In this Warm-up, students review how to find the area of a triangle given a pair of base-height measurements. The reasoning here prepares students to use division of fractions to solve area problems involving triangles later in the lesson.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time, followed by 1 minute of partner discussion. Before students begin, review the formula for the area of a triangle. Consider displaying a drawing of a triangle with one side labeled as a base and a corresponding height shown and labeled as such.
Student Lesson in Spanish
Activity
None
Find the area of Triangle A in square centimeters.
Show your reasoning.
A triangle labeled A with a vertical side. One vertex is to the left of the vertical side. A dashed horizontal line is drawn from the first vertex to the vertical side of the triangle and a right angle symbol is indicated. The dashed line and the vertical side are both labeled 4 and one half centimeters.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite a student to share a solution and reasoning. Record it for all to see. Ask if others used alternative ways of reasoning, and invite them to share their approaches (as many as time permits).
If any student wrote the fraction as 4.5 before performing any operations, consider discussing how the calculations are alike and how they are different.
Tell students that they will solve more problems involving the area of triangles in this lesson.
14.2
Activity
Standards Alignment
Building On
Addressing
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
In this activity, students apply their knowledge of division of fractions to find a unknown length given the area of a triangle and a fractional base or height.
The formula for the area of a triangle presents a different multiplication situation than students have seen in this unit—there are three factors at play. This means that to find a unknown factor, at least one additional step is needed. Students are likely to approach this in a number of ways. They may:
Draw a duplicate of the triangle and compose a parallelogram with the same base and height, double the given area (to represent the area of the parallelogram), and then divide by the known length to find the unknown length.
Without drawing, multiply the given area by 2 to find the value of , and then divide it by the known length.
Perform division twice, such as divide the area by and then by the known base or length, or vice versa.
Multiply the known length and first, so there are only two factors to work with. (For example, to find in , they may write and then .)
Monitor for these or other approaches as students work. Select students who use different strategies, and ask them to share later.
Launch
Keep students in groups of 2. Give students 3–4 minutes of quiet work time and 2 minutes to discuss their responses and complete the activity with their partner. Keep the formula for the area of a triangle and a labeled drawing of a triangle displayed.
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for finding the unknown base or height in each triangle before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language
Activity
None
The area of Triangle B is 8 square units. Find the length of . Show your reasoning.
A triangle labeled B has a horizontal side on the bottom of the triangle and a vertex above the horizontal side. A dashed line is drawn from the vertex to the horizontal sideand a right angle symbol is indicated. The horizontal side is labeled lowercase b and the dashed line is labeled eight thirds.
The area of Triangle C is square units.
What is the length of ? Show your reasoning.
A triangle labeled C has a horizontal side at the top of the triangle and a vertex below the horizontal side and to the left. A horizontal line extends from the horizontal side and to the left. A vertical dashed line is drawn from the bottom vertex to the extended horizontal line and a right angle symbol is indicated. The dashed line is labeled h and the horizontal side of the triangle is labeled 3 and three fifths.
Activity Synthesis
Ask 2–3 previously selected students to share their solutions and reasoning. Consider starting with students who reasoned concretely (such as by duplicating the triangle to compose a parallelogram) and following with those who reasoned symbolically (such as only by manipulating expressions or equations).
Clarify that there is no single way to solve problems such as these. Point out specific points in the solving process where division of fractions enabled them to complete their reasoning. Emphasize that what we learned about fractions and operations in this unit can help us reason more effectively about problems in other areas of mathematics.
MLR8 Discussion Supports. Invite students to repeat their reasoning using mathematical language: "Can you say that again, using words such as ‘compose,’ ‘parallelogram,’ ‘base,’ ‘height,’ and ‘side length’?” Provide all students with an opportunity to produce this language by inviting students to chorally repeat these words or phrases 1–2 times. Advances: Speaking, Conversing
14.3
Activity
Standards Alignment
Building On
Addressing
6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas and to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
In this activity, students find the volume of a prism with fractional edge lengths. In an earlier grade, students learned that the volume of a rectangular prism with whole-number edge lengths can be found by computing the number of unit cubes that can be packed into the prism. Here, the cubes have a unit fraction (such as or ) for their edge lengths. Students calculate the number of such cubes that can be packed into a rectangular prism and use that number to find the volume in a standard unit of volume measurement (cubic inches, in this case).
By reasoning repeatedly with small cubes (with -inch edge lengths), students notice regularity (MP8): that the volume of a rectangular prism with fractional edge lengths can also be found by directly multiplying the edge lengths in inches.
This activity works best when each student has access to physical cubes because students will benefit from manipulating the cubes and building prisms by hand.
If physical manipulatives are not available, consider using the digital version of the activity. In the digital version, students use an applet to build rectangular prisms from cubes with edge lengths of unit. The applet allows students to measure the edge lengths of the resulting prisms.
The digital version is adapted from an applet made in GeoGebra by Susan Addington.
Activity Synthesis
The purpose of the discussion is to highlight that the volume of a rectangular prism can be found by:
Packing it with cubes with no gaps and multiplying the number of cubes by the volume of each cube.
Multiplying the edge lengths of the prism.
Display a completed table for all to see. Give students a minute to check their responses. Invite a few students to share how they determined the volume of the prisms.
If no students mention using the number and volume of the -inch cubes, ask them to observe the relationship between the values in the last two columns of the table (the number of cubes with -inch edge lengths and the volume of prisms in cubic inches). Highlight that the volume can be found by calculating the number of cubes and multiplying it by , because each small cube has a volume of cubic inch.
If no students mention finding the product of the edge lengths in inches, draw students’ attention to the values in the first three columns of the table and in the last column. (Looking at the whole-number values in the second, fourth, and sixth rows of the table may be particularly helpful.) Make sure students see that the volume of each prism can also be found by multiplying its side lengths in inches.
Lesson Synthesis
To summarize ways to find the volume of a rectangular prism, display and read the following pair of problems:
A rectangular prism measures 1 inch by 2 inches by 3 inches. What is its volume?
A rectangular prism measures inch by 1 inch by inches. What is its volume?
Give students a minute to consider how to solve each problem, then ask questions such as:
“How are two problems different?” (The prisms are different sizes. One prism has edge lengths that are whole numbers. The other has edge lengths that are fractions.)
“Can we find the volume of both prisms by seeing how many cubes can be packed into them?” (Yes)
“Can we use 1-inch cubes for both prisms?” (No, we’ll need to use -inch cubes for the second prism.)
“Is there another way to find the volume of each prism?” (Yes, we can multiply the edge lengths.)
If time allows, consider summarizing how division can be used to find an unknown base or height of a triangle. Display and read the following pair of problems:
A rectangle has an area of 9 square inches and a height of inches. How long is its base?
A triangle has an area of 9 square inches and a height of inches. How long is its base?
Give students a minute to consider how they can solve each problem, then ask questions such as:
“How would you find the base of the rectangle?” (Divide 9 by ) “Why does it make sense to divide?” (Because )
“Would dividing 9 by also give the base of the triangle? Why or why not?” (No, because the area of the triangle is .)
“How can we find the base of the triangle?” (Divide 9 by and then divide the result by , or multiply it by 2.)
Student Lesson Summary
If a rectangular prism has edge lengths of 2 units, 3 units, and 5 units, we can think of it as 2 layers of unit cubes, with each layer having unit cubes in it. So the volume, in cubic units, is:
To find the volume of a rectangular prism with fractional edge lengths, we can think of it as being built of cubes that have a unit fraction for their edge length. For instance, if we build a prism that is -inch tall, -inch wide, and 4 inches long using cubes with a -inch edge length, we would have:
A height of 1 cube, because .
A width of 3 cubes, because .
A length of 8 cubes, because .
The volume of the prism would be , which is 24 cubic units.
How do we find its volume in cubic inches? We know that each cube with a -inch edge length has a volume of cubic inch, because . Since the prism is built using 24 of these cubes, its volume, in cubic inches, would then be , which is 3 cubic inches.
The volume of the prism, in cubic inches, can also be found by multiplying the fractional edge lengths in inches:
Standards Alignment
Building On
Addressing
6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that because of is . (In general, .) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Some students may be unsure how to find the unknown length in a triangle because finding the area of a triangle involves two operations. Encourage students to write an equation that shows the relationship between the unknown base, the known height, and the known area of the triangle. Ask students if there is any calculation they can perform first so that they are working with only one operation instead of two.
Launch
Display the image of the 1-inch cube for all to see. Ask students:
“This cube has an edge length of 1 inch. What is its volume in cubic inches?” (1 cubic inch.) “How do you know?” ()
“How do we find the volume of a cube with an edge length of 2 inches?” (. We can pack the cube with eight 1-inch cubes, so its volume is , or 8, cubic inches.)
If no students mentioned using the 1-inch cube to find the volume of the 2-inch cube, bring it up. Consider telling students that we can call a cube with an edge length of 1 inch a “1-inch cube.”
Arrange students in groups of 3–4. Give each group 20 cubes and 2 minutes to complete the first set of questions. Ask them to pause for a brief class discussion afterward.
Invite students to share how they found the volume of a cube with -inch edge lengths and the prism composed of 4 stacked cubes. For the -inch cube, if students do not mention one of the two ways shown in the Student Response, bring it to the students’ attention. For the tower, if they don’t mention multiplying the volume of a -inch cube, which is cubic inch, ask if that is a possible way to find the volume of the prism.
Next, give students 8–10 minutes to complete the rest of the activity.
MLR2 Collect and Display. Circulate among the groups to listen for and collect the language that students use to explain how they reason about the volume of a -inch cube and other rectangular prisms. On a visible display, record words and phrases, such as “length,” “width,” “height,” “cubes packed into the prism,” “cubic inches,” “multiply the number of cubes by the volume of 1 cube,” and “multiply the edge lengths.“ Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading
Activity
None
Your teacher will give you cubes that have edge lengths of inch.
Here is a drawing of a cube with edge lengths of 1 inch.
How many cubes with edge lengths of inch are needed to fill this cube?
What is the volume, in cubic inches, of a cube with edge lengths of inch? Explain or show your reasoning.
Four cubes are piled in a single stack to make a prism. Each cube has an edge length of inch. Sketch the prism, and find its volume in cubic inches.
Use cubes with an edge length of inch to build prisms with the lengths, widths, and heights shown in the table.
For each prism, record in the table how many -inch cubes can be packed into the prism and the volume of the prism.
