Student Page 1.4.3 Four Fours
Now that you've had a chance to verify and play with expressions for the numbers 1 through 12 created using three threes, use your creativity to find expressions for other numbers using four fours. The Four Fours Problem provides you additional opportunities to think about the order of operations and the mathematical operations you have studied in previous courses. Study the problem presented below.
How can you make the numbers 1 through 20, 24, 30, and 100 using 4 fours? Record all of the expressions you find for each number in the table. Work alone for at least 5 minutes before you share your results with your group. If you get stuck, consult Figure 1.4.2.1 for inspiration.
Use the number 4 exactly four times to write expressions equivalent to the numbers from 0 through 20. You may use any of the mathematical symbols below. A few examples are provided.
Arithmetic operations: \(+, -, \times, \div\)
Parentheses: ( )
Decimal points or percents: e.g. \(4.4\text{,}\) \(.44\text{,}\) 44%
Exponentiation: ^ (to the power of), e.g. \((4 + 4)^4\)
Square root: e.g. \(\sqrt{4}\)
Factorial: ! (\(4! = (4)(3)(2)(1) = 24\)
Concatenation: e.g. 44, 444
Repeating decimal: \(. \overline{4} = \frac{4}{9}\)
1.
Complete the table. Find at least one expression using 4 fours for each number. Use the Order of Operations; include grouping symbols as needed. Check your responses using a graphing calculator. Some examples for \(n = 4\) are provided. For the numbers whose solutions are shown in the table, find at least one more expression that works.
| Number | Four Fours Equation | Number | Four Fours Equation |
|---|---|---|---|
| 0 | 12 | ||
| 1 | \(1 = \left(\frac{4}{4} \right) ^{44}\) | 13 | |
| 2 | 14 | ||
| 3 | 15 | \(15 = \frac{44}{4} + 4\) | |
| 4 | 16 | ||
| 5 | 17 | ||
| 6 | 18 | ||
| 7 | \(7 = 4 + 4 - \frac{4}{4}\) | 19 | |
| 8 | 20 | ||
| 9 | 24 | \(24 = 4! \cdot \left(\frac{\sqrt{4} \times \sqrt{4}}{4} \right)\) | |
| 10 | 30 | ||
| 11 | 100 |
2.
What strategies did you use to find solutions?
3.
Find more than one solution for at least 3 of the numbers. Record them in the table.
4.
Which of your solutions are not dependent on the number 4? For example \(1=\ \left(\frac{4}{4}\right)^{44}\) is also true when each 4 is replaced with 2, 6, or 9. Highlight these solutions in your table.
Share the expressions you found using Four Fours with your group. Compare your work with your group members. Do you agree that each expression is correct? Circle any expressions you question; put a checkmark next to expressions with which you agree. Resolve any differences of opinion. When asked to do so, share some of your group's expressions with the class.