The Main Idea 

Problem-Solving Strategy

1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the internet. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations. For example, with numerical values, if the problem says “Sara has [latex]12[/latex] apples and gives [latex]4[/latex] to her friend, how many apples does she have left?”, the numbers [latex]12[/latex] and [latex]4[/latex] are numerical values that you’ll need to work with. For units of measurement, if you’re given distances in both kilometers and miles, make sure to convert them to the same unit. Finally, keywords that indicate mathematical operations could be words like “divide” in a statement like “If you divide [latex]100[/latex] by [latex]5[/latex], how many groups will you have?”, where the word “divide” tells you that you need to use division.
2. Identify what you are looking for. Determine what the problem is asking you to find or solve. Look for phrases like “Find,” “Calculate,” or “Determine.” Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Break It Down. Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. It may be helpful to first restate the problem in one sentence before translating.
5. Solve the equation. Once you have a plan in mind, solve the problem step by step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.
6. Check the answer in the problem. Make sure it makes sense. Re-read the problem, check your calculations, and assess whether the answer aligns with the question’s requirements.
7. Answer the question with a complete sentence and correct units.

The Main Idea 

Types of Problems

1. Solve: Here, our task is typically to uncover the value of an unknown. Take for instance, the equation “[latex]2x + 3 = 7[/latex]“. Our mission is to find the value of x by isolating it on one side of the equation.
2. Simplify: When faced with a complex expression, like “[latex](3x^2)^2[/latex]“, our job is to tame it into a simpler form. In this case, our tamed form would be “[latex]9x^4[/latex]“.
3. Calculate: These types of problems are like a treasure hunt, where we are given a map and we need to find the treasure – a specific number. For example, “Calculate the area of a circle with a radius of [latex]3[/latex]“. We use the formula for the area of a circle ([latex]πr^2[/latex]) to find our treasure.
4. Graph: Graphing problems usually involve plotting a function or equation on a coordinate plane. For example, “Graph the function [latex]y = 2x - 1[/latex]“. We would find several values of [latex]y[/latex] for different [latex]x[/latex]-values and plot those points on the graph.
5. Multi-step: These problems are like a math buffet, offering a bit of everything. They require some planning and a good understanding of the order of operations. For example, if we’re asked to “Solve for [latex]x[/latex] in the equation [latex]2x + 3 = 7[/latex], then calculate the value of [latex]y[/latex] in the equation [latex]y = 3x - 2[/latex]“, we’d first solve for [latex]x[/latex] and then use that value to calculate [latex]y[/latex].

The Main Idea 

Problem-Solving Approaches

• Break It Down: Scary-looking problems often aren’t that bad — they’re usually just collections of easier problems. The first trick is to break-up big problems into smaller, more digestible parts.
  • For example, consider this problem: “A zookeeper sees [latex]50[/latex] heads and [latex]140[/latex] legs among the monkeys and peacocks in his zoo. How many monkeys and peacocks are there?” This might sound complicated, but let’s break it down:
    • • Every animal (monkey or peacock) has [latex]1[/latex] head. So, the [latex]50[/latex] heads mean we have [latex]50[/latex] animals.
      • Monkeys have [latex]2[/latex] legs, peacocks have [latex]4[/latex]. So if all [latex]50[/latex] animals were monkeys, we would have [latex]100[/latex] legs.
      • But we have [latex]140[/latex] legs, which is [latex]40[/latex] more than [latex]100[/latex]. Since each peacock has [latex]2[/latex] extra legs compared to a monkey, the [latex]40[/latex] extra legs mean we have [latex]20[/latex] peacocks ([latex]40 ÷ 2 = 20[/latex]).
      • So, since we have [latex]50[/latex] animals in total, the remaining [latex]30[/latex] must be monkeys.
• Try It Out: Some problems don’t seem to have a straightforward solution. In these cases, good old trial and error can be a lifesaver.
  • For example, if you’re asked “What’s the value of [latex]x[/latex] in the equation [latex]5^x = 625?[/latex]” you might think about complex logarithmic equations, but trying a few values for [latex]x[/latex] could give you the answer quicker.
    • • If [latex]x = 3[/latex], then [latex]5^x = 5^3 = 125[/latex]. Not enough.
      • If [latex]x = 4[/latex], then [latex]5^x = 5^4 = 625[/latex]. That’s it!
• Pattern Finding: Mathematics is full of patterns! Spotting these can make problem-solving super easy.
  • For instance, if you’re asked “What’s the [latex]6[/latex]th term in the sequence: [latex]3, 6, 12, 24,...[/latex]?” identifying a pattern can help solve it.
    • • Here, it seems each term is twice the one before it. So, the [latex]6[/latex]th term is [latex]24*2 = 48[/latex].
• Reason It Out: Using logical reasoning can be a potent problem-solving strategy. This involves forming a logical chain of thoughts to find a solution.
  • For instance, consider this problem: “If every triangle is a polygon, and every polygon has at least three sides, does every triangle have at least three sides?” You can use logic to figure this out.
    • • We can reason logically that since every triangle is a polygon, and every polygon has at least three sides, it follows that every triangle must also have at least three sides.