Four 4s math lesson

1. The Four 4’s Challenge
This challenge asks students to construct each number from 1 to 50 using exactly four 4’s and no other
numerals. A number like 44 would use up two of the 4’s. Students are allowed to use any non-numeric
math symbols to combine the 4’s into the result.
Levels Grades 3 through Algebra.
Topics Order of Operations, Numerical Expressions, Number Sense, Fractions, Decimals, Problem Solving
Goals
• Students will increase their number sense as they try to reach different target numbers.
• Students will develop an understanding of the need for order of operations conventions and will
learn some of those conventions.
• Students will be exposed to exponents, square roots, fractions, decimals, and factorials.
Pre-requisite Knowledge Students should be familiar with basic arithmetic operations (adding, sub-
tracting, multiplying, and dividing) with whole numbers. Understanding of operations with decimals
or fractions is helpful for some numbers.
Materials and Preparation
• A large piece of paper or board where solutions and names of the contributors can be displayed.
• Notebooks and pencils for the students to work out and record their answers.
• Four function and scientific calculators may be helpful for the order of operations discussion.
Helpful Hints
• The students are not allowed to use fewer than four 4s. They must use exactly four 4s.
• Help students to realize that evaluating the same expression in a different order can sometimes lead
to a different result. This should lead to a discussion about the necessity of agreeing about the
order in which operations are performed. Students should start to learn about the appropriate use
of parentheses through this activity. See the notes below for more discussion about subtleties that
may emerge during these discussions.
• If students start to run out of ideas, introducing notation such as exponents, square roots, factorials,
decimals, floors, and ceilings one at a time can help them to make additional progress. Using repeating
bars is also permissible.
• Introduce students to the factorial operation. 4! is read “four factorial” and that it means 4 × 3 ×
2 × 1 = 24. This is a good way to make a larger number with only one four.
• A few of the solutions benefit from noting that
4
.4
= 10 and that
√
4
.4
= 5.
• Older students may find it helpful to note that .4 =
4
9
, though this approach is not necessary to
obtain solutions for 1 through 50. One way this fact could be used, for example, would be
4
.4
= 9.
• The ceiling function is basically a “round up function”. The ceiling of 5.2 is 6, the ceiling of 5.8
is also 6, and the ceiling of 5 is 5. The ceiling function is written like brackets with only the top
horizontal bars, as shown in these examples: ⌈4.4⌉ = 5. ⌈.4⌉ = 1. The floor function is the “round
down function”. The floor of 5.2 is 5, the floor of 5.8 is also 5, and the floor of 5 is 5. The floor
function is written like brackets with only the bottom horizontal bars, as shown in these examples:

2. ⌊44.4⌋ = 44. ⌊
√
44⌋ = 6. Ceiling and floor functions are not necessary to find solutions from 1 to
50. However, if you decide to change the problem so that students must use four 7s, these functions
come in handy. I commonly make a point of introducing ceiling and floor functions when working
with 6th through 9th grade students as a way to prepare them to work with algebraic formulas in
spreadsheets later in the course. The ceiling and floor functions are good tools for students who are
learning mathematical modeling because it is often necessary to round answers up or down to the
nearest whole number for real-world applications.
• Many different solutions to the four 4s challenge are easily located on the internet, so beware of this
if students might be tempted to search for answers in that way. At this time, the four 7s challenge
does not have solutions posted on the internet.
Notes on Order of Operations
Students should begin learning about order of operations during this activity. Look for student expressions
that can provide a good springboard for this discussion. For example, if a student creates the expression
4 − 4 ÷ 4 + 4, some students might think that this produces 4, while others might think that it makes 7.
If a simple four function calculator is used to check this, it might agree that this expression makes 4. A
scientific calculator might say that it makes 7. Which one is right?
Discuss the idea that the answer depends on whether we work out the subtraction or the division first.
Order of Operations conventions have been established to help people agree which way we will evaluate these
kinds of expressions. According to the Order of Operations, the expression above should give 7 because
multiplication and division operations are always performed before addition and subtraction operations
(unless parentheses are used, in which case we work inside the parentheses first).
The Order of Operations rules tell us to evaluate expressions in the following order:
1. Parentheses
2. Exponents
3. Multiplication and division
4. Addition and subtraction
This summary does not really give students a complete understanding of how order of operations works.
Here is a more complete discussion.
1. Parentheses. We always begin evaluating expressions by working inside parentheses. Note that
fraction bars, radicals, and exponents all come equipped with invisible assumed parentheses. This is
important when using a scientific calculator or computer, because it is often necessary to fill in the
assumed parentheses so that the machine gets the correct answer. For example 44+4
4 has invisible
parentheses around the expression in the numerator, so that we must enter (44 + 4) ÷ 4 when using
a scientific calculator in order to get the correct answer of 12. If we type 44 + 4 ÷ 4, we will get 45
instead. The expression 4
√
4
4+4 has invisible parentheses in the denominator, so we would need to type
4
√
4 ÷(4+4) to get the correct answer of 2. If we type 4
√
4 ÷4+4, the calculator will give the answer
8 instead. Similarly, the expression
√
44 +
√
4
.4 has invisible parentheses that keep everything under
the radical bar together. In a scientific calculator, we would enter
√
(44 +
√
4 ÷ .4) to obtain the
answer 7. Some calculators have a square root key that must be pressed after the quantity under the
radical. If in doubt, experiment with some expressions where you know what the answer should be.
2. Exponents and radicals. Note that towers of exponents are evaluated right to left so 223
is equal to
28 = 256, not 43 = 64. Note that while (22)3 does equal 26, 223
= 28.

3. 3. Multiplication and division in order from left to right. Note that division has the same level of
precedence as multiplication, so 16 ÷ 2 × 2 is equal to 16, not 4. Students in 6th grade and higher
should learn to avoid using × and ÷ symbols, instead using multiplication dots or parentheses for
multiplication and fractions for division.
4. Addition and subtraction in order from left to right. Again, subtraction has the same level of
precedence as addition, so 6 − 3 + 2 is 5, not 1.

4. The Four 4’s Challenge
Welcome to the four 4’s challenge! Your mission, should you choose to accept it, is to construct each
number from 1 to 50 using exactly four 4’s and no other numerals. A number like 44 would use up two of
your 4’s. You are allowed to use any non-numeric math symbols to combine your 4’s into the result.
Remember to record your solutions in your notebook. Each solution is worth one point. If you find more
than one way to make a number, you can get a point for each different method. You can also earn points
by recording any strategies you come up with, by describing things you learn, or by describing the math
concepts you are using.
The Four 4’s Challenge
Welcome to the four 4’s challenge! Your mission, should you choose to accept it, is to construct each
number from 1 to 50 using exactly four 4’s and no other numerals. A number like 44 would use up two of
your 4’s. You are allowed to use any non-numeric math symbols to combine your 4’s into the result.
Remember to record your solutions in your notebook. Each solution is worth one point. If you find more
than one way to make a number, you can get a point for each different method. You can also earn points
by recording any strategies you come up with, by describing things you learn, or by describing the math
concepts you are using.
The Four 4’s Challenge
Welcome to the four 4’s challenge! Your mission, should you choose to accept it, is to construct each
number from 1 to 50 using exactly four 4’s and no other numerals. A number like 44 would use up two of
your 4’s. You are allowed to use any non-numeric math symbols to combine your 4’s into the result.
Remember to record your solutions in your notebook. Each solution is worth one point. If you find more
than one way to make a number, you can get a point for each different method. You can also earn points
by recording any strategies you come up with, by describing things you learn, or by describing the math
concepts you are using.
The Four 4’s Challenge
Welcome to the four 4’s challenge! Your mission, should you choose to accept it, is to construct each
number from 1 to 50 using exactly four 4’s and no other numerals. A number like 44 would use up two of
your 4’s. You are allowed to use any non-numeric math symbols to combine your 4’s into the result.
Remember to record your solutions in your notebook. Each solution is worth one point. If you find more
than one way to make a number, you can get a point for each different method. You can also earn points
by recording any strategies you come up with, by describing things you learn, or by describing the math
concepts you are using.