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Study Guide
7th Grade Math Skills
06/01/2015
Divide By Decimals: Story Problems
In a division problem, the dividend is the number that is to be divided. The divisor is the number that is
divided into the dividend. The quotient is the answer to the division problem. In this skill, students are asked
to divide a whole number by a decimal number within the context of a real world problem.
Example 1: Divide. Round your answer to the nearest tenth, if necessary.
Step 1: When dividing any number by a decimal, the decimal point in the divisor must be moved to the
right the same number of spaces as there are decimal places (so that it becomes a whole number). The
decimal point must be moved the same number of places in the divisor and dividend. In this case, move
the decimal point one place to the right in the divisor and on place to the right in the dividend.
Step 2: Write the new division problem, making sure to write the decimal point straight above its current
location, and begin the division process. Since 1 and 12 cannot be divided by 68, determine the number
of times that 68 will go into 120 (1 time, with 52 left over). Write the 1 above the 0 in 120, and subtract
68 from 120.
Step 3: Bring the next zero straight down to turn the 52 into 520. The number 68 will go into 520 seven
times with 44 left over. Write the 7 to the right of the 1c6, and subtract 476 (68
7 = 476) from 520.
Step 4: Add another zero after the decimal point under the division bar and bring it straight down to turn
the 44 into 440. The number 68 will go into 440 six times with 32 left over. Write the 6 beside the
decimal point, and subtract 408 (68
6 = 408) from 440.
Step 5: The problem stated to round the answer to the tenths place, so the division must be taken out to
the hundredths place in order to determine how to round the number. Add another zero after the last
zero under the division bar, then bring it straight down to turn the 32 into 320. The number 68 will go
into 320 four times with 48 left over. Write the 4 next to the 6 and subtract 272 (68
4 = 272) from
320.
Step 6: Round 17.64 to the tenths place. The 4 in the hundredths place makes the 6 stay the same, so
17.64 rounded to the tenths place would be 17.6. NOTE: If the 4 had been a 5 or greater, the 6 would
have been rounded to a 7.
Answer: 17.6
Once the student is comfortable dividing whole numbers by decimal numbers, he or she will be ready to
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divide these numbers in the context of story problems.
Example 2: A quart of paint covers a 62.5 square foot area. If Jason wants to paint a wall that is 160
square feet, how many quarts of paint will he use?
(1) 160 ÷ 62.5 = ?
(2) 160 ÷ 62.5 = 2.56
Step 1: Read the problem to determine what operations are needed. To find the number of quarts of
paint used, divide 160 square feet by 62.5 square feet.
Step 2: Perform the division and simplify.
Answer: 2.56 quarts
Example 3: A bag contains 20 cups of sugar. If Alex needs 1.75 cups of sugar to make a batch of fudge,
how many batches of fudge can he make with the bag of sugar? Round your answer to the nearest
whole number, if necessary.
Step 1: Read the problem to determine what operations are needed. To find the number of batches of
fudge that can be made with 20 cups of sugar, divide 20 cups by 1.75 cups.
Step 2: Perform the division and simplify.
Step 3: Round the answer to the correct place value (nearest whole). The number 11.4 can be rounded
to 11.
Answer: 11 batches
An activity that can help reinforce this concept is to measure out 10 cups of sugar into a bowl. Give the
student a ? cup measuring utensil and explain that ? cup is equal to 0.25 cups. Next, have the student
count how many scoops of sugar it takes to empty the bowl of sugar. Finally, explain that when 10
cups of sugar are divided into 0.25-cup scoops, the number of 0.25 scoops is equal to 40. Therefore, 10
? 0.25 = 40.
Area of Triangle - B
This skill requires the student to find the area of a triangle, which is one half the area of a rectangle that has the
same base and height.
The area of a rectangle can be found by multiplying its base by its height.
If a rectangle is divided in half along its diagonal, each triangle formed is half the area of the rectangle.
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A regular triangle has three 60-degree angles, and sides that are all the same length. The area of a
regular triangle is also one half the area of a rectangle with the same base and height.
The formula for the area of a triangle is:
Remember: Units for area are always squared. Examples: ft2 , in.2 , m2
Example 1: What is the area of a triangle if its base is 13 ft and its height is 10 ft?
Step 1: Write the formula for the area of a triangle.
Step 2: Substitute 13 ft for the base and 10 ft for the height in the formula.
Step 3: Find the product of 13 ft and 10 ft.
Step 4: Multiply 130 by 1/2 (or divide 130 by 2).
Answer: 65 ft2
Example 2: The side of a hay storage building is in the shape of a triangle. What is the area of the side
of the building if its base is 10 ft, its height is 12 ft, and the lengths of the other two sides are 13 ft?
Step 1: Write the formula for the area of a triangle. As you can see from the formula, the lengths of the
other two sides, besides the base, are not needed to calculate the area.
Step 2: Substitute 10 ft for the base and 12 ft for the height in the formula.
Step 3: Find the product of 10 ft and 12 ft.
Step 4: Multiply 120 by 1/2.
Answer: 60 ft2
An activity that can help reinforce the concept of area of a triangle is to show students several examples
of triangles with the height and lengths of all three sides given. Ask them to write down the equations
that would allow them to find the area of each triangle, reminding them that the only two dimensions
they need are the height and base.
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Similar Figures - C
Similar figures are figures that have the same shape, but not necessarily the same size.
Imagine that you have reduced or enlarged a figure in a photocopy machine - the figure has the same
shape, but not the same size. In similar figures:
• corresponding angles are congruent (or equal)
• corresponding sides are proportional.
Similar triangles have the same angles, but the length of the sides are shorter or longer. However, the
length of the sides must be proportional. This means that if one of the sides is twice as long as the
corresponding side of the other triangle, then all the sides must be twice as long as the corresponding
sides of the other triangle.
Example 1: If we have a triangle with side lengths of 2, 3, and 4 and a larger similar triangle with the
shortest side equal to 6, what is the length of the other two sides of the triangle?
Solution: First, draw the figures so you can visualize the problem. Since the shortest side of the first
triangle is 2, we know that 2 x 3 is 6, so the other sides are 3 times the sides of the first triangle. The
other two sides are 9 and 12 (3 x 3 = 9 and 4 x 3 = 12).
We can also solve these types of problems using proportions.
Example 2: A rectangle has width equal to 10 feet and length equal to 16 feet. A similar rectangle has
length equal to 22 feet. What is its width of the second rectangle?
First, draw the figures so you can visualize the problem.
Step 1: Set up the proportion to solve for y.
Step 2: Cross multiply.
Step 3: Divide each side of the equation by 16.
Answer: 13.75 feet
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Accuracy - B
Accuracy and precision in measurement can be extremely important. There are two systems of measurement
that are commonly used: the metric system and the U.S. customary (or standard) system.
The Metric System:
The meter is the basis of length measurements in the metric system. Here is a basic breakdown of the metric
system of length.
1,000 millimeters (mm)
100centimeters (cm)
10 decimeters (dm)
1 dekameter (dam)
1 hectometer (hm)
1 kilometer (km)
=
=
=
=
=
=
1 meter (m)
1 meter (m)
1 meter (m)
10 meters (m)
100 meters (m)
1,000 meters (m)
The gram is the basis of weight measurements in the metric system. Here is a basic breakdown of the metric
system of weight.
1,000 milligrams (mg)
100centigrams (cg)
10 decigrams (dg)
1 dekagram (dag)
1 hectogram (hg)
1 kilogram (kg)
=
=
=
=
=
=
1 gram (g)
1 gram (g)
1 gram (g)
10 grams (g)
100 grams (g)
1,000 grams (g)
The liter is the basis of capacity measurements in the metric system. Here is a basic breakdown of the metric
system of capacity.
1,000 milliliters (ml)
100centiliters (cl)
10 deciliters (dl)
1 dekaliter (dal)
1 hectoliter (hl)
1 kiloliter (kl)
=
=
=
=
=
=
1 liter (l)
1 liter (l)
1 liter (l)
10 liters (l)
100 liters (l)
1,000 liters (l)
The U.S. Customary System:
The foot is the basis of length measurements of the U.S. customary system. Here is a basic breakdown of the
U.S. customary system of length.
12 inches (in) = 1 foot (ft)
1 yard (yd) = 3 feet (ft)
The pound is the basis of weight measurements of the U.S. customary system. Here is a basic breakdown of the
U.S. customary system of weight.
16 ounces (oz) = 1 pound (lb)
1 ton (ton) = 2,000 pounds (lb)
The gallon is the basis of capacity measurements of the U.S. customary system. Here is a basic breakdown of
the U.S. customary system of capacity.
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16 cups (C) = 1 gallon (gal)
8 pints (pt) = 1 gallon (gal)
4 quarts (qt) = 1 gallon (gal)
Comparing the Two Systems:
Length:
1 inch (in) = 2.54 centimeters (cm)
1 foot (ft) = 0.3048 meters (m)
1 yard (yd) = 0.9144 meters (m)
Weight:
1 ounce (oz) = 28.3495 grams (g)
1 pound (lb) = 453.59 grams (g)
Capacity:
1 cup (C)
1 pint (pt)
1 quart (qt)
1 gallon (gal)
=
=
=
=
0.2366
0.4732
0.9463
3.7853
liters
liters
liters
liters
(l)
(l)
(l)
(l)
Example 1: Choose the measure that is most precise.
A. 7.22 cm
B. 72 mm
C. 0.072 m
D. They are all of equal precision.
Step 1: It will be easiest to compare the measurements if they are all converted to the same unit.
• Convert 7.22 cm into meters by dividing by 100, since there are 100 cm in 1 meter.
7.22 cm ÷ 100 = 0.0722 m
• Convert 72 mm into meters by dividing by 1,000, since there are 1,000 mm in 1 meter.
72 mm ÷ 1,000 = 0.072 m
Step 2: Compare the three measures.
7.22 cm = 0.0722 m
72 mm = 0.072 m
0.072 m
Solution: Since 7.22 cm is carried out to more decimal places when the measurements are converted to the same
unit, it is the most precise.
Answer: A
Example 2: Choose the best estimate for the capacity of a cereal bowl.
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A. 2 l
B. 2 oz
C. 2 C
D. 2 gal
Solution:
•
•
•
•
2 liters is 1 large bottle of soda, so it is too large to be the capacity of a cereal bowl.
2 ounces are less than one cup, so it is much too small to be the capacity of a cereal bowl.
2 cups is less than two liters, but more than 2 ounces, so it is a possible capacity of a cereal bowl.
2 gallons is even larger than 2 liters, so it is definitely too large to be the capacity of a cereal bowl.
Answer: C
Spatial Relationships - B
Spatial relationships include geometric transformations. A transformation is a mapping of a point or shape to a
new location or orientation. Transformations include reflections, rotations, dilations, or translations. All
transformations except for dilations preserve the original size and shape of the image.
First, we'll define each of the transformations mentioned above:
A translation is sliding of a figure from one location to another. (Also called a "slide.")
For example:
A dilation is the image of a figure similar to the original figure. It can be thought of as shrinking or
enlarging a figure.
For example:
A reflection is flipping a figure across a line, just as you would reflect your hand in a mirror.
For example:
A rotation is the movement of a figure in a circular motion around a point. If you drew a figure on a
piece of paper, put the paper on the desk, and turned the paper, you would have a rotation.
For example:
It might be helpful to draw figures on a piece of paper, and rotate them to illustrate rotation of figures.
Example 1: What transformation was performed on the following figure?
The answer is a rotation.
We can use a coordinate plane to show where the parts of a shape are. If we draw the x- and y-axes, we
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divide the coordinate plane into four parts, each called a quadrant. The quadrants are numbered as
follows:
If we reflect a figure (triangle ABC) over the x-axis, what are the coordinates of the reflected figure?
(Use "prime" notation A' to identify the image. A' can be read "A prime.")
A translation moves a figure to another location. Below is triangle ABC moved 3 units left and 5 units
down. We can find the coordinates as follows:
Conversion: Variable Expressions/Words
Learning to convert variable expressions into words or words into variable expressions is an important problem
solving skill. In this tutorial, actual solutions to the problems will not be determined.
It is important for the student to understand the difference between an expression and an equation.
Expressions are variables or combinations of variables, numbers, and symbols that represent a mathematical
relationship. Expressions do not have equal signs, but can be evaluated or simplified.
Example: y - 6
Equations are expressions that contain equal signs. They can be solved, but not evaluated.
Example: n + 5 = 9
Writing Variable Expressions To Represent Word Phrases:
To represent a word phrase or story problem using algebraic symbols, there are three steps that the
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student should follow.
Step 1: Choose a variable to represent the unknown quantity, or use the variable provided. Some
problems will tell the student which variable to use.
Step 2: Look for key words in the phrase or story problem that indicate the use of a particular
operation. The chart below shows several key words with their corresponding operations.
Step 3: Use the chosen variable, relevant information, and operation key words to set up the variable
expression.
Example 1:
Translate the following story problem into an expression.
In August, a realtor sold 6 less than 4 times the amount of homes she sold in January. How many
homes did the realtor sell in August? Let h = the number of homes sold in January.
(1) h = the number of homes sold in January
(2) The key words are less than and times.
(3)
Step 1: Identify the variable.
Step 2: Look for key words and use the chart above to determine the necessary operations. Less than
implies subtraction and times implies multiplication.
Step 3: Set up the variable expression. Be sure to double check that all parts of the word phrase have
been represented. Also, it is important to recognize that subtraction, although usually stated first in the
word phrase, is normally placed at the end of the expression. This can be tricky for many students since
they are taught to read from left to right.
Answer: 4h - 6
Example 2:
Translate the following word phrase into an expression.
nine less than ten times the sum of a number, y, and three
(1) y = the number
(2) The key words are less than, times, and sum of.
(3)
Step 1: Identify the variable.
Step 2: Look for key words and use the chart above to determine the necessary operations. Less than
implies subtraction, times implies multiplication, and sum of implies addition (usually as a quantity
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written in parentheses).
Step 3: Set up the variable expression. Be sure to double check that all parts of the word phrase have
been represented. Also, it is important to recognize that subtraction, although usually stated first in the
word phrase, is normally placed at the end of the expression.
Writing Word Phrases To Represent Variable Expressions:
A similar process is used to write word phrases from variable expressions. In most cases, the student
will be provided with a choice of word phrases as opposed to actually having to create one. This is
because a variety of different phrases could accurately represent one expression. The student has two
options.
Option 1: The student can review the answer choices and apply the same three steps listed above to
determine the corresponding variable expression.
Option 2: The student can review the variable expression determining the operations used, values, and
order in which it is set up. This information will guide the student to the correct word phrase.
Example 3: Translate the following expression into words.
(A) the difference of half the peanuts, p, and four
(B) four plus twice the number of peanuts, p
(C) the sum of one-half the number of peanuts, p, and four
(D) one-half the sum of the number of peanuts, p, and four
Solution:
The student should rule out option (A) because it mentions difference which implies subtraction. This
description would represent the expression ?(p) - 4.
The student should rule out option (B) because it mentions twice which implies multiplying by 2 instead
of ?. This description would represent the expression 2p + 4.
The student should rule out option (D) because it represents taking one half of the sum, which means
you have to add the number, p, and four first, and then multiply by one-half. This description would
represent the expression ?(p + 4).
Answer: The correct answer is option (C).
Convert: Fahrenheit to Celsius
Converting degrees Fahrenheit to degrees Celsius is the process of changing a temperature reading from the
U.S. customary system to the metric system. In the customary system, the Fahrenheit scale, water freezes at
32º F and water boils at 212º F. In the metric system, the Celsius scale, water freezes at 0º C and water boils at
100º C. Have the student examine the chart below in order to gain a better understanding of how the two
systems relate to one another.
When changing from degrees Fahrenheit to degrees Celsius, it is necessary to use the following formula.
Remembering order of operations, the student will first find the difference of a given Fahrenheit
temperature and 32, and then multiply that result by 5/9. This will result in the Celsius temperature.
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Example 1: Convert 17º F to º C. Round your answer to the nearest tenth, if necessary.
Step 1: Substitute the Fahrenheit value into the conversion formula.
Step 2: Working in the parentheses first, calculate 17 - 32. The result is - 15.
Step 3: Multiply (5/9) and (- 15). Reduce by dividing - 15 and 9 by 3. This will make the
multiplication easier. The result is - 25/3.
Step 4: Convert - 25/3 into a decimal by dividing the numerator (- 25) by the denominator (3). The
result is - 8.33333 which is equal to - 8.3 repeating. Rounding to the nearest tenth would yield - 8.3.
Answer: - 8.3º C
As an additional activity, place a Fahrenheit thermometer outside where it can easily be seen from the
house. Have the student take two readings a day, one in the morning and one at night. Then, have the
student convert the temperatures to the Celsius scale. This will allow the student to work with actual
measurements and track the weather over a given interval of time. If a thermometer is unavailable, the
high and low temperature can be read from the newspaper, television news, or the Internet.
Convert: Celsius to Fahrenheit
Converting degrees Celsius to degrees Fahrenheit is the process of changing a temperature reading from the
metric system to the U.S. customary system. In the customary system, the Fahrenheit scale, water freezes at
32º F and water boils at 212º F. In the metric system, the Celsius scale, water freezes at 0º C and water boils at
100º C. Have the student examine the chart below in order to gain a better understanding of how the two
systems relate to one another.
The formula for converting degrees Celsius(C) to degrees Fahrenheit(F) is:
The following are examples of problems converting temperatures given in Celsius to Fahrenheit.
Example 1: Convert 15º C to º F.
Step 1: Substitute 15 in for C.
Step 2: Multiply 9/5 and 15.
Step 3: Add 27 to 32.
Answer: 59 º F
Example 2: A digital thermometer reads 21.4º C. What is the equivalent Fahrenheit temperature?
Round your answer to the nearest degree.
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.
Step 1:
Step 2:
Step 3:
Step 4:
.
Substitute 21.4 in for C.
Multiply 9/5 and 21.4
Add 38.52 and 32.
Round your answer to the nearest degree.
Answer: 71º F
As an additional activity, place a Celsius thermometer outside where it can easily be seen from the
house. Have the student take two readings a day, one in the morning and one at night. Then, have the
student convert the temperatures to the Fahrenheit scale. This will allow the student to work with
actual measurements and track the weather over a given interval of time. If a thermometer is
unavailable, the high and low temperature can be read from the newspaper or the Internet.
Experimental Probability
Probability is the ratio of the number of times a certain outcome can occur to the number of total possible
outcomes. The probability of an event cannot be smaller than zero or larger than one. A probability of zero
means there is no chance of an event occurring and a probability of one means that an event is certain to occur.
Example 1:
If there are 6 green marbles, 3 orange marbles, 2 blue marbles, and 1 black marble in a bag, what is the
probability that an orange marble will be blindly pulled from the bag first?
Solution:
Start with the number of marbles in the bag. There is a total of 12. Then, figure out how many
marbles are orange. There are 3. The probability ratio is 3/12, or, in reduced form, 1/4.
Answer:
At times, experimental probability questions involve working with a standard deck of cards. Therefore,
it is important for the student to know the composition of a deck of cards. A standard deck contains
fifty-two cards. It also contains four different suits: spades, clubs, diamonds, and hearts. Each suit has
thirteen cards. The suits of diamonds and hearts are red cards and the suits of spades and clubs are black
cards.
Example 2:
If you were to draw a playing card from a standard deck of 52 cards, what is the probability of drawing a
3 of diamonds?
Solution:
Since there are 52 cards in the deck, and there is only one 3 of diamonds, the probability is 1/52.
Answer:
Experimental Probability is found by gathering data through observation or experimentation. The ratio
to determine experimental probability is:
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Experimental probability is not determined using theoretical data. It is strictly the result of the
experiments that are performed. For example, if a coin is flipped, the mathematical probability of
getting heads is ?, as it is equally likely to get heads or tails. It is unlikely that if an actual experiment
was performed, the results will come out to be exactly ?. If a coin is flipped ten times, it might land on
heads seven times, not five as would be expected mathematically. It is important to remember that the
more times an experiment is performed, the closer the result will be to the true mathematical (called
theoretical) probability.
Example 3:
Jill was playing cards with her friends and was dealt two pairs twice during the thirteen games that they
played. Based on this information, what is the experimental probability that Jill will be dealt two pairs
on the next hand?
Solution:
The experimental probability is the ratio of the number of times Jill was given two pairs, 2, to the
number of times she played the game, 13.
Answer:
Example 4:
Joon kept track of the minutes that he worked late at his job last month in the following chart.
Based on this information, what is the experimental probability that he will work 30 minutes late the
next day he has to work?
Solution:
The experimental probability is the ratio of the number of times Joon worked 30 minutes late, 5, to the
total number of days when he was keeping track of his hours, 21.
Answer:
An activity that may help to reinforce this skill could be to buy a large bag of some type of different
colored, small candies. Empty the candy into a large paper bag. Have the students choose their
favorite color of those candies. Then have them pick out 10 candies and count out the number of
candies of that color and the total number of candies that they pull out. From this information, help
them determine the experimental probability that the next candy that they pull out will be their favorite
color. Replace the candy into the bag. Repeat the experiment two more times, the next time pulling
out 5 candies and the third time pulling out 20 candies.
Measures of Central Tendency: Apply
The study of statistics involves looking at the measures of central tendency. Measures of central tendency are
numerical values used to describe the overall clustering of data in a set. They include mean, median, and
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mode. Although range is a measure of spread, it is often included when learning measures of central tendency.
This study guide will focus on the application of these measures within real world problems.
It is important to review the definitions and calculation procedures for determining mean, median, mode, and
range with the student. This will help to prepare the student before progressing to real world problems.
The mean of a group of numbers is found by adding the numbers and then dividing the sum by the number of
addends (items in the group). It is also referred to as the average.
The median is the middle number, or the mean of the two middle numbers, of a group of numbers ordered
sequentially.
The mode of a group of numbers is the number that occurs most often. Sometimes the mode is referred to as
the most typical number. If every number only occurs once, there is no mode. If more than one number
occurs with the same frequency (other than once), there can be multiple modes.
The range of a group of numbers is the difference between the highest number and the lowest number. The
range is the number that tells how widespread the data is.
Example 1: Find the mean of 15, 10, 5, 20, 5.
Solution:
(1) 15 + 10 + 5 + 20 + 5 = 55
(2) 55 ? 5 = 11
Step 1: Find the sum of the numbers.
Step 2: Divide by 5, the total number of addends (items in the group).
Answer: The mean is 11.
Example 2: Find the median of 9, 1, 3, 5, 0, 7.
Solution:
(1) 0, 1, 3, 5, 7, 9
(2) 3 and 5
(3) 3 + 5 = 8
8?2=4
Step 1: Put the numbers in order from least to greatest.
Step 2: Determine the middle number(s). Since there is an even amount of numbers, there are two middle
numbers. If there is an odd amount, there will be just one, and that will be the median.
Steps 3: Determine the mean of 3 and 5 by adding the two numbers together and dividing by 2.
Answer: The median is 4.
Example 3: Find the mode of 0, 6, 7, 9, 6, 4, 3.
Solution:
(1) 0, 3, 4, 6, 6, 7, 9
(2) 0, 3, 4, 6, 6, 7, 9
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Step 1: It may be easiest to list the numbers in order, allowing the student to clearly see numbers that repeat.
Step 2: Determine the number that occurs most often.
Answer: The mode is 6 because it occurs twice and the other numbers only occur once.
REMEMBER: There can be zero, one, or multiple modes.
Example 4: Find the range of 9, 5, 16, 8, 2, 3.
Solution:
(1) 16 - 2 = 14
Step 1: Subtract the lowest value in the set from the highest value.
Answer: The range is 14.
Using Measures of Central Tendency Within Real World Problems:
A common mistake students make is calculating the wrong measure of central tendency because they have
forgotten which one is which. A couple of easy tricks to help the student remember these definitions are:
•The word mode has the same amount of letters and starts the same way as the word most. Therefore, mode
should remind the student to find the number that occurs the most.
•The word median contains the prefix 'med', just like the word medium, which is in the middle of small and
large. Also, it may help if the student remembers that a median on a street is the area in the middle of the
lanes. Therefore, median should remind the student to find the number that is located in the middle.
In this skill, students will be required to find and use the most appropriate measure of central tendency in a real
world scenario.
Example 5:
The average high temperatures in London, England are given for each month of the year: 44º F, 45º F, 50º F,
55º F, 61º F, 67º F, 71º F, 72º F, 66º F, 59º F, 50º F, and 46º F. Which measure is most appropriate to use to
make the temperature appear the lowest?
(1) range: 72 - 44 = 28
(2) mean: 44 + 45 + 50 + 55 + 61 + 67 + 71 + 72 + 66 + 59 + 50 + 46 = 686. 686 ÷ 12 =
57.167.
(3) median: 44, 45, 46, 50, 50, 55, 59, 61, 66, 67, 71, 72. 55 + 59 = 114. 114 ÷ 2 = 57.
(4) The mode is 50.
(5) 57.167 > 57 > 50 > 28
Step 1: Determine the value of the range. Subtract the smallest number from the largest number: 28º .
Step 2: Determine the value of the mean. Add all the numbers in the data set together, to get 686. Then
divide this by the number of addends in the data set, in this example, 12. The mean, when rounded to the
nearest thousandth is 57.167º .
Step 3: Determine the median. Write all of the numbers in order and determine the middle value. Because
there is an even number of values, it is the mean of the middle two numbers, 55 and 59. The median is equal to
57º .
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Step 4: Determine the mode. The number that occurs most often in the set is 50 because it is the only number
in the set given more than once. Therefore, the mode is equal to 50º .
Step 5: Compare the four values to determine which has the lowest value and is the most appropriate. In this
case, the lowest value was 28 which was the range, but the range is not the most appropriate measure to use
because it is a measure of the spread of the temperatures rather than a measure of the data between the highest
and lowest numbers. Also, the range is much lower than any of the temperatures in the data set. The next
lowest number is 50, the mode.
Answer: The mode is the most appropriate measure to use to make the temperature appear the lowest.
Example 6:
The number of grams of carbohydrates from a variety of candy bars are given: 28, 25, 27, 25, 36, 38, 59 and
32. Which measure best represents the number of grams of carbohydrates in the candy bars?
(1) range: 46 - 25 = 21
(2) mean = 28 + 25 + 27 + 25 + 36 + 38 + 59 + 32 = 270. 270 ÷ 8 = 33.75
(3) median = 25, 25, 27, 28, 32, 36, 38, 59. 28 + 32 = 60. 60 ÷ 2 = 30.
(4) mode = 25
Step 1: Determine the range (21).
Step 2: Determine the mean (33.75).
Step 3: Determine the median (30).
Step 4: Determine the mode (25).
Solution: Now, the numbers must be interpreted to determine the one that best represents the data.
The range is smaller than the other numbers in the data set, so it does not the best represent the data.
The mode is the smallest number in the data set, so it does not the best represent the data.
The mean is made higher by the 59 (which is an outlier: a number that is much larger or much smaller than the
other numbers in the data set), so it does not the best represent the data.
The median has exactly four numbers on each side of it and is not affected by the 59, so it best represents the
data.
Answer: The number of grams of carbohydrates is best represented by the median.
Example 7:
The number of gold medals that the United States has won in the last several Olympics games are 34, 36, 45,
33, 34, 83, 36, 37, 44, and 40. Which measure will show how widespread this data is?
Answer: The range provides information about the spread of data.
As an additional activity, have the students calculate the mean, median, mode, and range of a variety of grocery
or department store receipts. If the receipts are long and contain too much data, restrict the calculations to just
certain items. Finally, have the students determine which measure of central tendency to use if they want to
make the prices appear the highest or the lowest, if they want to explain how widespread the prices are, and if
they want the most typical value on the receipt.
Area of Parallelogram - B
A parallelogram is a quadrilateral (a four-sided figure) with two pairs of parallel and congruent sides. Area is
the measure, in square units, of the interior region of a two-dimensional figure.
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To find the area of a parallelogram, multiply the length of the base (b) by the height (h). The base is
one of the sides of the parallelogram. The height is the length of the segment going from the base at a
right angle (or perpendicular) to the opposite side. Here is the formula:
Area of a parallelogram = base
height
Example 1: Find the area of the parallelogram.
Solution:
The formula for the area of a parallelogram is Area = base
height. The height of this parallelogram
is 3.5 cm and the base is the length of the side that the height is perpendicular to, in this case, 8 cm.
Therefore, the area of the parallelogram is 3.5 cm
8 cm = 28 cm2
Answer: 28 cm2 .
One way to help the student reinforce the concept of finding the area of parallelograms is to use a ruler
to draw a few parallelograms. Have the student measure the base and height of the parallelograms and
then calculate the area using the formula given above. Also, try to find parallelograms in real world
figures (such as those in designs) that can be measured so the area can be computed.
Equivalent Fractions - B
A fraction is made up of two parts: a numerator and a denominator. The numerator is the number on the top of
the fraction and the denominator is the number on the bottom of the fraction. For example, in the fraction 4/5, 4
is the numerator and 5 is the denominator. A mixed number is a combination of a whole number and a fraction.
An improper fraction is a fraction in which the numerator is larger than or equal to the denominator. An
improper fraction can be rewritten as a mixed number or as a whole number.
Example 1: Find the missing number.
Solution: Convert 4 2/3 into an improper fraction by multiplying the whole number by the denominator
(4 x 3 = 12), then adding that product to the numerator (12 + 2 = 14). 4 2/3 can be written as 14/3.
The missing number is 14.
Example 2: Find the missing number.
Step 1: Replace the question mark with the variable, 'N'. Then cross multiply to begin the process of
determining the missing number. Multiply the denominator of the first fraction by the numerator of the
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second fraction (N x 36 = 36N). Next multiply the denominator of the second fraction by the numerator
of the first fraction (12 x 12 = 144).
Step 2: Place an equal sign between the two products. Divide both sides of the equation by 36.
144 ÷ 36 = 4
Answer: N = 4
Coordinate Geometry - C
A coordinate graph is used to name the position of points. The x-coordinate (horizontal) is listed first and the
y-coordinate (vertical) is listed second. For example, the coordinate pair (3, 2) is at the horizontal position 3 and
the vertical position 2.
It may be helpful to use graph paper to develop a coordinate graph. Help the student plot points on the
graph and determine the coordinate pair.
Example 1: What is the ordered pair for point J?
Answer: (2, -2) because the point J is 2 units over and 2 units down.
Circle Graphs - B
A graph is a visual aid used to show and compare information. A circle graph, or pie chart, is often used to
show data given in percentages. An interesting method for increasing the student's understanding of graphs is
to help him or her develop a graph for a school project or event.
Have the student create a circle graph for the events that typically occur during a normal day. Remind
the student that the total for a circle graph or pie chart is always 100% or one whole. So, if he or she
spends half of his or her day at school, then half (50%) of the circle would be filled with the title
"School." Similarly, if one-quarter (25%) of his or her day is spent at the soccer field, then a quarter of
the pie would be titled "Soccer." The following is an example:
Example 1: This circle graph represents how Chelsea spends her time each day. Use the circle graph to
answer the question.
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How much of her time does Chelsea spend practicing volleyball and shopping?
(1) Volleyball Practice = 15%, Shopping = 5%
(2) 15% + 5% = 20%
Step 1: According to the circle graph, Chelsea spends 15% of her time at volleyball practice and 5% of
her time shopping.
Step 2: To find the total amount of time, add the percentages together.
Answer: Chelsea spends 20% of her day practicing volleyball and shopping.
Example 2: The circle graph below shows the distribution of types of donuts prepared each morning
by a local donut shop. Use the circle graph to answer the question.
What percentage of the prepared donuts contain jelly?
(1) Glazed = 32%, Chocolate = 14%, Apple = 11%, Peanut = 8%, Cream = 22%
(2) 32% + 14% + 11% + 8% + 22% = 87%
(3) 100% - 87% = 13%
Step 1: List all of the existing percentages shown in the circle graph.
Step 2: Since the percentages in a circle graph must always add up to 100%, begin by adding the
existing percentages.
Step 3: Next, subtract the total from step 2 (87%) from 100%.
Answer: 13% of the prepared donuts contain jelly.
Example 3: The circle graph below shows the distribution of the types of dinner rolls served each
night at the Bread Basket Restaurant. Use the circle graph to answer the question.
If the desire for poppy seed rolls increases by 5%, which of the following graphs could accurately show
this change?
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Choice (A) is not the answer because although poppy seed increased to 20%, none of the other types
decreased. Therefore, the total of the graph is now over 100%.
Choice (B) is the answer because the new percentages reflect the increase in the desire for poppy seed
rolls, while there is also a decrease in the desire for sourdough. Therefore the percentages add up to
100%.
Choice (C) is not the answer. Although poppy seed increased to 20%, the others did not adjust
accordingly. Therefore, the total of the graph is now over 100%.
Choice (D) is not the answer. Although poppy seed increased to 20%, the others did not adjust
accordingly. Therefore, the total of the graph is now over 100%.
Answer: Choice B.
Subtract Decimals: Hundred Thousandths
Subtracting decimal numbers with more than one decimal position (columns of numbers) is very similar to
subtracting whole numbers. Subtracting decimal numbers requires the ability to regroup (carry, borrow, or
rename) when the number being subtracted is greater than the other number.
Example: 7.36925 - 4.8217 =
Step 1: Write the problem vertically. Remember to line up the decimal points and place a zero at the end
of 4.8217 to hold the hundred thousandths place.
Step 2: Subtract the numbers in the hundred thousandths column (5 - 0 = 5). Place the 5 in the hundred
thousandths place.
Step 3: Before the numbers in the ten thousandths column can be subtracted we must "borrow" or
"trade" from the thousandths column. Cross out the 9 and make it an 8, then make the 2 in the ten
thousandths column a 12. Now, subtract the numbers in the ten thousandths column (12 - 7 = 5). Place
the 5 in the ten thousandths place.
Step 4: Subtract the numbers in the thousandths column (8 - 1 = 7). Place the 7 in the thousandths place.
Step 5: Subtract the numbers in the hundredths column (6 - 2 = 4). Place the 4 in the hundredths place.
Step 6: Before the numbers in the tenths column can be subtracted, we must "borrow" or "trade" from
the ones column. Cross out the 7 and make it a 6, then make the 3 in the tenths column a 13. Now,
subtract the numbers in the tenths column (13 - 8 = 5). Place the 5 in the tenths place.
Step 7: Subtract the numbers in the ones column (6 - 4 = 2). Place the 2 in the ones place.
Answer: 7.36925 - 4.8217 = 2.54755
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Divide Decimals by Whole Number
Dividing a decimal number by a whole numberis very similar to dividing whole numbers. The decimals point
must remain in the same position in the answer.
Example: Solve 18.9 divided by 9 = ?
Step 1: Write the problem in long division format.
Step 2: Division follows the same format as with whole numbers. 9 goes into 18 two times because 9 x 2
= 18. Place 2 in the ones position. Subtract 18 from 18 resulting in 0. Bring down the 9.
Step 3: Place the decimal point.
Step 4: 9 goes into 9 one time because 9 x 1 = 9. Place 1 in the tenths position. Subtract 9 from 9
resulting in 0.
The answer is 2.1.
Story Problems: Decimals
Decimal story problems (word problems) require students to read passages, determine the question being asked,
identify the elements needed to solve the problem, decide on the correct operation or operations (addition,
subtraction, multiplication, division), and find a solution.
Decimal story problems are especially difficult because students must master addition, subtraction,
multiplication, and division of decimals before story problems can be understood.
When the student has mastered decimal calculations, it will be helpful to begin to talk with him or her
about the different ways decimals are used in real world scenarios: taxes, interest, measurements,
money, etc. Next, encourage the student to write his or her own decimal story problems by using the
following pattern: 1) Choose an operation: addition, subtraction, multiplication, or division. 2)
Choose a decimal scenario: purchasing merchandise or measuring. For instance, if the student
chooses addition of decimals in a purchasing scenario, he or she could write a decimal story like this
one.
Example: Mary Ann spent $3.26 on two candy bars. Michelle bought a pack of gum for $.85 and a
magazine for $2.95. How much money did Mary Ann and Michelle spend?
Step 1: Rewrite the problem vertically. Always line up the decimal points.
Step 2: Add the numbers in the hundredths position (6 + 5 + 5 = 16). Write the 6 in the hundredths
position. Carry the 1 to the next column (tenths).
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Step 3: Add the numbers in the tenths column, including the number carried over from the previous
column (1 + 2 + 8 + 9 = 20). Write the 0 in the tenths position. Carry the 2 to the next column (ones).
Step 4: Bring the decimal point down.
Step 5: Add the numbers in the ones position, including the number carried over from the previous
column (2 + 3 + 2 = 7). Write the 7 to the left of the decimal point. Bring down the dollar sign.
Answer: $3.26 + $0.85 + $2.95 = $7.06
Fractions: Estimation - C
An estimate is an approximate calculation. In other words, an estimate is close to the actual answer, but not
exact. When estimating fractions, it is important to understand how fractions work. The number on top of the
fraction is called the numerator and the number on the bottom of the fraction is called the denominator.
When estimating a fraction, one of the most important concepts is the ability to determine whether a
fraction is larger than or smaller than one-half, one-fourth, or three-fourths. This ability allows you to
"round" a fraction to the closest "common" fraction -- 1/4, 1/2, 3/4, or 1.
Example 1:
To solve this problem, it may be helpful to draw the following pictures.
The first picture represents 5/12 of a pizza. The second picture represents 1/2 of a pizza. Since 5/12 of a
pizza is close in value to 1/2 of a pizza, we can estimate that Charlie has 1/2 of a pizza.
We have estimated that Charlie has 1/2 of a pizza. Now we need to determine how much pizza each
friend will get. To do this, we divide 1/2 by 2.
Step 1: Write the division problem out. Remember to place a denominator of 1 under the whole number
2 to make the whole number a fraction.
Step 2: To divide two fractions, the rule is "flip the second fraction and multiply." When we flip 2/1 we
get 1/2. Now we multiply 1/2 by 1/2 to get 1/4. Remember, when multiplying fractions, multiply the
numerators together then multiply the denominators together.
Example 2:
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To solve this problem, it may be helpful to draw the following pictures.
The first picture represents 5/6 of a cup. Since 5/6 is closest to 3/4 cup, we can estimate that 5/6 is 3/4
cup. The second picture represents 2/9 and the third picture represents 1/4. Since 2/9 and 1/4 are close to
the same amount, we can estimate that 2/9 is 1/4.
We have estimated that 5/6 is 3/4 and 2/9 is 1/4. To determine the approximate amount of raisins that
Dehlia will need, we add 3/4 and 1/4. 3/4 + 1/4 = 1
Answer: Dehlia will need about 1 cup of raisins to make both recipes.
Expressions: Addition
Expressions are number sentences which do not have equal signs, but need to be evaluated or simplified.
Example: y - 6
An equation is a number sentence that does have an equal sign.
Example: y - 6 = 14
Example 1: Evaluate the expression x + 23, when x = 5.
5 + 23
Solution: Substitute the value 5 in place of x in the expression.
Answer: 28
Example 2: For x = -7, find 2x + -12.
(1) 2(-7) + -12
(2) -14 + -12
(3) -26
Step 1: Substitute -7 in for the value of x.
Step 2: Multiply 2 by -7 and rewrite the expression with the new value.
Step 3: Add -14 and -12.
Answer: -26
Example 3: Write a mathematical expression to represent the following:
The sum of a number and 23.
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Solution: Remember that "sum" is the answer to an addition problem, so the expression is x + 23.
Answer: x + 23
Expressions: Multiplication
Expressions are number sentences which do not have equal signs, but need to be evaluated or simplified.
Example: y - 6
An equation is a number sentence that does have an equal sign.
Example: y - 6 = 14
Example 1: Evaluate the expression below for x = 18.
5 + 3x
(1) 5 + 3(18)
(2) 5 + 54
(3) 59
Step 1: Replace x with 18.
Step 2: Multiply 3 by 18 to get 54.
Step 3: Add 5 and 54.
Answer: 59
Example 2: Write a mathematical expression that represents the following word expression.
four times a number less 6
(1) four times x less 6
(2) 4x less 6
(3) 4x - 6
Step 1: Replace the words "a number" with a variable (x was chosen)
Step 2: "four times x" can be written as 4x. Make this replacement.
Step 3: "less 6" can be written as "- 6." Make this replacement and the expression is complete.
Answer: 4x - 6
Expressions: Story Problems
Story problems for equations (word problems) require students to read passages, determine variables, write
equations, and solve.
Expressions are number sentences which do not have equal signs, but need to be evaluated or simplified.
Example: y - 6
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Equations are number sentences which contain equal signs.
Eample: n + 5 = 9
Example: The length of John's model airplane is 10 inches more than twice the width. The width is 15 inches.
What is the length of John's model airplane?
(1) L = 2W + 10
(2) L = 2(15) + 10
(3) L = 30 + 10
(4) L = 40
Step 1: The length is equal to two times the width plus 10. Develop a formula with W = Width and L =
Length.
Step 2: Replace W with 15.
Step 3: Calculate the right side of the equation by adding 30 and 10.
Answer: The length of John's model airplane is 40 inches.
Distance/Rate/Time
The formula for solving distance (D), rate (R), and time (T) is:
D=RxT
Students should be able to read story problems, decipher two elements of the distance formula, plug the
elements in the distance formula, and solve.
The following are examples of problems requiring the D = R x T formula.
Example 1: If Carson drove 55 miles per hour on the freeway for 495 miles, how long did he drive?
Step 1: Determine the distance, rate, and time values and substitute them into the D = R x T formula. D
= 495, R = 55, and T = ?
Step 2: Solve for T (Time) by dividing both sides of the equation by 55.
Answer: Carson drove for 9 hours.
Example 2: Stan ran for 164.5 hours. He ran a total of 2500 miles. What was the rate of speed that Stan
ran? (Round miles per hour to the nearest hundredth).
Step 1: Determine the distance, rate, and time values and substitute them into the D = R x T formula. D
= 2500, R = ? and T = 164.5.
Step 2: Solve for R (Rate) by dividing both sides of the equation by 164.5.
Step 3: Round the answer to the nearest hundredth.
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Answer: Stan ran at a rate of 15.20 miles per hour.
Bar Graphs - D
A bar graph is a visual aid used to show and compare data. A bar graph has rectangular bars of various lengths
which represent specific information.
An interesting method for increasing the student's understanding of graphs is to help him or her develop
a graph for a school project or event, such as a magazine sale. The following is an example of a bar
graph.
A double bar graph is used to display two sets of data. The following is an example of a double bar
graph. It compares the amount of miles that Sarah ran each day to the amount of miles that Sally ran
each day.
Example 1: On which days did Sarah run more miles than Sally?
Answer: Sarah ran more miles than Sally on Tuesday and Wednesday. On these days, the bar
representing Sarah's mileage is higher than the bar representing Sally's mileage.
Example 2: How many total miles did Sally run?
(1) On Monday, the bar representing Sally's mileage goes to 4 miles.
(2) On Tuesday, the bar goes to 2.5 miles.
(3) On Wednesday, the bar goes to 1 mile.
(4) On Thursday, the bar goes to 3.5 miles.
(5) On Friday the bar goes to 6 miles.
(6) Add these five numbers together: 4 + 2.5 + 1 + 3.5 + 6 = 17.
Answer: Sally ran a total of 17 miles.
Tables - E
Tables are created to communicate information visually. Students are expected to interpret data from tables.
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A creative method for improving the student's data interpretation skills is to use actual tables from
magazines or books. Help the student understand the information provided in these tables.
The following is a sample multiplication table. The top row contains even numbers. The left column
contains odd numbers.
Example 1: What is 6 times 3?
Select a number from the top row (6) and a number from the left column (3). Follow the top number
down and the left column number across until the two meet (18).
Answer: 18.
The following table shows the average amount of students attending Mr. Ranney's math class. The top
row lists the different months, September through January. The left column lists the period numbers 1
through 6.
Example 2: Use the table above to answer the question. Which class period had the highest average in
October?
Look at the column beneath October. The highest number, 35, is from period 6.
Answer: Period 6.
Example 3: Use the table above to answer the question. Which is lower, the attendance in period 2 in
January or the attendance in period 4 in November?
(1) Select period 2 from the left column and January from the top row. Follow the left column across
and the top row down until they meet at 32.The attendance in period 2 in January was 32.
(2) Select period 4 from the left column and November from the top row. Follow the left column across
and the top row down until they meet at 28.
The attendance in period 2 in January was 32.
The attendance in period 4 in November was 28.
Answer: Since 28 is less than 32, the lower attendance was in period 4 in November.
Exponential Notation - C
An exponent is a number that represents how many times the base is used as a factor. The base number 5 to the
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3rd power translates to 5 x 5 x 5 which equals 125. 5 to the 3rd power is not 5 x 3. To perform operations with
exponents, all exponential properties must be understood.
Have the student find the equivalent whole number forms of these exponential numbers:
Scientific Notation:
Scientific notation is based upon exponential properties and is used to communicate very large or very
small numbers. Scientific notation deals with significant digits. The most significant digit in a number is
the first non-zero digit in the number (reading from left to right). To write a number using scientific
notation, place the decimal point to the right of the most significant digit and count the digits between
the new placement of the decimal point and the old placement of the decimal point. The number of
places that the decimal point moved will be represented by a power of 10.
Example 1: Write 123,000,000 using scientific notation.
Step 1: Determine where the decimal point is in the number to be written using scientific notation.
Step 2: Place the decimal point to the right of the most significant digit and count the number of places
the decimal point was moved.
Step 3: Write all of the significant digits (with the decimal in the new postion) and multiply by 10 to a
power. The power on the ten is the number of places that the decimal point was moved. The power is
positive because the decimal point was moved from the right to the left.
Answer:
To take a number out of scientific notation, move the decimal point the same number of places as the
exponent in the power of ten.
Example 2:
Answer: 62,900,000 (move the decimal 7 places to the right)
Example 3: Find the equivalent form.
Step 1: Write the number in the first set of parentheses in standard form and rewrite the expression.
Step 2: Write the number in the second set of parentheses in standard form and rewrite the expression.
Step 3: Add 6,000 and 200 to get 6,200.
Answer: 6,200
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Scientific Notation
Scientific notation is a condensed way to write very large or small numbers without including each digit.
Scientific notation is a number written as the product of a number between 1 and 10 and a power of 10.
To write a large number using scientific notation, count the digits (from right to left) to be represented
by a power of 10. 123,000,000 can be written in scientific notation as 1.23 x 10 to the 8th power. To
write a small number, count the digits from left to right. To undo scientific notation, move the decimal
point the same number of places as the exponent in the power of ten.
Example 1:
Answer: 62,900,000 (move the decimal 7 places to the right)
Example 2:
Answer: The missing exponent would be 4.
Radicals and Roots
Mastering roots and radicals is an essential step toward learning advanced mathematics concepts. A radical
sign looks like a check mark with a line attached to the top. The radical sign is used to communicate square
roots.
The following rules are required to perform operations with roots and radicals.
1. If x multiplied by x equals y, then x is a square root of y. For example, 6 multiplied by 6 is 36, so 6
is a square root of 36. In fact, 36 is called a perfect square because its square root, 6, is a whole
number. Most algebra text books contain a table of perfect squares.
2. -3 and 3 are both square roots of 9 because -3 x -3 = 9 and 3 x 3 = 9. 3 is referred to as the principal
square root because it is the positive square root of 9.
3. To find the simplest radical form of a radical expression, factor the number under the radical sign (the
radicand). The square root of 45 could be factored to be the square root of 9 multiplied by the square
root of 5. The square root of 9 multiplied by the square root of 5 can be simplified further by finding
the square root of 9. The result is 3 (the square root of 9) multiplied by the square root of 5.
Example 1: Find the equivalent form.
Solution: Multiply the numbers under the radical symbols.
Example 2: What symbol would best replace the ? in the given statement?
A. =
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B. <
C. >
There are two methods that can be used to solve this problem. Each method is shown and explained
below.
Solution Method 1:
Step 1: Simplify the square root of 12 by making it the square root of 6 x 2.
Step 2: Further simplify the square root of 6 x 2 by making it the square root of 3 x 2 x 2. (If you
multiply 3 x 2 x 2, you will get 12.)
Step 3: The square root of 3 x 2 x 2 becomes 2 times the square root of 3, because the square root of 2 x
2 is 2 and the 3 must remain under the square root symbol.
Step 4: Two times the square root of 3 can also be written as the square root of 3 plus the square root of
3.
Step 5: Now, we can make a comparison. We know that the larger a number is, the larger that number's
square root will be. We can determine that the square root of 5 plus the square root of 7 will be greater
than the square root of 3 plus the square root of 3 because 5 and 7 are both larger than 3.
Answer: C
Solution Method 2:
Step 1: Estimate the square root of 5, the square root of 7, and the square root of 12. This estimation can
be done using a calculator.
Step 2: Add together the 2.24 and the 2.65 to get 4.89.
Step 3: Replace the question mark with the > symbol because 4.89 is greater than 3.46.
Answer: C
Example 3: Solve for the value of x.
Step 1: The square root of 36 is 6 because 6 x 6 = 36.
Step 2: Subtract 6 from each side of the equation to isolate the square root of x.
Step 3: The square root of x is equal to 8. We can replace the x with 64, since the square root of 64 is 8.
Step 4: Since the square root of 64 equals 8, the value of x is 64.
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Answer: x = 64
Scale Drawing - B
A scale drawing represents an object's actual proportions, but in a smaller size. The scale is a ratio that
compares the measurement on a map or drawing to the actual measurement.
It may be helpful to have the student develop his or her own scale drawing for either a room or your
home.
Example 1:
The scale of this drawing is: 2 centimeters equal 5 meters. What is the width of the bedroom?
Solution: One method is to use the ratio of the scale to determine the unknown length. We can use the
ratio to write a proportion, using a variable to represent the width of the bedroom. From the scale
drawing, we know the scale width of the bedroom is 6 cm. So we write the proportion as follows:
Step 1: Write the appropriate proportion. Let w represent the width of the bedroom.
Step 2: Write the cross products. Multiply w by 2 and multiply 5 by 6.
Step 3: Rewrite the equation with the new values.
Step 4: Divide both sides of the equation by 2 to isolate the w.
Step 5: 30 ÷ 2 = 15
Answer: The width of the bedroom is 15 meters.
Example 2: The blueprints for Mr. Fitzpatrick's new house in New York have a scale of
the area of his library if its dimensions on the blueprint are
Solution: To find the area, the length and width must be known.
Calculate the length.
Step 1: Write the appropriate proportion. Let L represent the length of the library.
Step 2: Write the cross products. Multiply L by 1/8 and multiply 2 by 5/8.
Step 3: Rewrite the equation with the new values.
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What is
Step 4: Multiply both sides of the equation by 8/1 to isolate the L.
Step 5: Length = 10 feet
To calculate the width:
Step 1: Write the appropriate proportion. Let w represent the width of the library.
Step 2: Write the cross products. Multiply w by 1/8 and multiply 2 by 1 1/2.
Step 3: Rewrite the mixed fraction 1 1/2 as an improper fraction, 3/2.
Step 4: Multiply, then rewrite the equation with the new values.
Step 5: Multiply both sides of the equation by 8/1 to isolate the w.
Step 6: Width = 24 feet
Answer: 240 square feet
Properties - D
Students must be able to solve for a missing value in a given equation. Understanding properties such as the
order of operations is the key to correctly solving these problems.
Please review the following rules with the student:
1. Multiplication by 0: the product of any integer and 0 equals 0.
-3 x 0 = 0
3x0=0
2. Associative Property of Addition: (a + b) + c = a + (b + c).
(1 + 2) + 3 = 1 + (2 + 3)
3. Associative Property of Multiplication: (a x b) x c = a x (b x c).
(1 x 2) x 3 = 1 x (2 x 3)
4. Reciprocals: two numbers are reciprocals if their product equals 1.
5. Commutative Property of Addition: a + b = b + a
1+2=2+1
6. Commutative Property of Multiplication: a x b = b x a
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1x2=2x1
7. Order of Operations:
A. When calculations for a given expression or equation require both addition and multiplication, the
rule is to multiply first and add second.
(3)(2) + 3 = ?
6+3=?
6+3=9
B. The number outside the parentheses is multiplied with each number within the parentheses:
x(y + z) = xy + xz.
(1) 3(x + y)
(2) 3(x) + 3(y)
(3) 3x + 3y
C. If a given expression contains both parentheses and brackets, calculations should be completed
working from the innermost parentheses or bracket outward.
The following are sample questions using the above properties.
Example 1: Which answer best completes the number sentence?
5 = (5 x 4) + (5 x 6)
A.
B.
C.
D.
x (4 + 6)
+ (4 + 6)
x (20 + 30)
+ (4 x 6)
Answer: A (because of rule 7B)
Example 2: Which one of the following best completes the number sentence?
(2.3 + 3.1) x 5.6 = ?
A.
B.
C.
D.
3.5 x 5.6
7.2 x 5.6
9.1 x 5.6
5.4 x 5.6
Answer: D (because of rule 7A)
Example 3: What is the value of n in the following statement?
13 x (3.4 x 0) = n
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A.
B.
C.
D.
44.2
0
13
3.4
Answer: B (because of rule 1)
Inequalities - A
An inequality is a number sentence that uses "is greater than", "is less than", or "is not equal to" symbols. For
example, 6n > 4 is a number sentence with an inequality symbol.
It may be useful to review the inequality symbols.
Example 1: Solve for y: 8y > 40
Get the variable being solved for (y) on one side of the inequality and the whole number on the other.
To do this, divide both sides by 8.
The correct answer is that y is greater than 5.
Inequalities can be represented as a value on a number line. The following number line represents the
inequality
Example 2: Which inequality represents the value shown on the number line below?
A. n < 3
B. n > 3
C. n = 3
The answer is A. n < 3 because the dot on the number line is open.
Multiple-step Story Problems - E
These problems are designed to test a student's ability to interpret data from story (word) problems. Answers
are found by solving equations with multiple operations.
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It may be helpful to develop a series of multiple-step word problems that relate to the student's activities, such
as allowance. The following is a step-by-step example of a multiple-step story problem.
Example 1: On Saturday, Stella earned $3.50 for each hour of work. She earned $3.25 for each hour of work on
Sunday. She worked 5 hours each day. How much money did she earn for both days?
(1) $3.50 x 5 = ?
$3.25 x 5 = ?
(2) $3.50 x 5 = $17.50
$3.25 x 5 = $16.25
(3) $17.50 + $16.25 = $33.75
Step 1: Develop 2 separate equations. One to find the earnings on Saturday, and one to find the earnings on
Sunday.
Step 2: Find the products of the two equations.
Step 3: Add the two products together.
Answer: Stella earned $33.75.
Example 2: Saman ate 3 times as many cookies as Alli. Alli ate 5 cookies less than Josh. Josh ate 10 cookies.
How many cookies did Saman eat?
(1) 10 - 5 = 5
(2) 5 x 3 = 15
Step 1: Since Alli ate 5 cookies less than Josh, subtract 5 from 10 to determine the number of cookies she ate.
Step 2: Now that we know how many cookies Alli ate, we can determine the number of cookies Saman ate by
multiplying 5 by 3.
Answer: Saman ate 15 cookies.
Probability/Statistics - C
Probability is the ratio of the number of times a certain outcome can occur to the number of total possible
outcomes. For instance, if the probability that an event will happen is 7 out of 10, this probability could be
expressed as 7/10, 7:10 or 70%. Statistics is the study of numerical data. This data is collected, classified, and
analyzed to provide a meaningful presentation.
One of the best ways to introduce the student to probability and statistics is to apply them to activities he or she
enjoys. If the student likes to play marbles, have him or her put marbles of different colors in a bag. Then,
help him or her use probability calculations to figure out the likelihood of certain events. (Example: pulling a
blue marble.)
Example 1: If there are 6 green marbles, 3 orange marbles, 2 blue marbles, and 1 black marble in a bag, what is
the probability that either an orange or black marble will be blindly pulled from the bag first?
Solution: Start with the number of marbles in the bag. There are 12. Then, figure out how many marbles are
either orange or black. There are 4 (3 orange and 1 black). The probability ratio is 4/12, or, in reduced form,
1/3.
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The probability that either an orange or a black marble will be blindly pulled out of the bag first is 1/3.
Example 2: Michelle conducted a poll to find out how many of her classmates lived in apartments and how
many lived in houses. In the poll, Michelle found that 12 out of 25 classmates lived in apartments. There are
250 people in her class. How many classmates can Michelle expect to live in apartments?
Solution: We'll let 'x' represent the number of classmates expected to live in apartments.
Since 12 of 25 live in apartments, 12 is to 25 as 'x' is to 250. We can represent this as a proportion:
12/25 = x/250
We can use algebra to solve by cross multiplication:
(12)(250) = 25x
3,000 = 25x (Divide each side by 25)
x = 120
Michelle can expect that 120 of her classmates live in apartments.
Venn Diagram
A Venn diagram is a visual aid used to show relationships between numbers, sets of numbers, figures, etc.
Venn diagrams are a series of circles which overlap when elements being diagrammed fall into common
categories.
Statements relating to Venn diagrams either use specific numbers or the words "all," "some," or "none."
The Venn diagram below represents how many students participated in hockey and soccer. Use the
following Venn diagram to answer the following questions.
Example 1: How many students participated in both hockey and soccer?
Since there are 2 people in the overlap between soccer and hockey, the answer is 2.
Example 2: How many students participated in soccer only?
Since there are 5 people in the soccer portion that does not overlap with the hockey portion, the answer
is 5.
Example 3: How many students participated in hockey?
Since there are 5 people in the entire hockey circle, the answer is 5.
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Mean/Median/Mode/Range - A
The study of statistics involves looking at the measures of central tendency. They are the mean, median, and
mode.
The mean of a group of numbers is found by adding the numbers and then dividing the sum by the number of
addends (items in the group). It is also referred to as the average.
The median is the middle number, or the average of the two middle numbers, of a group of numbers ordered
sequentially.
The mode of a group of numbers is the number that occurs the most often.
The range of a group of numbers is the difference between the highest number and the lowest number.
Example 1: Find the mean of 15, 10, 5, 20, 5.
(1) 15 + 10 + 5 + 20 + 5 = 55
(2) 55 ÷ 5 = 11
Step 1: Find the sum of the numbers.
Step 2: Divide by the total number of addends (items in the group).
The mean is 11.
Example 2: Solve for the median of 9, 1, 3, 5, 0, 7.
(1) 0, 1, 3, 5, 7, 9
(2) 3 and 5
(3) 3 + 5 = 8
(4) 8 ÷ 2 = 4
Step 1: Put the numbers in order from smallest to largest.
Step 2: Determine the middle number(s). Since there is an even amount of numbers, there are two middle
numbers. If there is an odd amount, there will be just one.
Steps 3 and 4: Determine the mean of 3 and 5 by adding the two numbers together and dividing by 2.
The median is 4.
Example 3: Solve for the mode of 0, 6, 7, 9, 6, 4, 3.
The mode is the number which occurs most often in a group. For this group of numbers the mode is 6 because
it occurs twice and the other numbers only occur once. NOTE: There can be more than one mode or there can
be no mode at all.
Example 4: Find the range of 9, 5, 16, 8, 2, 3.
(1) 16 - 2 = 14
The range is the difference between the highest number in a group and the lowest number in a group.
16 - 2 = 14
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The range is 14.
Irrelevant/Missing Info - A
Students are presented with story problems that test their ability to read word problems, identify and utilize
information pertinent to the question being asked, and disregard irrelevant information.
The following is an example of a story problem that contains irrelevant information.
Example: Cameron is twice as old as Gerardo. Chris is half Gerardo's age. Jelena, Cameron's 25 year old sister,
is 7 years older than Cameron. Based on this information, how old is Gerardo?
Solution: Make a list of the ages.
1. Cameron's age = 2 x Gerardo's age
2. Chris's age = 1/2 x Gerarado's age
3. Jelena's age = Cameron's age + 7
The question requires Gerardo's age to be found. The only age we are given is Jelena's, 25. We can determine
that Cameron is 18 because we know that he is 7 years younger than Jelena. Because we know that Cameron is
18, we know that Gerardo is 9 since twice Gerardo's age equals Cameron's age. Therefore, Gerardo is 9 years
old. Chris's age was the irrelevant piece of information.
Many word problems contain irrelevant information. Help the student practice breaking down information the
story problems in his or her math textbook. As he or she learns this skill, he or she will become better at
solving word problems.
Perimeter - C
Perimeter is the measurement of the distance around a figure.
To calculate the perimeter of a figure, add the lengths of all the sides of the figure.
Example 1: What is the perimeter of a figure that has four sides measuring 3 inches, 7 inches, 3 inches
and 7 inches?
P = 7 + 3 + 7 + 3 = 20
Answer: 20 inches
Example 2: What is the perimeter of the figure?
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Step 1: Add the length of each side together.
Step 2: Since the length of each side includes a fraction, it may be easier to add the numbers vertically.
Find the common denominator. The common denominator is 12 because 2 x 3 x 4 = 12. Rewrite the
fractions so they all have the common denominator (12).
Step 3: 30/12 is an improper fraction because the numerator (top number) is larger than the denominator
(bottom number). Twelve will divide into 30 two times with 6 left over. Add the 2 to the 35 and now the
fraction is 37 6/12. Six and 12 can both be divided by six, so the final answer is 37 1/2.
Answer: 37 1/2
Area of Trapezoid
The area of a trapezoid is the number of square units needed to cover the surface of the figure.
The following is the formula needed for calculating the area of a trapezoid:
Example 1: Solve for the area of a trapezoid with bases equal to 6 meters and 8 meters, and height equal
to 4 meters.
Step 1: Apply the amounts given in the problem to the formula.
Step 2: Add the numbers within the parentheses.
Step 3: Multiply the whole numbers.
Step 4: Perform calculations to find the answer.
Answer: 28 square meters
Example 2: Find x if the area of the trapeziod is 73.5 centimeters squared.
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Step 1: Apply the given values to the formula for the area of a trapeziod. (NOTE: This time you are
given the area of the trapeziod.)
Step 2: Add the numbers within the parentheses.
Step 3: Perform the multiplications on the right side of the equation.
Step 4: Multiply both sides of the equation by 2. Simplify.
Step 5: Divide both sides of the equation by 21.
Answer: x = 7
Area of Circle
The area of a circle is the number of square units needed to cover the surface of the figure.
The following is the formula needed for calculating the area of a circle:
Pi is approximately equal to 3.14. The symbol for Pi is The radius is the length from the center of the
circle to the outside edge. The diameter is the line segment that connects two points on the outside edge
of the circle and passes through the center of the circle. The length of the diameter is twice the length of
the radius.
Example 1: Solve for the area of a circle with a radius equal to 4 meters.
(1) Area = 3.14 x (4 x 4)
(2) Area = 3.14 x 16
(3) Area = 50.24
Step 1: Apply the amounts given in the problem to the formula.
Step 2: Multiply the numbers within the parentheses.
Step 3: Perform calculations to find the answer.
A semicircle is half of a circle. The area of a semicircle is exactly half of the area of a circle with the
same radius.
Example 2: What is the area of the following semicircle? Round your answer to the nearest hundredth.
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Step 1: The diameter of the semicircle is 13 inches, so the radius is 13 inches divided by 2 (6.5 inches).
Step 2: Determine the area of a circle with radius 6.5 inches.
Step 3: Divide the area of the circle by 2 to find the area of the semicircle with radius 6.5 inches.
Step 4: Round 66.3325 to the nearest hundredth.
The area of the semicircle is 66.33 square inches.
Volume of Rectangular Prisms
Volume is the measurement of a three-dimensional figure's interior space. Volume is measured in cubic units.
The formula for calculating volume of a rectangular prism is:
Volume = length x width x height
Example 1: Find the volume of a rectangular prism with length = 6 inches, width = 4 inches,
height = 2 inches.
(1) Volume = 2 x 4 x 6
(2) Volume = 48 cubic inches
Step 1: Apply the amounts given in the problem to the formula.
Step 2: Perform calculations to find the answer.
Answer: 48 cubic inches
Example 2: What is the height of the rectangular prism with volume =10 cubic meters, length = 2
meters, and width = 1 meter?
(1) 10 = (2)(1)(h)
(2) 10 = 2(h)
(3) 5 = h
Step 1: Substitute the known values into the formula for the volume of a rectangular prism.
Step 2: Multiply 2 by 1 by h to get 2(h).
Step 3: Divide both sides of the equation by 2 to get that h = 5.
Answer: 5 meters
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Volume of Triangular Prism
Volume is the measurement of a three-dimensional figure's interior space. Volume is measured in cubic units.
A triangular prism has a triangular base and three lateral faces.
The formula for calculating the volume of a triangular prism is:
Example 1: Find the volume of a triangular prism with the length of the triangular face equal to 5
meters and the height of the triangular face equal to 2 meters. The height of the prism is equal to 4
meters.
Step 1: Find the area of the base (triangle). Apply the known amounts from the problem to the formula.
Step 2: Perform calculations to find the area of the base.
Step 3: Find the volume of the triangular prism. Apply the known amounts from the problem to the
formula.
Step 4: Perform calculations to find the answer.
Volume of Cylinders
Volume is the measurement of a three-dimensional figure's interior space. Volume is measured in cubic units.
A cylinder is a solid with two bases that are congruent circles.
The formula for calculating volume of a cylinder:
Remember, pi is approximately 3.14. The symbol for pi is Example 1: Solve for the volume of a
cylinder with the radius equal to 4 meters and a cylinder height equal to 10 meters.
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Step 1: Find the area of the base (circle). Apply the known amounts from the problem to the formula.
Step 2: Perform calculations to find the area of the base.
Step 3: Find the volume of the prism. Apply the known amounts from the problem to the formula.
Step 4: Perform calculations to find the answer.
Answer: 502.4 cubic meters
Angles - B
An angle is created by two rays with the same endpoint. That endpoint is called the vertex.
An interesting method for improving the student's understanding of angles is to have him or her draw the
various types of angles. Then, develop a series of flash cards. On one side of the card, draw the
figure. On the other side of the card, write the name. The following are definitions to help get you
started:
Obtuse Angle - an angle with a measure greater than 90 degrees and less than 180 degrees
Right Angle - an angle with a measure equal to 90 degrees
Acute Angle - an angle with a measure greater than 0 degrees and less than 90 degrees
Adjacent angles - two angles with a common vertex and a common side
Complementary angles - two angles whose measures have a sum of 90 degrees
Supplementary angles - two angles whose measures have a sum of 180 degrees
Vertical angles - opposite angles formed by two intersecting lines; vertical angles are congruent
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Triangles - A
A triangle is a polygon with three sides.
The following are common triangles used at this level:
Scalene Triangle - a triangle with three unequal sides
Isosceles Triangle - a triangle with at least two equal sides
Equilateral Triangle - a triangle with three equal sides
Right Triangle - a triangle with one right (90º ) angle
It is also important for the student to understand that the sum all of the angles in a triangle equals 180º .
Example 1: If one angle of a triangle is 10º and a second angle is 45º , what is the measure of the third angle?
(1) 10º + 45º = 55º
(2) 180º - 55º = 125º
Step 1: Add the measures of the two known angles.
Step 2: Since the measures of the three angles of a triangle add up to 180º , subtract 55º from 180º .
Answer: The measure of the third angle is 125º .
An alternate method for determining the measure of the third angle of a triangle is to set up an equation.
Example 2: A triangle has two angles, each measuring 47º . What is the measure of the third angle?
(1) 180º = 47º + 47º + x
(2) 180º = 94º + x
(3) 180º - 94º = 94º + x - 94º
(4) 86º = x
Step 1: Since the sum of the angles of a triangle equals 180º , let 180º = 47º + 47º + x where x represents the
measure of the third angle.
Step 2: Combine like terms (47º +47º = 94º )
Step 3: To isolate x, subtract 94º from both sides of the equation.
Step 4: Simplify both sides of the equation.
Answer: The measure of the third angle is 86º .
Lines/Segments/Points/Rays
At this level, the lines skill involves the following concepts: lines, rays, points, and segments.
It is important for the student to understand the following definitions:
Point - a specific location on a figure, usually on a line or plane
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Line - a straight path extending in both directions with no endpoints. A line AB is denoted as
Segment - a part of a line that ranges from one point to another. A line segment AB is denoted as
Ray - a part of a line that continues forever in one direction from its endpoint. A ray BA with endpoint B
is denoted as
Skew lines - lines that are not intersecting and are not parallel.
Parallel lines - lines in the same plane that do not intersect. Parallel lines have no points in common.
Perpendicular lines - two lines that intersect and form right angles. Perpendicular lines have only one
point in common.
Point of intersection - the point at which lines cross.
Transversal - a line that intersects 2 or more lines.
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One method for improving the student's understanding of these concepts is to have him or her develop a
series of flash cards with the definition on one side and the written name on the back.
Polygons - B
A polygon is a closed shape formed by three or more sides. For example, a triangle is a polygon with three
sides and a quadrilateral is a polygon with four sides.
It may be helpful to verify that the student is familiar with different polygons commonly taught in this
grade. The following is a list of polygons and their definitions.
A line segment is a part of a line that is bounded by two endpoints.
A diagonal is a line segment that joins two vertices of a polygon, but is not a side of the polygon.
Congruency - C
Congruent figures have the same shape and size.
It may be beneficial to verify that the student understands the definition of similar and congruent figures.
To help understand congruence, draw two triangles exactly the same shape and size (use graph paper or
a copy machine to ensure the figures are the same). Draw an additional triangle of a different shape and
size. Cut out each figure. Arrange the figures on a table and ask the student to find the congruent
triangles. Remember, two figures are congruent if they are the same shape and size.
Symmetry:
A figure is said to have a line of symmetry when it can be folded in half along that line so that the two
halves are congruent. One artistic way to help the student understand symmetry is for you to draw half
of a simple shape (such as a square, circle, triangle, or heart) on a piece of paper and have the student
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draw the other half. Fold the paper on the halfway line and compare to see if both sides match.
A vertical line of symmetry crosses through the figure vertically. A horizontal line of symmetry crosses
through the figure horizontally. A diagonal line of symmetry crosses through the figure at a diagonal.
The following three figures show the lines of symmetry (dotted lines):
Symmetry - B
To determine lines of symmetry, imagine a line cutting a figure into two parts. If the pieces were placed on top
of each other, the two parts would be exactly the same. The line where the figure was cut is called the
line of symmetry. Figures that can be divided into two parts that are exactly the same are called symmetric
across a line.
Imagine folding a figure cut out of a piece of paper in half so the figure fits perfectly over the other half of the
figure, this is called a figure that is symmetric.
Below are examples of figures that when divided into two parts across the line of symmetry would fit
together exactly. Notice that some figures have more than one line of symmetry.
It may be helpful to cut figures out of construction paper and try to decide how many lines of symmetry
there are in the figure. Then, fold or cut the figure across the determined lines of symmetry to check for
symmetry.
Example 1: How many lines of symmetry are in the following figure?
The answer is 2. The figure has a vertical and horizontal line of symmetry.
Place Value: Decimals - B
In place value, the student must determine what a digit in a given number represents. In place value with
decimals, the students must determine the place value for digits to the right of the decimal place.
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It may be helpful to review the diagram below to help determine decimal place value.
In the number 32.4578, the 4 represents 4 tenths.
Example 1: What does the digit 8 mean in 57.929384?
The answer is hundred thousandths.
Example 2: In what place is the underlined digit?
143.680259
Answer: The underlined digit is in the millionths place.
Graphing Equations - A
Students must graph an equation (such as y = 3x - 5) on the coordinate plane.
Example 1: Graph: y = x + 3
Step 1: When given an equation, such as y = x + 3, the first step is to make a table of ordered pairs.
Select a list of values for x and then calculate the values of y.
Step 2: Graph the ordered pairs you calculated in Step 1. Remember, the x-value moves left (negative)
or right (positive) from zero and the y-value moves up (positive) or down (negative) from the x-value.
If you connect the points, this equation will create a straight line.
In order to determine whether a set of ordered pairs is a solution to the graph of a line, you substitute
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those values in for the variables.
Example 2: Which of the following points is a solution to the graph of the line, y = 2x + 4?
A. (2,7)
B. (1,6)
C. (1,3)
Substitute the x- and y-values from the ordered pairs into the equation y = 2x + 4.
(2, 7): 7 = 2(2) + 4
7 = 8
false
(1, 6): 6 = 2(1) + 4
6 = 6
true
(1, 3): 3 = 2(1) + 4
3 = 6
false
The answer is: (1, 6) is a solution to the graph because its values of x and y make the equation true.
An equation that represents a line has an infinite number of solutions because there are two variables in
the equation. If one of the variables is fixed, then there will only be one solution to the equation.
Example 3: If the equation y = 3x + 1creates a straight line, how many solutions to the equation are
there?
Solution: There are an infinite number of solutions because for every value of x, a different value of y
can be found.
Example 4: If 2y = x - 4 and x = 1, how many solutions to the equation are possible?
Solution: There is only one possible solution because if x = 1, 2y = 1 - 4, so y = -3/2. For each value of
x, there is only one value for y.
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