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Diane Ahmann Math Tasks/Investigations Number /Operations Tasks: 1. The total of three consecutive integers is 384. What are the three integers? 2. Divide one half of one half of one half by one half of one fourth of one half. 3. How many numbers between 0 and 1000 have a sum of fifteen when the digits are added? 4. When area codes were first developed, the middle digit could only be a one or a zero and the first digit could not be a one or a zero. How many area codes could there be? 5. Find the difference between the least and greatest prime factors of 33,660. Algebra Tasks: 6. The sum of a number and 8 is equal to twice the number’s value. What is the number? 7. Samantha has 312 baseball cards. She wants to sort her cards into sheets that hold 24 cards each. How many sheets will she need? 8. A parking lot is filled with 260 cars. 1/3 of the cars are SUV’s. 1/2 are sub-compact cars. The remaining cars are luxury cars. How many of the cars on the lot are luxury cars? 9. Four lollipops and two sticks of gum cost one dollar. The cost of 2 lollipops and 3 sticks of gum is 70 cents. How much does one lollipop cost? How much does one stick of gum cost? 10. Jenny can send texts at a rate of 32 words per minute. If she sends 8 texts at 80 that average words each, how long would it take her to send all 8 texts? Geometry Tasks: 11. How many different rectangles can be made if you use the numbers on a clock’s face as the vertices? 12. Draw the right triangle DCF when angle C is 90⁰, angle F is 42⁰, and the hypotenuse is 12m. Label all angle and side measurements. 13. Find the area of the shape below: 14. Find the volume of the shape below: a=3.2 cm, b=7cm 15. Find the missing values for the rectangular prism: Length: 12 Width: 6 Height: ________ Surface area: ________ Volume: 720 Logical Thinking/Problem Solving Tasks: 16. A man needs exactly five gallons of gas to refill his gas tank. He has one empty three gallon gas can and one empty seven gallon gas can. How can he use the two cans to fill his tank? 17. Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe. The canoe can only carry 100kg. Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40 kg, and they have 20kg of supplies. How do they get across? 18. You are on an island and there are three crates of fruit that have washed up in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges. Each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and oranges". You know that NONE of the crates have been labeled correctly - they are all wrong. If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly? 19. There are three people (Alex, Brook and Cody), one of whom is a knight, one a knave, and one a spy. The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth. Alex says: "Cody is a knave." Brook says: "Alex is a knight." Cody says: "I am the spy." Who is the knight, who the knave, and who the spy? 20. You are about to leave for holiday, but you forgot socks! You race back to your room, but all the lights are off, so you can't see the color of the socks. Never mind, because you remember that in your drawer there are ten pairs of white socks, ten pairs of black socks, and eleven pairs of blue socks, but they are all mixed up. How many of your socks do you need to take before you can be sure to have at least one matching pair? Descriptions: 2. Divide one half of one half of one half by one half of one fourth of one half. Task number two requires students to work with fractions in a complex manner. First (a) the students must decipher the language of the question to create a numerical representation of the task. Students must then (b) apply the principles of fraction multiplication to reduce each half of the equation. Finally, (c) students must apply the principles of fraction division to solve (d). (a) 1/2×1/2×1/2 (b) 1/8 1/2×1/4×1/2 1/16 (c) 16 (d) 2 8 This task can be used to solidify students’ understanding of working with fractions. It is a cumulative task that requires language decoding as well as a firm understanding of multiplying and dividing fractions. It can be used to review before a unit test or to activate prior knowledge before a new unit is introduced. I would hope that in solving a task like this that students would realize that neither word problems nor fractions are as scary as most students think they are. Improving familiarity of these tasks will solidify the skills students will need for more complex math problems. 3. How many numbers between 0 and 1000 have a sum of fifteen when the digits are added? Task number three requires students to problem solve by building upon the information given. A student must identify the various numbers that will create a sum of fifteen and realize that any combinations that use double digits will not work: 1&14 = 114 = 1+1+4 =6. Once they know that the only usable two digit combinations are 6 & 9 and 7 & 8, they can begin to identify which numbers qualify to the constructs of the question: 069, 096, 609, 690, 906, 960 & 078, 087, 708, 780, 807, 870. They will also have to identify the three digit combinations that work: 1, 5 & 9; 1, 6 & 8; 2, 4 & 9; 2, 5 & 8; 2, 6 & 7; 3, 4 & 8; 3, 5 & 7; 4, 5 & 6. If students follow the pattern from the first half, they can realize that these numbers also create six combinations. Otherwise, they would have to write them out. Finally, by using digits multiple times, they can find the combinations: 1, 7 & 7; 3, 3 & 9; 3, 6 & 6; 4, 4 & 7. These numbers can only be combined three ways because of the repetition. Finally, using 5, 5 & 5 works. If students have correctly identified the combinations, they can add 6+6+6+6+6+6+6+6+6+6+3+3+3+3+1=73. This task can be a great warm up to working with multi-step problems and for logical thinking. This problem can be quite complex even though it utilizes simple addition and/or multiplication. I would hope that students would learn to reason through all aspects of a problem without getting to an answer too quickly. For this task it would be easy for students to forget that the numbers can be rearranged to make all of the numbers. They might come up with an answer as small as 15 if they forget to use the numbers in multiple combinations. If students make this mistake, I would expect them to take that mistake and apply the lesson learned in the future building their logical skills. 9. Four lollipops and two sticks of gum cost one dollar. The cost of 2 lollipops and 3 sticks of gum is 70 cents. How much does one lollipop cost? How much does one stick of gum cost? Task number nine requires that students set up a system of equations. First they must recognize that there are two pieces of information that each constitute and equation: 4L+2G=1D and 2L+3G=70C. Students must also decipher 70 cents and one dollar as either .70D & 1D or 70C & 100C so the equations have the same variables. From here, students can solve one of two ways: 1) they can a) solve for one equation in terms of one variable then b) substitute the equation into the variable in the other equation and solve for one variable or 2) they can c )multiply one equation by a negative value to allow for d) subtracting the like terms and solve for one variable. Using either solution, they can then 3) plug in the answer into the original equation to solve for the other variable. 1) a)4L+2G=100 = 2G=100-4L = G=50-2L; plug into 2L+3G=70: b)2L+3(50-2L)=70 = 2L+150-6L=70 = 80=4L = L=80/4=20 2) c) (-2)(2L+3G=70) = -4L-6G=-140; subtract from 4L+2G=100 d) –4G=-40 or G=10 3) [for 1] 4(20)+2G=100 = 80+2G=100 = 2G=20 = G=10 OR [for 2] 4L+2(10)=100 = 4L+20=100 = 4L=80 = L=20 The multiple approaches to this task allow students to solve math problems in multiple ways. The ability to utilize more than one method is vital to the development of critical thinking skills. This type problem would be used to help students explore the way they think about math and their approaches to problems. It is the culmination of basic algebra problems and the applications for solving them. I would expect students to learn how to decipher the word problem and critically analyze the best way for them to solve the problem. 11. How many different rectangles can be made if you use the numbers on a clock’s face as the vertices? Task number eleven requires students to visualize, draw models, and use logic and reasoning to ensure that all possibilities are covered. They must also recognize that at some points the rectangles begin to repeat each other. This task would be used to give students exposure to elements of geometry that can be found in everyday life. Also, I would use tasks like these to activate students’ logic and evaluation skills. I would hope that students learn to use multiple reasoning and ordering skills to solve problems. The visualization and deduction needed to complete this task can help students gain the ability to get a mental picture of geometric figures and their connectivity to each other and the world. 17. Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe. The canoe can only carry 100kg. Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40 kg, and they have 20kg of supplies. How do they get across? For task seventeen, students will have to use logic and critical thinking. Although these tasks do not have direct math procedurals in them, they are crucial to the thought processes involved in math. I would use this type of task as a daily or weekly warm-up. Also, I might use them for a bonus question on a test. Although students use mathematical thought processes, sometimes it is nice to have a problem with “no math” in it to ease the anxiety that some students feel with math word problems. I would hope that students learn to reason in a logical manner and attack other problems with these skills. This type of thinking can cross all subject lines and help students throughout their academic lives. This task can have a variety of solutions, one possible answer is: Chris and Dusty cross, Dusty comes back. Alex rows crosses, and Chris returns. Chris and Dusty cross again, Dusty returns. Brook crosses with the Supplies, and Chris returns. Chris and Dusty cross again for the last time.