Download Study Guide 5 gr numerat whole numbers 01/21/2010 Divide Whole

Study Guide
5 gr numerat whole numbers
01/21/2010
Divide Whole No: Story Problems - D
Story problems, also called word problems, relate division of whole numbers to actual situations. Operational
symbols, such as the division symbol (÷ ), are replaced with text. For example, "If there are 18 pencils and 9
pencil boxes, how many pencils are in each box?" The student must determine that division is required to
perform this problem. 18 ÷ 9 = 2 The answer is 2 pencils.
Story problems are often a very difficult area for students to master. It may be useful for you to
confirm that the student is comfortable with division skills. Then, create equations that relate to his or
her daily activities, such as friends or food. Help the student determine the correct formulas.
Example: There are 437 cookies in a jar. There are 30 children in the class.
How many cookies will each child get? How many cookies will be left over?
Step 1: Determine that division is required. Write the problem in long division
format.
Step 2: Find the quotient of 43 and 30 (1). Put the 1 in the tens place. Multiply 30 by 1 and write the
product (30) below 43. Subtract 30 from 43 (13). Bring the 7 down next to the 13, to make 137.
Step 3: Find the quotient of 137 and 30 (4). Put the 4 in the ones place. Multiply 30 by 4 and write the
product (120) below 137. Subtract 120 from 137 (17). Write the remainder (17) beside the ones.
Answer: Each child will receive 14 cookies. There will be 17 cookies left in the jar.
Odd/Even - C
An odd number is a whole number that is not a multiple of two, such as 3. An even number is a whole number
that is a multiple of two, such as 4.
It may be helpful to discuss the concept of odd and even numbers with the student. Once he understands the
theory, develop a list of whole numbers and help the student determine whether they are odd or even.
Remember, the number in the ones column determines whether the number is even or odd.
Examples of even numbers: 2, 4, 6, 8, 10, 12,...
Examples of odd numbers: 1, 3, 5, 7, 9, 11,...
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Equivalent Forms: Decimal/Mixed Fract.
Fractions can be written in decimal number format, and vice versa. For example, 1/4 = 0.25.
It may be advantageous to concentrate on either fractions or decimals. Do not introduce a new area
until one has been completely mastered. Once the student has mastered either fractions or decimals,
begin to introduce its equivalent form. Develop a series of fractions and decimals and help the student
find the equivalent forms.
The following examples will help get you started:
To write a fraction as a decimal number, simply divide the numerator (the top number) by the
denominator (the bottom number).
Example 1: Write the fraction 5/8 as a decimal.
(1) 5 ÷ 8 = ?
(2) 5 ÷ 8 = 0.625
Step 1: Divide the numerator by the denominator.
Step 2: Complete the division problem.
Answer: 0.625
Example 2: Write the mixed number 2 3/4 as a decimal.
(1) whole number: 2; fraction: 3/4
(2) 3/4 = 3 ÷ 4 = 0.75
(3) 2 3/4 = 2.75
Step 1: Separate the mixed number 2 3/4 into a whole number and a fraction. The whole number will
always remain a whole number, but the fraction can be changed into a decimal.
Step 2: Write the fraction 3/4 as a decimal by dividing the numerator by the denominator.
Step 3: Put the whole number and the decimal back together to get the complete decimal number.
Answer: 2.75
Example 3: Which of the following is another way to write 7.38?
A.
B.
C.
D.
738/10
7 3810
7 38/100
7 38/1000
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Solution:
(1) whole number: 7; decimal: 0.38
(2) 0.38 = "thirty-eight hundredths" = 38/100
(3) 7.38 = 7 38/100
Step 1: Separate 7.38 into its parts: whole number and decimal number.
Step 2: Since the 8 is in the hundredths place, we can state 0.38 as "thirty-eight hundredths", which can
be written as 38/100.
Step 3: Put the whole number and the fraction back together to get the mixed number.
Answer: C.
It may be necessary to completely reduce a fraction. A fraction is said to be in lowest terms when the
greatest common factor of the numerator and denominator is 1.
Example 4: Determine another way to write 24/36.
Solution: Determine the greatest common factor of 24 and 36. The greatest common factor (GCF) of
two or more numbers is the largest number that will divide into all of the numbers without remainders.
The GCF of 24 and 36 is 12. Divide the numerator and the denominator by the GCF. 24 ÷ 12 = 2 and 36
÷ 12 = 3.
Answer: 2/3
Story Problems: Multiple Operations
Multiple-step word problems test a student's ability to interpret data from a written word problem. Answers are
found by solving equations with multiple operations. The following is a step-by-step example of a
multiple-step story problem.
Example 1: Mr. Peterson's class earns 12 points per day for every day they are well-behaved. The class
needs 96 points to earn a pizza party for the next month. If they are well-behaved for 6 days during the
next 2 weeks, how many more days do they need to behave to earn the pizza party?
Solution:
(1) 12
6 = 72
(2) 96 - 72 = 24
(3) 24 ÷ 12 = 2
Step 1: Determine the number of points the students will earn from being good for 6 days.
Step 2: Subtract the number of points earned (72) from the original number of points needed (96) to
determine the number of points still needed (24).
Step 3: Divide the number of points still needed (24) by 12 (the number of points it is possible to earn in
one day) to determine the number of days the students will need to be well-behaved.
Answer: 2 days
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An activity to help reinforce the concept of solving story problems with multiple operations is to have
the student develop a series of multiple-step word problems that relate to his or her activities, such as an
allowance or hobbies. Then have the student solve the problems.
Add Whole No: Story Problems - D
Story problems, also called word problems, relate addition of whole numbers to actual situations. Operational
symbols, such as the addition (+) symbol, are replaced with text. For example, "If Jill had 2 apples and Jack
gave her 2 more apples, how many apples would Jill have now?" The student must determine that addition is
required to solve this problem. (Answer: 4 apples)
Story problems are often very difficult for children to master. It may be beneficial for you to verify that
the student is comfortable with addition skills. Then, create humorous addition story problems and help
the student determine the correct formulas.
Example: My father has been losing weight. In May, he lost 25 pounds. In June, he lost 7 pounds.
In July, he lost 12 pounds. How many pounds did he lose in all?
Step 1: This problem requires addition. Since we want to determine the total amount of weight lost, we
need to add 25, 7, and 12. Write a vertical equation.
Step 2: Add the numbers in the ones column (5 + 7 + 2 = 14). Write the 4 in the ones place and carry the
1 to the tens column.
Step 3: Add the numbers in the tens column (1 + 2 + 1 = 4). Write the 4 in the tens place.
Answer: My father lost 44 pounds.
Subtract Whole No: Story Problems - D
Story problems, also called word problems, relate subtraction of whole numbers to actual situations.
Operational symbols, such as the subtraction (-) symbol, are replaced with text.
Story problems are often very difficult for students to master. It may be beneficial for you to confirm that the
student is comfortable with subtraction skills. Then, create equations that relate to real life, such as sports or
school. Help the student determine the correct formulas.
Example 1: Alabaster High School had 345,263 yearbooks printed. They sold 299,999 of the yearbooks.
How many of the yearbooks did they have left?
Solution: The student must determine that subtraction is necessary to solve the problem and determine the
necessary equation.
345,263 - 299,999 = ?
Answer: 45,264
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Multiply Whole No: 3+ Digits by 2-Digit
Multiplying multiple-digit numbers often requires regrouping (carrying, trading, renaming). Regrouping
occurs when the product is equal to or greater than ten in a column.
The following is a step-by-step example of a multiple-digit multiplication problem.
Solve: 237 x 56=?
Step 1: Rewrite the problem vertically.
Step 2: Multiply 237 by 6. Write the product (1422).
Step 3: Place a 0, as a place holder, below the product of Step 2 in the ones position.
Step 4: Multiply 237 by 5. Write the product (1185) to the left of the 0.
Step 5: Add the two products (1422 + 11850) to determine the answer. Insert a comma after the
thousands place.
The correct answer is 237 x 56 = 13,272.
Multiply Whole No: Story Problems - D
Story problems, also called word problems, relate multiplication of whole numbers to actual situations.
Operational symbols, such as a multiplication (x) symbol, are replaced with text. For example, "If there are
1,505 students and each student should receive 12 books, how many books are needed?" The student must
determine that multiplication is required to solve the problem. The answer is 18,060 books.
Many students find story problems very challenging. It may be useful for you to confirm that the student is
comfortable with multiplication skills. Then create real life situations that relate to his or her daily activities
and help the student determine the correct formulas.
Example 1: A telemarketing sales person calls 6,721 homes per week.
In 34 weeks, how many homes would the telemarketing sales person call?
Solution: The student must determine that multiplication is needed to solve the problem and write the necessary
equation.
6,721 x 34 = ?
Answer: 228,514 homes
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Divisibility/Multiples/Factors - A
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 because 1 x 8 = 8
and 2 x 4 = 8.
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.
A prime number is a positive integer which is not 1 and has no factors except 1 and itself. For example, 3 is a
prime number because its only factors are 1 and 3.
A composite number is a number that has more than two factors.
It may be helpful to create a divisibility game. On a small piece of paper, write the number 1. Continue on
other pieces of paper up to 100. Put the 100 pieces of paper in a bag or hat. Have the student draw two pieces
of paper from the hat. Help the student determine if one number is a factor or is divisible by the other number.
Example 1: The student pulls the number 56 and the number 7. The number 56 is divisible by 7 with no
remainder.
56 ÷ 7 = 8
A similar game can be created for factors and multiples. On a small piece of paper, write the number 1.
Continue on other pieces of paper up 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.
Example 2: The student pulls the number 24.
Factors of 24: 1, 2, 3, 4, 6, 8, 12,24
Multiples of 24: 24, 48, 72, 96, 120. . .
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 3: As the student selects numbers from the bag, have him or her determine if the numbers are prime
numbers.
Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.
Read Numerals - C
Students must be able to read and write numerals.
It may be helpful to develop a list of numbers. Help the student correctly read and write each number. For
example, the number "234,567" could be written as "two hundred thirty-four thousand, five hundred
sixty-seven" or "2 hundred thousands, 3 ten thousands, 4 thousands, 7hundreds, 6 tens, 7 ones."
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Expanded Notation - D
Expanded notation is a format for writing numbers so that each digit shows a place value. For example,
753 = 700 + 50 + 3.
It may be helpful to discuss the concept of expanded notation with the student. Once he or she understands the
theory, develop a series of numbers and help the student create expanded notation for each number. The
following samples may be helpful:
Number
5,672
113
289
19
3
Expanded Notation
5,000 + 600 + 70 +
100 + 10 +
200 + 80 +
10 +
2
3
9
9
3
Example 1: Find the expanded form: 726
A.
B.
C.
D.
700 + 20 + 6
7+2+6
700 + 26
700 + 20 + 2
The answer is A. 700 + 20 + 6.
Example 2: What is another way to write 500 + 70 + 7?
A.
B.
C.
D.
777
5, 707
577
5,777
The answer is C. 577.
Example 3: What number is expressed by (9 x 1000) + (7 x 100) + (3 x 10) + (4 x 1)?
A.
B.
C.
D.
9,734
90,734
4,379
900,734
To solve this problem, you multiply within parentheses:
(9 x 1000) = 9000
(7 x 100) = 700
(3 x 10) = 30
(4 x 1) = 4
Write this number in standard notation. The result is 9,734.
The answer is A.
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Rounding and Estimation - C
Rounding and estimation are used to express a number to the nearest ten, hundred, thousand, and so forth.
An interesting method for improving the student's rounding and estimation skills is to create a list of numbers.
Help the student round each number. Remember, numbers less than 5 are rounded down, while numbers 5 or
greater are rounded up (in both cases, you are looking one place to the right of the place value you wish to
round).
34 rounded to the nearest ten is 30.
37 rounded to the nearest ten is 40.
In order to round decimal numbers to whole numbers, we look at the digit in the tenths place. If the digit is less
than 5, drop the decimal part. If the digit is 5 or more, drop the decimal part and round up.
Example 1:
7.328 rounded to the nearest whole number is 7
8.74 rounded to the nearest whole number is 9
In estimating an answer, we round the numbers we are operating with in order to determine a simpler answer.
The examples below illustrate this process.
Example 2: Which of the following formulas should be used to estimate 36.3 x 8.9?
A. 37 x 8
B. 37 x 9
C. 36 x 9
D. 36 x 8
Solution: Round both numbers to the nearest whole number.
36.3 rounds to 36
8.9 rounds to 9
The answer is C.
Example 3: Which of the following formulas should you use to estimate 7,849 x 3,434?
A.
B.
C.
D.
7,000 x 3000
7,000 x 4000
8,000 x 3,000
8,000 x 4000
Solution: We round both numbers to the nearest thousand.
7,849 rounds to 8,000
3,434 rounds to 3,000
The answer is C.
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Number Lines - C
A number line is a line with equally spaced points marked by numbers. Students are given problems regarding
calculating the sum and difference of points, as well as determining the value of a specific point.
An interesting method for improving the student's understanding of number lines is to develop a series
of number lines. Help the student plot specific points on the number line.
Example 1: On the following number line, what is the difference between Point A and Point B?
A.
B.
C.
D.
8
7
10
6
Solution: There are two ways to find the answer. We can count over from A to B (we would count 7).
The answer is 7. Or, we can find the number of the points and subtract. Point A is at 40 and Point B is
at 47. 47 - 40 = 7. The answer is 7.
Example 2: Point X is at what fraction?
A.
B.
C.
D.
5/12
10/12
11/12
9/12
The answer is 10/12. We count over from 0 to 10/12.
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