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1) If a class of children is separated into groups of 5 children, 2 children will be left over. If the class is separated into groups of 6 children, 3 children will be left over. What is the smallest number of
children the class could have?
2) There is an even number between 200 and 300 that is divisible by 5 and also by 9. What is the number?
3) A book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit 1 appear in the page numbers?
1) Method 1 - List multiples of 5, each increased by 2: A = 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, . . .
List multiples of 6, each increased by 3: B = 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, . . .
List A and list B have 27, 57, 87, . . . in common. The smallest number of children in the class is 27.
Method 2 - Each multiple of 5, increased by 2, ends either in 2 or 7. Each multiple of 6, increased by 3, must be an odd number. Then the only possibilities for the class size are: 7, 17, 27, 37, . .
Divide each by 6. The first number that has a remainder of 3 is the required number. That number is 27.
Method 3 - Algebra: Let A and B be numbers such that 5A + 2 = 6B + 3. Subtract 2 from each member of the equation. Then 5A = 6B + 1. Since 6B + 1 is an odd number, 5A is also odd. 5A
may be 5, 15, 25, 35, 45, . . . Divide each of these numbers by 6. The first number that has a remainder of 1 is 25: 25/6 = 4 R1. So B = 4. Since 6B + 3 is the class size, and B = 4,
the class size is 6 x 4 + 3 = 2.
2) Since the number is divisible by 2, 5, and 9, it is also divisible by the product 2 x 5 x 9 = 90. The multiple of 90 between 200 and 300 is 270.
3) Consider the frequency of appearance of the digit "1" in each of the places.
units place: the digit "1" appears once in every ten. Since 500 has 50 tens, the digit "1" will appear 50 times in the units place.
tens place: the digit "1" appears ten times in every hundred. Since 500 has 5 hundreds, the digit "1" will appear 50 times in the tens place.
hundreds place: the digit "1" will appear 100 times in the hundreds place (100, 101, 102, ..., 199)
The digit "1" will appear a total of 200 times in the page numbers.
Here are some challenging word problems that are similar to what 4th and 5th graders experience at a Math Olympiad. Math team members can use these for extra practice and everyone else can use them to help extend your problem solving skills. Solutions are at the bottom of this post.
1) Suppose today is Tuesday. What day of the week will it be 100 days from now?
2) The four-digit numeral 3AA1 is divisible by 9. What does A represent?
3) A purse contains 4 pennies, 2 nickels, 1 dime, and 1 quarter. Different values can be obtained by using one or more coins in the same purse. How many different values can be obtained?
1) Every 7 days from "today" will be Tuesday. Since 98 is a multiple of 7, the 98th day from today will be Tuesday. Then the 100th day from today will be Thursday.
2) If the number is divisible by 9, then the sum of its digits is divisible by 9. The digit sum is 3 + A + A + 1 = 4 + 2A. The digit sum cannot by 9, otherwise A = 2 1/2. So 4 + 2A = 18 which produces
A = 7.
3) The largest amount that can be made is $0.49. Using the given set of coins, any amount from $0.01 to $0.49 can be made. Therefore there are 49 different amounts that can be made.
Kindergarten- We looked at dot representations of 6, sang songs to practice counting to 100 and the months of the year. We solved a couple visual problems dealing with patterns. Students matched up dominoes with the correct numerals.
1st Grade- We sang a song that they had to add one more to given numbers. We did a number talk using ten frames. We worked together to solve two story problems using the Bar Model Drawing Method.
2nd and 3rd Grade-We worked on the counting on strategy when adding 2 to a number. We played a racing game to help us practice this strategy. We also worked on finding missing addends when given one addend and the sum. We solved a subtraction problem comparing two amounts using the Bar Model Drawing Method.
4th and 5th Grade- We worked on the doubling strategy in multiplication. Students played a tic-tac-toe game to practice the strategy. We used Model Drawing to solve a multiplication problem.
September Math Camp
A camp designed for mathematically talented students interested in competitive mathematics
September 26-29, 2016
9:00 am to 2:00 pm
Walton High School
September Math Camp is designed to introduce 4th through 6th grade students talented in mathematics to competitive mathematics. Students will
· Explore math topics
· Practice solving ciphering questions quickly
· Learn problem-solving strategies
· Experience practice tournaments
Early Bird registration before September 1 is $125.
September 1 to September 16, registration is $150.
Late Registration (after September 16) is $175
Registration must completed online at www.waltonmathteam.com. Please make checks payable to Walton High School and send payment to Jennifer West at Walton High School, 1590 Bill Murdock Rd., Marietta, GA 30062. A spot for your student will be reserved upon payment. You will be notified through e-mail when your child has been accepted. If registrations are full, your check will be returned to you through mail.
Please send lunch with your student. All other supplies will be provided.
Reservations are limited to 60 students, so make your reservations early.
This event is a fund-raiser for Walton High School Math Team and will be operated by its members and faculty sponsors. No refunds will be given.
Need more information? Contact Laura Speer at email@example.com.
Let me know if you would like to volunteer to help with the math lab. There are things that can be done from home or at Lewis.
Kindergarten- As always we stared with number talks and visual problem solving. We sang a song and moved while practicing counting to 10 and we sang and danced to a song about the months of the year. Students were broken into 2 groups to play memory with a small group and teacher where they matched numerals to dot representations of that number.
1st Grade- We changed up our routine for this lesson so that we could have a middle school helper for each small group of kids to help them learn a new game. The game is played like Uno but allows them to practice their subitizing skills. After the middle school students had to leave for school we did our number talks and problem solving for the day. We sang and danced to a song that teaches about coins.
2nd & 3rd Grade- We practiced the counting on strategy with adding 1 and 2 more. Students played a game similar to Chutes and Ladders that allowed them to practice the strategy of counting on. We worked together to write a constructed response to the word problem we solved in lesson 6.
4th & 5th Grade- We extended the using our 10 facts for our 5 facts strategy by applying the strategy to bigger numbers. (Ex. 46 X5= 230 because 46X10=460 and 1/2 of 460 is 230). Students practiced this strategy with a game called "Bump". We worked together to write a constructed response to the word problem we solved in lesson 6.