1 Dot: Can you make 8 using 13 and 5 or 15 and 9? How?
2 Dots: Can you make 8 using 11/1/3 or 3/12/7? How?
3 Dots: Can you make 24 using 13/8/3 or 18/13/9? How?
How can you make 24?
1 Dot: 1, 4, 8, and 8
2 Dots: 7, 2, 8, and 3
3 Dots: 7, 2, 10, and 2
1) A bag contains 500 beads, each of the same size, but in 5 different colors. Suppose there are 100 beads of each color, and I am blindfolded. What is the fewest number of beads I must pick to be absolutely sure there are 5 beads of the same color among the beads I have picked blindfolded?
2) A number has a remainder of 1 when divided by 4, a remainder of 2 when divided by 5, and a remainder of 3 when divided by 6. What is the smallest number that has the above properties?
3) A total of fifteen pennies are put into four piles so that each pile has a different number of pennies. What is the smallest possible number of pennies that could be in the largest pile?
1) Suppose I select 5 beads blindfolded. The 5 beads could contain1 bead of each color. Suppose I select 5 beads three more times with the same result. I will then have a total 20 beads consisting of 4 beads of each color. The selection of the 21st bead guarantees that there are now at least 5 beads of the same color among the 21 beads.
2) Let N be the required number. Notice that when N is divided by 4, 5, or 6, the remainder is 3 less than the divisor in each case. If N is increased by 3, this new number N + 3 will be divisible by 4, 5, and 6. The smallest number that N + 3 can be is the least common multiple of 4, 5, and 6 which is 60. Therefore the required number is 57.
3) Method 1: The different ways 15 pennies can be distributed into 4 piles are (1, 2, 3, 9), (1, 2, 4, 8), (1, 2, 5, 7), (1, 3, 4, 7), (1, 3, 5, 6), (2, 3, 4, 6). Thus, the number of pennies in the largest pile of each distribution may be 6, 7, 8, or 9. The smallest of these is 6.
Method 2: To find the smallest possible number of pennies in the largest pile, make the sum of the numbers of pennies in the other three piles as large as possible. This occurs when the first three piles contain 1, 3, and 5 pennies, or 2, 3, and 4 pennies. In either case the fourth pile will contain 6 pennies.
1) There are many numbers that divide 109 with a remainder of 4. List all two-digit numbers that have that property.
2) If a number ends in zeros, the zeros are called terminal zeros. For example, 520,000 has four terminal zeros, but 502,000 has just three terminal zeros. Let N equal the product of all natural numbers from 1 through 20: N = 1 x 2 x 3 x 4 x . . . x 20. How many terminal zeros will N have when it is written in standard form?
3) The XYZ club collected a total of $1.21 from its members with each member contributing the same amount. If each member paid for his or her share with 3 coins, how many nickels were contributed?
1) If 4 is subtracted from 109, the result is 105. Then each of the two-digit numbers that will divide 109 with a remainder of 4 will divide 105 exactly. Thus, the problem is equivalent to finding all two-digit divisors of 105. Since the prime factors of 105 are 3, 5, and 7, the divisors are 3 x 5, 3 x 7, and 5 x 7, or 15, 21, and 35.
2) Multiplication by 10 produces an additional terminal zero when a product is written in standard form. If 1 x 2 x 3 x . . . x 20 is written as a product of prime factors, it will contain four 5s and more than four 2s among many factors. Then part of the product can be written as: (5 x 2) (5 x 2) (5 x 2) (5 x 2) which can also be represented as 10 x 10 x 10 x 10. Therefore there are four terminal zeros in the product.
3) Since 121 = 11 x 11, the club has 11 members and each contributed $0.11. Each $0.11 share was paid in 3 coins which had to be 2 nickels and 1 penny. Then the 11 members contributed a total of 22 nickels.
Primer Version for 2nd & 3rd Graders:
1 Dot: Which makes 5: 6 and 1 or 8 and 2? How?
2 Dots: Which makes 3: 8/11/7 or 2/12/7? How?
3 Dots: Which makes 24: 21/1/3 or 13/20/9? How?
Intermediate Version for 4th and 5th Graders:
How do you make 24 using each number once and only once?
1 Dot: 3/12/6/10
2 Dots: 1/4/6/4
3 Dots: 2/6/4/8
The foundation bought this game for the math lab and the kids LOVE it so I wanted to share it with the parents in case you are looking for a gift idea. I have played it with 2nd-5th grade. I have played the same type game with the kindergarten and first graders as a whole group but they have not played the actual game with a partner like the older kids. This game is good for all elementary school kids, you just change the types of questions you ask about the number based on age and ability. I am putting a picture of some resources we use when playing the game below. I can email you the file to print out if you want them.
1) The product of 7 x 7 may be written as 72, 6 x 6 x 6 as 63, and 5 x 5 x 5 x 5 as 54. Let A = 25, B = 34, C = 43, and D = 52. Write A, B, C, and D in order of their values beginning with the smallest value at the left.
2) The cost of mailing a letter first-class is $0.29 for the first ounce and $0.23 for each additional ounce. A letter weighs exactly N ounces where N is a natural number, and the total mailing cost is $1.90. What is the value of N?
3) A group of 30 bikers went on a trip. Some rode bicycles and the others rode "tandems." (A tandem is a bicycle that is ridden by 2 people at the same time). If the total number of bicycles and tandems was 23, how many tandems were used?
1) A = 2 x 2 x 2 x 2 x 2 = 32; B = 3 x 3 x 3 x 3 = 81; C = 4 x 4 x 4 = 64; D = 5 x 5 = 25. Answer: D, A, C, B; or 52, 25, 43, 34; or 25, 32, 64, 81.
2) Since the cost of the first ounce was $0.29, the total cost of the remaining ounces was $1.61. Since the cost of each of the remaining ounces was $0.23 per ounce, the number of remaining ounces was $1.61/$0.23 = 7. The total number of ounces in the letter was 8.
3) If there were no tandems on the trip, only 23 people could have gone on the trip. Then 7 people of the 30 would have been left out. There must have been 7 tandems for the 7 extra people.
1 Dot: Which numbers can be used to make 7: 13 and 8 or 16 and 9? How?
2 Dots: Which numbers can be used to make 4: 8/7/12 or 7/8/3? How?
3 Dots: Which numbers can be used to make 24: 6/20/8 or 1/21/4? How?
How can you make 24 using all 4 numbers?
1 Dot: 6, 1, 9, and 12
2 Dots: 2, 1, 4, and 7
3 Dots: 3, 2, 3, and 5
We are so thankful for the iPads in the Math Lab. We used them daily in November to enhance what we are working on in the lab! I want to share some of the Apps that we have used because several of the students have expressed they want to add some of the apps on their personal devices. I am only listing the ones that would be good for individual use and not ones that require a teacher setting up the activities. Some of the apps we have on the iPads are Math Slide (Place Value for younger students and Basic Facts for older students), Prodigy, Sushi Monster, MADS 24, Twenty48 Solitaire, Math Math, KaKooma, KenKen, SpeedDrill, Ten Frame Mania (for primary students), and Math Learning Center Manipulatives (this is a great resource to have so that your student can use virtual manipulatives at home when completing homework).
What are we doing in the lab this month-
Kindergarten: We are continuing to work on decomposing numbers (Ex. 5 can be broken into 0 and 5, 1 and 4, 2 and 3).
1st Grade: We are continuing to work on 10 More and 10 Less. (10 more than 43 is 53 and 10 less than 43 is 33)
2nd Grade: We are working on using double facts to solve near double facts in addition (Ex. I know that 4+4=8 so 3+5 is also 8)
3rd Grade: We are working on building on double facts to help us in subtraction (Ex. I know that 5+5=10 so 11-5 will be one more than 10-5 so it is 6)
4th Grade: Triple Double when Multiplying by 8 (Ex. 32 X8=, I know 32+ 32=64, 64+64=128, and 128+128=256 so 32X8=256)
5th Grade: Doubling and Halving (Ex. 14X5 is the same as 7 X 10 so both are equal to 70)
2nd-5th grade will have the 2nd Quarter 24 Tournament in December. The 24 card game makes a great stocking stuffer if they are in stock (sometimes they are hard to find in stock, especially the primer version).