Lesson 2.01:

Processing Skills and Problem Solving

To become a confident and capable problem solver, you need to work on more than just finding the right answer. Great problem solvers think on multiple levels—they develop their own strategies, reflect on their thinking, and stay honest about what they know and don’t know.

But even the best strategies won’t help much if your processing skills—like accuracy, speed, and confidence—aren’t strong. These skills give you the freedom to explore and experiment, especially when working under time pressure. While they aren’t the only key to success, they can make a big difference—and using them incorrectly can hold you back.

Problem Solving is not math, but many problems do have numbers in them. This lesson will focus on these core skills. You’ve likely seen them before, but they’re easy to overlook or misapply. This is your chance to sharpen your skills, spot any weak spots, and build the confidence you need to tackle problems with clarity and control.

Proportional Reasoning

Proportion

A proportion is an equation that shows two ratios are equal.

Proportional Reasoning

Proportional reasoning is the ability to compare two quantities using ratios and to solve problems involving equivalent ratios

Why is it important?

Cooking (e.g., doubling a recipe)
Maps and scale drawings
Converting units (e.g., inches to centimeters)
Shopping (e.g., comparing prices)

Example 1: Simple Ratio

Problem:
If 3 apples cost $6, how much do 5 apples cost?

Solution:
Set up a proportion:

Cross-multiply:

Answer: 5 apples cost $10.

Example 2: Real-World Application

Problem:
A map uses a scale of 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, how far apart are they in real life?

Solution:
Set up a proportion:

Cross-multiply:

Answer: The cities are 175 miles apart.

Compound Units

Compound unit

A compound unit is a unit made by combining two or more different units. These are often used to describe rates or ratios.

Examples

Speed: kilometers per hour (km/h), meters per second (m/s)
Density: grams per cubic centimeter (g/cm³)
Fuel efficiency: miles per gallon (mpg)
Unit cost: dollars per item ($/item)

Why are they useful?

Compound units help us describe how one quantity changes in relation to another. For example:

Speed tells us how far something travels in a certain amount of time.
Unit cost tells us how much one item costs.

Example 1: Speed

Problem: A car travels 180 kilometers in 3 hours. What is its speed?

Solution:

Example 2: Unit Cost

Problem: A pack of 6 pens costs $9. What is the cost per pen?

Solution:

Example 3: Density

Problem: A block of metal has a mass of 240 grams and a colume of 60 cm cubed. What is the density?

Solution:

Finding Averages

What is average?

An average, or mean, is a way to describe a set of numbers with a single value that represents the "typical" number in the group

Why are averages important?

Averages help us:

Summarize data
Compare performance
Make decisions (e.g., budgeting, grading, sports stats)

Example 1: Test scores

Problem:
A student scores 82, 76, 91, and 87 on 4 tests. What is their average score?

Solution:

Example 2: Missing value

Problem:
The average of five numbers is 72. Four of the numbers are: 68, 75, 70, and 74. What is the missing fifth number?

Solution

Parsing Relationships Algebraically

Imagine you're given this statement:

"For every book Emma reads, Liam reads twice as many, and together they read 36 books."

Can you figure out how many books each person read? This is where parsing relationships algebraically comes in.

In real-world problem solving, relationships are rarely simple. Consider this:

"A company’s profit is directly proportional to the square of its advertising budget, but only after a fixed threshold is met. Below that threshold, the profit remains constant."

This is no longer a simple linear equation—it’s a piecewise function with conditional logic. Let’s learn how to model it.

Example 1: Conditional Relationship

Statement:
A tutor charges $30 per hour for the first 5 hours in a week, and $25 for each additional hour.
Let:
h = total hours tutored in a week
C(h) = total cost

Then:

Example 2: System of Equation

Statement:
Three friends—Ava, Ben, and Carlos—share a sum of money. Ava has twice as much as Ben. Carlos has $10 more than Ava. Together, they have $160
Let:
a = Ava’s amount
b = Ben’s amount
c = Carlos’s amount

Then:

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