Math 221 A1 Fill-In the Digits 1. 3-Digit Sums Using each of the digits 1 through 9 exactly once, fill in the boxes below so that the sum is correct: _____ _____ _____ | || || | |____ ||___ ||___ | _____ _____ _____ + ||___ ||||_ ||||_ || __________________________ _____ _____ _____ | || || | |____ ||___ ||___ | 2. 1-9 Triangle Puzzle ' ' $ $ &%&% ' $ ' $ &% &% ' $ ' $ &% &% ' ' $ $ ' $ ' $ ' ' $* * $ &%&% &% &% * * &%&% a) Using each of the digits 1 through 9 exactly once, fill in the circles of a triangular arrangement similar to the one above so that the sums of the four digits along each side is the same. b) Find as many different solutions as you can in which the side sum is 20. (How should you define "differ- ent"? Are some solutions more different than oth- ers?) c) Can you find a solution in which the sum on each side is 15? 18? 21? 22? Math 221 A2 Procedures and Concepts 3 . 1 1. What is __ _ __? Show how to compute it. 4 . 2 3 . 1 2. Write a story problem for which __ _ __ would be the ap- 4 . 2 propriate computation to find the solution. 3. Find the area of the triangle below: /\ /| \ / | \ / | 5 \ / | \ /____|_______________\ 8 4. Why does your method work in problem 3? 5. Find 435 18 without using a calculator. 6. In the following partial computation, why do we "bring down the 5"? __________2________ 1 8 ) 4 3 5 _3____6____ 7 5 7. In school you were probably taught that "division by zero is not allowed". Why not? Math 221 A3 Some Problems to Solve Work on the problems on this sheet in two phases: 1. First read through all the problems and make a list of ways you might attempt to solve the problem. (Keep in mind the heuristics we have seen.) Jot down some notes for each problem, then move to the next one. 2. Then go back and (attempt to) solve the problems. For each of the problems, explain the reasoning you use to solve the problem. Whether you solve the problem com- pletely or not, keep a list of things you tried, patterns you observed, conjectures you made, facts you discovered, etc. Be sure to note any problem solving strategies you used that did not occur to you when you first read the problem (before working on it). You may work the problems in any order. 1. What is the last digit of 399 ? 2. How many rectangles (of any size) are there in the picture below? ______________________________________ | | | | | | | | | | | | | |___|__|__|__|__|__|__|__|___|__|__|__| 3. Jan is having a party, to which she invited 24 people. She plans to serve the meal on card tables, arranged in a long rectangle, with each table pushed up against the next. If the card tables are only large enough to seat one person on a side, how many tables will Jan need? 4. How many ways are there to cover the large rectangle be- low using "dominoes". (A dominoe is a rectangle that is 1 square wide and 2 squares tall.) ______________________________________ _* *___ | | | | | | | | | | | | | |* * | |___|__|__|__|__|__|__|__|___|__|__|__| Dominoes: |* *___| | | | | | | | | | | | | | |* * | |___|__|__|__|__|__|__|__|___|__|__|__|___ |* *___| | | | |___|__| [Hints: That large rectangle is pretty large, perhaps you should try a some smaller examples first. How can you use smaller examples to help you with the original problem?] 5. A pail with 40 marbles in it weighs 175 grams. The same pail with 20 marbles in it weighs 95 grams. How much does the pale weigh alone? How much does one marble weigh alone? 6. How many ways are there to make change for 27 cents us- ing (any number of) pennies, nickels, dimes and quarters. Math 221 A4 Some More Problems 1. What is the last digit of 1399 ? 2. A regular pentagon is a figure with five equal (straight) sides and five equal angles. How many diagonals are there in a regular pentagon? (The diagonals are the line seg- ments that connect one corner with another but are not sides. All together they look like a star.) Can you generalize this for other polygons? 3. How many ways are there to make a path connecting ad- jacent letters in the diagram below so that the path is labeled with the alphabet (each letter exactly once and in order)? A ABA ABCBA ABCDCBA ABCDEDCBA ABCDEFEDCBA ABCDEFGFEDCBA ABCDEFGHGFEDCBA ABCDEFGHIHGFEDCBA ABCDEFGHIJIHGFEDCBA ABCDEFGHIJKJIHGFEDCBA ABCDEFGHIJKLKJIHGFEDCBA ABCDEFGHIJKLMLKJIHGFEDCBA ABCDEFGHIJKLMNMLKJIHGFEDCBA ABCDEFGHIJKLMNONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUVUTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUVWVUTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUVWXWVUTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUVWXYXWVUTSRQPONMLKJIHGFEDCBA ABCDEFGHIJKLMNOPQRSTUVWXYZYXWVUTSRQPONMLKJIHGFEDCBA 4. There are 20 people in a room. If each person shakes hands with each other person exactly once, how many handshakes occur? 5. How many squares (of any size) are there on an 8 x 8 checkerboard? Math 221 A5 Find the Pattern 1. For each of the following patterns, o fill in the blanks, o describe the pattern using words, o decide if it is a repeating pattern or a growing pattern, o see if you can determine the 10th and 50th numbers in the pattern, o see if you can express the nth number in the pattern. (For some this may be very hard, for others it is much easier.) (a) 2; 4; 6; 8; ______; ______; : : : (b) ______; 5; 8; 11; 14; ______; ______; : : : (c) 5; 10; 20; 40; 80; ______; ______: : : (d) 3; 10; 7; 3; 4; 1; ______; ______; ______; : : : (e) 2; 5; 7; 12; 19; ______; ______; : : : 2. There is a handy "formula" that provides a short cut when we want to know the sum of consecutive numbers starting with 1: 1 + 2 + 3 + . . .+ n = (n)(n+1)__2. (Why is this formula true?) See if you can find similar "short-cuts" for the following sums: (a) a + (a + 1) + (a + 2) + . . .+ b. (adding all numbers from a to b.) (b) 1 + 3 + 5 + . . .+ n. (adding consecutive odd numbers) (c) 2 + 4 + 6 + . . .+ n. (adding consecutive even numbers) Can you prove any of these? For you, which of these were problems and which were exercises? 3. How many diagonals are there in a regular pentagon? (A regular pentagon is a figure with five equal (straight) sides and five equal angles.) Can you generalize this for other polygons? Math 221 A6 Set Operations 1. Let R be the set of red blocks, let T be the set of triangles, let L be the set of large blocks, let B be the set of blue blocks and let D be the set of diamond-shaped blocks. For each of the following sets, o locate the the set in terms of loops and A-blocks, o draw a picture (Venn diagram) shading in the loop regions that represent the set, o list the elements of the set, and o describe the elements (with words). (a) R \ T _________ (b) R \ T ___ ___ (c) R [ T ___ (d) R \ T (e) L \ B \ D (f) L \ (B [ D) (g) (L [ B) \ (L [ D) __________________ (h) (L [ D) \ B (Hint: something in your previous work might be useful here.) Find two examples among the sets above of one set being a subset of another. How do you write this using set nota- tion? Which of the four sets contain the large blue square? How do you write this? 2. Invent a new set operation. Give it a name, make up symbolism for it, explain what it does, and give a few examples. Can you express your new operation in terms of the "standard" operations on sets? 3. Repeat as much of problem 1 as makes sense using R = {1; 2; 3; 4; 5}, T = {0; 3; 6}, L = {1; 2; 3; 4; 5; 6}, B = {5; 6; 7; 8; 9; 10}, and D is the set of numbers between 0 and 10 (inclusive) that divide 10 evenly.