SECTION 1
Why Do We Need Scaling?
Have you ever doubled a recipe to make enough cookies for the whole class? That is scaling! People have been scaling things for thousands of years.
Long ago, builders needed to make plans for huge buildings. They drew tiny versions of those buildings on paper. Every measurement in the small drawing kept the same relationship to the real building. That idea is the heart of scaling.
3000 BC • Ancient Maps
500 BC • Greek Proportions
1500s • Scale Drawings
Today • Everyday Scaling
So here is the big question: how do you change every part of a problem by the same amount and keep everything fair? That is exactly what this lesson will teach you!
SECTION 2
Core Ideas of Scaling
Scaling means multiplying or dividing every number in a group by the same factor (a factor is just the number you multiply by). When you do this, the relationship between the numbers stays the same. Let's look at the main ideas.
Ratio
Scale Factor
Multiply BOTH Parts
Equivalent Ratios
KEY TAKEAWAY
SECTION 3
See How Scaling Works
Let's look at a picture that shows scaling in action. Imagine a recipe that uses 2 cups of flour and 3 cups of sugar. Watch what happens when we scale it up!
Look at the diagram above. When we multiply by 2, flour goes from 2 to 4 and sugar goes from 3 to 6. When we multiply by 3, flour goes from 2 to 6 and sugar goes from 3 to 9. Every time, both parts are multiplied by the same number. That is scaling!
SECTION 4
The Math Behind Scaling
Scaling uses a simple rule. You find the scale factor, then multiply both parts by it. Here are the key formulas.
FINDING THE SCALE FACTOR
If 3 grows to 12, the scale factor is 12 ÷ 3 = 4.
SCALING UP
Multiply by the scale factor to make both parts bigger.
SCALING DOWN
Divide by the scale factor to make both parts smaller.
ISEE Test Tip
SECTION 5
Scaling in Action — A Closer Look
Let's see how one ratio can be scaled to create many equivalent ratios. We will start with the ratio 3 to 5 and multiply by different scale factors.
| Scale Factor | First Number (starts at 3) | Second Number (starts at 5) |
|---|---|---|
| × 1 | 3 | 5 |
| × 2 | 6 | 10 |
| × 3 | 9 | 15 |
| × 4 | 12 | 20 |
| × 5 | 15 | 25 |
All rows show the same ratio (3 to 5) scaled up.
The diagram shows the two key steps. First, figure out what number the original was multiplied by. Then use that same number on the other part. It's like a matching game — whatever you do to one side, you must do to the other!
SECTION 6
Worked Example — Step by Step
Let's solve a full problem together, just like you would on the ISEE. Take it one step at a time!
A bakery sells 5 muffins for $10. At the same rate, how much would 20 muffins cost?
- 1
Step 1 — Write what you know
We know that 5 muffins cost $10. We want to find the cost of 20 muffins. - 2
Step 2 — Find the scale factor
Ask yourself: what do I multiply 5 by to get 20? Divide the new number by the original number: 20 ÷ 5 = 4. The scale factor is 4.Scale factor = 4 - 3
Step 3 — Multiply the other amount by the same factor
Now multiply the cost by the same scale factor: $10 × 4 = $40.Cost = $10 × 4 = $40 - 4
Step 4 — Check your answer
Does it make sense? 20 muffins is 4 times as many as 5 muffins, so the cost should be 4 times as much. $40 is 4 times $10. It checks out!20 muffins cost $40.
ISEE Strategy
SECTION 7
Scaling Strategies — What Works Best?
There are different ways to solve scaling problems. Let's compare them so you can pick the best approach on test day.
| Strategy | When to Use It | Example |
|---|---|---|
| Multiply Both Parts | When you can see a clear scale factor (one number divides evenly) | 3 to 6 is × 2, so 5 to ? is also × 2 → 10 |
| Find the Unit Rate | When you need the cost of 1 item first | 6 toys for $18 → 1 toy costs $3 → 4 toys cost $12 |
| Use a Table | When you need to scale in small steps | Build a table: 2→4→6→8, then read the matching value |
| Eliminate Wrong Answers | When the ISEE answer choices are spread apart | Estimate first, then cross out answers that are way too big or small |
BEST STRATEGY FOR THE ISEE
SECTION 8
From Scaling to Bigger Math Ideas
Scaling is the foundation for lots of math you will learn later. Here is a peek at how today's skill connects to future topics.
| What You Know Now | What Comes Next |
|---|---|
| Multiplying both parts of a ratio by the same number | In middle school, you will write proportions like 2/3 = 4/6 and solve for missing numbers using cross-multiplication. |
| Finding a unit rate (cost of 1 item) | Later, unit rates become slopes on graphs. A line that goes up 2 for every 1 across has a slope of 2. |
| Scaling a recipe up or down | Scientists scale chemical formulas to create the exact amount of a substance they need. |
| Reading a map with a scale | Architects use scale drawings every day to design buildings, bridges, and parks. |
Every time you scale on the ISEE, you are building skills for bigger math adventures. Keep practicing and you will be ready!
SECTION 9
Practice Problems
Time to practice! Try each problem on your own before looking at the answer. Remember: find the scale factor first, then use it on the missing number. Good luck — you've got this!
Question 1: CONCEPTUAL
- A. Add 2 to $6
- B. Multiply $6 by 2
- C. Divide $6 by 2
- D. Subtract 2 from $6
Show answer (B)
Question 2: BASIC CALCULATION
- A. 12
- B. 15
- C. 18
- D. 27
Show answer (C)
Question 3: INTERMEDIATE
- A. 8
- B. 12
- C. 14
- D. 16
Show answer (B)
Question 4: APPLIED
- A. 14 miles
- B. 17 miles
- C. 24 miles
- D. 28 miles
Show answer (D)
Question 5: CRITICAL THINKING
- A. 17
- B. 20
- C. 25
- D. 30
Show answer (C)
SUMMARY
Lesson Summary
Summary
Scaling means multiplying or dividing both parts of a ratio by the same number, called the scale factor. To find the scale factor, divide the new number by the original number. Then multiply the other part by that same factor to find the missing value. This keeps the ratio equivalent — fair and balanced.
On the ISEE, remember these tips: look for the number that changed, find the scale factor, apply it to the missing part, and always check if your answer makes sense. If you buy more items, the cost goes up. If you buy fewer, it goes down. Use estimation and process of elimination to cross out wrong answers. You are ready to scale like a pro!