Number Activities
Sums
a. Continue the table. What can you say about the sum of consecutive odd integers?
Draw a picture that shows this.
N
Sum
1
1
3
1+3 = 4
5
1+3+5 =9
Continue the table of the cubes of integers below. Add a column of the sum of the
consecutive cubes. Conjecture about the entries in this sum of consecutive cubes column.
Can you prove it?
N
N cubed
Sum
1
1
1
2
8
1+8=9
3
27
1+8+37 = 36
4
…
5
…
6
…
Other things to know:
1  1  12
1  3  4  22
1  3  5  9  32
1  3  5  7  16  42
1  1  13
3  5  8  23
7  9  11  27  33
13  15  17  19  64  43
1  1  12


1  8  27  36  6  1  2  3
1  8  27  64  100  10  1 2  3  4 
1 8  9  32  1 2
2
2
2
2
2
Note: 1, 3, 6, 10, … are triangular numbers.
Let me count the ways
Holton has 19 toothpicks. How many ways can he arrange them into three piles so that
each pile contains an odd number of toothpicks? For what other numbers can you make 3
piles so that each pile contains an odd number of toothpicks? Is there a pattern to the
number of ways they can be arranged?
Sequences
1.
a. Find the sum of two consecutive integers. Find the sum of the next two. Find
the difference. Repeat this for several sets of numbers. What do you notice? Now find
the sum of 3 consecutive integers and the sum of the next 3 consecutive integers. Find
the difference. Repeat this for several sets of numbers. What do you notice? Now try 4
integers. Generalize. Then explain why your generalization is true
b. Repeat the experiment of a, but this time use consecutive odd integers.
c. What if the numbers have a common difference of d?
2.
a. Express the number 90 as the sum of two or more consecutive integers in 5
different ways.
b. Find at least two strings of consecutive positive integers that add up to 150.
Digits
1.
a. Use the digits 1-9 to make two 3-digit numbers with a sum of 1332.
b. Use the digits 1-9 to make three 2-ditit numbers so that the sum of 2
of them is the third number
c. Use the digits 1-9 to make three 2-ditit numbers so that the difference
of 2 of them is the third number
d. Use the digits 1-9 to make three 2-ditit numbers so that the sum of 2
of them is the third number AND as large as possible.
e. Use the digits 1-9 to make three 2-ditit numbers so that the product
of 2 of them is the third number
2. Use the digits 0-5 to make two 3-digit numbers with the smallest difference.
3. Write down a number, find the sum of the squares of the digit to make a new number,
find the sum of the squares of the digits of the new number. Continue this process until
you reach a stopping point. Do this for the first 20 numbers. What do you observe? Try
using cubes instead of squares and see what happens.
4.
a. Write down a 2-digit number with the first digit larger than the
second. Reverse the digits for a second 2-digit number. Subtract
the smaller from the larger. Repeat several times with different
starting numbers. What do you notice about the results of the
subtraction? How are they related to the digits of the starting
number? Prove this with algebra.
b. Find all the 2-digit numbers that give a sum of 154 when the original
number and the number obtained by reversing its digits are added.
5. Write any three digits, then repeat those three digits to give a 6-digit number in the
form abc,abc. Divide your number by 7 and get a result. Is there a remainder? Divide
the result by 11: again was there a remainder? Take this second result and divide it by 13
and note the final result. Repeat for several other beginning three digits. Explain with
algebra.
6. Write the digits 1-8 in the squares so that no two consecutive numbers are next to each
other. Is there more than one solution?
Circle
Find several circles (lids) and make the ratio of circumference to diameter and so
determine an approximation of  .
Diagonal of a square
a. Make several squares and for each find the ratio of the diagonal to the side, and so
determine an approximation of 2 .
b. Starting with a unit square construct segments whose lengths are
2, 3, 5, 6 and 7 .