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CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
Week & Day Week 10 – Day 1
Concept/Skill Lowest Common Denominator
Standard 7 NS: 2.2 Add and subtract fractions by using factoring to find common
denominators.
Materials CAHSEE Math Released Item # 23
Overview for Week
This week students will be adding, subtracting and multiplying fractions. They need to
know how to find the lowest common denominator, change mixed numbers to improper
fractions, and reduce fractions to lowest terms. On Day 5 they will be writing an
equation to find the answer to a word problem that uses fractions.
ƒ Review the Lowest Common Denominator, also known as the Least Common
Denominator. When adding fractions students must make sure all of the denominators
(number on the bottom of the fraction) are the same. To do this they must first find the
Lowest Common Denominator or LCD. The LCD is the lowest number that both of
the denominators can be divided into without a remainder.
ƒ Have students review the steps to find the Lowest Common Denominator::
1. Factor each of the denominators into primes. (Prime factors are those numbers
that can only be divided by 1 or themselves)
2. Count the number of times each prime number appears in each of the
factorizations
3. For each prime number, take the largest of these counts.
4. Write down that prime number as many times as you counted for it in step #3
5. The lowest common denominator is the product of all the prime numbers written
down.
ƒ Have students do the guided practice on Day 1 of the student CAHSEE Math
Problem of the Day -- Week 10 worksheet.
In today’s problem students will be doing steps one through four:
Example:
What is the prime factored form of the lowest common denominator of
2
2
2
The prime factors of 8 are _____
_____
_____
2
2 _____
3
The prime factors of 12 are _____
____
2
2 ______
2 _____
3
The factored form of the Lowest Common Denominator are _____
______
Day 1 is continued on next page.
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Fall 2007
Week 10 – Teacher - Page 1 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
ƒ You may choose to have students do an additional practice problem for Day 1.
Additional Practice:
Which of the following is the prime factored form of the lowest common denominator of
A
3x7
B
3x3x7
C
7x9
D
21 x 9
The factors of 21 are: 3 x 7
The factors of 9 are: 3 x 3
The factors in the Lowest Common Denominator are 3 x 3 x 7
Since 9 is not a prime number answer C can not
be the correct answer.
ƒ Have students do the CAHSEE released test item on Day 1 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
23. Which of the following is the prime factored form of the lowest common
denominator of
A
5x1
B
2x3x5
C
2x5x3x5
D
10 x 15
Laura Westermeier, Ed.D.
SAUSD, Educational Services
The prime factors of 10 are 2 x 5
The prime factors of 15 are 3 x 5
The Lowest Common Denominator is 2 x 3 x 5
Fall 2007
Week 10 – Teacher - Page 2 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
Week Week 10 – Day 2
Concept/Skill Adding fractions
Standard 7 NS: 2.2 Add and subtract fractions by using factoring to find common
denominators.
Materials CAHSEE Math Released Item # 22
ƒ Review the parts of a fraction:
numerator (top)
1
denominator (bottom)
3
ƒ Have students review the steps for adding fractions:
1. Change the fractions so they each have the lowest common denominator (steps 1-5 of
Day 1) (Note: Remember to multiply the numerator of each fraction by the same
number that you used to multiply the denominator)
2. Add the numerators to find the answer. (Do not add the denominators)
3. Reduce the fraction to lowest terms by finding an equivalent fraction in which the
numerator and denominator are as small as possible.
ƒ Have students do the guided practice on Day 2 of the student CAHSEE Math
Problem of the Day -- Week 10 worksheet.
Example: What is
6
The Lowest Common Denominator is ______
To change
Does
3
to the LCD multiply the numerator and denominator both by ______.
Yes
have the LCD? ________
Rewrite the fractions with the LCD and add to find the answer:
3
6
+
1
6
=
4
6
Remind students not to add the denominator.
Can the answer be reduced? ________
Yes
2
To reduce the fraction divide both the numerator and denominator by _____
2
Write the reduced fraction _______
3
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Day 2 continued on next page.
Fall 2007
Week 10 – Teacher - Page 3 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
ƒ You may choose to have students do an additional practice problem for Day 2.
Additional Practice
Which fraction is equivalent to
A
B
The Lowest Common Denominator for 4 and 10 is 20.
Change 1/4 to 5/20.
Change 2/10 to 4/20.
C
D
ƒ Have students do the CAHSEE released test item on Day 2 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
22. Which fraction is equivalent to
A
B
The Lowest Common Denominator for 6 and 8 is 24.
Change 5/6 to 20/24.
Change 7/8 to 21/24.
C
D
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Note: This is an improper fraction, however, students
do not need to change it to a mixed number.
Fall 2007
Week 10 – Teacher - Page 4 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
Week Week 10 – Day 3
Concept/Skill Subtracting fractions
Standard 7 NS: 2.2 Add and subtract fractions by using factoring to find common
denominators.
Materials CAHSEE Math Released Item # 24
Have students review the Steps for subtracting fractions:
Note: The steps for subtracting and adding fractions are the same except students
subtract on step 2.
1. Change the fractions so they each have the lowest common denominator (steps 15 of Day 1) (Note: Remember to multiply the numerator of each fraction by the
same number that you used to multiply the denominator)
2. Subtract the numerators. (Do not subtract the denominators)
3. Reduce the fraction to lowest terms by finding an equivalent fraction in which
the numerator and denominator are as small as possible. To do this divide the
numerator and denominator by the same number.
ƒ Have students do the guided practice on Day 3 of the student CAHSEE Math
Problem of the Day -- Week 10 worksheet.
Example: What is
?
36
The Lowest Common Denominator is ______
To change
Does
12 = 2 x 2 x 3
36 = 2 x 2 x 3 x 3
LCD = 2 x 2 x 3 x 3
3
to the LCD multiply the numerator and denominator both by ______.
Yes
have the LCD? ________
Rewrite the fractions with the LCD and subtract to find the answer:
15
36
-
5
36
=
10
36
Remind students not to subtract the denominator.
Yes
Can the answer be reduced? ________
2
The numerator and denominator each needs to be divided by ______.
5
Write the reduced fraction _______
18
Day 3 continued on next page.
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Fall 2007
Week 10 – Teacher - Page 5 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
ƒ Have students do the CAHSEE released test item on Day 3 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
24. What is
A
B
The Lowest Common Denominator for 4 and 6 is 12.
Change 3/4 to 9/12.
Change 1/6 to 2/12.
C
D
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Fall 2007
Week 10 – Teacher - Page 6 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
Week Week 10 – Day 4
Concept/Skill Adding and subtracting fractions in problems with parentheses
Standard 7 NS: 1.2 Add, subtract, multiply, and divide rational numbers (integers,
fractions, and terminating decimals) and take positive rational numbers
to whole-number powers.
Materials CAHSEE Math Released Item # 4
ƒ Orally quiz the students to see if they can remember what to do when a problem has
parentheses. They should know to find the answer to the part inside the parentheses
and then use that number to complete the calculation.
ƒ Review the steps for fraction problems with parentheses:
1. Do the parts of the problem inside the parentheses first.
2. Once the answer is found for each part inside the parentheses then finish finding
the answer for the problem.
ƒ Have students do the guided practice test item on Day 4 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
Example:
( )+( ) =
First find the answer for each part inside the parentheses:
=
_________
=
_________
Now write the fraction for each parentheses:
1
4
+
2
3
12
What is the Lowest Common Denominator? ____
Rewrite the problem with the Lowest Common Denominator and then find the answer:
3
12
+
8
12
=
11
12
No
Can the answer be reduced? _________
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Day 4 continued on next page.
Fall 2007
Week 10 – Teacher - Page 7 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
ƒ Have students do the CAHSEE released test item on Day 4 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
First find the answer of the problem inside
the parentheses: ⅓ + ¼ = 4/12 + 3/12 = 7/12
Next find the answer for 11/12 – 7/12 which
is 4/12
Reduce 4/12 to its lowest terms which is ⅓
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Fall 2007
Week 10 – Teacher - Page 8 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
Week Week 10 – Day 5
Concept/Skill Multiplying fractions
Standard 7 NS: 1.2 Add, subtract, multiply, and divide rational numbers (integers,
fractions, and terminating decimals) and take positive rational numbers
to whole-number powers.
Materials CAHSEE Math Released Item # 7
ƒ Review the definitions of improper fractions and mixed numbers:
An improper fraction is a fraction that has a numerator larger than or equal to its
denominator. A proper fraction is a fraction with the numerator smaller than the
denominator.
A mixed number consists of an integer followed by a proper fraction.
ƒ Review changing an improper fraction to a mixed number: The improper fraction 8/5
can be changed to the mixed number 1 3/5 by dividing the numerator (8) by the
denominator (5). This gives a quotient of 1 and a remainder of 3. The remainder is
placed over the divisor (5).
ƒ Review changing a mixed number to an improper fraction: The mixed number 1 3/5
can be changed to the improper fraction 8/5 by multiplying the integer (1) by the
denominator (5) and adding that to the numerator.
ƒ Review the steps for multiplying fractions:
1. Change any integers or mixed numbers back to improper fractions.
2. Multiply the numerators and multiply the denominators.
3. Reduce the answer to lowest terms and, if possible, change it back to a mixed
number.
ƒ Have students do the guided practice test item on Day 5 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
Example:
Mary uses 1 ½ cups of sugar to make a gallon of lemonade. How may cups of sugar
does she need to make 2 gallons of lemonade?
Write the equation to solve the problem:
1 1/2 x
= cups of sugar needed
2
Change 1 ½ back to the mixed number: ______
Change the integer to an improper fraction: ____
Rewrite the equation and find the answer:
3
2
x
2
1
Change the improper fraction to an integer or mixed number:
Laura Westermeier, Ed.D.
SAUSD, Educational Services
6
2
=
3
Fall 2007
Week 10 – Teacher - Page 9 of 10
CAHSEE Math Problem of the Day -- Week 10
Teacher Notes & Answers
ƒ Have students do the CAHSEE released test item on Day 5 of the student CAHSEE
Math Problem of the Day -- Week 10 worksheet.
7. John uses ⅔ of a cup of oats per serving to make oatmeal. How many cups of
oats does he need to make 6 servings?
A
2⅔
B
4
C
5⅓
D
9
x
=
= 4
ƒ At the end of the week remind students to keep the student CAHSEE Math
Problem of the Day -- Week 10 worksheet to use as a study guide as they continue
to prepare for the CAHSEE.
Laura Westermeier, Ed.D.
SAUSD, Educational Services
Fall 2007
Week 10 – Teacher - Page 10 of 10