Unlocking Math Word Problems with the CUBE Strategy
Math word problems often feel like decoding a secret language. Students frequently struggle not with the arithmetic itself, but with understanding what the problem is asking, extracting the relevant information, and knowing which operations to use. This challenge can lead to frustration, errors, and a general dislike for math.
Fortunately, there's a powerful and systematic approach designed to simplify this process: the CUBE strategy. CUBE provides a clear framework, guiding you through each critical step of analyzing a word problem, ensuring you don't miss essential details or get sidetracked by irrelevant information. By breaking down complex problems into manageable parts, CUBE helps build confidence and significantly improves accuracy.
What is the CUBE Strategy?
CUBE is an acronym that stands for:
- Circle key numbers
- Underline the question
- Box mathematical keywords
- Evaluate and eliminate unnecessary information
While often presented as CUBE, many educators also extend it to CUBES, adding an 'S' for Solve and Check. We'll cover both aspects to give you a complete toolkit. This strategy is applicable across various grade levels and types of math problems, from basic arithmetic to algebra and beyond.
A Step-by-Step Guide to CUBE
Let's delve into each component of the CUBE strategy with practical advice and examples.
C: Circle Key Numbers
The first step is to identify and circle all the numerical values presented in the word problem. These numbers are the raw data you'll be working with. Don't just look for digits; sometimes numbers are written out as words (e.g., "four," "half a dozen").
Why it's important: Circling numbers helps you visually isolate the quantitative information. It prevents you from overlooking a crucial piece of data and prepares you to consider how these numbers relate to each other.
Example: "Sarah has 12 apples. She gives 3 apples to her friend, Tom, and buys 5 more apples at the store. How many apples does Sarah have now?"
- Circle: 12, 3, 5
U: Underline the Question
Next, read the problem carefully and underline the specific question it's asking you to solve. This is arguably the most critical step, as it defines your goal.
Why it's important: Underlining the question ensures you understand precisely what you need to find. Many errors occur because students solve for an intermediate step or a related quantity, rather than the actual question posed. It also helps you determine the final unit or type of answer required.
Example: "Sarah has 12 apples. She gives 3 apples to her friend, Tom, and buys 5 more apples at the store. How many apples does Sarah have now?"
- Underline: How many apples does Sarah have now?
B: Box Mathematical Keywords
This step involves identifying words or phrases that indicate specific mathematical operations. These keywords are your clues to determine whether you need to add, subtract, multiply, divide, or perform other operations.
Why it's important: Boxing keywords helps translate the verbal problem into a mathematical expression. It's like finding the verbs in a sentence that tell you what action to perform with the numbers.
Common Keywords and Their Operations:
- Addition (+): sum, total, in all, altogether, combine, increased by, more than, plus, added to, both, perimeter
- Subtraction (-): difference, how many more, how many less, less than, fewer than, take away, decreased by, remaining, left, minus, change
- Multiplication (x): product, times, by, each, every, groups of, area, volume, twice, double, triple, per (sometimes, depending on context)
- Division (÷): quotient, shared equally, divided by, per (often), split, each (when distributing), average, ratio
- Equality (=): is, are, was, will be, yields, results in
Example: "Sarah has 12 apples. She gives 3 apples to her friend, Tom, and buys 5 more apples at the store. How many apples does Sarah have now?"
- Box: gives (implies subtraction), buys 5 more (implies addition)
E: Evaluate and Eliminate Unnecessary Information
Word problems often include details that are interesting but irrelevant to solving the actual problem. Your task in this step is to critically evaluate all the information presented and eliminate anything that doesn't help you answer the underlined question.
Why it's important: Extraneous information can be distracting and lead to confusion or incorrect calculations. By identifying and crossing out these details, you simplify the problem and focus solely on what's necessary. This step also involves thinking about what operations you've identified and if you have enough information to proceed.
Example: "Sarah has 12 apples. She gives 3 apples to her friend, Tom, and buys 5 more apples at the store. Tom's favorite color is blue. How many apples does Sarah have now?"
- Evaluate/Eliminate: "Tom's favorite color is blue." (This information is irrelevant to the number of apples Sarah has.)
The "S" in CUBES: Solve and Check
While CUBE focuses on dissecting the problem, the "S" for Solve and Check completes the strategy, ensuring you arrive at the correct answer and verify your work.
S: Solve and Check Your Work
Once you've applied CUBE, you should have a clear understanding of the numbers, the question, the operations, and the relevant information. Now, it's time to perform the calculations and then review your solution.
- Formulate the equation(s): Based on your boxed keywords and circled numbers, write down the mathematical expression(s) needed to solve the problem.
- Solve: Perform the calculations carefully.
- Check:
Does your answer make sense? Is it reasonable in the context of the problem? For instance, if you're calculating the number of apples, a negative or fractional answer might indicate an error. Did you answer the question? Refer back to your underlined question. Recalculate: Briefly go through the steps again to catch any arithmetic mistakes. Use inverse operations: If you added, try subtracting your answer from the total to see if you get the original number.
Example (continuing Sarah's apples): "Sarah has 12 apples. She gives 3 apples to her friend, Tom, and buys 5 more apples at the store. How many apples does Sarah have now?"
- C: 12, 3, 5
- U: How many apples does Sarah have now?
- B: gives (subtract), buys 5 more (add)
- E: No unnecessary info in this simplified version.
- S (Solve):
Start with 12 apples. She gives 3 away: 12 - 3 = 9 apples. She buys 5 more: 9 + 5 = 14 apples. Answer: Sarah has 14 apples now.
- S (Check):
Does 14 apples make sense? Yes, she started with 12, gave some away, and got more, so a number close to 12 or slightly higher is reasonable. Did I answer the question "How many apples does Sarah have now?" Yes. * Recalculate: (12 - 3) + 5 = 9 + 5 = 14. Correct.
Applying CUBE: Walkthrough Examples
Let's walk through a couple more complex problems using the full CUBES strategy.
Example 1: Multi-Step Problem
"A baker made 240 cupcakes for a school fair. He sold 180 cupcakes in the morning. In the afternoon, he sold half of the remaining cupcakes. How many cupcakes does the baker have left?"
- C: Circle Key Numbers
240 (total cupcakes) 180 (sold in morning) * half (implies division by 2)
- U: Underline the Question
* How many cupcakes does the baker have left?
- B: Box Mathematical Keywords
made (initial quantity) sold (implies subtraction) remaining (implies subtraction result) sold half of the remaining (implies division by 2) * left (implies subtraction result, final quantity)
- E: Evaluate and Eliminate Unnecessary Information
* "for a school fair" - context, but not needed for calculation.
- S: Solve and Check
1. Remaining after morning sales: 240 (made) - 180 (sold) = 60 cupcakes. 2. Sold in the afternoon: "half of the remaining" means 60 ÷ 2 = 30 cupcakes. 3. Cupcakes left: 60 (remaining after morning) - 30 (sold in afternoon) = 30 cupcakes. * Answer: The baker has 30 cupcakes left.
- S (Check):
Does 30 make sense? Yes, he started with 240 and sold many, so 30 is a reasonable final amount. Did I answer the question? Yes. * Recalculate: (240 - 180) - ((240 - 180) / 2) = 60 - (60 / 2) = 60 - 30 = 30. Correct.
Example 2: Comparing Quantities
"Mark has three times as many toy cars as David. If David has 15 toy cars, and their friend Emily has 10 fewer cars than Mark, how many toy cars does Emily have?"
- C: Circle Key Numbers
three times (implies multiplication by 3) 15 (David's cars) * 10 (fewer cars)
- U: Underline the Question
* how many toy cars does Emily have?
- B: Box Mathematical Keywords
three times as many (multiplication) If (conditional, sets up known fact) * fewer cars than (subtraction)
- E: Evaluate and Eliminate Unnecessary Information
* No extraneous information in this problem.
- S: Solve and Check
1. Find Mark's cars: Mark has "three times as many toy cars as David." David has 15. So, Mark has 15 x 3 = 45 toy cars. 2. Find Emily's cars: Emily has "10 fewer cars than Mark." Mark has 45. So, Emily has 45 - 10 = 35 toy cars. * Answer: Emily has 35 toy cars.
- S (Check):
Does 35 make sense? Yes, Mark has many, and Emily has fewer than Mark, so 35 is reasonable. Did I answer the question? Yes. Recalculate: (15 3) - 10 = 45 - 10 = 35. Correct.
Tips for Mastering the CUBE Strategy
- Practice Regularly: The more you use CUBE, the more intuitive it becomes. Start with simpler problems and gradually move to more complex ones.
- Use Visuals: Don't be afraid to draw diagrams, charts, or models to represent the problem. Visualizing the information can often clarify relationships between numbers and quantities.
- Break Down Complexity: For very long or intimidating problems, apply CUBE to each sentence or clause if necessary. Sometimes a problem is a series of smaller problems.
- Don't Rush: Take your time with each step. Rushing often leads to overlooking critical details.
- Work Backwards: Sometimes, especially with multi-step problems, it helps to identify the final question and then think about what information you would need to answer that question, working backward to the given data.
The Benefits of Using CUBE
Implementing the CUBE strategy offers several significant advantages for students and anyone tackling math word problems:
- Improved Comprehension: CUBE forces you to read the problem actively and thoroughly, enhancing your understanding of what's being asked.
- **