Multiplying a Fraction by a Whole Number

Grade 4, Standard 4.NF.B.4

In this lesson:

• See a fraction as counted copies of a smaller piece
• Multiply any fraction by a whole number
• Solve real problems with fractional amounts

Your Learning Objectives for Today

By the end, you can:

1. See a fraction as counted unit-fraction pieces
2. Multiply a fraction by a whole number
3. Model the multiplication with number lines and area models
4. Connect repeated addition to multiplication
5. Solve word problems, converting improper fractions to mixed numbers

Is a Fraction Hiding a Multiplication Fact?

• Recall:
• Recall: 4 × 3 means "4 groups of 3"
• Could secretly be a multiplication fact, too?

How Many Fourths Make Three Fourths?

3 copies of : — what's the shorter way to write "add the same number three times"?

Three Jumps Land on Three Fourths

3 jumps of :

What Happens Past One Whole?

5 jumps of :

Same Idea, Shown as an Area

• Each rectangle splits into 4 equal-size columns
• Shade one column at a time, counting copies of
• Four columns fill the first rectangle completely
• A fifth shaded column spills into the second rectangle:

Quick Check: Name the Copies

• , , and — how many copies of which unit fraction, for each?
• Which of these are improper? What mixed numbers do they equal?

Say each multiplication fact out loud before moving on

Your Turn: Work the Fraction Backwards

Given :

• Write it as a single fraction
• Mark its location on a number line marked in fifths
• Is your answer more or less than one whole?

Three Servings of Two Fifths

• You eat of a pizza, then eat that same amount two more times
• Using addition you already know:
• Is there a shorter way to write "add the same fraction three times"?

Multiplication Written as a Chain of Copies

Two Ways to See

• Number line: 3 jumps of land on
• Area model: 3 rectangles, 2 of 5 parts shaded in each
• Both show 6 total copies of

The Rule: Multiply the Count, Not the Size

• Multiply the numerator by the whole number
• The denominator stays the same
• Why: the size of each piece hasn't changed — only how many you have

Two Wrong Paths, Same Wrong Answer

• Path A: multiply top and bottom by 3 →
• Path B: write 3 as , then multiply straight across →
• Both are wrong. Which piece size changed that shouldn't have?

Two More Products to Convert

• Both products are improper fractions — convert to mixed numbers
• Notice the simplifying step at the end of each

Simplify the Product All the Way

• The product is improper — convert to
• simplifies further to
• Don't stop at the first mixed number if it can simplify more

Predict the Answer Before You Compute

• Is proper or improper?
• Make your prediction, then compute to check

Your Turn: Compute and Explain

• Compute , then simplify all the way
• Convert your answer to a mixed number if needed
• In one sentence, explain why the denominator didn't change

The Rule Meets the Real World

• You already know:
• Word problems hide this structure: "each ___ gets ___, there are ___ of them"
• Today's process: identify the groups, model, equation, compute, then check it makes sense

Setting Up: Water at Practice

5 students each drink liter:

From 15 Eighths to a Sensible Answer

• Between and — so the total is between 1 and 2 liters
• liters
• If your answer just says " liters," what's missing?

Second Problem: Sugar for Four Batches

• Each batch needs cup; Maria makes 4 batches
• The answer is between and cups

Third Problem: A Whole-Number Answer

• A runner completes mile per lap, for 4 laps
• This time, the product simplifies exactly to a whole number — no mixed number needed

What If There Were Twenty Friends?

• 4 friends each eat of a sandwich
• Adding works and is correct
• But what if there were 20 friends? Would you still add 20 times?

Write the Equation, Not the Answer

A ribbon is cut into pieces yard long. You need 6 pieces.

• Write the multiplication equation — don't compute yet
• What are the two numbers being multiplied, and what do they represent?

Your Turn: Solve It Start to Finish

A painter uses gallon of paint on each wall. She paints 3 walls.

• Identify the groups and group size
• Draw a model, write the equation, and compute
• Bound your answer between two whole numbers, then interpret it in a sentence

Four Mistakes to Watch For

Multiplying both the numerator and denominator by the whole number

Writing a whole number as instead of

Leaving an improper fraction as the final word-problem answer

Solving with repeated addition but missing the multiplication shortcut

A Fraction Is a Multiplication Fact in Disguise

✓ Any fraction is copies of the unit fraction

✓ Multiplying by a whole number multiplies the count of copies, not their size

✓ Word problems with equal fractional groups are this same multiplication, in disguise

Coming Up: Fraction Times Fraction

• So far, you've always multiplied by a whole number of copies
• Next, the number of copies is itself a fraction — repeated addition won't work anymore
• You'll need a new tool: multiplying a fraction by a fraction