How are 34 , 0.75, and 75 all connected?
The answer you get when you multiply two or more numbers together.
Having the same value, even when written in different forms.
A number that uses a decimal point to show a value between whole numbers — tenths, hundredths, thousandths.
The value of a digit based on its position in a number. Moving left makes a digit 10× bigger.
Look at these three math problems:
Think About It
These three problems look totally different... but they're secretly connected by ONE powerful pattern. By the end of today, you'll be able to solve all three — and explain why the answers are related.
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Skill 1
Find a fraction of a whole number
34 of 12 = 9
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Skill 2
Connect fractions to decimals
0.75 × 12 = 9
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Skill 3
Use place value to go bigger
75 × 12 = 900
Same digits. Same pattern. The decimal point changes everything.
When you see the word "of" in a math problem with fractions, it means multiply.
The Strategy
Step 1: Divide the whole number into equal groups (use the denominator).
Step 2: Take the number of groups you need (use the numerator).
Let's split 12 into 4 equal groups, then take 3 of them.
34 of 12 = 9
Split 8 into 4 equal groups, then take 1 of them.
14 of 8 = 2
To find a fraction of a whole number:
1. Divide the whole number by the denominator (bottom number) to find one group.
2. Multiply that result by the numerator (top number) to find how many groups you need.
Example: 34 of 12 → 12 ÷ 4 = 3, then 3 × 3 = 9
What is 14 of 20?
Think: 20 ÷ 4 = _____, then take 1 group.
👍 Thumbs up when you have your answer!
20 ÷ 4 = 5, so 14 of 20 = 5
Every fraction has an equivalent decimal form. These two pairs are the stars of today's lesson:
1 ÷ 4 = 0.25
3 ÷ 4 = 0.75
If 34 of 12 = 9, then 0.75 × 12 must ALSO equal 9 — because they're the same value!
A fraction is just division. Watch: 3 ÷ 4 = ?
100-grid: 75 out of 100 squares filled = 0.75
34 = 75100 = 0.75 — Three different ways to write the same number!
We already know 34 of 12 = 9. Does 0.75 × 12 give us the same answer?
The Rule
If 34 of a number = some answer, then 0.75 × that number = the same answer. Always!
Fractions and decimals are equivalent:
14 = 0.25 34 = 0.75
So: 34 of 12 = 9 and 0.75 × 12 = 9 → SAME ANSWER
"Of" means multiply. The fraction and the decimal always give the same product.
If 14 of 8 = 2, what is 0.25 × 8?
Think: Are 14 and 0.25 equivalent?
👍 Thumbs up when you know!
0.25 × 8 = 2 — Same answer! Because 14 = 0.25
Look at these two numbers side by side:
0.75
75 hundredths
75
75 ones
What happened?
The digits are the same (7 and 5), but they moved two place value positions to the left. That's the same as multiplying by 100!
When you multiply a decimal by 100, the decimal point moves two places right:
So if you already know what 0.75 × 12 equals...
75 is 100 times bigger than 0.75, so the product is also 100 times bigger!
Instead of doing long multiplication, use the pattern!
You just solved 75 × 12 without long multiplication! The fraction gave you the answer — place value just scaled it up.
The ×100 Rule:
0.75 × 100 = 75 and 0.25 × 100 = 25
So: if 0.75 × 12 = 9, then 75 × 12 = 900 (× 100)
And: if 0.25 × 8 = 2, then 25 × 8 = 200 (× 100)
Multiply the decimal product by 100 to get the whole-number product.
You already know: 0.25 × 8 = 2
So what is 25 × 8?
👍 Thumbs up when you know!
25 × 8 = 2 × 100 = 200
We just learned 3 connected skills. In your own words, explain to your partner: Why does 75 × 12 give an answer that's 100 times bigger than 0.75 × 12?
Sentence starter: "75 × 12 is 100 times bigger because ___"
Fraction of a number:
Decimal multiplication:
Place value jump (× 100):
Fraction of a number:
Decimal multiplication:
Place value jump (× 100):
What We Just Learned
Three connected skills: fraction of a number → decimal multiplication → place value scaling (×100).
Up Next: We Do Together!
Now it's YOUR turn to help me solve some problems. I'll guide you through each step — you tell me the answers. Ready?
34 of 20 = ?
Think: 20 ÷ 4 = ___, then × 3
20 ÷ 4 = 5, then 5 × 3 = 15
0.75 × 20 = ?
Same answer as the fraction!
0.75 × 20 = 15
75 × 20 = ?
× 100 from the decimal answer!
15 × 100 = 1,500
14 of 32 = ?
Think: 32 ÷ 4 = ___
32 ÷ 4 = 8
0.25 × 32 = ?
0.25 × 32 = 8
25 × 32 = ?
8 × 100 = 800
Look at what we've solved so far. What pattern do you notice about the decimal answer and the whole-number answer?
0.75 × 20 = 15 → 75 × 20 = 1,500
0.25 × 32 = 8 → 25 × 32 = 800
Sentence starter: "I notice that the whole-number answer is always ___ times bigger because ___"
34 of 40 = ?
40 ÷ 4 = ___, then × 3
40 ÷ 4 = 10, then 10 × 3 = 30
0.75 × 40 = ?
0.75 × 40 = 30
75 × 40 = ?
30 × 100 = 3,000
14 of 50 = ?
Think: 50 ÷ 4 = ___
50 ÷ 4 = 12.5 (that's okay!)
0.25 × 50 = ?
0.25 × 50 = 12.5
25 × 50 = ?
12.5 × 100 = 1,250
A school has 28 students in 5th grade. 34 of the students brought their own lunch today. How many students brought lunch?
34 of 28 = 21 students
Check: 0.75 × 28 = 21 ✓
Place value bonus: 75 × 28 = 21 × 100 = 2,100
What We Practiced Together
We solved 4 full chain problems as a class: 34 of 20, 14 of 32, 34 of 40, and 14 of 50.
Now: You Try on Your Own!
Complete each chain independently. Use the reveal buttons to check your work after you've tried each step.
Step 1: Divide by denominator, multiply by numerator
Step 2: 14 = 0.25 and 34 = 0.75 → same answer as step 1
Step 3: Multiply the decimal answer by 100 → that's the whole-number product
34 of 24 = _____
24 ÷ 4 = 6, then 6 × 3 = 18
0.75 × 24 = _____
0.75 × 24 = 18
75 × 24 = _____
18 × 100 = 1,800
14 of 36 = _____
36 ÷ 4 = 9
0.25 × 36 = _____
0.25 × 36 = 9
25 × 36 = _____
9 × 100 = 900
34 of 52 = _____
52 ÷ 4 = 13, then 13 × 3 = 39
0.75 × 52 = _____
0.75 × 52 = 39
75 × 52 = _____
39 × 100 = 3,900
A farmer has 84 apple trees. 34 of the trees produced fruit this year.
84 ÷ 4 = 21, then 21 × 3 = 63 trees
0.75 × 84 = 63 ✓
75 × 84 = 63 × 100 = 6,300
This time, I'll give you the whole-number product. Can you work backwards to find the fraction?
If 75 × 16 = 1,200...
What is 0.75 × 16? What is 34 of 16?
1,200 ÷ 100 = 12
So 0.75 × 16 = 12 and 34 of 16 = 12
If 25 × 44 = 1,100...
What is 0.25 × 44? What is 14 of 44?
1,100 ÷ 100 = 11
So 0.25 × 44 = 11 and 14 of 44 = 11
Today's three skills are really one connected idea:
Fractions, decimals, and whole numbers are all connected through place value. When you understand one, you can figure out all three!
In the bottom of your notebook page, write one sentence explaining what you learned today about how fractions, decimals, and place value are connected when you multiply.
14 of 36 is _____, so 25 × 36 = _____
14 of 36 = 9, so 25 × 36 = 9 × 100 = 900
34 of 84 is _____, so 75 × 84 = _____
84 ÷ 4 = 21, then 21 × 3 = 63, so 75 × 84 = 63 × 100 = 6,300