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Dear Parents/Students,
In the summer time, many necessary mathematical skills are lost due to the absence of daily exposure. The loss of skills may result in a lack of success and unnecessary frustration for students as they begin the new school year. The purpose of this math assignment is to set the stage for instruction for the 2017-2018 school year. Packets are to be downloaded, printed out, and worked on neatly in the packet or on a separate piece of paper. Additionally, students should attempt all problems without calculators. The completed packet is due on the first day of school during math class and will be worth 30 points.
The packet are a review of previously taught concepts. Each concept includes a quick review and practice. Some might even include review videos students can access. Additional help can be found at www.khanacademy.org. These skills are required to be successful in the upcoming year. We will be briefly reviewing this information on the first day of school, and then moving into the class curriculum.
Thank you,
The High School Math Team
Geometry Summer Packet
12
Addition and Subtraction of Fractions
with the Same Denominator To add or subtract fractions, the denominators MUST be the same.
Example 1:
?51
53
=−
513
51
53 −
=−
52
=
Because both fractions have the same denominator, you may subtract the numerators and keep the denominator.
Example 2:
?97
95
=+
975
97
95 +
=+
912
=
931=
311=
Because both fractions have the same denominator, you may add the numerators and keep the denominator. Always change improper fractions to a mixed number. Reduce, when possible.
Add or Subtract as indicated. 1.
83
84+
2.
101
107−
3.
484
489
487
++
4. 373
3740
−
5.
134
1310
+
6.
1717
1711
179
++
7. 36
34
32
−+
8.
61
65
67
+−
9.
139
137+
13
Addition and Subtraction of Fractions
with Different Denominators Remember: In order to add or subtract fractions, the denominators MUST be the same.
Example:
?83
32
=+
LCM = 24 Find the LCM
2416
88
32
=×
+ 249
33
83
=×
2425
Write the problem vertically. Find the equivalent fractions with the LCM as a denominator. Add the fractions with the same denominator.
2411
2425
= Remember to write as a mixed number and reduce when possible!
Add or Subtract:
1) 78 +
34
2) 78 -
34
3) 1112 +
1718
4) 37 +
25
5) 1524 -
1027
6) 712 +
516
7) 1627 -
524
8) 114 +
38
9) 114 +
2318
10) 298 +
97
11) 21335 - 1
514
12) 23 +
121 -
27
14
Subtraction of Fractions with Borrowing Example 1: Example 2:
?3117 =− ?
652
315 =−
Note: There are two common methods; DO NOT mix the steps of the methods!
Method 1 Example 1
7 = 336
- 311 =
311
325
Subtraction with Borrowing Write problem vertically Cannot subtract fraction from whole without finding common denominator.
Borrow one whole from 7 and express as .LCDLCD ⎟
⎠⎞
⎜⎝⎛ =
331
Subtract numerators and whole numbers. Example 2
625
315 =
684=
- 652
652 = =
652
212
632 =
Write problem vertically and find LCD Cannot subtract 5 from 2.
Borrow one whole from 5, ⎟⎠
⎞⎜⎝
⎛66
4 and add ⎟⎠⎞
⎜⎝⎛ +
=6
264625 .
Subtract numerators and whole numbers; reduce as needed.
Method 2 Example 1:
7 = 321
- 311 =
34
325
317
=
Subtraction Using Improper Fractions Write the problem vertically. Convert the whole numbers and mixed numbers to improper fractions using the LCD.
Subtract ⎟⎠⎞
⎜⎝⎛ −
3421 and convert improper fraction to
mixed number. Example 2:
625
315 =
632
=
- 652
652 = =
617
232
615
=
212
232 =
Write problem vertically and find the LCD. Change the mixed numbers to improper fractions. Subtract the numerators. Convert to a mixed number. Reduce.
15
Subtract:
1) 5 - 213
2) 7 - 1 16
3) 10 - 4 56
4) 3 58 - 2
78
5) 1 18 -
34
6) 3 512 - 1
1516
7) 8 - 6 45
8) 4 38 - 3
56
9) 17 - 4 59
10) 5 518 - 1
34
11) 5 27 - 3
38
12) 18 - 1 716 -
712
16
Multiplication of Fractions Example:
653
103×
Note: LCD is not needed to multiply fractions.
65)36(
653 +×=
Change mixed numbers to improper fractions
210231
623
103
××
=× Before multiplying, reduce by dividing any numerator with any denominator with a common factor. (3 and 6 have a common factor of 3)
2023
210231
=××
Multiply numerators and denominators
2031
2023
= Convert improper fractions to mixed numbers.
Multiply: 1)
32
214 ×
2) 411
513 ×
3) 9116 ×
4) 211
612 ×
5) 1571
1110
×
6) 15534 ×
7) 922
833 ×
8) 173234 ×
9) 54
879 ×
10) 411
1097 ×
11) 154
73118 ××
12) 83
651
513 ××
17
Division of Fractions
Example:
832
432 ÷ OR
832
432
Note: One fraction divided by another may be expressed in either way shown above. Also, LCD is not needed to divide fractions.
411
432 = and 8
19832 =
Convert mixed numbers to improper fractions
198
411
819
411
×=÷
Invert the divisor ⎟⎠⎞
⎜⎝⎛
819 . (Turn the fraction after the
division sign upside down)
191211
194811
××
=××
Reduce if possible. (4 and 8 have a common factor)
1922
191211=
××
Multiply numerators and denominators
1931
1922
=
Convert to a mixed number and reduce if needed.
Divide these fractions. Reduce to lowest terms!
1) 56 ÷
12
2) =÷73
43
3) 3 ÷ 1 25 =
4)
3121
=
5) 12 ÷ 6 =
6) 2 14 ÷ 3 =
7) 3 17 ÷ 2
514 =
8) 2
58
1 78
9) 4 12 ÷ 1
34 =
18
Some Fraction Word Problems
Example 1: One day Ashley biked
43 of a mile before lunch and
87 of a mile after lunch. How far
did she cycle that day? Note: this problem is asking you to add the distances traveled.
87
43+
87
86+
851
813
=
To add fractions, find a LCD (8). Add the numerators; keep the denominators. Convert improper fraction to a mixed number; reduce if needed. Ashley cycled
851 miles that day.
Example 2: A tailor needs
413 yards of fabric to make a jacket. How many jackets can he make
with 2119 yards of fabric?
Note: this problem is asking you to divide.
413
2119 ÷
413
239
÷
1123
134
239
××
=×
313=
To divide fractions, convert mixed numbers to improper fractions. Invert the divisor and reduce if possible, (39 and 13 have a common factor, as do 2 and 4). Multiply numerators and denominators. The tailor can make 3 jackets from
2119 yards of fabric.
19
Solve the following problems. 1. An empty box weighs
412 pounds. It is then filled with
3216 pounds of fruit. What is
the weight of the box when it is full?
2. Yanni is making formula for the baby. Each bottle contains 526 scoops of formula.
The formula container holds 320 scoops of formula. How many bottles of formula can Yanni make?
3. Miguel bought 412 pounds of hamburger,
511 pounds of sliced turkey, and 2 pounds
of cheese. What was the total weight of all of his purchases?
4. Sheila had 8 yards of fabric. She used 412 yards to make a dress. How much fabric
does she have left?
5. A father leaves his money to his four children. The first received 31 , the second
received 61 , and the third received
52 . How much did the remaining child receive?
(Hint: You can think of father’s money as one whole.)
6. Find the total perimeter (sum of the sides) of an equilateral triangle, (triangle with
equal sides), if each side measures 412 inches.
1
Topic 1: Order of Operations
Topic 1 Exercises:
Evaluate each expression
1. 2. 3.
4. 5. 6.
7. 8. [ ] 9.
Evaluate each expression when , , ,
, and
10. 11. 12.
13.
14. 15.
16.
17.
18.
2
19. (
)
(
)
20. (
) (
)
Topic 2: Distributive Property and Combining Like Terms
Topic 2 Exercises:
Simplify each expression. If not possible, write simplified.
1. 2.
3.
4. 5. 6.
7. 8. 9.
10.
11. 12.
13. 14.
Write an algebraic expression for each verbal expression, then simplify.
15. Six times the difference of and , increased by .
3
16. Two times the sum of squared and squared, increased by three times the sum of squared and squared.
Topic 3: Writing Equations
Topic 3 Exercises:
Translate each sentence into an equation or formula.
1. Three times a number minus twelve equals forty.
2. One-half of the difference of and is 54.
3. Three times the sum of and 4 is 32.
4. The area of a circle is the product of and the radius, , squared.
5. WEIGHT LOSS Lou wants to lose weight to audition for a part in a play. He weighs 160 pounds now. He wants
to weigh 150 pounds.
a. If represents the number of pounds he wants to lose, write and equation to represent this situation.
b. How many pounds does he need to lose to reach his goal?
Translate each equation into a sentence.
4
6. 7.
8.
9.
Topic 4: Solving Equations, Ratios and Proportions
Topic 4 Exercises:
Solve each Equation. It is possible for the equation to be an identity or have no solution.
1. 2. 3.
4. 5. 6.
7.
8. 9.
10. 11. 12.
13. 14. 15.
5
16. 17. [ ] 18.
19. 20.
Topic 4 Exercises continued:
Solve each equation or formula for the variable indicated.
21. 22. 23.
24. 25. 26.
27.
28.
29.
Topic 4 Exercises continued:
Evaluate each expression.
30.
31.
32.
6
33.
34.
35.
36. Josh finished 24 math problems in one hour. At that rate, how many hours will it take him to complete 72
problems?
Topic 5: Graphing Linear Equations, Slope, and Direct Variation
Topic 5 Exercises:
7
Topic 5 Exercises continued:
Find the slope that passes through each pair of points.
8
Find the value of so that the line passing through the given points has the given slope.
Topic 5 Exercises continued:
Suppose varies directly as . Write a direct variation equation that relates to . Then solve.
7. 9. 8.
10. 11. 12.
13. 14. 15.
9
16. If when , find when .
17. If when , find when .
18. If when , find when .
19. If
when
, find when
.
Topic 6: Writing Equations of Lines
Standard Form
Standard Form , where A and B are coefficients and C is a constant
** After finding point-slope form, manipulate the equation to put the equation in slope-intercept form and standard form
Watch this Video!
https://www.khanacademy.org/math/trigonometry/graphs/line_equation/v/point-slope-and-standard-form
Topic 6 Exercises:
Write the equation of the line passing through the given points in:
a.) Point-slope form
b.) Slope-intercept form
c.) Standard form using integers
(follow the example in the video. Each problem has 3 solutions)
10
1. 2.
3. 4.
5. 6.
Topic 6 Exercises continued:
Write the equation of the line in slope-intercept form that passes through the given point and is parallel
to the given line.
11
7. 8. 9.
Write the equation of the line in slope-intercept form that passes through the given point and is
perpendicular to the given line.
10. 11. 12.
Topic 7: Solving and Graphing Inequalities
Topic 7 Exercises:
Solve each inequality and graph the solution on a number line.
1. 2. 3.
12
4. 5. 6.
7. 8.
9.
10.
Topic 7 Exercises Continued:
13
Topic 8: Solving Systems of Linear Equations
11. 12. 13.
14. 15. 16.
17. 18. 19.
14
Topic 8 Exercises:
15
Topic 8 Exercises continued: Use substitution to solve each system of equations.
7. {
8. {
9. {
10. {
11. {
12. {
16
Watch this Video!
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/solving-systems-addition-
elimination/v/solving-systems-by-elimination-2
Topic 8 Exercises continued: Use elimination to solve each system of equations.
13. {
14. {
15. {
16. {
17. {
18. {
19. The length of Sally’s garden is 4 meters longer than 3 times the width. The perimeter of her
garden is 72 meters. What are the dimensions of Sally’s garden?
