Same as when dividing
negative a number on both
sides;
𝑎 𝑏
=
−𝑐 −𝑐
What I need to know
Differentiates algebraic expressions, equations and inequalities.
Illustrates linear equation and inequality in one variable (M7AL-IIh-4)
General Instruction
Write your answers on a clean sheet of paper.
What I Know
A. Write X if it is an expression, Y if it is an equation and Z if it is an inequality.
1. x – 3 = 20
4. 3x ≥ 27
2. x + 4y
5. x + 16 = 37
3. -4x < 24
B. Write LEQ if it is a linear equation in one variable and LIQ if it is a linear inequality in one
variable.
1. 6 = 6𝑦 − 12
4. −5𝑧 − 5 = 5
2. 4𝑥 > 5
5. -4y  12
3. 2𝑦 = 14
What’s In
Choose the correct expression in terms of x.
1. the product of a number and twenty
20x
20 + x
2. seven increased by thrice a number
3x + 7
3x - 7
2x
2+x
x+4
x-4
x+3
x-3
3. two times a week
4. a number decreased by four
5. Ana’s age is subtracted by three
What’s more
Determine the following statement or example if algebraic expressions, equations or
inequalities. Write A if algebraic expression, B if equation and C if inequalities.
1. A mathematical sentence indicating that two expressions are equal.
2. Used to express relationships between variable and quantities.
3. It contains the symbol >, <, ≥ or ≤.
4. 3y – 9 is an example of
.
5. What mathematical sentence is 3x + 10 = 15?
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What I Can Do
A. Classify the following and list down on the blank provided.
3x - 5
9x ≠ 10
6y + 24 = 15
x – 10 = 15
4x + 3 ≤ 12
7z - 13
2x < 15
2y + 4
3x + 16 = 20
ALGEBRAIC EXPRESSION
EQUATIONS
INEQUALITIES
B. Make a number line to represent the following inequalities
1. x < 1
2. y ≤ 8
3. m ≠ - 5
Assessment ACTIVITY 2
Choose the letter of the correct answer.
1. It has either infinitely many solutions or no solution.
a. Algebraic Expression
b. Linear Equation
c. Linear Inequality
2. It can be a term or a collection of terms separated by addition or subtraction operators.
a. Algebraic expression
b. Equation
c. Inequality
3. A mathematical statement that shows two numbers or two expressions are equal is called
.
a. Algebraic expression
b. Equation
4. Which of the following are the symbols of inequality?
c. Inequality
a. =, <, ≠
b. =, >, <
c. >, <, ≠
5. Which of the following is an example of linear equation in one variable?
a. 3x + 7 > 10
b. 3x + 7 =10
c. 3x + 7y = 10
6. Which of the following is the graph of x ≠ 1
.
a.
b.
c.
7. Which of the following is not an example of inequality?
a. 2x > 14
b. 2x = 14
8. What shade of the circle if the given inequality is x ≥ 1?
a. shaded
b. non-shaded
c. 2x ≥ 14
c. half-shaded
Additional Activities
A. Give 2 examples each of Algebraic Expression, Equation and Inequality.
ALGEBRAIC EXPRESSIONS
1.
2.
EQUATIONS
INEQUALITIES
B. Translate the graphical representation to an inequality using the variable x.
1.
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2.
3.
4.
5.
Answer Key
What I know
A.
1. Y
2. X
B.
1. LEQ
2. LIQ
What’s In
1. 20x
2. 3x + 7
What’s More
1. B
2. A
What Can I Do
ALGEBRAIC EXPRESSION
1. 3x - 5
2. 7z - 13
3. 2y + 4
EQUATION
1. 6y + 24 = 15
2. x – 10 = 15
3. 3x + 16 = 20
INEQUALITIES
1. 9x ≠ 10
2. 4x + 3 ≤ 12
3. 2x < 15
Assessment
1. c
2. a
Additional Activities
1. x ≤ 4 2. x > (-3)
3. Z
4. Z
5. Y
3. LEQ
4. LEQ
5. LIQ
3. 2x
4. x - 4
5. x – 3
3. C
4. A
5. B
3. b
3. x < 5
4.c
4. x ≠ (-1)
5. b
6. a
5. (-2) > x < 2
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8. a
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References
Department of Education. Mathematics MELCs. p. 305.
Nivera, Gladys C,Ph.D. Grade 7 Mathematics Patterns and Practicalities(Makati City:
SalesianaBOOKS by Don Bosco Press, Inc.,2013) pp.200-203 , 296-307.
Gamboa, Job D., Elementary Algebra (Lipa City, Batangas: United Eferza Academic Publications
Co.,2010) pp.128-129, 259-264 .
“Expressions, Equations and Inequality,” Google, last modified September 10, 2020,
http://www.differencebetween.net/language/difference-between-inequalities-and-equations/
“Linear Inequalities in One Variable,” Google, last modified September 9, 2020,
https://byjus.com/maths/linear-equation-in-one-variable/
“Linear Inequalities in One Variable,” Google, last modified September 9, 2020,
https://2012books.lardbucket.org/books/beginning-algebra/s05-08-linear-inequalities-one-variab.html
Prepared by:
Ma. Reva G. Castro
SST – I, FGNMHS
Vanessa A. Villanueva
SST – I, FGNMHS
Illustrated by:
Rommel G. Salem
SST – III, FGNMHS
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MATHEMATICS 7
Quarter 2 – Module 8:
Find Solution of Linear Equation or Inequality in One Variable
Name
Week 8
Section
Date
Background Information
What’s
New Guess
and Check
Direction: Encircle the number which satisfies the equation or inequality.
–1
6
10
–5
–4
x–1>4
–2
3
6
3x = –12
2x – 8 = 6
x + 10 = 5
7
2
14
– 12
3
6
–4
2x + 5 ≥ 9
5
1
2
–7
–9
–9
6
–1
5
What I need to know
Finds the solution of linear equation or inequality in one variable. (M7AL-III-1).
General Instruction
Write your solutions and answers on a clean sheet of paper.
What is It
Finding Solutions to Linear Equations and Inequalities
A. Solving Linear Equations
Solving an equation means finding the values of the unknown (such as x) so that the
equation becomes true. Although you may solve equations using Guess and Check, a more
systematic way is to use the properties of equality as the following examples show.
Example 1. Solve x – 4 = 8.
Solution:
x–4=8
x–4+4=8+4
Given
Addition Property of Equality (APE)
Added 4 to both sides
x = 12Checking the solution is a good routine after solving equations.
The Substitution Property of Equality can help. This is a good practice for you to
check mentally.
If x = 12
then, x – 4 = 8
12 – 4 = 8
8=8
✓ Since 8 = 8 is true, then the x = 12 is a correct solution to the equation.
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Example 2. Solve x + 3 = 5.
Solution:
x+3=5
x + 3 + (–3) = 5 + (–3)
Given
Addition Property of Equality (APE)
(Added –3 to both sides)
x=2
Example 3. Solve 3x = 75.
Solution:
3x = 75
3x • 1 = 75 • 1
3
3
Given
Multiplication Property of Equality (MPE)
(Multiplied to both sides)
x = 25
Note also that multiplying to both sides of the equation is the same as dividing by 3, so the
following solution may also be used:
3x = 75
Given
3𝑥
75
𝟑
=
Multiplication Property of Equality (MPE)
(Divided both sides of the equation
𝟑
x = 25
In examples 1-3, we saw how the properties of equality may be used to solve an equation
and to check the answer. Specifically, the properties were used to “isolate” x, or make one
side of the equation contain only x.
REMEMBER:
Whatever you do to one side of the equation, do it on the other side of the equation.
MORE EXAMPLES:
1. Solve 2x – 10 = – 2.
Solution: 2x – 10 = – 2
2x – 10 +10 = – 2 +10
2𝑥 = 8
2𝑥
=
𝟐
2. Solve
𝑥
8
Given
(APE) Added 10 to both sides
Simplified
(MPE) Divided 2 to both sides
𝟐
𝑥=4
Answer
+ 4 = 5.
2
Solution:
𝑥+
2
𝑥
2
4 =5
Given
+4–4=5–4
(APE) Subtracted 4 to both sides
=1
Simplified
2=1•2
(MPE) Multiplied 2 to both sides
𝑥
2
𝑥•
2
𝑥=2
Answer
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3𝑥 = – 6
3𝑥
=
𝟑
−6
Simplified
(MPE) Divided 3 to both sides
𝟑
𝑥 = −2
Answer
B. Solving Linear Inequalities
Properties of Inequalities
The following are the properties of inequality. These will be helpful in finding the solution
set of linear inequalities in one variable.
1. Trichotomy Property
For any number a and b, one and only one of the following is true:
a < b, a = b, or a > b.
This property may be obvious, but it draws our attention to this fact so that we can
recall it easily next time.
2. Transitive Property of Inequality
For any numbers a, b and c, (a) if a < b and b < c, then a < c, and
(b) if a > b and b > c, then a > c.
3. Addition Property of Inequality (API)
For all real numbers a, b and c:
(a) If a < b, then a + c < b + c, and
(b) If a > b, then a + c > b + c.
Observe that adding the same number to both a and b will not change the
inequality. Note that this is true whether we add a positive or
negative number to both sides of the inequality.
4. Multiplication Property of Inequality (MPI)
For all real numbers a, b and c, then all the following are true:
(a) If c > 0 and a < b, then ac < bc;
(b) If c > 0 and a > b, then ac > bc.
(c) If c < 0 and a < b, then ac > bc;
(d) If c < 0 and a > b, then ac < bc.
Observe that multiplying a positive number to both sides of an inequality does
not change the inequality. However, multiplying a negative number to both sides of
an inequality reverses the inequality.
POINTS TO REMEMBER:
• Subtracting numbers. The API also covers subtraction because subtracting a number
is the same as adding its negative.
• Dividing numbers. The MPI also covers division because dividing by a number is the
same as multiplying by its reciprocal.
• Do not multiply (or divide) by a variable. The MPI shows that the direction of the
inequality depends on whether the number multiplied is positive or negative. However, a
variable may take on positive or negative values. Thus, it would not be possible to
determine whether the direction of the inequality will be retained not.
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Example 1: Solve x + 11 ≥ 23.
Solution:
x + 11 ≥ 23
x + 11 + (–11) ≥ 23 + (–11)
x ≥ 12
Given
API (Added –11 to both sides)
Example 2: Solve 5x < –15.
Solution:
5x < –15
(5x) • 𝟏 < (–15) • 𝟏
𝟓
Given
MPI (Multiplied 1 to both sides)
𝟓
5
x < –3
Note also that multiplying to both sides of the equation is the same as dividing by 5, so
the following solution may also be used:
5x < –15
Given
5𝑥
−15
Divided 5 to both sides
<
▪
𝟓
𝟓
x < –3
Example 3: Solve 3x – 7 > 14.
Solution:
3x – 7 > 14
3x – 7 + 7 > 14 + 7
3𝑥
>
𝟑
Given
Addition Property of Inequality (API)
(Added 7 to both sides)
Divided 3 to both sides
21
x>7
𝟑
What I Know
Activity 1: Cross-puzzle!
Direction: Solve the following linear equation. Show your solution.
1
2
3
ACROSS
4
1. 20 – x = 8
5
4.
6
2.
𝑥
– 21 = 22
5
𝑥
4
DOWN
3. 15 – x = 3
= 28
6. 2x – 102 = 204
5.
7. 3x – 38 = 22
6.
𝑥
11
7
–7=5
2𝑥
+ 8 = 10
11
What’s In
Activity 2: Smash it!
Direction: Match the linear inequalities in the mallet to its corresponding solution in the
stone with different shapes. Draw the stone below the knife.
x<4
x≤1
x ≥ –10
x>1
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1.
2.
3.
4.
5.
What’s More
Activity 3: Color the
Match!
Direction: Find the solution of linear equations or inequalities. Color the solution with the
same color of its corresponding linear equation/inequality.
Given: x + 5 > 9
Given: 4 + 3x = 7
x<–6
solution
x=–8
solution
YELLOW
GREEN
Given:
x=–1
solution
𝑥
Given: x – 3 < – 9
+5=
3
4
x>4
solution
ORANGE
BLUE
Given: 𝑥 + 3 ≥
4
2
Given: 2x + 8 = 6
x≥2
solution
x=1
solution
VIOLET
RED
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What I Can Do
Activity 4. Direction: Answer the following real-life problems below.
1. Margie is 3 times older than Lyka. In 15 years, the sum of their ages is 39 years. Find
their present ages.
2. Kevin wants to buy some pencils at a price of P4.50 each. He does not want
to spend more than P55.00. What is the greatest number of pencils can
Kevin buy?
Assessment
Activity 5: Choose
Wisely!
Direction: Find and write the solution of the following equations/inequality.
Encircle your answer from the choices provided.
2. x – 4 < – 1
1. 4 + 3x = 7
a. x = 1
b. x = –1
a. x < – 5
4. 2x ≥ –10
a. x ≥ 5
b. x < 3
b. x ≥ – 5
a. x = 28
b. x = – 12
8. 𝑥 + 2 > 8
5
a. x > 2
a. x ≤ – 3
9.
3
b. x = –1
a. x = 6
b. x = 9
6. x + 5 ≤ 2
5. 20 + x = 8
7. 𝑥 – 20 = – 25
a. x = – 25
3. 2x – 3 = 15
𝑥
2
b. x > 18
b.x ≤ 7
– 6 = – 10
a. x = – 8
b. x = – 2
Additional
activities Direction:
Riddle Me This!
Solve for “x”. Write the corresponding letter in the box below that matches your answer.
1. 10 + x = 15
2. 3 + 2x > – 5
3. 2x – 4 = 4
4. 3x + 2 < 8
5. 9x = – 18
6. 2x + 7 ≥ 9
7. x + 5 ≤ – 3
8. 5 + 3x = 14
9. 3x – 9 ≤ 12
10. 11 + 2x = 31
A
B
C
E
G
What do ghosts serve for dessert?
x≥1
x=4
x<2
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I
K
M
R
S
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Answer Key
What’s New
What I know
What’s In
What’s More
1.
2.
3.
Assessment
4.
5.
What Can I Do
1.
2.
Lyka is 6 years old and Margie is
18 years old at present.
Kevin can buy at most 12 pencils
Additional Activities
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MATHEMATICS 7
Quarter 2 – Module 8:
Solves Linear Equation or Inequality in One Variable Involving Absolute
Value by: (a) Graphing; and (b) Algebraic Methods
Solves problems involving equations and inequalities in one variable.
Name
Week 8
Section
Date
Background Information
What’s
New CLASSIFY
Inequalities
Equations
ME!
-4x < 24
x+4>7
x + 16 = 37
2x + 5 = 20
x – 9 = -15
3x ≥ 27
-4x – 6 ≤ 13
What is It
Equations and Inequalities Involving Absolute Value
Absolute Value
In this section, we used the concept of absolute value to describe precisely how to
operate with positive and negative numbers. At that time, we gave a geometric description of
absolute value as the distance between a number and zero on the number line. For example,
using vertical bars to denote absolute value, we can state that |−3| = 3 because the distance
between -3 and 0 on the number line is 3 units. Likewise, |2|=2 because the distance between
2 and 0 on the number line is 2 units. Using the distance interpretation, we can also state that
|0|=0 (Refer to figure
|−3| = 3
|2|=2
1)
-3
-2
-1
0
1
2
| 0| = 0
Figure 1
More formally, we define the concept of absolute value as follows:
Definition of Absolute Value
𝑎 𝑖𝑓 𝑎 > 0,
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Applying the above definition, we obtain the following results:
|6| = 6
|0| = 0
|−7| = 7
Note the following ideas about absolute value:
1. The absolute value of any number except zero is always positive.
2. The absolute value of zero is zero
3. A number and its opposite have the same absolute value.
We summarize these ideas in the following properties.
Properties of Absolute Value
The variables a and b represent any real
number.
Note: a – b and b – a are opposites of each other
Absolute Value in Equation
Example 1. Solve |𝑥 − 7| =
3
Solution:
This equation, according to the definition of absolute value, expresses the fact
that x – 7 must be 3 or -3, since in either case the absolute value is 3.
When x – 7 = 3, we have x = 10; and
when x – 7 = -3, we have x = 4.
We see that there are two values of x which solve the equation: x = 4, 10
Hence, the solution set is {4, 10}
Example 2. Solve |2𝑥 − 3| = 5
Solution:
The two possibilities of the given equation are
2x -3 = 5
2x – 3 = -5
or
2x = 5 + 3
2x = -5 + 3
2x = 8
2x = -2
x=4
x = -1
Hence, the solution set is {4, −1}
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Formally, we have the following properties related with the Equations and Inequalities
involving absolute value.
ax – b = c or (ax – b) = -c
Property 1
Absolute Value in Inequalities
Inequalities involving absolute value can also be represented on a number line as follows:
-4
-3
-2
-1
0
1
2
3
4
Since |𝑥| represents the distance of x from the origin, the reader can easily see that
the condition that |𝑥| < 4 is equivalent to the condition that x be any number in the interval
extending from -4 to +4.
Or, using the symbol for intervals, x must lie in the interval (-4, 4). Sometimes, a
statement such as |𝑥| < 4 is used to denote value signs, the statement
-4 < x < 4
is equivalent to |𝑥| < 4. In a similar way, the inequality |𝑥 − 3| < 5 means that x – 3 must
lie in the interval ( -5, 5 ). We could also write
-5 < x – 3 < 5
This consists of two inequalities, both of which x must satisfy. If we add 3 to each
member of the double inequality above, we obtain
-2 < x < 8.
Therefore x must lie in the interval ( -2 , 8 )
-2
-1
0
1
2
3
4
5
6
7
8
The above examples suggest the additional properties which can be used to solve equations
and inequalities that involve absolute value.
Property 2
Example 1.
Solution:
Solve |𝑥 + 1| < 3 and graph the solution set.
From (Property 2), the given inequality is equivalent to
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-3 < x + 1 < 3,
Now subtract 1 to each term, we have
-4 < x < 2
Hence, the solution set is (-4, 2), with the graph as shown.
-4
-3
Example 2.
3 Solution:
-2
-1
1
2
3
4
Solve and graph |𝑥| ≤
|𝑥| ≤ 3
|𝑥| ≤ 3
-4
0
-3
|𝑥| ≥ -3
and
|𝑥| ≥ -3
-2
-1
0
1
2
3
4
|𝑥| ≤ 3
The solution set consists of all the numbers greater than or equal to -3, and less than or equal
to 3. The graph is the intersection of the two arrows; this is the heavy line segment between 3 and 3.
Solving problems involving equations and inequalities in one variable
In solving problems, the first step is to represent the quantities by symbols. In many
cases, it is sufficient to use one variable to represent related quantities. The independent
quantity, which is the quantity that is not described in the problem, is represented by a variable.
The related quantities are represented in terms of the same variable.
Example 1.
In each situation, represent the quantities by symbols.
a) Roel’s average grade is 2 point higher than that of Ryan.
If x = Ryan’s average grade (independent quantity),
x + 2 = Roel’s average grade
b) Mrs. Tapang is four times as old as her daughter Michaela. Represent their
present ages, their ages 5 years ago, and their ages in 5 years.
Present Ages
Ages 5 Years Ago
Ages in 5 Years
Michaela
x
x–5
x+5
Mrs. Tapang
4x
4x - 5
4x + 5
c) One number is 5 more than twice another number.
If y = first number, then 2y + 5 = second number
After the related quantities are represented by symbols, the next step is to form the equation
or the relationship of the quantities.
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Express the following statements as equations:
Example 2.
a) If twice a number is decreased by 6, the result is the number increased by
4.
If x = the number, the equation is 2x – 6 = x + 4
b) One number is 5 more than another number. Four times the larger number
is the smaller number plus 26.
Let x = smaller number
x + 5 = larger number
Equation: 4(larger) = smaller + 26
4(x+5) = x + 26
The third step in solving word problems is to solve the derived equation and to find the
other quantities asked in the problems. The final step is to check the answer by verifying
if the conditions stated in the problems are met.
In short, the solution of a word problem consists of four parts, namely:
( 1 ) representation
( 3 ) solution
( 2 ) equation
( 4 ) check
A. Word problem involving equations in one variable
Example:
One number is more than another number. Four times the larger number is 26
more than the smaller number. What are the numbers?
( 1 ) Representation:
Let x = smaller number
x + 5 = larger number
( 2 ) Equation:
4(larger) = smaller + 26
4(x + 5) = x + 26
( 3 ) Solution:
4x + 20 = x + 26
4x – x = 26 – 20
3x = 6
x = 2 smaller number
x + 5 = 7 larger number
( 4 ) Check:
4(larger) = smaller + 26
4 ( 7 ) = 2 + 26
28 = 28
A. Word problem involving inequalities in one variable
Problems containing expressions such as less than (or not less than), greater than (or not
greater than), at least, and at most can be solved by inequalities.
Study the meanings of the following statements:
Statement
x is at least 5
x is at most 5
x is not less than 5
Symbol
x≥5
x≤5
x≥5
Meaning
at least 5 means 5 or greater
at most 5 means 5 or less
not less than 5 means 5 or greater
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x≤5
x is not greater than 5
not greater than 5 means 5 or less
Example:
The sum of a number and 5 is at most 10. What is the number?
Solution:
Let x = the number
Represent the unknown quantity
x + 5 ≤ 10
x ≤ 10 – 5
Form the inequality
Solve the inequality
x≤5
The number is 5 or less.
What I Need to Know
Solves linear equation or inequality in one variable involving absolute value by: (a) graphing
and (b) algebraic methods (M7AL-Iii-j-1) and solves problems involving equations and
inequalities in one variable (M7AL-Ilj-2)
General Instruction
Write your solutions and answers on a clean sheet of paper.
What I Know
Activity 1: Connect
me!
Direction: Connect which graph below matches each equation or inequality that involves
absolute value.
●
●
1. |𝑥| =
a.
-∞
-4
0
4
∞
4
●
● b.
2. |𝑥| <
-∞
-4
0
4
∞
4
●
● c
3. |𝑥| >
4
4. |𝑥| ≤
4
●
●
d.
-∞
-4
0
4
∞
-∞
-4
0
4
∞
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What’s In
Activity 2. Translate me!
A.
1.
2.
3.
4.
5.
Translate the following statements into algebraic equations.
Two times a number is ten.
Five more than a number is eight.
Six less than a number is twenty-four.
The sum of nine and a number is fourteen.
The difference of seven and three times a number is ten.
What’s More
Activity 3. I CAN
SOLVE!
A. Solve the following given equations.
1. |2𝑥 − 6| = 12
2. |2𝑥 − 3| = 9
3. |2𝑥 + 2| = 16
B. Find the solution set of each inequality and draw the graph.
1. |𝑥| > 3
2. |𝑥 − 2| < 4
What I Can Do
Activity 4.
Direction: Answer the following real-life problems below.
1. In a class of 45 students, the number of girls is twice the number of boys. How many
girls are there in the class?
(1) Representation:
(2) Equation:
(3) Solution:
(4) Check:
2. A piece of wire that is not less than 18 meters long is to be cut into three pieces. The
second piece is to be twice as long as the first piece, and the third piece is to be 3
meters longer than the second piece. What will be the shortest length of the first piece?
a. Solution:
Assessment
ACTIVITY 1 Direction:
SOLVE ME!
A. Solve each equation.
1. |𝑛| = 4
2. |4𝑥| = 20
3. |𝑦| – 3 = 5
B. Find the solution set of each inequality and draw the graph.
1. |𝑥| < 9
2. |𝑛| – 2 ≤ 4
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x = 4, x = -4
x = 8, x = -2
x = 2, x = -2
x = -3, x = 3
x = 8, x = -2
x = 10, x = -8
x = 8, x = -2
x = 4, x = -4
x = 4, x = -2
1. |𝑥 + 5 | = 3
E. x = 10, x = -8
2. |2𝑥| – 6 = 10
R. x = 4, x = -4
3. |5𝑥| = 20
T. x = -2, x = -8
4. |3𝑥 − 3| = 9
B. x = -3, x = 3
6. |𝑥| = 2
K. x = 4, x = -2
5. |− 9𝑥| = 27
H. x = 2, x = -2
7. |2𝑥 − 2| = 18
O. x = 8, x = -2
x = -2, x = -8
x = 10, x = -8
References
Ma. Luisa V. Villano, Lowela B. Mupas, Josephine L. Sy Tan, and Simon L. Chua, D. T., Phoenix Math
for the 21st Century Learners (K to 12) Grade 7 2 nd Edition (Quezon City: Phoenix Publishing House,
Inc, 2016), pp.317 - 321
Lutgarda S. Sundiam, Integrated Mathematics Grade 7 (Quezon City: New Horizon, 2012), pp.
167,189,193
Grade 7 Learner’s Material (LM), accessed September 12, 2020
http://educationalprojams.weebly.com/uploads/1/0/2/3/102353042/math_vii_learners_materials q3_.pdf
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Answer Key
What’s New!
Equations
• 2x + 5 = 20
• X + 6 = 37
• X – 9 = -15
Inequalities
• -4x < 24
• X+4>7
• 3x ≥ 27
• -4x – 6 ≤ 13
What’s I know
1.C
2.A
3.D
4.B
What’s In!
A.
1. 2x = 10
2. x + 5 = 8
3. x – 6 = 24
4. x + 9 = 14
5. 7 – 3x = 10
B.
1. x > 4
2. ½ m < 16
3. n ≥ 10
4. y ≥ 25
5. p + 8 ≤ 11
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What’s more
A.
1. The solutions are 9 and -3.
2. The solutions are 6 and -3.
3. The solutions are 7 and -9.
B.
1. x > 3 ,
x < -3
2. x < 6 ,
x > -2
What can I do
1.Representation:
Let n = number of boys
2n = number of girls
Equation:
n + 2n = 45
Solution:
3n = 45
n = 15
2n = 30
Check
N + 2n = 45
15 + 30 = 45
45 = 45
2. Represent the unknown
quantities
Let x = the first piece
2x = the second piece
2x + 3 = the third piece
Form the inequality
x + 2x + 2x + 3 ≥ 18
Solve the inequality
5x + 3 ≥ 18
5x ≥ 18 – 3
5x ≥ 15
x≥3
The shortest length of the first
piece is 3 meters.
Assessment
A.
1. n=4, n= -4
2. x=5, x= -5
3. y=8, y= -8
B.
1. x<9, x>-9
-15 -12
-9
-6
-3
-2
0
0
3
6
2. n≤6, n≥-6
-10 -8
-6
-4
2
4
6
8
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