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Study Guide
Grade 7 Math-a-thon
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.
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).
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,
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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:
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.
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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 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
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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
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?
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(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 .
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?
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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.
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
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 ycoordinate (vertical) is listed second. For example, the coordinate pair (3, 2) is at the horizontal position 3 and
the vertical position 2.
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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.
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.
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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?
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
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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.
Divisibility/Multiples/Factors - B
Divisibility occurs when one number is divided by another and the remainder is zero. For example, 15 divided
by 3 is 5.
Factors are numbers that when combined in a multiplication equation give the product. The factor of a number
is a whole number that divides it exactly. For example, 1, 2, 4, and 8 are factors of 8. A common factor of two
or more numbers is a factor of all the numbers. The greatest common factor (GCF) is the greatest number in a
list of common factors of the numbers.
A multiple is a number that is the product of a given number and a whole number. For example, the multiples
of 3 are 3, 6, 9, 12, etc. The common multiple of two numbers is any number that is a multiple of both numbers.
The least common multiple (LCM) of two or more numbers is the least number in the list of their common
multiples.
It may be helpful to create a divisibility game. On small pieces of paper, write the numbers 1 to 100. Put the
100 pieces of paper in a bag or a hat. Have the student draw two pieces of paper from the hat. Help the student
determine if one number is divisible by the other number. A strong knowledge of the multiplication tables will
be helpful.
A similar game can be created for factors and multiples. On small pieces of paper, write the numbers 1 to 100.
Put the 100 pieces of paper in a bag or hat. Have the student draw one piece of paper from the hat. Help the
student determine the factors and multiples of the number.
It may be helpful to note that there are an infinite amount of multiples for a specific number, but there are only a
certain amount of factors for the same number.
Example 1: Find the least common multiple (LCM) of 4 and 6.
Step 1: Determine the multiples of 4 and 6. (Multiply 4 times 1, 2, 3, 4, etc. and the same with the number 6)
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36,. . . .
multiples of 6: 6, 12, 18, 24, 30, 36,. . . . . .
Step 2: List the common multiples.
common multiples of 4 and 6: 12, 24, 36,. . . .
Step 3: The least common multiple is the least number in the list of common multiples, which is the number 12.
Answer: The LCM of 4 and 6 is 12.
Example 2: Find the greatest common factor (GCF) of 24 and 56.
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Step 1: Determine the factors of 24 and 56.
factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24
factors of 56 are: 1, 2, 4, 7, 8, 14, 28, and 56
Step 2: List the common factors of 24 and 56.
Common factors of 24 and 56 are: 1, 2, 4, and 8
Step 3: The greatest common factor is the greatest value in the list, which is the number 8.
Answer: The GCF of 24 and 56 is 8.
Compare Whole Number Equations - B
Comparing whole number equations involves determining which side of the equation is larger in value than the
other. Knowledge of the greater than (>), less than (<), and equal to (=) signs is needed for this skill.
It may be helpful to review the ordering symbols with the student.
Example 1: 11,025 ÷ 105 ? 70 x 2
(1) 11,025 ÷ 105 = 105
(2) 70 x 2 = 140
(3) 105 ? 140
Step 1: Simplify the expression on the left.
Step 2: Simplify the expression on the right.
Step 3: Rewrite the mathematical sentence with the new numbers and determine which symbol to place
between the two numbers.
The answer is: 11,025 ÷ 105 < 70 x 2.
Example 2: 200,000 ÷ 20 ? 20,000 ÷ 2
(1) 200,000 ÷ 20 = 10,000
(2) 20,000 ÷ 2 = 10,000
(3) 10,000 ? 10,000
Step 1: Simplify the expression on the left.
Step 2: Simplify the expression on the right.
Step 3: Rewrite the mathematical sentence with the new numbers and determine which symbol to place
between the two numbers.
The answer is: 200,000 ÷ 20 = 20,000 ÷ 2.
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Multiple Operations: Whole Numbers - B
Multiplication skills at this level include operations with multi-digit numbers (237 x 56) which require
regrouping (carrying, borrowing, renaming) when the product of the numbers in the ones or tens (hundreds,
thousands, etc.) position are equal to or greater than ten. Necessary division skills include long division,
remainders, and two-digit quotients.
When performing multiple operations on expressions, it is important to remember that operations inside
parentheses are completed first. If there are two sets of parentheses, then the set that comes first when reading
from left to right is completed first.
Example 1: (237 x 56) ÷ 12 = ?
(1) 237 x 56 = 13,272
(2) 13,272 ÷ 12 = ?
(3) 13,272 ÷ 12 = 1,106
Step 1: Multiply the numbers inside the parentheses. (237 x 56)
Step 2: Rewrite the problem with the new value in place of the parentheses.
Step 3: Divide 13,272 by 12.
Answer: 1,106
Whole Numbers: Story Problems
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. The student's textbook may have word problems that he or she can
use to practice these five steps. Remember, some story problems have extra information.
One of the best ways to understand story problems is to write them. Help the student come up with a story
problem like the one below:
Sam went to the store with $81. At the store, he saw Michelle, who owed him $16. Michelle had $19.00 more
than what she owed Sam. Michelle paid Sam what she owed him, and Sam bought a model car set for $31.
How much money did Sam go home with?
The answer is $66.
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 = ?
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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.
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.
Solution: Remember that "sum" is the answer to an addition problem, so the expression is x + 23.
Answer: x + 23
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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
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?
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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.
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.
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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.
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
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:
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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
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.
Place Value: Whole Numbers - C
Each digit of a number is given a value for the position it is located. For example, in the number 56 the place
value of the 6 is ones. The place value of the 5 is tens.
It may be helpful to develop a place value chart like the one below to help the student visualize place
value.
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Example 1: What does the digit 5 mean in 7,235,689?
Solution: Help the student fill in the place value chart with the correct digits (as seen in the place value
chart below). Then use the chart to determine the meaning of the indicated number.
Since the 5 is in the thousands column, the 5 in 7,235,689 means 5 thousands.
Example 2: In which place is the underlined number?
123,458,079
Solution: Help the student fill in the place value chart with the correct digits. Then use the chart to
determine the meaning of the indicated number.
The 3 is in the millions place.
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.
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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
The range is 14.
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
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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
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
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.
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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.
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.
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
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of a simple shape (such as a square, circle, triangle, or heart) on a piece of paper and have the student
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):
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.
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
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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
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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