Mathematics · Grade 7 · Surface Area and Volume · Term 3

Density and Mass

Understanding the concept of density and using it to solve problems involving mass and volume.

About This Topic

Density measures mass per unit volume, given by the formula density equals mass divided by volume. In Grade 7 mathematics, students in the Spatial Sense strand apply this concept within the Surface Area and Volume unit to solve problems. They calculate densities for regular and irregular objects, rearrange the formula to find missing values, and use proportional reasoning to compare densities.

This topic connects mathematics to physics and engineering. Students predict if objects float or sink by comparing their density to water's density of 1 gram per cubic centimetre. Real-world examples include why ships float, how submarines dive, and material selection in construction. These applications strengthen problem-solving skills and introduce unit conversions, such as cubic centimetres to millilitres.

Active learning suits density best because students handle concrete materials to measure mass with balances and volume by displacement or formulas. Experiments reveal patterns that formulas alone cannot convey, while group predictions and tests build collaboration and correct misconceptions through evidence.

Key Questions

1. Explain the relationship between density, mass, and volume.
2. Predict whether an object will float or sink based on its density.
3. Analyze how density is used in various scientific and engineering applications.

Learning Objectives

• Calculate the density of regular and irregular objects using provided mass and volume measurements.
• Rearrange the density formula to determine the mass or volume of an object when two of the three variables are known.
• Compare the densities of different substances to predict whether they will float or sink in water.
• Analyze real-world scenarios to explain how density influences the design of objects and structures.

Before You Start

Measuring Mass and Volume

Why: Students need to be able to accurately measure the mass of objects using a balance and the volume of regular solids using formulas, and the volume of liquids.

Introduction to Formulas and Variables

Why: Students should have prior experience substituting values into simple formulas and understanding the concept of variables representing unknown quantities.

Key Vocabulary

Density	A measure of how much mass is contained in a given volume. It is calculated by dividing mass by volume.
Mass	The amount of matter in an object. It is typically measured in grams (g) or kilograms (kg).
Volume	The amount of space an object occupies. It is measured in cubic centimetres (cm³) or millilitres (mL).
Displacement	The volume of liquid a submerged object pushes aside. This method is used to find the volume of irregular objects.

Watch Out for These Misconceptions

Common Misconception

Density is the same as mass or weight.

What to Teach Instead

Density accounts for volume, so a large light object can have lower density than a small heavy one. Hands-on measuring activities let students compare objects of similar mass but different volumes, revealing the formula's role through direct calculation and observation.

Common Misconception

Objects float if lighter than water, regardless of volume.

What to Teach Instead

Buoyancy depends on average density compared to water. Float-or-sink tests in small groups help students see that shape and air pockets affect overall density, correcting the idea through evidence from trials.

Common Misconception

All metals sink because they are heavy.

What to Teach Instead

Steel ships float due to lower average density from hollow structure. Building simple models in pairs demonstrates how volume impacts density, shifting focus from mass alone.

Active Learning Ideas

Pairs: Density Calculation Lab

Partners select classroom objects, measure mass using a balance and volume by water displacement or ruler. They calculate density with the formula and record in a shared table. Discuss which objects might float based on results.

30 min·Pairs

Small Groups: Float or Sink Challenge

Groups predict if five objects will float or sink, then test in a water tub and measure densities to verify. Chart predictions versus outcomes and explain discrepancies using the density formula. Share findings with the class.

45 min·Small Groups

Whole Class: Density Rainbow Column

Prepare layers of liquids with different densities like oil, water, syrup. Class adds objects to observe positions and calculates average densities. Discuss how this models ocean layers or fuel tanks.

35 min·Whole Class

Individual: Engineering Design Brief

Students design a floating raft from recyclables, calculate its density, and test load capacity. Adjust design based on trials and document mass, volume, density changes.

50 min·Individual

Real-World Connections

• Naval architects use density calculations to ensure that large ships, made of heavy materials, can float by displacing a volume of water whose mass is greater than the ship's mass.
• Engineers select materials for construction based on their density. For example, lighter, less dense materials might be chosen for high-rise buildings to reduce structural load, while denser materials might be used for foundations.
• Submarine designers manipulate the density of the submarine by taking in or expelling water to control buoyancy, allowing it to dive or surface.

Assessment Ideas

Quick Check

Provide students with three objects (e.g., a small rock, a piece of wood, a metal bolt) and their masses. Have students measure the volume of each object using water displacement. Ask them to calculate the density of each object and record it on a worksheet.

Exit Ticket

On an exit ticket, ask students to write the formula for density. Then, present a scenario: 'An object has a mass of 50g and a volume of 25cm³. Will it float or sink in water? Explain your reasoning.'

Discussion Prompt

Pose the question: 'Imagine you have two balls of the exact same size, one filled with feathers and one filled with sand. Which ball is denser and why? How does this relate to the mass and volume of each ball?' Facilitate a class discussion to solidify understanding.

Frequently Asked Questions

How do you explain the density formula to Grade 7 students?

Present density as mass squeezed into a volume, using the formula density = mass / volume. Start with familiar examples like comparing sugar cubes to cotton balls of same volume but different masses. Practice rearranging for mass or volume with word problems tied to Ontario curriculum expectations, ensuring students master proportional relationships.

What real-world applications of density should Grade 7 students explore?

Discuss ships and icebergs floating despite high material density due to overall lower average density. Include submarines adjusting ballast for density changes, hot air balloons rising on low-density air, and recycling sorting by density. These tie math to science and engineering, aligning with curriculum connections to everyday problem-solving.

How can students predict if an object floats or sinks?

Compare the object's density to water's 1 g/cm³; less than 1 floats, more sinks. Measure or research density, considering average for composite objects. Classroom tests reinforce this, building confidence in predictions through data collection and analysis.

Why use active learning for teaching density and mass?

Active approaches like measuring real objects and conducting buoyancy tests make the abstract formula concrete. Students discover relationships through trial and error, collaborate on predictions, and revise ideas based on evidence. This boosts retention, addresses misconceptions immediately, and aligns with inquiry-based Ontario math expectations for deeper understanding.