The Four 4 S Project

The Four 4’s Project

Developed by: Ben Forman

Adapted from: The Four 4’s Problem

Lesson Overview: This purpose of this lab Subject area(s): Mathematics is to provide an interesting context for students to apply a wide range of Audience: 9th-12th Grade mathematical operations PA Science Education Standards Topic: An interesting problem that Addressed: 1. develops a student’s understanding of the full gamut of mathematical operations Philadelphia Core Curriculum Topics Suggested Time: Flexible (Anywhere from Addressed: 1. 1 class period to several weeks) Depending on how the lab is structured and how much time a student is given, this lab can be condensed into a single class period or can be stretched out over several weeks

1. Specific Objectives: i. Students will briefly review the basic mathematical operations such as addition, subtraction, multiplication, and division. ii. Students will learn new mathematical operations such as square roots, exponents, and factorial. iii. Students will start working competitively and/or collaboratively in groups and/or individuals on the Four 4’s Problem.

2. Materials List: This handout, a pencil, and scratch paper. i. Obtaining Materials and Lesson Prep: Print out from computer

3. Duration/Time Required i. Prep time: 5 minutes ii. Implementing exercise during class: The rest of the class period and/or future class periods

4. Costs i. Est. Base Cost: $0 ii. Est. Additional Cost per Student: $0

5. Safety Information: None

6. Math/Science Context: This lab teaches students some new mathematical functions and draws on their knowledge of more familiar ones. It allows students to think about numbers and functions in a creative and dynamic way. 7. Motivation and Incentive for Learning: The Four 4’s problem is incredibly interesting and entertaining. Students may stumble their way onto some answers while struggling for others. The lab allows for students to collaborate on the solutions they have reached or compete over who can achieve the most solutions the quickest.

8. Vocabulary: Factorial, Exponents, Square Roots

9. Methods/Procedures for students: i. Pre-investigation work: None ii. Investigation work: None

10. Assessment: Evaluate students by the volume and creativity of their solutions. Also, make sure student is familiar and adept at using the new functions such as factorial, exponents, and square roots.

11. Extension Ideas: Expand the types of functions used, solving for negative numbers, solving for numbers greater than 100, and finding multiple ways to achieve one solution.

12. Scalability: Scalability is the strength of this lab. The most basic student can achieve many of the solutions using only addition, subtraction, multiplication, and division. On the other hand, more advanced students are able to incorporate many of the new more complicated functions to achieve a wider range of solutions.

13. Evaluation/Reflection by Fellows and Teachers of how it went:

14. Teaching Tips: The Four 4’s Problem

Developed by: Ben Forman

Adapted from: The Four 4’s Problem

Introduction

Today’s lesson prepares students to tackle an interesting math problem — achieve all numbers 0 through 100 using the digit 4 four times. To prepare students to solve this problem, this lab begins with a review of the fundamental mathematical operations and then expands to some slightly more complicated and unfamiliar ones.

Objectives

While it is doubtful that most students will figure out the solutions to all the numbers, this fascinating problem will demonstrate to them the powerful and surprising capabilities of numbers. Depending on how the teacher wants to organize the lab, students may compete in groups on the problem, or instead work individually and later collaborate.

Materials -Handout -Pencils -Scratch Paper

Vocabulary -Square Root -Exponents -Factorial

Methods:

Part I – Designed to refresh student’s knowledge of mathematical operations such as addition, subraction, multiplication, and division.

Part II- Designed to expand on student’s knowledge of mathematical operations such as square roots, exponents, and factorials.

Part III- Allows students to utilize their knowledge and apply it to the Four 4’s problem.

\ The Four 4’s Problem

Today’s lesson is short. Why? Because we want to jump right into a really exciting problem. Most of your learning today will be from yourself and your peers rather than from your teacher. Let’s start with some basic review and then learn a few new tips before moving on to the problem.

Think about all the things we can do with numbers:

Addition Subtraction

5 + 5 = 10 5 – 5 = 0

Multiplication Division

5 x 5 = 25 5 / 5 = 1

What else can we do with numbers?

Let’s learn 3 new mathematical operations that will help us in our problem. Exponents:

Have you ever seen a number at the top right of another number? If so, this is called an exponent. The number on the left is called the “base” and the number on the upper right is called the “exponent”.

Source: http://www.mathsisfun.com/images/8-squared.gif

This expression is solved by taking the base and multiplying it by itself the number of times that the exponent indicates. For example, if the base is 8 and the exponent is 2, as in the example above, we can solve by taking 8 x 8 = 64.

Try it yourself: 2 3 =

Base: Exponent:

Base: Exponent: Square Roots:

The concept of a “square root” is similar to an exponent. If you are asked what the square root of 4 is, someone is asking you “what times itself equals 4?” Hmm…what times itself equals 4? Two! 2 x 2 equals 4. Therefore, the square root of 4 equals 2.

Let’s try one more: What is the square root of 9? This is the same as asking “what times itself equals 9?” Well, 2 x 2 = 4 and 4 x 4 = 16. Let’s try 3. 3 x 3 equals 9! Therefore, the square root of 9 is 3.

Square roots are marked by funny looking signs that look like this: 

Let’s try a few problems to make sure you understand the meaning of a square root:

36 =

64 =

100 =

Factorials:

You probably haven’t heard of a factorial, but it happens to be a very useful mathematical operation. It’s marked by an exclamation point! So if someone asks you to solve 3!, you must multiply 3 x 2 x 1. In other words, a factorial of a number is the product of all the positive integers equal to or less than that number. So 4! = 4 x 3 x 2 x 1 = 24

Try a few on your own (feel free to use your calculator, but make sure to write out your calculation first):

Now for the fun part! The Four 4’s Problem! What if I were to tell you that the number 4 is special?

It is possible to make every number between 0 and 100 using the digit 4 exactly four times. That means you can use addition, subtraction, multiplication, division, exponents, factorials, square roots, fractions, decimal points, and every other mathematical formula you know. The catch is that you can only use the digit 4 and you can only use it four times.

Let me help get you started: 4 + 4 + (4 x 4) = 22

Feel free to write that equation below next to the number 22.

It doesn’t matter where you start. Just try and get as many numbers as you can. There are many different ways to get a number but this problem only asks for one.

Think creatively and remember that getting every number is possible!

0 51 1 52 2 53 3 54 4 55 5 56 6 57 7 58 8 59 9 60 10 61 11 62 12 63 13 64 14 65 15 66 16 67 17 68 18 69 19 70 20 71 21 72 22 73 23 74 24 75 25 76 26 77 27 78 28 79 29 80 30 81 31 82 32 83 33 84 34 85 35 86 36 87 37 88 38 89 39 90 40 91 41 92 42 93 43 94 44 95 45 96 46 97 47 98 48 99 49 100 50