Appendices

1. Take a piece of notebook paper and fold it hot dog style, then fold it hot dog style again. Open and see how the paper has 4 sections. Shade the 3 of the 4 sections using this pattern: ////
2. Take the same sheet of paper and fold it hamburger style, then repeat folding the top to the bottom 2 more times. Open and see how the paper made 8 sections. Shade one 1 of the 8 sections using this pattern:\\\\\\\\
3. How many sections have both the //// and \\\\\\\\ shading? Out of how many? (Notice that this is what multiplying fractions represents: a piece of a piece.)

Example. The shaded portion of each of the following circles represent the fractions [latex]\frac38[/latex] and [latex]\frac34[/latex], respectively.

1. How would you visually depict each of the following?
   1. [latex]\frac{1}{2} \cdot \frac{2}{3}[/latex]
   2. [latex]\frac{1}{6} + \frac{4}{6}[/latex]
   3. [latex]\frac{5}{8} - \frac{1}{4}[/latex]
2. 4. [latex]\frac{4}{9} \cdot \frac{3}{16}[/latex]
   5. [latex]\frac{7}{8} \div \frac{2}{9}[/latex]
   6. [latex]\frac{21}{24} \cdot \frac{16}{18}[/latex]
   7. [latex]\frac{4}{9} \div \frac{42}{18}[/latex]
   8. [latex]\frac{4}{5} \cdot \frac{1}{8}[/latex]
   9. [latex]1 \frac{4}{9} \div 2 \frac{1}{4}[/latex]
   10. [latex]\frac{1}{8} + \frac{2}{5}[/latex]
   11. [latex]7 \frac{3}{5} - \frac{1}{2}[/latex]
   12. [latex]\frac{5}{9} - \frac{1}{6}[/latex]
   13. [latex]\frac{2}{9} + \frac{2}{3}[/latex]
3. 14. [latex]\frac{2}{3} \cdot \frac{12}{19} = \frac{24}{57}[/latex]
   15. [latex]\frac{4}{5} + \frac{2}{3} = \frac{6}{8} = \frac{3}{4}[/latex]
   16. [latex]1 \frac{2}{3} - \frac{4}{5} = -1 \frac{2}{5}[/latex]
   17. [latex]2 \frac{1}{3} \div \frac{4}{5} = \frac{7}{3} \div \frac{4}{5} = \frac{3}{7} \cdot \frac{4}{5} = \frac{12}{35}[/latex]

4. Solve.
   18. Dolores sold candy by the pound. The first person who came into her store bought [latex]2\frac{1}{4}[/latex] pounds of gumdrops. The second person who came into the store bought [latex]5\frac{3}{4}[/latex] pounds of jelly beans. How many pounds of candy did Dolores sell to her first two customers?
   19. In Language Arts the class was assigned to write a team paper. Each team was to write [latex]5\frac{1}{2}[/latex] pages for the combined paper. When all the teams put their papers together, there were a total of [latex]44[/latex] pages. How many teams had worked on this paper?
   20. Dennis’s new dog weighs [latex]127\frac{3}{4}[/latex] pounds. His friend’s dog weighs [latex]94\frac{1}{2}[/latex] pounds. How much more does Dennis’s dog weigh than his friend’s dog?
   21. On their vacation trip last summer the Pace family got [latex]30\frac{1}{2}[/latex] miles per gallon of fuel on their trip from home. On the return trip they only got [latex]17\frac{7}{8}[/latex] miles per gallon. How many more miles per gallon of fuel did they get on their way out than on their way home?
   22. Jerry bought [latex]12\frac{1}{2}[/latex] pounds of candy to share with his friends. If he gave [latex]2\frac{1}{2}[/latex] pounds each to three friends, how much candy does he have left?
   23. Joy and her friend wanted to do something di!erent. They decided to go to the track and see how many times they could run around the track without stopping. When they finally finished, their stopwatch showed they each ran a total of [latex]23\frac{2}{5}[/latex] minutes. If on average Joy also knew it took them [latex]4\frac{1}{2}[/latex] minutes to run around the track once, how many times did they make it around the track?

1. Perform the operation
   1. [latex]\frac{3}{8} + \frac{3}{8}[/latex]
   2. [latex]5\frac{3}{8} + 2\frac{7}{8}[/latex]
   3. [latex]\frac{5}{9} + \frac{1}{3}[/latex]
   4. [latex]7\frac{5}{6} - 2\frac{3}{10}[/latex]
   5. [latex]\frac{2}{7} \cdot \frac{3}{5}[/latex]
   6. [latex]10 - 5\frac{4}{9}[/latex]
   7. [latex]\frac{5}{12} + \frac{7}{8}[/latex]
   8. [latex]7\frac{1}{3} - 3\frac{6}{7}[/latex]
   9. [latex]11\frac{1}{3} \div 4\frac{1}{2}[/latex]
   10. [latex]12\frac{7}{9} + 3\frac{11}{12}[/latex]
   11. [latex]\frac{7}{9} - \frac{1}{6}[/latex]
   12. [latex]5\frac{1}{3} \div 2\frac{1}{4}[/latex]
   13. [latex]3\frac{1}{2} \cdot 4\frac{2}{3}[/latex]
   14. [latex]8\frac{2}{5} - 4\frac{3}{4}[/latex]
   15. [latex]4\frac{2}{5} + 2\frac{1}{5}[/latex]
   16. [latex]4 - 1\frac{7}{8}[/latex]
2. Find the errors and correct them.
   17. [latex]\frac{3}{4} + \frac{1}{2} = \frac{4}{6} = \frac{2}{3}[/latex]
3. Solve.
   18. Deb had [latex]4\frac{3}{4}[/latex] yards of leopard print fabric. She wanted to make a skirt that called for [latex]2\frac{7}{8}[/latex] yards of fabric. How much will she have left over?
   19. Traci bought [latex]1 \frac{1}{4}[/latex] yards of yellow ribbon, [latex]2 \frac{5}{6}[/latex] yards of pink ribbon and [latex]3 \frac{1}{2}[/latex] yards of purple ribbon. How many yards of ribbon did she buy altogether?
   20. Emily is buying [latex]\frac25[/latex] pounds of beans for dinner. Ricky is buying [latex]\frac74[/latex] of the amount of beans that Emily is buying. How many pounds of beans is Ricky buying? Express your answer in simplest form.
   21. How much chocolate will each person get if 3 people share [latex]\frac12[/latex] pound of chocolate equally?
   22. How many [latex]\frac34[/latex] cup servings are in [latex]\frac23[/latex] of a cup of yogurt?
   23. Sheila was making cupcakes for the school parent-student dinner. Sheila knew that she had put in [latex]13\frac23[/latex] ounces of ingredients she had to mix together in the blender. After she was finished, she poured the mix into [latex]2\frac14[/latex] ounce cupcake tins. How many of these cupcakes would one batch of this mix make? How much of the mix would be left over?
   24. One day Mr. Saunders walked [latex]4\frac12[/latex] miles to one store and [latex]2\frac15[/latex] miles to another store. Then he walked [latex]7\frac56[/latex] miles again to a third store. How far did he end up walking altogether?
   25. For one of their Science experiments, a class used [latex]10\frac12[/latex] quarts of water. If there were 7 groups doing the experiment and each group used the same amount of water, how any quarts of water did each group use?
   26. Robert and his father raked up [latex]17\frac14[/latex] bags of leaves from their yard. Robert’s father has a machine that shreds the leaves to almost nothing. If Robert has already thrown in [latex]8\frac12[/latex] bags, how many more does he have to throw in to shred them all?

1. Simplify the expressions.
   1. [latex]5 - 3 + 72(10 - 2)[/latex]
   2. [latex]2 + 2(3 - 6) - (1 + (1 - 4)^2)[/latex]
   3. [latex]\sqrt{5^2 - 3^2}[/latex]
   4. [latex]|4 - 8| - 9 \cdot (2 - 3)[/latex]
   5. [latex]\frac{7^3 + 2(4)}{17 - 3}[/latex]
   6. [latex]1 + \frac{\frac{2}{5} + \frac{1}{3}}{2^3}[/latex]
   7. [latex]-3(5 + 1)^2[/latex]
   8. [latex]48 \div 3 \cdot 2 - 4^2 + 5[/latex]
   9. [latex]\frac{(4 + 3 \cdot 2) + (3 \cdot -1)}{-1^2}[/latex]
   10. [latex]\frac{\frac{3}{5} + \frac{1}{3}}{2^3}[/latex]
   11. [latex]3 + 2(4 + (-6)) + (1 + (-2)^2)[/latex]
   12. [latex]2(4 - |-5|) - (1 + 4)^2[/latex]
   13. [latex]\frac{1}{2} \cdot 8 \div 6[/latex]
   14. [latex]\frac{2}{5} + \frac{4}{3} \div 2[/latex]
   15. [latex](\frac{1}{2})^2 + \frac{1}{5}[/latex]

2. Solve the equations.
   16. [latex]5(x - 2) = 4x - 10[/latex]
   17. [latex]2x - 5x = 3(2x - 4)[/latex]
   18. [latex]10x + (4 - x) = 7x + 2(3x + 5)[/latex]
   19. [latex]3 - (2 - 7x) = 2x - (-5x - 1)[/latex]
   20. [latex]\frac{3(t - 7)}{2} = t - 6[/latex]
   21. [latex]\frac{6}{7} + x = \frac{13}{7}[/latex]
   22. [latex]\frac{(x - 2)}{3} = x - 1[/latex]
   23. [latex]\frac{2}{5}x - \frac{5}{7}x = x - 1[/latex]
   24. [latex]10x - (4 - 3x) = 7x + 2(3x + 5)[/latex]
   25. [latex]4 - \frac{1}{3}(2 - 7x) = 2(x - 1)[/latex]
   26. [latex]\frac{4(x + 9)}{3} = x[/latex]
   27. [latex]\frac{6}{11} + x = \frac{11}{7}[/latex]

3. Simplify or solve.
   28. [latex]\frac{x}{3} + \frac{1}{4}x - 1[/latex]
   29. [latex]\frac{x}{3} + \frac{1}{4}x = 1[/latex]
   30. [latex]\frac{x}{4} + 3x[/latex]
   31. [latex]5x - (x - 3) = 7 + 2(x + 6)[/latex]
   32. [latex]\frac{3(x - 3)}{2} -x[/latex]
   33. [latex]\frac{3(x - 3)}{2} = x[/latex]

1. Simplify the expressions.
   1. [latex]5 - (7 - 3)[/latex]
   2. [latex]\sqrt{16} - 3[/latex]
   3. [latex](8 + 7)5[/latex]
   4. [latex]\frac{7}{10} + (-\frac{2}{3})[/latex]
   5. [latex]\frac{10 - 6}{9 - 7}[/latex]
   6. [latex](\frac{1}{3})(-6)^2 + 3[/latex]
   7. [latex]-7(2) + 8(\frac{1}{2}) - \sqrt{25}[/latex]
   8. [latex]\frac{(5 + 3 \cdot 5) + 3(-1)}{(-1)^2}[/latex]
   9. [latex](\frac{1}{5} + \frac{6}{5})^2[/latex]
   10. [latex]\frac{\frac{7}{8} - \frac{1}{9}}{3}[/latex]
   11. [latex]\frac{(\frac{2}{7})^2}{\frac{2}{21}}[/latex]
   12. [latex]\frac{(\frac{1}{3} - \frac{3}{4})^2}{\frac{2}{3}}[/latex]
   13. [latex]\frac{\frac{2}{3} - \frac{1}{4}}{\frac{2}{5} + \frac{1}{4}} \div 2[/latex]
   14. [latex]\frac{1}{3} + \frac{(\frac{1}{2} - \frac{3}{4})^2}{(-\frac{1}{2})^2}[/latex]
   15. [latex]\frac{(\frac{1}{4} - \frac{2}{3})^2}{2^3} + \frac{1}{2} \div \frac{3}{8}[/latex]
2. Solve the equations
   16. [latex]\frac{x}{-2} = -9[/latex]
   17. [latex]6x + 5x = -31 - 2[/latex]
   18. [latex]-3(4 - x) = 12[/latex]
   19. [latex](7x - 5) - (x - 4) + (3x - 8) = 3x - 5[/latex]
   20. [latex]8.54x - 5.71x + 3.8x = -39.78[/latex]
   21. [latex]5(x + 2) - \frac{2}{3}x + \frac{x}{2} = 6x - 11[/latex]
   22. [latex]-9x - 1 + 7x = -3 - 10[/latex]
   23. [latex]7 - x = 10[/latex]
   24. [latex]36 = (5c + 4) - (2c + 4)[/latex]
   25. [latex]x = -x + 2[/latex]
   26. [latex]\frac{1}{2}x = -x + 2[/latex]
   27. [latex]5x = 3x + (x + 2)[/latex]
   28. [latex]\frac{3}{5}x = 1[/latex]
   29. [latex]\frac{3}{4}x - 2x = 7[/latex]
   30. [latex]\frac{1}{2}x - x = 7(x + 2)[/latex]
   31. [latex]\frac{x}{2} + \frac{5}{6} = \frac{x}{3}[/latex]
   32. [latex]\frac{x}{2} + 2x = \frac{1}{3}[/latex]
   33. [latex]-3(x - 4) = 12[/latex]
   34. [latex](3x - 4) - (x - 1) + (2x - 15) = 3x - 7[/latex]
   35. [latex]3.5x - 5.75x + 1.75x = -1.5[/latex]
   36. [latex]x - \frac{5x}{2} + 7 = \frac{x}{4}[/latex]
   37. [latex]\frac{x}{2} + \frac{5}{5} = \frac{2x}{3}[/latex]
   38. [latex]14 = 5x - (3x - 2)[/latex]
   39. [latex]\frac{x - 5}{9} = -3[/latex]
   40. [latex]\frac{x}{4} + 2 = \frac{4}{8}[/latex]
   41. [latex]-4x = \frac{5(1 - x)}{6}[/latex]
   42. [latex]\frac{1}{2}x - 3x = 7(x + 5)[/latex]
   43. [latex]3x + 1 = 2(x - 3) + x[/latex]

1. Perform the indicated operations.
   1. [latex](3a + 2a^2 + 3a^3) - (a + 2a^2 - 3a^3)[/latex]
   2. [latex](3a + 2a^2 + \frac{3}{5}a^3) + (6a - \frac{2}{5}a^2 - 3a^3)[/latex]
   3. [latex](\frac{2}{7}a + 2a^2 + \frac{3}{5}a^3) - (a - \frac{2}{7}a^2 - 3a^3)[/latex]
   4. [latex](4x - 3)^2[/latex]
   5. [latex](x - 3)(x^2 + 3x + 9)[/latex]
   6. [latex](r - 6)(r - 3) - (3r + 2r^2 - 7)[/latex]

2. Answer the following.
   7. Write the polynomial that represents the perimeter of the triangle.
   8. Find the polynomial that represents the area of the shaded region
   9. Write the polynomial that represents the perimeter of the parallelogram in the figure.
   10. Write the polynomial that represents the total number of square feet for the floor plan below.

3. Factor out the greatest common factor (GCF) from the expressions.
   11. [latex]6y^2 + 6[/latex]
   12. [latex]xy^2 - 3y^2[/latex]
   13. [latex]3x^2 + 6x[/latex]
   14. [latex]16x^2y^2 - 28xy^2 + 48y^5[/latex]
   15. [latex]25x^3y^4 - 65x^2y^5 - 80x^4y^3[/latex]

1. Perform the indicated operations.
   1. [latex](3x + 2) + (3x - 2)[/latex]
   2. [latex](3x + 2) - (3x - 2)[/latex]
   3. [latex](3x + 2)(3x - 2)[/latex]
   4. [latex]2(3x + 2)[/latex]
   5. [latex](3x + 2)^2[/latex]
   6. [latex](3x + 5)(4x - 7)[/latex]
   7. [latex](3x^2y + 2) + (3xy - 25)[/latex]
   8. [latex](3x + 4) + (4x - 5)[/latex]
   9. [latex](x^3 + 7x^2 - x + 9) + (5x^2 + 4x - 5)[/latex]
   10. [latex](8x + 3) - (x + 4)[/latex]
   11. [latex](3x + 5)(6x - 7)[/latex]
   12. [latex](x + 5)(x + 2)[/latex]
   13. [latex](5x^2 + 8x - 4) - (2x^2 + 6x - 5)[/latex]
   14. [latex](x + 3)^2[/latex]
   15. [latex](x^2 - 3x + 7) + (2x^2 + 5x - 6)[/latex]
   16. [latex](x - 4)(x + 5) - (3x + 2x^2 - 30)[/latex]
2. Answer the following.
   17. Write the polynomial that represents the total volume of the rectangular prism below. Use the formula: [latex]V\;=\;\text{length}\;\cdot\;\text{width}\;\cdot\;\text{height}[/latex]
       
   18. A countertop pictured below will have a hole drilled to hold a utensil holder. The area of the entire countertop is given by [latex]3x^2+12x+4[/latex]. The area of the hole is given by [latex]x^2+2x+1[/latex]. Write an expression for the area of the countertop that remains after the hole is drilled (the shaded region).
       
3. Factor out the greatest common factor (GCF).
   19. [latex]6x^2 + 15x[/latex]
   20. [latex]36x^3 + 90x^5 - 72x^7[/latex]
   21. [latex]2x^2 - 12x[/latex]
   22. [latex]14x^2 - 12x + 8[/latex]
   23. [latex]14x^3 - 12x^2 + 8x[/latex]
   24. [latex]32x^2 - 16x[/latex]
   25. [latex]32x^2 - 16x + 8[/latex]