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1-2-3-4 Puzzle
1-2-3-4 Puzzle
Intent
1-2-3-4 Puzzle and its companion activity, Uncertain Answers, help students gain insight into the need for rules for order of operations and provide additional experience with the algebraic logic of graphing calculators.
Mathematics
Order of operations is a set of conventions that facilitate mathematical communication. By convention, arithmetic problems are worked out according to the following precedence rules:
- Simplify expressions within parenthesesbefore combining them with expressions outside the parentheses.
- Within parentheses (or where no parentheses exist), do operations in this order: (1) Apply exponents to their bases. (2) Multiply and divide as the operations appear from left to right. (Neither operation has precedence over the other.) (3) Add and subtract as the operations appear from left to right. (Neither operation has precedence over the other.)
These precedence rules have been established to remove the ambiguity from the meaning of such written expressions as 3·7+2², which might otherwise be evaluated by multiplying 3 by 7, adding 2, and then squaring the result to obtain 529. Using the precedence rules above, the value of this expression is 25, because 2²=4, 3·7=21, and 21+4=25.
Progression
This open-ended exploration highlights the importance of order-of-operations rules for communicating mathematically and gives students an opportunity to explore order of operations on the graphing calculator. The activity is also an ideal time to establish the conventional order of operations.
Approximate Time
30 minutes
Classroom Organization
Groups
Doing the Activity
As you begin planning for this activity, keep in mind these three components, which work well together: this activity, the companion activity Uncertain Answers (in which students examine order of operations on the graphing calculator), and a brief lecture on order of operations. One sequence is to have students begin the exploration of 1-2-3-4 Puzzle, conduct the lecture, and assign Uncertain Answers for homework.
Many students enjoy this challenging puzzle. There are lots of ways to use 1, 2, 3, and 4 to generate each answer from 1 to 25. For example, 1+2+3+4=10 and 3+2·4-1=10. It isn’t necessary that all students find an expression for every number from 1 to 25; rather, just ask them to find as many 1-2-3-4 expressions as they can for 1 to 25. You might hang a poster in the room, with the numbers from 1 to 25 on it, and invite students to add new 1-2-3-4 expressions to it at any time.
Some key points to consider when orchestrating this activity:
- You might need to clarify the meaning of factorial and its position within the rules for order of operations. It has priority over the other operations, including exponentiation. For instance, 2·3! means 2·(3!) not (2·3)! Using the ^ notation for exponents, 2^3! is interpreted as 2^(3!) not (2^3)!.
- If any of the following graphing calculator basics were not discussed during Calculator Exploration, this activity may present opportunities to raise them. (It is not a requirement that all students know all these techniques at the conclusion of this activity.)
- Editing an expression that has been entered incorrectly (rather than starting over), including use of the Insert key
- Entering an exponent using the ^ key
- Using parentheses and recognizing that braces (the { and } keys) and brackets (the [ and ] keys) do not work like parentheses
- Copying the last entry
- Using the answer to the last calculation as part of a new calculation
- Some students may have learned the acronym PEMDAS as a way to remember the order of operations. Unfortunately, this memory device reinforces the common misconception that multiplication is performed before division, and addition before subtraction. Reiterate that within each pair of operations, the operations are performed from left to right. For instance, in the expression 12÷6·2, the division is performed first.
- There is nothing wrong with inserting parentheses that aren’t strictly required. People often do this in order to avoid any chance of confusion. For example, one might write the expression 5·3+5·7 as (5·3)+(5·7). Not only is the latter expression harder to misinterpret, it’s also easier to see the intent at a glance.
This is an easy activity to engage students in. Begin by simply asking someone to volunteer a numeric expression using each of the numbers 1, 2, 3, and 4 and any operations they would like. Record their suggestion, and ask the class to calculate the result. Here, or whenever the possibility for multiple interpretations of an expression arises, is a good time to begin discussion of order of operations and the use of parentheses.
Ask for two or three more expressions, again instructing the class to calculate the results, and then wonder aloud, Do you think we could create an expression for every number from 1 to 25?
In their groups, students can productively explore for 15 or more minutes very easily. At some stage—possibly after a break to introduce order of operations more formally and introduce the activity Uncertain Answers—gather the class to review the activity instructions. Reading the instructions will give students more ideas about operations they can use. Many students won’t have thought to use a square root, and few will be familiar with factorials.
Discussing and Debriefing the Activity
Students will be interested in the discussion of this activity in order to see expressions for numbers they haven’t figured out yet. Have volunteers share expressions for answers that other students haven’t found.
During the discussion, opportunities will arise to clarify order-of-operations rules. As they present themselves, ask the volunteer or the class to help rewrite the solution in the conventional form.
You might ask questions like the following to encourage volunteers to also talk about how they found their 1-2-3-4 expressions.
What methods did you use to find your expressions?
Did you proceed in numeric order or did you jump around?
Did you get an expression for one number by adjusting the expression for another?
Did you use any patterns that you saw in the expressions?
Order of Operations
Though many students have been exposed to order-of-operations rules, we treat the topic here as if some have not.
To introduce the topic, you might write arithmetic expressions involving several operations, such as those that follow, on the board, and ask students to work on their own to find the value of each expression.
| 4+5·3+1 | 3+4² |
| 10+2-4+3 | 2+4·3² |
| 2+3(5+4) | 12÷4-3 |
Some students may remember and apply the order-of-operations rules to get the correct answers, while others may never have learned the rules or may have forgotten them. Ask students to share their results for one or two of the expressions, and go through the details of their computations to demonstrate how the expressions can be interpreted in more than one way. Then point out that it would create great difficulties if more than one answer were correct. Mathematicians, scientists, and everyone who deals with numbers must communicate in writing, so there is a need for a set of rules that will govern how to interpret any apparent ambiguity in a problem.
Tell students that, by convention, arithmetic problems are worked out according to these rules:
- Simplify expressions within parentheses before combining them with expressions outside the parentheses.
- Within parentheses (or where no parentheses exist), do operations in this order:
- Apply exponents to their bases.
- Multiply and divide as the operations appear from left to right. (Neither operation has precedence over the other.)
- Add and subtract as the operations appear from left to right. (Neither operation has precedence over the other.)
Post the rules so that you and students can refer to them when needed. You might use a shortened version, such as the one that appears in Uncertain Answers.
Key Questions
Do you think we could create an expression for every number from 1 to 25?
What methods did you use to find these expressions?
Did you proceed in numeric order or did you jump around?
Did you get an expression for one number by adjusting the expression for another?
Did you use any patterns that you saw in the expressions?
Comments, questions, feedback, criticisms?
Send feedback
- E-mail the author of the module, 1-2-3-4 Puzzle
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This work is licensed by Interactive Mathematics Program under a Creative Commons Attribution License (CC-BY 2.0).
Last edited by Christine Osborne on May 28, 2008 1:40 pm GMT-5.