Squares and Cubes
In this lesson, students learn about perfect squares and perfect cubes. They see that these names come from the areas of squares and the volumes of cubes with whole-number side lengths. Students find unknown side lengths of a square given the area or unknown edge lengths of a cube given the volume. To do this, they make use of the structure in expressions for area and volume (MP7).
Students also use exponents of 2 and 3 and see that in this geometric context, exponents help to efficiently express multiplication of the side lengths of squares and cubes. Students learn that expressions with exponents of 2 and 3 are called squares and cubes, and see the geometric motivation for this terminology. (The term “exponent” is deliberately not defined more generally at this time. Students will work with exponents in more depth in a later unit.)
In working with length, area, and volume throughout the lesson, students must attend to units. In order to write the formula for the volume of a cube, students look for and express regularity in repeated reasoning (MP8).
Note: Students will need to bring in a personal collection of 10–50 small objects ahead of time for the first lesson of the next unit. Examples include rocks, seashells, trading cards, or coins.
- Generalize a process for finding the volume of a cube, and justify (orally) why this can be abstracted as $s^3$.
- Include appropriate units (orally and in writing) when reporting lengths, areas, and volumes, e.g. cm, cm2, cm3.
- Interpret and write expressions with exponents 2 and 3 to represent the area of a square or the volume of a cube.
Let’s investigate perfect squares and perfect cubes.
Prepare sets of 32 snap cubes for each group of 2 students.
- I can write and explain the formula for the volume of a cube, including the meaning of the exponent.
- When I know the edge length of a cube, I can find the volume and express it using appropriate units.
We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).
In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).
We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).
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