Problems of the Week. Visual. Data/Investigations. Multi-Step. Problem-Solving Strategies. Geometry/ Measurement. What Do You Say?

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1 Problems of the Week Visual Draw a picture to answer the following question based on the information given: Margaret walked 8 blocks south and blocks east from her home to school. After school she walked 2 blocks north and block east to the park. From the park she traveled block north and 2 blocks west to visit her friend. How far does Margaret have to walk to return home? Jack Bean-Stalk has a school teacher who knows a few magic tricks of her own. She is watching her magic beans grow according to this pattern. Day # of beans On which day will she have enough magic beans to give at least 5 to each of her 20 students? The Gum Machine Karen and Diane are having an argument. They both know that the machine is loaded with gum balls of 6 different colors. Karen feels lucky. She thinks she can get each color by using only 6 pennies. Diane feels it will take at least 5. Let s assume that the machine is loaded with an equal number of each color. How many pennies do you think are needed to get one of each color? Do you agree with either Karen or Diane? Use a fair number cube to experiment and find your answer. Geometry/ Measurement An aquarium is 6 inches long, 24 inches wide and 6 inches tall. The aquarium is filled with distilled water to a level of 2 inches. If a cubic foot of distilled water weighs 62.4 pounds, how many pounds of water are in the aquarium? Express your answer as a decimal correct to the nearest tenth. From Mathcounts School Handbook Problem-Solving Strategies Four students are called to sit in four chairs on a stage. The students are Bob, Brenda, Carol, and Don. Bob and Brenda are cousins. If the students are called up in random order and take seats in the order in which they are called, what is the probability that Bob and Brenda will sit next to each other? Johnny has trouble finding the LCD when subtracting fractions. You want to help him. Write the steps Johnny needs to follow in order to find the LCD. Grade 8 Page

2 *See answer key. This section should not be duplicated for the students.. (5 + )5 2. (2)(5)(0)(7)(8). Write.0 as a fraction. 4. Write. as a fraction. 5. Solve for y: y = 6. True or False: = Write 25,000 in scientific notation Simplify: 2x + 5y 7x + 2xy Mental Math = 2. 0% of 00 = = = 5. 2[(4 + ) (2 7)] = = (0) = = = = Write answers here: Grade 8 Page 2 Grade 8

3 Visual N Home Answer Key Mental Math W S 2 blocks west 5 blocks north Answer: Day 2 E Geometry/Measurement Answer: 74.4 pounds 8 blocks Boyfriend F 2 blocks blocks 2 blocks block block Park School False x + 2xy + 5y This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Answers will vary depending upon experiments. Problem-Solving Strategies Strategy: make an organized list or use a formula. B BR C D B BR D C B C BR D B C D BR B D BR C B D C BR BR B C D BR B D C BR C B D BR C D B BR D B C BR D C B C BR B D C BR D B C B BR D C B D BR C D BR B C D B BR D BR C B D BR B C D C BR B D C B BR D B BR C D B C BR 24 possible 2 they sit together Answer 2/24 = / Answers will vary; however, students could mention using the prime factorization of each denominator to determine the LCD. Grade 8 Grade 8 Page

4 Problems of the Week Visual Nine one inch cubes are put together to make a T. If the complete figure is painted red, how many cubes have exactly 4 red faces? 2 An estate valued at $62,000 is left by a will as follows: to each of two grandchildren a certain sum, to the son twice as much as to the two grandchildren together, and to the widow $2,000 more than to the son and grandchildren together. How much goes to each? Geometry/ Measurement Imagine a new type of lottery. To make the four-digit number, digits are randomly chosen from, 2, and 4, and no digit is repeated. The order of the digits is important. If your lottery number is 24, what is the probability that you will win? NCTM Math Journal Sept/Oct 996 Problem-Solving Strategies Working Backwards Find the starting number on this flow chart: A square is folded in half to form a rectangle. If the resulting rectangle has a perimeter of 5 inches, what is the area of the original square??? 7 Will these two expressions always be equal? Explain. x y x y Grade 8 Page 4

5 2 *See answer key. This section should not be duplicated for the students.. In what quadrant would you find the ordered pair ( 2, )? 2. Identify the mode of the data: {5, 7,, 7,, 5, 6, 7} falls between what two consecutive integers? Mental Math Write the next number in this pattern:, 9, 27, 8, =? 5 9. True or false: 2 7 = Order these numbers from least to greatest:, 0,,, 2 Write answers here: = Compare:? Compare: 5 28? List all the factors of Explain why is a prime number = 0. Half of a dozen Grade 8 Page 5

6 Visual Answer: 5 Answer Key Mental Math. III Answer: The grandchildren each get $5,000; the son gets $20,000; and the widow gets $2,000. Geometry/Measurement Answer: 25 inches and True 0. 2 This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Answer: 4 2 = ,, 2, 0, > Problem-Solving Strategies Answer: 56 6.,, 2, 0, 7. (, 2,, 4, 6, 9, 2, 8, 6) 8. It has only factor The two expressions are not always the same. For example, if x = and y = 5 ( 5) = 8 while 5 = 2 Grade 8 Page 6

7 Problems of the Week Visual Consider the shape shown. Can you keep the perimeter the same and make the area smaller? Larger? If so, draw the picture. NCTM Magazine During Bully's first week working in a china shop he breaks a vase that costs $45.89, two glasses that cost $6.25 each, and a platter that costs $5.49. If he makes $5.25 an hour, how many hours will he have to work to pay for the things he broke? Geometry/ Measurement Tom has some cubes that are 2 inches on each side. How many of them will fit into a box that is a cube with each side 4 inches long? For the set (, 4, 5, 8, x) the mean, median, and mode all have the same value. What is the value of x? Problem-Solving Strategies Make a Diagram Martinez s A- Cars and Trucks is having a super sale. José and Alex are setting up the lot. Mr. Martinez gives them a diagram of the lot and these directions. Put the 4-door car in front of the van. Put the jeep between the truck and the van. Put the sports car to the left of the 2-door and 4-dor cars. How did Mike and Don set up the lot? Will the expression 5 x than 5? Explain. always be smaller Grade 8 Page 7

8 *See answer key. This section should not be duplicated for the students.. What is 25% of 20? 2. (2 0 ) ( 0 4 ). Solve for x: 2x = Multiply three tenths by two tenths. Mental Math 6. Compare 2 and x 2 + y + 7x 2 8. Solve for x: x 2 = Express.08 as a fraction in simplest form. 0. If you simplified ( ) (4)( 7)( 8)(2)( 5), would your answer be positive or negative? Convert.5 to a fraction.. Find the area: m 5m Write answers here: m 2 m Find the area: 5. = x Solve for x Three loaves of bread cost $.87. How much do two loaves cost? 7. Find 55% of out of 40 is equivalent to out of cm Grade 8 Page 8

9 Visual Answer: 4 Geometry/Measurement Answer: 8 Answer Key Mental Math = x 2 + y 8. 5, = 0. positive 2 25 This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Answer: 5 Problem-Solving Strategies T J V SC 2D 4D or SC 2D 4D T J V square meters 4..4 square centimeters $ Answers will vary: out of 20, 2 out of 80 etc No. If x is less than one, x will divide into 5 more than 5 times. Also, if x is negative, 5 x negative and therefore less than 5. will be Grade 8 Page 9

10 Problems of the Week Visual Mr. Ellen had the following exercise on the overhead projector for his class to do when they came in Monday morning. What was his message to the class? Look at or draw this number line and fill in the letters given in the instructions. The first one is done for you. 4 M M to I is +4 O 2 to O is +4 I to O is 8 O to G is 6 O to N is +7 G to D is + N to N 2 is +2 D to G 2 is +8 N 2 to O 2 is 8 G 2 to R is 4 Jamal was hired to do some lawn work for $40 a day, but on the condition that for every day he was idle, he should forfeit $5.50. At the end of 20 days he had $ How many days had Jamal been idle? I Geometry/ Measurement A prism with a square base has a height of 45 centimeters; the volume is 7,200,000 cm. Find the area of the base. A television station plays songs and 4 commercials every 20 minutes. For one 20 minute segment the director chooses three different songs to be played and the producer chooses four different commercials. The producer then decides that the order will be song, commercial, two songs, three commercials. How many possible arrangements are there for this 20 minute segment? Problem-Solving Strategies Martinez s A- Cars and Trucks is having a super sale. What would be the total price of all the vehicles if the least expensive is $0,000 less than the most expensive, the van costs $5,500 more than the jeep which costs the same as the 4-door, the truck costs $25,000, the sports car costs $0,000, and the 2-door is the least expensive and costs $2,000 less than the 4-door? = 0 Explain how you know your answer is correct. Grade 8 Page 0

11 4 *See answer key. This section should not be duplicated for the students What is the LCM of 8 and 2?. What is the GCF of 8 and 2? 4. What is the complement of a 0 angle? 5. (0.0) 2 6. Write in standard form. 7. Multiply: x 2 (2x + ) Solve for k: 2 = 8 k 0. Write.75 as a fraction. Mental Math = 2. Find the perimeter: m 4 m 5m 4m. Find the measure of angle A. 4. 5% of 4 = = 6. Express 7.4% as a fraction and a decimal. 7. Solve for n: 2n = [48 (2 + 4)] 6 = 9. Solve for x: x + 2 = Multiply ; round the answer to the nearest hundredth. A Write answers here: Grade 8 Page

12 Visual Answer: GOOD MORNING Answer: 6 days idle 4 $40 = = 9 $467 Geometry/Measurement Answer: 60,000 cm 2 Answer Key Mental Math , x + x This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." 44 possible arrangements x 4 x 2 x x x 2 x = 44 Problem-Solving Strategies Answer: $46, meters and , = The two numbers are equivalent, so if we multiply one of them by ten by increasing the exponent, then we must also divide it by ten by shifting the decimal point in 2.5 one place to the left. Grade 8 Page 2

13 Problems of the Week Visual Show how to slice this cube so that the cross section is a trapezoid. 5 At a local store, the price of all televisions was reduced 25%. Store employees are given an additional 20% discount on sale prices. What would be the cost a store employee would pay for a television originally priced at $60? Mathcounts Handbook A math class earned the following set of scores on a test: Create a box plot from these scores. Geometry/ Measurement A cube has a surface area of 294 cm 2. What is its volume? Problem-Solving Strategies Logical Reasoning Suppose two identical tanks are to be filled with water. The first tank begins filling 5 minutes before the second tank begins filling. The first tank can be filled in 0 minutes. The second tank can be filled in 20 minutes. How much time will pass before both tanks hold the same amount of water? A poll asks students questions: Do you play tennis? Do you swim? Do you ride a bike? The results are tallied, and the percentage of students who participate in each sport is found. Why should these data not be presented in a circle graph? Grade 8 Page

14 5 *See answer key. This section should not be duplicated for the students.. Simplify: Write. as a percent.. Find the volume of a cube with sides measuring 2 centimeters. 4. Find the surface area of a cube with sides measuring 2 centimeters. 5. Which distance is greater: 5. km or 500 m? 6. Round to the nearest hundredth. 7. Solve for x: 2x + = 7 8. What is the mean for the following set of numbers {7, 4, 7}? 9. What is 80% of 50? 0. Simplify: x 7y + 4x 2y Mental Math. How many inches are in 9 yards? =. Four pencils and one pen cost $.25. Two pencils and three pens cost $2.25. What is the cost of one pen? = 2 5. Write the next two numbers in this pattern:, 4, 7,,, 6. Using the distributive property, write an expression equivalent to 4(x + ). 7. Express 00,000 in scientific notation. 8. Find the area of a rectangle 6" by 4". 9. Write in simplest form: 0 6 Write answers here: Which is greater: 2 or? Grade 8 Page 4

15 Visual Answer: $26 Answer Key Mental Math %. 8 cm cm km x = x 9y 5 This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Geometry/Measurement Answer: 4 cm Problem-Solving Strategies Answer: Both tanks will be half full fifteen minutes after the first tank starts to fill or ten minutes after the second tank starts to fill inches $ = 5. 8, (4 x) + (4 ) sq. in Each student may participate in more than one or in none of the sports. The percentages might not add up to be exactly 00% Grade 8 Page 5

16 Problems of the Week Visual 6 How many cubes are in the figure? Be sure to count the hidden cubes. Margie had an average of 87 on her first five test scores. On the next two tests she did better. She made point higher on the seventh test than she did on the sixth. Her average on all seven tests was now 90. What were her two new test scores? In Pollville, records are kept on the previous mayors. Here are the ages of the last six mayors when they were voted into office: Compare the mean and median. Which is a more representative measure of the average age? Why? Geometry/ Measurement In a triangle, the largest angle is 29 degrees more than the smallest. The other angle is degrees larger than the smallest. What is the smallest angle in the triangle? Problem-Solving Strategies Draw a Diagram There are 6 tennis players in a single elimination tournament. How many tennis matches will be played during the tournament? A student s scores on 4 math tests were 89, 92, 95, and 85. What will he need to make on the 5th test to have an average of 9? Is this possible? Grade 8 Page 6

17 6 *See answer key. This section should not be duplicated for the students.. Evaluate if x = : x Mental Math 2. Solve for z: z = 7 2. (2 ) 2 4. Simplify: 2x(x y) 5. What percent of a day is8 hours? 6. Find the area of the shaded portion of the rectangle: 7. True or false: 7 < ( 8) 9. Convert 4 milligrams to grams m 0 0 m 5 5 m m. Write all prime numbers between 5 and is to 5 as 4 is to. Write answers here:. 2.. Express as a decimal , Evaluate 2a + 7b if a = 5 and b = Find the prime factorization of Find the perimeter of a rectangle 2.5 cm by.5 cm. 0. Complete the table: Input Output Grade 8 Page 7

18 Visual Answer: 5 Answer: 97 and 98 Geometry/Measurement Answer: 46 degrees Answer Key Mental Math or x 2 6xy 5. 75% square meters 7. False, the two are equal. 8. true g This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Answer: Problem-Solving Strategies Answer: 5 The median is less sensitive to the outlier, , 9, or , cm 0. 5 Answer: 04. This is possible if there is a bonus problem on the test. Grade 8 Page 8

19 Problems of the Week Visual Add a square to the figure so that the area is 6 square units and the perimeter does not change. 7 Pebble Pebble! A pebble is thrown into a cylindrical water pool feet in diameter causing the water to rise inches. What is the volume of the pebble in cubic inches? Give an example of data that would show a () positive relationship, (2) negative relationship, () or no relationship when displayed on a scatter plot. Geometry/ Measurement The surface area of a cube is 486 square inches. What is the number of inches in the length of one of the edges? From Mathcounts School Handbook Problem-Solving Strategies Guess and then Check Jack has a total of 0 nickels and quarters. If the value of the coins is $.70, how many quarters does he have? You are given a scatter plot and told to estimate and draw a line of best fit. Explain how you do this. Grade 8 Page 9

20 7 *See answer key. This section should not be duplicated for the students.. Solve for x: x + 7 = 2. Write % as a decimal. Mental Math. How many quarters equal $20.75? 4. A six-pack of soda sells for $2.28. What is the cost per can? 5. Write 5 4 as a percent. 6. Convert.2 0 to standard form. 7. Simplify: 0 8. Solve for x: 4 x 8 = is between what two consecutive integers? 0. True or False: (0.2)(0.8) = 0.6. What is the mode for the following set of scores: {75, 62, 7, 88, 7, 88, 72, 77}? (8 + 2) =. Give the 6th term in the sequence, 4, 9, is 5% of what number? 5. Simplify: (2x 2 ) 6. 75% of 9.00 = 7. Add four to the multiplicative inverse of 2 and simplify. 8. Draw an obtuse angle and estimate its measure. 9. For the given parallelogram determine the measure of angle x. xx Write answers here: What percent of 4 is 7? Grade 8 Page 20

21 Visual Answer: Answer Key Mental Math. x = Answer: π (8) ~.5 cubic inches Geometry/Measurement Answer: 9 inches $ , and 5 0. True This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Answers will vary.. Positive: amount of money earned and spent 2. Negative: age of car and the car's value (The car is not a classic.). No: Your height and month of birth x Problem-Solving Strategies Answer: 6 quarters 8. Answers will vary ( 90 < measure of angle <80 ) 9. 25% 0. 75% Answer: The line should follow the general path of the points. The line should have an equal number of points above and below it and should pass through as many of the points as possible. Grade 8 Page 2

22 Problems of the Week Visual The dot patterns show the first three square numbers. 8 Show the next two square numbers. Use exponents to write each number. Simplify the following expression: x + 4(y + 8x) Write the property that allows you to complete each step (out to the side). On five tests (on which scores could range anywhere from 0 to 00, inclusive) Johnny had an average of exactly 88. Find the lowest score Johnny could have received on one test. Problem-Solving Strategies Geometry/ Measurement In the diagram, lines m and n are parallel. The measure of angle is 44 more than the measure of angle 2. Find the smaller of the two angles. Make a Table The sum of two brothers ages is 8. One brother is ten years older than the other. How old is each brother? 2 m n x Estimate the measure of the indicated angle in degrees. Explain or show how you got your estimate. Grade 8 Page 22

23 8 *See answer key. This section should not be duplicated for the students True or False: 6 0 = 0 Mental Math. Solve for x: x 4 6 = 6 4. Express in standard form. 5. In which quadrant does the ordered pair (, 4) lie? 6. Convert 5,000,000 millimeters to kilometers. 7. Write 489,000 in scientific notation Find the mean of 4, 6,, 2, Find the median of the numbers in Problem #.. What are the next 2 numbers in the pattern?, 2, 4, 8 6,, 4. Order from least to greatest:.5,.05,.500, Find the measure of angle x: 6. Simplify: 2x 2 y + 7x + 2x 2 y xy 2 7. An elevator started at the first floor and went up 8 floors. It then came down floors and went back up 6. At what floor was it stopped? 8. (x 2 y) 9. Find the area of a circle with radius of 8 cm. 0. What does area mean? x 80 Write answers here: Grade 8 Page 2

24 Visual x + 4(y + 8x) = x + 4y + 4 8x Distributive Property = x + 4y + 2x Substitution = x + 2 x + 4y Commutative Prop. of Add. = x + 2x + 4y Multiplicative Identity = ( + 2)x + 4y Distributive Property = x + 4y Substitution Answer Key Mental Math False , II 6. 5 km This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." Geometry/Measurement Answer: The smaller angle is 68 degrees = Problem-Solving Strategies Answer: 4 years, 24 years , , 0.49, 0.5, x 2 y + 7x xy x 6 y cm 2 0. Number of square units needed to cover a surface 5 Sample explanation: ( 90 ) = 5 Grade 8 Page 24

25 Problems of the Week Visual Using Color Cubes. Make a rectangle that is 4 red, 4 blue, 8 yellow 9 and the rest green. 2. Make a rectangle that is 5 yellow, 2 blue and 0 red.. Make a rectangle that is blue, 4 red, 6 yellow and the rest green. How many cubes of each color are in the above rectangles? Is there more than one possible solution? You have to wrap gift boxes for your club project. Each box is 4.5 inches wide, 7 inches long, and 4 inches tall. What size rectangle do you need to cut to cover the box? Gift wrap paper comes in sheets which are 0 inches wide and 6 inches long. How should you cut your rectangles to make the best use of the paper? Geometry/ Measurement If each circle has a diameter of 50 feet, find the distance around the racetrack? The data values below are the speeds (in kilometers per hour) of ten trucks traveling on Interstate , 65, 70, 60, 60, 70, 7, 70, 59, 46 Interpret and construct a box plot. Analyze the data. Compute the mean. How does the mean compare with the median? Problem-Solving Strategies Guess and Check Using the digits 0,, 2,, 4, 5, to fill in the boxes. Make a subtraction problem with the smallest possible answer. Explain in words how you would solve: n 8 = 2 Grade 8 Page 25

26 9 *See answer key. This section should not be duplicated for the students.. ( 4) 2. Simplify: 4(2x 4). If a $6 shirt is on sale for 25% off, what is the sale price? Mental Math If 7 out of 0 students like pizza, how many students should like pizza in a class of 0? 7. Solve for x: x 2 = x Compare perimeter and circumference = 6. Determine the measure of angle x. 4. Simplify: (2 0 )(4 0 8 ) x 60 Write answers here: How much less is the square root of 64 than 6 2 4? (20 6) (6 + 2) + 9. Find the mean, median, and mode for the following set of data: { 0.2, 0.4, 0., 0.6,.2,. } 0. What is the sum of the first 5 square numbers? Grade 8 Page 26

27 Visual There is more than one solution possible depending on the total number of cubes used to make each rectangle. A possible solution for each follows:. 2 red, 2 blue, yellow, green 2. 2 yellow, 5 blue, red. 4 blue, red, 2 yellow, green The smallest sheet of paper that will work is.5 in. 7 in. If you cut the large sheet of paper into four sheets, each 8 5, the least paper will be wasted. Geometry/Measurement Answer: is about 57 feet about 20 yards Answer Key Mental Math. 2. 8x 6. $ , Distance around a polygon vs. the distance around a circle This section provides an oral drill for sharpening one's mental computation. Before each session give these directions, "Number your paper from to 0. Write your answer as the questions are called out. Each question will be repeated one time only." LE: 46 UE: 80 LQ: 60 UQ: 70 Median: 67.5 Mean: 65. Problem-Solving Strategies Answer: They are very close = = 2 6 = 9. mean: 0.6, median: 0.5, mode: none Add eight to both sides of the equation. n = 20 Grade 8 Page 27

Minute Simplify: 12( ) = 3. Circle all of the following equal to : % Cross out the three-dimensional shape.

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