Mission 4 · Unit 2

Fraction Operations

6.NS.A.1 · Unit 2

Today's objective: Divide fractions by fractions and explain the result with a model.

Need a hint?

Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The school garden club is building a raised flower bed that is 3/4 of a meter wide. They need to fill 2/3 of the bed with sunflowers and divide the rest equally among 3 types of herbs. Your team must figure out how much space the sunflowers get (multiply fractions), how much is left for herbs, and how wide each herb strip is (divide a fraction by a whole number). Use area models to prove every answer.

Team Roles

Facilitator Reads the garden problem aloud, ensures the team tackles multiplication first and then division, and tracks the timer.

Model Builder Draws area models for the multiplication (2/3 x 3/4), labels rows and columns, shades the overlap, and sketches the division model.

Precision Checker Verifies that the area model has the correct number of rows and columns, checks that fractions are simplified, and confirms the herb strip width.

Reporter Prepares the defense: explains what 2/3 x 3/4 means in context, shows both models, and describes how division by 3 splits the leftover strip.

Investigation

The Problem

The garden bed is 3/4 meter wide. The sunflower section is 2/3 of the bed. The remaining space is split equally among 3 herb types.

Your tasks:

Multiply: 2/3 x 3/4 = ? (How wide is the sunflower section?)
Subtract: 3/4 - sunflower width = herb section width.
Divide: herb section width / 3 = width of each herb strip.
Draw an area model for the multiplication.
Draw a diagram showing the division of the herb section.

Step-by-Step Investigation Guide

Set up the area model for 2/3 x 3/4 Draw a rectangle. Divide it into 4 rows (for fourths) and 3 columns (for thirds). This gives you 12 equal cells.
Why do we use rows for one fraction and columns for the other?
Shade the overlap Shade the first 3 rows (that is 3/4 of the rectangle). Shade the first 2 columns (that is 2/3 of the rectangle). The overlap (dark area) shows 2/3 x 3/4.
How many cells are in the overlap? How many cells are there total? What fraction is that?
Multiply with the algorithm and simplify 2/3 x 3/4 = (2 x 3)/(3 x 4) = 6/12 = 1/2. The sunflower section is 1/2 meter wide. Does the area model match?
What shortcut could you use before multiplying? (Hint: cancel common factors.)
Find the herb section width Total bed width - sunflower width = herb width. So 3/4 - 1/2 = 3/4 - 2/4 = 1/4 meter for herbs.
How did you convert 1/2 to fourths? Why is a common denominator needed for subtraction?
Divide the herb section by 3 Each herb gets: 1/4 / 3 = 1/4 x 1/3 = 1/12 meter wide. Draw the herb strip divided into 3 equal parts and label each 1/12.
Why does dividing by 3 become multiplying by 1/3? Can you show this on a diagram?
Check: do all parts add up? Sunflower (1/2) + 3 herbs (3 x 1/12 = 3/12 = 1/4) = 1/2 + 1/4 = 2/4 + 1/4 = 3/4. That matches the total bed width!
Why is checking that parts add up to the whole a good way to verify your work?

Language Support

Key Vocabulary

Multiply fractions: Find a part of a part (2/3 of 3/4 means 2/3 x 3/4)

Area model: A rectangle divided into rows and columns to show multiplication

Simplify: Write a fraction in its smallest form (6/12 = 1/2)

Reciprocal: The fraction flipped upside down (reciprocal of 3 is 1/3)

Divide fractions: Multiply by the reciprocal (1/4 / 3 = 1/4 x 1/3)

Of: In math, "of" often means multiply (2/3 OF 3/4 = 2/3 x 3/4)

Sentence Frames

"2/3 of 3/4 means I need to find 2/3 x 3/4, which equals ___."
"In the area model, the overlap has ___ cells out of ___ total, so the answer is ___."
"To divide ___ by 3, I multiply by the reciprocal: ___ x 1/3 = ___."
"I know my answer is correct because all the parts add up to ___."

Multiple Representations

Area Model

Rectangle with rows and columns. The overlap shows the product of two fractions.

Equation

2/3 x 3/4 = 6/12 = 1/2. Show the algorithm and simplification.

Strip Diagram

Draw the bed as a strip. Mark the sunflower section, then divide the rest into 3 herbs.

Timed Lab Phases

Launch phase

Click a phase, then press Start.

03:00

Read the garden problem aloud. Assign roles. Underline the key numbers: 3/4 meter, 2/3 of the bed, 3 herb types. Discuss: what operations will you need?

What does "2/3 of the bed" mean? Is "of" telling us to add, subtract, multiply, or divide?

Before moving on: every team member knows the three calculations needed (multiply, subtract, divide).

Model Builder draws the area model for 2/3 x 3/4: rectangle with 4 rows and 3 columns. Shade the overlap. Count the shaded cells and write the fraction.

How many total cells? How many in the overlap? Does your count match the algorithm?

Before moving on: area model drawn, labeled, and the product written as a simplified fraction.

Subtract to find herb width (3/4 - 1/2). Divide herb width by 3 to find each strip. Draw the strip diagram. Precision Checker verifies: do all parts add to 3/4?

What is 1/4 divided by 3? Show why dividing by 3 is the same as multiplying by 1/3.

Before moving on: all three answers found, strip diagram complete, and sum-check verified.

Reporter prepares: explain what 2/3 x 3/4 means in the garden context, walk through the area model, show the strip diagram for division, and present the sum-check.

Why is the sunflower section (1/2 m) less than the total bed (3/4 m)? Does that make sense?

Before moving on: Reporter can present in under 60 seconds with both models shown.

Extension Challenge

The garden club decides to add a second bed that is 5/6 meter wide. They want 3/5 of it for tomatoes. How wide is the tomato section? Draw the area model (5 columns, 6 rows) and simplify. Then find each herb strip if the remaining space is split among 4 herb types.

"What If?" Variations

the bed was 1 full meter wide instead of 3/4? How would the sunflower width change?
they wanted 3/4 of the bed for sunflowers instead of 2/3? Would the herb strips get wider or narrower?
there were 4 herb types instead of 3? How would each strip width change?

Real-Life Connection

Carpenters multiply fractions when cutting wood. If a board is 3/4 foot wide and you need 2/3 of it, you calculate 2/3 x 3/4 to know exactly where to cut. Getting the fraction wrong wastes material.

Defense Preparation

Defense Questions

Show your area model. How many cells are in the overlap, and what fraction does that represent?
What does 2/3 x 3/4 mean in the garden story? Explain in words, not just numbers.
How did you find each herb strip width? Show the division step.
How did you verify that all parts add back to 3/4?
What shortcut could simplify 2/3 x 3/4 before multiplying? (Cross-canceling)

Sentence Starters for Your Defense

"The sunflower section is ___ meter wide because 2/3 of 3/4 equals ___."
"In the area model, the overlap has ___ out of ___ cells, which simplifies to ___."
"Each herb strip is ___ meter wide because we divided ___ by 3."
"We verified by adding: ___ + ___ = ___, which matches the total bed width."

What Makes a Strong Defense

Area ModelCorrect rows, columns, and overlap shaded

DivisionClear reciprocal step with diagram

VerificationParts add back to the whole

Exit Product

Submit a one-page lab note with:

Area model for 2/3 x 3/4 with cells labeled and overlap shaded
Multiplication algorithm with simplification (6/12 = 1/2)
Subtraction step: 3/4 - 1/2 = 1/4
Division: 1/4 / 3 = 1/12, with strip diagram
Verification: 1/2 + 3(1/12) = 1/2 + 1/4 = 3/4
Three-sentence defense: claim, model evidence, and verification check

Work Space

Area Model (2/3 x 3/4):

Multiplication and Simplification:

Subtraction (herb section):

Division (each herb strip):

Verification (parts add to whole):

Defense: