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Challenge 4-6/Ages 9-11

AwardPresented to MathWorld Interactive by Hervey Hurricanes from Medford, Massachusetts, USA

Question #1

1. We know the question asks for us to express all the ways we can express the number 2. It does not specify whether the number is positive , negative,....

2. We are looking for all the possible combinations of operations that would allow the final result to be expressed by the number 2.

3. We decided that by attaching the different operations I.E. +, -, *, /, <, >, =, square root, % ... we could solve this problem.

4. Infinite combinations can be derived that express the number 2 as an integer, rational number, prime number, even number, decimal, square root, cube root, fraction, mixed number, mean, median, mode, real number, natural number, negative number, and positive number. Example: 1. The number: 2. As a percentage, .25% of 800. As addition: 1+1. As subtraction: 3-1. As a Square root of 4. As a cube root of 8.

5. By coming up with practice problems. Example: (200/2) - 98= 2

6. How many ways can you express the number 1/2 using only whole numbers?

Question #2.

1. We know that the squares are congruent (all the same).   We can also see that some of the square are contiguous.   A rectangle is a four sided plane closed figure having opposite sides equal and parallel and four right angles. A square can be a rectangle.

2. How many rectangles do we see in this diagram.

3. We counted and traced all the possible combinations of one square and two or three contiguous squares.

We figured out how many squares are attached by one or more sides to another square.

Some of us did not see the rectangles in all possible directions, so we did it again.

4. There are 10 contiguous squares of 2 and 4 contiguous squares of 3, 2 contiguous squares of 4 and 9 single squares. For a total of 25 rectangles.

5. Count them, trace them in different colors, check each others work.

6. How many triangles would you have if each square in the diagram was cut in 1/2 diagonally?

Question #3.

1. We have five penny's in total. One penny is lucky four are not.

2. How many times would you have to reach into your pocket to get the lucky penny? Probability

3. We did an experiment using four penny's and one dime as the lucky penny and charted the results.   We learned the formula for probability. probability = # of favorable outcomes (lucky penny) -------------------------------- # of possible outcomes ( all the penny's)

4. 1 out of 5, or 20% of the time.

5. Repeated the experiment. Compared work.

6. You have five nickels dimes and quarters, and one silver dollar what is the probability when you reach into your pocket you pick the silver dollar?

Question #4.

1. There is a pack of six sodas at a cost of $1.68, and another pack of eight sodas at a cost of $2.16.

2. In each pack of sodas how much does one soda cost. Which one cost less per soda?

3. Divide $1.68 by 6, and divide $2.16 by eight.

4. The six pack cost 28 cents per soda, The eight pack cost 27 cents per soda, therefor the eight pack is a betters buy.

5. Rechecked each others work on paper and with a calculator.

6. If each soda is eight ounces how much per ounce is each pack?

Question #5.

1. There are 5,280 feet in one mile. There are 12 inches in a foot.

2. How many inches to a mile? How many feet in 27 miles?

3. Multiply 5280 feet by 12 inches for part one. Multiply 5280 feet times 27 miles for part two.

4. There are 63360 inches in one mile. (5280 x 12)   There are 142,560 feet in 27 miles. (5280 x 27)

5. Rechecked each others work on paper and with a calculator.

6. How many inches are there in 27 miles?

pencil

AwardPresented to MathWorld Interactive by The Mentally Challenged from Shoreham, New York, USA

 Question 1:

1) What we know: We have to express the Number 2.

2) What we’re looking for: Express the Number 2 as many ways as possible.

3) Strategies: We brainstormed and used Guess and Check.

4) Solutions: There are an infinite number of ways.

Arithmetic: example - 2X1, 4/2, 2+0

Roman Numerals: II

Arabic Numerals: 2

Other Languages: deux, duo

Picture symbols: (any two objects)

Other number systems: Base-2, Base-5

5) Check: We tried problems and researched our answers.

6) Extension: How many ways can you express the number 2 using the number 5?

Question 2:

1)What we know: Squares are rectangles. Rectangles can be traced out of the

figure.

2) What we’re looking for: How many rectangles can be traced.

3) Strategies: Drawing a diagram, tracing the figure, Make a systematic list

4) Solutions: 9-1x1’s, 10- 1X2’s, 4- 1X3’s, 2- 2X2’s = Total 25 rectangles.

5) Check: Re-did the problem and checked to see if the answer was the same.

6) Extension: Draw 2 diagonals through every square. How many triangles can be

traced?

Question 3:

1) What we know: You have 5 pennies in your pocket. One is lucky.

2) What we’re looking for: What is the Probability you will pull out your lucky

penny?

3) Strategies: Using fractions and math to solve the problem.

4) Solutions: 1 out of 5 chance.

5) Check: I put pennies in my pocket and pulled them out.

6) Extension: You have 5 tokens in your pocket. One of them is blue, 2 are yellow,

one is green, and one purple. What percent of each color are there in your pocket and

what is the probability of getting each color?

Question 4.

1) What we know: A 6 pack of soda costs $1.68 and an 8 pack costs $2.16

2) What we’re looking for: Which is the better buy

3) Strategies: Multiplication, comparing, division and subtraction: 1.68/6 = .28,

.28 X 8 = 2.24, 2.24>2.16

4) Solutions: An 8-pack is cheaper.

5) Check: one can of a 6-pack costs more than one can of an 8-pack.

6) Extension: A 7-pack of soda costs $1.90 and a 9-pack of soda costs $2.43.

which is the better buy?

Question 5:

1) What we know: There are 5,280 feet in a mile. There are 12 inches in a foot.

2) What we’re looking for: How many inches in a mile? How many feet in 27

miles?

3) Strategies: Multiplication and division. 5,280 X 12 = 63,360, 5280 X 27 =

142,560

4) Solutions: There are 63,360 inches in a mile. There are 142,560 feet in 27

miles.

5) Check: 63,360/5280 = 12, 142,560/5280 = 27

6) Extension: How many inches are there in 27 miles?

 pencil

AwardPresented to MathWorld Interactive by Hillel Rules from North Miami Beach, Florida, USA

 The answer to question 1 from Hillel Rules:

1. We know that I have to say the number two in as many ways as I can.

2. How many ways to tell two.

3. Write all he problems that you can use to express two.

4. Two can be expressed in an infinite amount of ways.

5. We went over all the problems that we thought of over again.

6. How many ways ca you express the number 4.

Question: There are 5,280 feet in one mile. How many inches are there in one

mile? How many feet are there in 27 miles

#2 - 1. the facts are that we need to find out how many rectangles are in the figure.

2. we are looking for as many possible rectangles are in the figure.

3. at first we found few, since we didnt know squares are rectangles. we came up with numbers like 8,10,11. Then when we found out squares are rectangles we got in the low twenties and we i finally got 25. we found many rectangles made by other rectangles connected to each other.

4.are final answer was 25.

5.we checked by giving the others problem and seeing if they got the same answer. most did.

6. if you added another row of rectangles above the last row and one rectangle left, how many recdangles would you now have in total?

#3-1. I KNOW THAT THERE ARE FIVE PENNIES AND THAT ONE OF THEM IS MY LUCKY PENNY

2. I AM LOOKING FOR MY LUCKY PENNY OUT OF THE FIVE PENNIES IN MY POCKET

3. I USED MENTAL MATH. I TOOK THE FIVE PENNIES AND KNEW THAT ONE OF THEM IS MY LUCKY ONE. I MADE A FRACTION. THE NUMERATOR IS ONE BECAUSE ONE IS THE NUMBER I AM LOOKING FOR, THE DENOMINATOR IS FIVE BECAUSE THERE ARE FIVE ALL TOGETHER IN MY POCKET.

4. 1/5

5. THIS IS A SIMPLE PROBLEM, SO I KNEW I WAS RIGHT. ONE OUT OF FIVE IS THE ANSWER. THIS KIND OF MATH IS CALLED PROBABILITY.

6. I HAVE 6 NICKELS. TWO OF THEM ARE BUFFALO NICKELS. WHAT IS THE PROBABILITY THAT I PULL OUT A REGULAR NICKEL?

Answer To Problem Number 4

1)I know that a 6-pack of soda costs $1.68 I know that an 8-pack of soda costs $2.16

2)It is looking for which of the 2 packs is the better buy

3)What i did in order to solve it was :

I did $1.68 divided by 6 = $0.28

Then i did $2.16 divided by 8 = $0.27

So in the eight pack i figured it was

was $0.27 per can , and in the six pack

it was $0.28 per can

4)the 8 pack was a better buy

5)I checked it because 0.28 times 6 = 1.68

and 0.27 times 8 = 2.16 .So i figured that

the 8 pack was a better buy .

6)If there is a 10-pack of cereal that costs $26.98 , and there is a 12 pack that costs

$30.77 , which is a better buy

#5-1) I know that there are 5,280 feet in a mile and 63,360 inches in one mile.

2) The question is asking you to find how many inches in one mile and how many

feet in 27 miles

3) I multiplied 12 times 5.,800 and got 63,660 but then i checked my answer

and found it to be 63,360 then I multiplied 5,2800 times 27 and got 142,600

but then checked it and got 1,710,720 then I knew the answers I checked them

one more time by calculator and found them right

4) There are 63,360 inches in one mile and 1,710,720 feet in 27 miles

5) I multiplied to get answers then I checked by doing them over and found

them to be wrong so I checked it again

6 (There are 5,280 feet in one mile how many inches are in 250 miles? How many

feet in 1,250,000 miles

in the second part of question i read how many inces in 27 miles so my first

answer in part two of my answer the answer is 142,600 like i said firstly

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