The 24 Game (+-x/^!) with a Target Number of 24

After playing hundreds of rounds of Target Number last week, where the target number was randomly selected from 1-10, this week we increased the challenge with The 24 Game. Here the target number always is 24 and each game card has four numbers. The object of the game is to make 24 using all four numbers on a card. You can add, subtract, multiply, or divide and use each number only once. 

The number 24 is special since it is the only whole number less than 30 that has eight factors. 1 and 24, 2 and 12, 4 and 6, and 8 and 3. 

For example, if you choose a card with the numbers 3 7 7 7, one solution would be to add the numbers, 3 + 7 + 7 + 7 = 24. 

The great challenge is to find alternative solutions (7 + 3 + 7 + 7 = 24 is not an alternative solution since this is only using the commutative property to switch the order of addition). One alternative solution might be to focus on the number 3 and thinking about what number 3 needs to make 24; since 3 x 8 = 24, can we make 8 with the numbers 7 7 7? Try dividing 7 by 7 to get 1 and then add that 1 to 7 to get 8; now multiply 3 x 8 = 24.

First and second graders should focus on one dot cards (easiest), third and fourth graders on two dot cards (medium difficulty), and fifth and sixth graders on three dot cards (most challenging).

For the older children, I added the complexity of allowing students to also use exponents and factorials if they so choose. For example, if your four numbers are 2 3 5 8, a relatively easy solution might by (8 x 2 = 16) and (3 + 5 = 8) and adding 16 + 8 = 24; however, you might try 2 to the 3rd power (2x2x2=8); then subtract 5 from 8 (8-5=3) and then multiply 8x3=24. Factorials are when you multiply the number by each number before it until you reach 1; for example, 3 factorial (3! = 3 x 2 x1 = 6); 4 factorial or 4! = 4 x 3 2 x 1 = 24. So, with the same numbers 2 3 5 8, you could use 3! = 6 and then try to make 4 with the remaining 2 5 8; let’s take 2^5=32 then 32/8 = 4 and finally 6 x 4 = 24.

The best strategy is to choose a number on the card and turn the other three numbers into what you need to make 24. Use pairs of numbers and multiply them to generate numbers that may be added or subtracted to make 24. 

Another advanced method to generate 24 is with fractions. Since division and fractions are the same thing, look at fractions that can be helpful such as 3/2 or 1.5. Since 16 x 3/2 = 24, if you have a 2 3 4 4 you could divide 3 by 2 to make 3/2, then multiply 4 by 4 to make 16 and then multiply those two results to make 24. Since 32 x 3/4 = 24, if you have a 3 4 4 8, you could divide 3 by 4 to get 3/4, multiply 4 x 8 to make 32, and then multiply those two results to make 24. Since 64 x 3/8 = 24, if you have a 3 8 8 8, you could divide 3 by 8 to get 3/8, multiply 8 x 8 to make 64, and then multiply those two results to make 24.

The children should use the attached pdf and create as many solutions as possible, showing their work with parenthases. For example, if they have a card with the numbers 3 4 5 7, they should show one solution as (3 x 4) = 12 and (5 + 7) = 12 and 12 + 12 = 24. Write the solution next to the card on the pdf.