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Someone said, "A picture is worth a thousand words." Turning the words of a problem into a picture or a diagram can help you "see" the problem. By using the part of your brain that visualizes a situation or object, you may see relationships or information that helps you solve the problem. When someone tells you a story, try turning the words into a motion picture or a cartoon. When reading a descrip- tion, try "seeing it in your mind's eye." If you can do these things, this strategy may be for you! Try using a picture or make a diagram to solve this problem:
Strategy of the Month
2. How many 2 x 5 tiles are needed to cover this floor?
Answer: _____________________
In the restaurant there are 12 square tables. Only one person can sit on each side. What is the greatest number of people that can be seated if the tables are pushed end to end into one large rectangle?
3. At 9:00 a.m., I went to the Ol' Fishin' Hole to fish. There is a three fish per hour limit. If I need 20 fish for a cook-out tomorrow, at what time will I probably have my 20 fish?
Answer: ____________________
4. Mary has three skirts, two blouses, and either black or white shoes that she likes to wear to school. How many days can she go without repeating the same combination of skirt, blouse, and shoes?
Answer: ____________________
1. How many 2's must be multipled to- gether for the product to be a number between 100 and 200?
Answer: ____________________
MathStars Home Hints
Every year you grow and change in many different ways. Get someone to help you measure and record these data about your- self. Be sure to save the information because we will measure again in two months! How tall are you? _____________________
How much do you weigh? ______________
What is the circumference of your head?
_______________________
Setting Personal Goals
Problem solving is what you do when you don't know what to do. Being a good problem solver will help you be ready to live and work in our changing world. Computers can do computations but people must tell the computers what to do. Good problem solvers know how to make plans and use many different strategies in carrying out their plans. They use all of their past experiences to help them in new situations. We learn to swim by getting in the water; we learn to be good problem solvers by solving problems!
5. How many cubes do you think it will take to make a cube that is twice as high as one cube?
Answer: ____________________
Three times as high?
Answer: ____________________
Four times as high?
Answer: ____________________
6. If a cat catches seven mice in four days, how many mice should it catch in 16 days?
Answer: ____________________
7. At the end of the soccer tournament, each team captain shakes hands with every other team captain. If there were eight teams in the tournament, how many handshakes were there?
Answer: ___________________
x?
? = __________
9. Julia spent 1/3 of her birthday money. Then she lost 1/2 of the rest. She now has $10 left. How much did she get for her birthday?
Answer: _____________________
- (8, 27, 64) Students can use unifix cubes to model the problems, especially if abstract visualization is
difficult. Twice as high will be 2 x 4, three times as high 3 x 9, and four times as high would be 4 x 16.
- (28) If he catches seven mice in four days, students can expansd to twice as many in eight days or, 14 mice. Doubling again, he can catch 28 in sixteen days. Another approach would be to create a table that increases by four days at each step ( i.e. 7 mice in 4 days, 14 mice in 8 days, 21 mice etc.).
- (28) Encourage students to make a chart to determine the combinations of handshakes. They should note that when coach X shakes hands with coach Y it counts as a handshake for coach Y. Another strategy is modeling using the vertices of an octagon.
- (11) Examining the problem students may initially determine that the missing multiplier is greater than ten (432 x 10 = 4320) How much greater than 10? 4752 - 4320 = 432 So one greater than ten or eleven. Another strategy could involve straight division.
- ($30) Working backwards is an excellent strategy to use in this situation. If losing half left her with $10, then she had $20 before that happened. If spending one-third left her with $20, then that must be two-thirds. She had three-thirds or all to start with and that must be $30.
- (12) A tree diagram is one strategy which students may employ to solve this problem. Colored markers, cubes or other manipulatives can also help students model the situation. Black shoes Blouse # Skirt #1 White Shoes
Blouse # Black Shoes
White Shoes Blouse# Skirt #2 Black Shoes White Shoes Blouse # Black Shoes
White Shoes
Blouse # Skirt #3 Black Shoes White Shoes Blouse # Black Shoes
Strategy of the Month
Your brain is an organizer. It organizes infor- mation as it stores that information. When a problem involves many pieces of information, your brain will have an easier time sorting through it if you make an organized list. A list helps you be sure you have thought of all of the possibilities without repeating any of them. Like drawing a picture or making a diagram, making an organized list helps your brain "see" the problem clearly and find a solution. Try making an organized list to solve this problem:
Tickets for the concert cost $12 for adults or teenagers and $6 for children. If the group has $60, how many adults or teenagers and how many children could go?
1. RIDDLE ME THIS:
I'm thinking of a number. It is odd. It's between 1 and 100. It's higher than 20. It is smaller than the answer to 6 x 6. It is a multiple of 5. The sum of its digits is 7.
Answer: _____________________
2. Hank had an average of exactly 84% after taking two tests. On the third test, he scored 96%. Find his average for all three tests.
Answer: ____________________
3. What day of the week was yesterday, if five days before the day after tomorrow was Wednesday?
Answer: ____________________
S M T W Th F S
5. Complete the following number pattern:
14 28 18 36 26 52 42 84 ___ ___ ___
Answer: __________________
6. You know that the perimeter of a certain rectangle measures 22 in. If its length and width each measure a whole number of inches, how many different areas (in square inches) are possible for this rectangle?
Answer: ___________________
About these newsletters...
The purpose of the MathStars Newsletters is to challenge students beyond the classroom
setting. Good problems can inspire curiosity about number relationships and geometric
properties. It is hoped that in accepting the challenge of mathematical problem solving,
students, their parents, and their teachers will be led to explore new mathematical
horizons.
As with all good problems, the solutions and strategies suggested are merely a sample of
what you and your students may discover. Enjoy!!
Discussion of the problems...
- (25) Students should satisfy one condition of the riddle at a time and eliminate numbers as the information is given. They can then be challenged to write riddles of their own to share with classmates.
- (88) Be sure all students have a good understanding of average. Unifix cubes can be used to demonstrate the "equal"ing or "even"ing of values when an average is computed. The formula will also be of help. Since 84 is his average after two tests the sum of those two scores is 2 x 84 or 168. When the third score is added the new average becomes 264 ÷ 3 or 88.
- (Friday) Encourage students to use the calendar to try out their conjectures. A similar problem : What will tomorrow be if yesterday was the 2nd day of the week? [Wednesday]
- (74, 148, 138) Start with an easier problem such as 2, 4, 3, 5, 4, 6, 5,... This is especially helpful if students have not seen series with two or change factors +2 -1 +2 -1 +2 -
- (5) Since the perimeter of the rectangle measures 22 inches, the sum of a single length and a single width must be half 22 or 11 inches. Students can make a table of possible lengths, widths and corresponding areas.
length 10 9 8 7 6 width 1 2 3 4 5 area 10 18 24 28 30
- ( a.) at least 3 black-eyed peas, 4 red beans and 5 lima beans; b.) 8 black-eyed peas, 5 red beans and 5 lima beans; c.) 2 black-eyed peas, 2 red beans and 4 lima beans) Trial and error, guess and check are good strategies for this problem.. Tri-colored manipulatives will help students model their conjectures. This is also a good introduction to ratio and proportion.
- (4 days) If 2 hens lay 3 eggs in 4 days, then 8 hens (four times as many) will lay 12 eggs in 4 days. Students can use drawings, charts or manipulatives to act out or model the situation.
- ($15.75) Students need to see the relationship between the regular price of $20 and the discount of 1/ 4 off. One-fourth of $20 is $5. When the regular price is reduced by $5 the new price become $15. Now for the tax. Tax is paid at the rate of 5 cents for each of 15 dollars for a total of 75 cents. Adding, the customer pays $15.75. If the tax is computed first, then it is $1, added to $20 gives us $21. Now take 1/4 off and the cost is still $15.75. This is a nice illustration of the commutative law for multiplication.
- (Alice - 7, Joe - 14, Mickey - 28) Students need to begin with a careful reading of the problem and attention to details. A diagram would be helpful. Seeing that Alice gets the least, and that other portions are based on hers, will help students establish a method of attack. When guess and check are tried the total, 49, becomes the goal. Teachers may wish to lead students to this type of problem with a simpler example: Mary jumped 30 times before she missed a step in the jump rope contest. Sally jumped twice that number and Sam jumped half of what Mary jumped. How many times did Sally and Sam jump before they missed a step? [Sally - 60, Sam - 15]
Setting Personal Goals
Communicating mathematically means that you are able to share your ideas and under- standings with others orally and in writing. Because there is a strong link between lan- guage and the way we understand ideas, you should take part in discussions, ask questions when you do not understand, and think about how you would explain to someone else the steps you use in solving problems.
MathStars Home Hints
Memorizing number facts will save you time. Flash cards are one way to learn new facts, but you also might try these ideas:
- play dice or card games in which you need to _add, subtract, multiply, or divide.
- learn new facts using ones you already know_ _(7+7 =14 so 7+8=15).
- learn facts that are related to each other_ (7x6=42, 6x7=42, 42 ÷ 6=7, 42 ÷ _7=6).
- make a list of the facts you need to memorize_ _and learn 5 new facts each week.
- Spend 5-10 minutes every day practicing facts._
5. Michael was supposed to multiply a number by 5. By mistake, he divided the number by 5 instead. His answer was 5. What should have been the correct answer?
Answer: ____________________
6. The fifth grade is going on a field trip to the zoo. The zoo requires that for every 15 students, there must be one chaperon. If there are 194 students going on the trip, how many chaperones will be needed?
Answer: ___________________
7. Find the volume for each building.
BANK OFFICE
HOSPITAL
Bank _______________________
Hospital _____________________
Office ______________________
8. What is the greatest six-digit number in which the thousands place is twice the digit in the tens place? What is the least number?
Answer:
Greatest number ______________
Least number ________________
About these newsletters...
The purpose of the MathStars Newsletters is to challenge students beyond the classroom
setting. Good problems can inspire curiosity about number relationships and geometric
properties. It is hoped that in accepting the challenge of mathematical problem solving,
students, their parents, and their teachers will be led to explore new mathematical
horizons.
As with all good problems, the solutions and strategies suggested are merely a sample of
what you and your students may discover. Enjoy!!
Discussion of the problems...
- [120cm] The diagram of the hexagon is not shown because students need to recall that a hexagon has six sides. Regular indicates all sides are congruent. Further application could involve finding the perimeter of a regular triangle, quadrilateral, pentagon, and octagon.
- (2/7, 4/7, 25%, 50%) Students need to observe that segment AD is divided into seven equal parts. If segment AE represents a whole, and it is divided into eight equal parts, then AB is one-fourth or 25%. Considering AC as a whole, AB is half or 50%.
- (60) The solution is the least common multiple (LCM) of 2, 3, 4, 5, and 6. Listing multiples of each and comparing is an excellent strategy.
6--->6, 12, 18, 24,.. 48, 54, 60 , 66,... So the LCM is 60.
- (yes) Even though the cat can run faster than the mouse, in one second the mouse will be half-way to his holeand the cat will be half-way to the mouse's former position. In two seconds the mouse will be at his hole while the cat will be where the mouse was at the start or 20 meters from the hole. This is a good problem for students to act out.
- (125) The first question the student needs to ask is " if N ÷ 5 = 5, what is N?" When they have arrived at 25 for the original multiplicand, then 25 x 5 will give the correct answer.
- (13) This problem is a good application of mathematics and the importance of common sense or logic in its use. Division (194 ÷ 15) will give (12), the number of groups of 15 for which a chaperone is needed, however the remainder or left-over students, (14) will also require a chaperone, therefore 13 caperones are needed.
Strategy of the Month
Noticing patterns helps people solve problems at home, at work, and especially in math class! Math has been called "the study of patterns," so it makes sense to look for a pattern when you are trying to solve a problem. Recognizing patterns helps you to see how things are orga- nized and to make predictions. If you think you see a pattern, try several examples to see if using the pattern will fit the problem situation. Looking for patterns is helpful to use along with other strategies such as make a list or guess and check. How can finding a pattern help you solve this problem? A palindromic number is onewhich reads the same backwards as forwards. How many 3-digit palindromic numbers are there?
1. Use the numbers 4 through 12 to fill in the circles. The numbers on each straight line must add up to 21.
2. Your mother and father decide to change your allowance. You are given the choice:
a. They will pay you $10 a week
- or- b. They will pay you one cent the first week, two cents the second week, four cents the next week, and so on, doubling your allowance each week for a year.
Which will give you the most money?
Why?
30m 3m
20m
3. A rectangle lot 30m by 20m is surrounded on all four sides by a concrete walk 3m wide. If you need to concrete only the sidewalk, how much concrete will you need? (surface area)
Answer: ____________________
Setting Personal Goals
If your goal is to become a more responsible student, it means that you:
- actively participate in class.
- complete your assignments.
- have everything you need in class.
- ask for help when you do not understand.
- be willing to investigate new ideas.
MathStars Home Hints
Set aside a special time each day to study. This should be a time to do homework, to review, or to do extra reading. Be organized and have a special place in which to work.This place needs to have a good light and to be a place where you can concentrate. Some people like to study with quiet music; others like to sit at the kitchen table.You need to find what works for you!
Remember that when you are reviewing or working on solving problems it may help to study in a group.
4. Label the correct measurements on the marked angles in the square.
5. What are the two least likely sums to be
rolled on two regular dice? Why?
Answer: __________________
6. Move only two discs and turn the triangle upside down. (Draw arrows to show how to move them.)
7. The figure below is constructed of equilateral triangles and rectangles. Label the ten unmarked segments with their correct lengths.
30
90
100
- (336 square meters) There are several approaches to this problem. Students may compute the area of the entire lot and of the unpaved portion. Subtracting will give them the paved area. Another strategy would be to divide the paved area into four rectangles and add their areas.
Students may use protractors or logic to solve this problem.
- (2 and 12 because there is only one way to get these sums) Students can best illustrate this situation with a chart of possible outcomes.
30^30
Sometimes mathematical ideas are hard to think about without something to look at or to move around. Drawing a picture or using objects or models helps your brain "see" the details, organize the information, and carry out the action in the problem. Beans, pennies, tooth- picks, pebbles, or cubes are good manipulatives to help you model a problem. You can use objects as you guess and check or look for patterns. Try using objects to help you solve this problem:
What happens to the volume of a rectangular prism if the width is tripled?
Strategy of the Month
1. If x = 4 and y = 2, then:
3 x + y = ___________ and
4 y – 2 x = __________
2. Graph the factors of 45 and 54 in the Venn diagram below.
3. If there are two computers for every 40
students at Elm Elementary, how many computers do they have for the 440 students attending school?
Answer: __________________
4. Write a number in the triangle that will
make the answer 50.
X4 ÷2 +6 = ___
2/3 yard
1 yard
1 yard
2 yard
5. Find the area of the flower bed below in square feet.
Answer: ___________________
About these newsletters...
The purpose of the MathStars Newsletters is to challenge students beyond the classroom
setting. Good problems can inspire curiosity about number relationships and geometric
properties. It is hoped that in accepting the challenge of mathematical problem solving,
students, their parents, and their teachers will be led to explore new mathematical hori-
zons.
As with all good problems, the solutions and strategies suggested are merely a sample of
what you and your students may discover. Enjoy!!
Discussion of the problems...
- (14, 0) If students have not explored using unknowns and variables, concrete objects will be of help. Put four objects in a bag and two in another. Show that 3x means three bags (with four in each) or 12 objects etc.
- The diagram should be similar to the following:
Be sure students understand the rationale for the common factors placement in the intersection of the Venn diagram.
- (22 computers) If students have not yet mastered long division, there are other ways to solve this problem using number sense. Using diagrams or manipulatives the number of groups of 40 can be determined, followed by counting two computers per group. Another approach is to reason that two computers for 40 students means one computer for twenty. Then the number of groups of twenty can be calculated.
- (22) Students can work backward to show that (50 - 6) x 2 ÷ 4 = 22. Some students may use guess and check or rely on their numbersense to solve the problem.
- (15 square feet) The flower bed can be divided into two rectangles with either a horizontal or vertical line. Suggest that students change the measurements to feet before beginning.
- (88) Since chickens have two legs and the other animals four legs each, this is a good opportunity for students to write expressions that illustrate the order of operations and the distributive and associative properties. 2 x 20 + 4 x 4 + 4 x 8 = 2 x 20 + (4 x 4 + 4 x 8) = 2 x 20 + (4 x [4 + 8]) = 2 x 20 + 4 x 12 =
40 + 48 = 88.
- (Snobhops are sets of numbers which can be the lengths of the sides of triangles, i.e., the sum of any two is greater than the third.) Students may wish to verify that snobhops always form triangles and that the non-snobhops cannot form triangles. Answers will vary for the student-generated snobhop but they should be ready to prove it fits the rule.
- (7) Remind students that a Venn diagram is a very useful tool for sorting information in problems of this type. After they have completed the study groups for math and science, students should note that they have accounted for only 23 of the 30 students. This leaves seven who studied neither math nor science. Study Group
Math Science
- (rectangle; 5 inches) If students have difficulty visualizing the figure, teachers may wish to use an empty paper towel roll to demonstrate that indeed a rectangle is the result. Knowing the area and the distance around the top (4 inches) students should be able to determine the length as, Area = length x width or 20 = 4 x.
