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Section 1.5 Four Fours Problem, Extended

Subsection 1.5.1 Overview

The section expands upon the ideas in Section 1.4 by exploring other ways to use fours to make other numbers.

Student Page 1.5.2 Organizing Your Work: Ways to Use 1, 2, or 3 Fours to Make Other Numbers

In mathematics, it is often helpful to create an organized list. Organizing Your Work: Ways to Use 1, 2, or 3 Fours to Make Other Numbers helps you organize expressions you have found using fewer than 4 fours.

1.

As you worked on the Four Fours Problem, you likely found many ways to combine fewer than 4 fours. Use this table to organize what you've found. Find other ways to combine 1, 2, or 3 fours to make other numbers. For each column, order the combinations you find least to greatest.

Using 1 Four Using 2 Fours Using 3 Fours
     
     
     
     
     
     
     
     
     
     
     
     
     
     

2.

Combine some of the expressions you found in the table above to make numbers using four fours. How can this organized list help you find solutions for some of the numbers in Four Fours?

3.

What is the smallest number greater than zero you can make using four fours? What strategies did you use to find this number?

4.

What is the largest number you can make using four fours? What strategies did you use to find this number?

Share your solutions with your group. Verify each other's work. Get all possible answers on the board for the numbers 1 through 20, 24, 30, and 100. If there are any numbers for which solutions haven't been found, work on these together.

Once remaining questions about the Four Fours Problem are resolved, if time remains, work on Flexibility with Numbers: Four Numbers Problem in your small group.

Homework 1.5.3 Homework

2.

The Flexibility with Numbers: Four Numbers Problem extends Four Fours, this time using four copies of the same number. Your group will choose one of the numbers 1, 2, 3, 5, 6, 7, 8, or 9 (no group may choose the same number as another group). This is your first group assignment. Most of the work on this group assignment will occur outside of class. Exchange contact information and arrange times to get together electronically or in person to work on this assignment.

Your group's chosen number is .

The completed Flexibility with Numbers: Four Numbers Problem is due in class on .

4.

Once you complete the Flexibility with Numbers: Four Numbers Problem, you will also submit a reflection about how well your group worked together to solve the Flexibility with Numbers: Four Numbers Problem. Answer the following questions to reflect on your work as a group. Turn in this reflection on the day you submit your group report for the Flexibility with Numbers: Four Numbers Problem.

(b)

Did each group member come prepared to group meetings ready to contribute individual work to the group discussion?

(c)

Did some group members only work on the problem when you were together? If so, characterize their contributions.

(d)

What steps can your group take to help your group work more productively together?

(e)

Are you satisfied with your group? If not, what would need to happen for you to be satisfied with your group?

(f)

Would you like to stay with your current group? If you want to switch groups, with whom would you like to work?

Student Page 1.5.4 Flexibility with Numbers: Four Numbers Problem

Use the same number exactly four times to write expressions equivalent to the numbers 0 through 20. You may use any of the mathematical symbols below. A few examples for the number 4 are provided. Choose the number you will use and tell your instructor. You may choose one of the numbers: 1, 2, 3, 5, 6, 7, 8, 9. (Note that you may not use the number 4.)

  • 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 writing solutions as expressions. Use the Order of Operations; include grouping symbols as needed. Check your responses using a calculator. Some examples for the number 4 are provided. Your group's number is .

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.

4.

Which of your solutions are not dependent on the number you chose? Highlight these solutions in your table.

Activity 2. Summarizing Your Work on the Four Numbers Problem.

Use Reflecting on Your Group's Work on the Four Numbers Problem to guide your group's reflection on your work on Flexibility with Numbers: Four Numbers Problem. Look for similarities in how numbers were created. Keeping the Order of Operations in mind, check each other's solutions to see if you agree. Resolve disagreements.

The Four Fours activities were designed to help you:

  • Have fun with mathematics,

  • See how mathematics can be a puzzle to play with,

  • Experience how flexible problem solving can be (There were many expressions for each number, allowing you to use your creativity to find them.),

  • Persevere in mathematics when you get stuck,

  • Review and remember the Order of Operations,

  • Recall and use many mathematical operations (+, \(-\text{,}\) √ó, \(\div\text{,}\) !, exponentiation),

  • Become more familiar with your graphing calculator;s operation, and

  • Experience generalization; several numbers could be expressed in the same way using any single digit integer.

One example of generalization in the Flexibility with Numbers: Four Numbers Problem is: \(2=\frac{1+1}{\sqrt1\sqrt1}=\frac{4+4}{\sqrt4\sqrt4}=\frac{5+5}{\sqrt5\sqrt5}=\frac{8+8}{\sqrt8\sqrt8}=\frac{9+9}{\sqrt9\sqrt9}\text{.}\) This equation works in the same way for 2, 3, 6, and 7. If you replace each numeral with 0! = 1, the number 2 can be represented in this way for 0! and all integers 1 through 9. To generalize this equation, we let n represent any single digit non-zero integer and replace the numerals in the equation with n. The equation becomes \(2=\frac{n+n}{\sqrt n\sqrt n}\) indicating that as long as each n in this equation is replaced by the same number, the equation is always equal to 2. The letter n is a variable. It takes the place of a number, allowing us to be free of having to show the equivalence using each number as we did at the beginning of this paragraph. This is a very powerful idea.

Student Page 1.5.5 Reflecting on Your Group's Work on the Four Numbers Problem

1.

Share your Four Numbers responses with your group. On your group member's sheet, highlight any expressions you find particularly interesting and any expressions you are unsure of so that you may discuss them together.

(a)

Do you agree with each of the solutions each group member found? If not, resolve any difficulties. Do not erase any of a student's original work. Instead, note any expressions over which group members had differences of opinion on the back of this page. Also indicate how you resolved them.

(b)

Look for similarities in how numbers were created. What do you notice?

2.

For which numbers did you and your group members have the most numerical expressions?

(a)

Write the numbers and as many different responses as you can in the table below.

Number Four Numbers Expression Number Four Numbers Expression
       
       
(b)

What made these responses easy to find?

3.

For which numbers did you and your group members have the smallest number of numerical expressions?

(a)

Write the numbers and the different responses group members found in the table below.

Number Four Numbers Expression Number Four Numbers Expression
       
       
(b)

What made these responses difficult to find?

4.

Were there any numbers for which your group found no numerical expressions using four numbers? Write the numbers below. We will work on these together as a class.