Scaling and Area
10 min
Teacher Prep
Setup
Required Preparation
Prepare to distribute the pattern blocks, at least 18 blue rhombuses, 18 green triangles, and 10 red trapezoids per group of 3–4 students. For the Are You Ready for More? prepare to distribute 7 yellow hexagons per group.
If there are not enough pattern blocks for each group to have a full set, consider rotating the blocks of each color through the groups. Using real pattern blocks is preferred, but the Digital Activity can be used if physical manipulatives are unavailable.
For the digital version of the activity, acquire devices that can run the applet.
Narrative
By now, students understand that lengths in a scaled copy are related to the original lengths by the scale factor. Here they see that the area of a scaled copy is related to the original area by the square of the scale factor.
Students build scaled copies of a single pattern block, using blocks of the same shape to do so. They determine how many blocks are needed to create a copy at each specified scale factor. Each pattern block serves as an informal unit of area. Because each original shape has an area of 1 block, the (scale factor)2 pattern for the area of a scaled copy is easier to recognize.
Students use the same set of scale factors to build copies of three different shapes (a rhombus, a triangle, and a hexagon). They notice regularity in their repeated reasoning and use their observations to predict the number of blocks needed to build other scaled copies (MP8).
If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use several applets to build scaled copies of a single pattern block. The applets organize the placement of the blocks as they are added. The digital version may help students build the shapes quickly and accurately so they can focus more on the number of pattern blocks it takes to make the scaled copy rather than on how the blocks are arranged.
Launch
Arrange students in groups of 3–4. Distribute pattern blocks and ask students to use them to build scaled copies of each shape as described in the task. Each group would need at most 16 blocks each of the green triangle, the blue rhombus, and the red trapezoid. If there are not enough for each group to have a full set with 16 each of the green, blue, and red blocks, consider rotating the blocks of each color through the groups, or having students start with 10 blocks of each and ask for more as needed.
Give students 6–7 minutes to collaborate on the task and follow with a whole-class discussion. Make sure all students understand that “twice as long” means “2 times as long."
Student Task
Your teacher will give you some pattern blocks. Work with your group to build the scaled copies described in each question.
-
How many blue rhombus blocks does it take to build a scaled copy of Figure A:
-
Where each side is twice as long?
-
Where each side is 3 times as long?
-
Where each side is 4 times as long?
-
-
How many green triangle blocks does it take to build a scaled copy of Figure B:
-
Where each side is twice as long?
-
Where each side is 3 times as long?
-
Using a scale factor of 4?
-
-
How many red trapezoid blocks does it take to build a scaled copy of Figure C:
-
Using a scale factor of 2?
-
Using a scale factor of 3?
-
Using a scale factor of 4?
-
Sample Response
-
- Three scaled copies of an equilateral triangle. First labeled 4 blocks. Second labeled 9 blocks. Third labeled 16 blocks.
- Three scaled copies of a red trapezoid. First figure labeled 4 blocks. Second figure labeled 9 blocks. Third figure labeled 16 blocks.
Activity Synthesis (Teacher Notes)
Display a table with only the column headings filled in. For the first four rows, ask different students to share how many blocks it took them to build each shape and record their answers in the table.
| scale factor | number of blocks to build Figure A | number of blocks to build Figure B | number of blocks to build Figure C |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 10 | |||
| s | |||
| 21 |
To help students notice, extend, and generalize the pattern in the table, guide a discussion using questions such as these:
- “In the table, how is the number of blocks related to the scale factor? Is there a pattern?” (The number of blocks is equal to the scale factor times itself.)
- “How many blocks are needed to build scaled copies using scale factors of 5 or 10? How do you know?” (25; 100; 5⋅5=25, 10⋅10=100)
- “How many blocks are needed to build a scaled copy using any scale factor s?” (s2)
- “If we want a scaled copy where each side is half as long, how much of a block would it take? How do you know? Does the same rule still apply?” (41 because 21⋅21=41)
If not brought up by students, highlight the fact that the number of blocks it took to build each scaled shape equals the scale factor times itself, regardless of the shape (look at the table row for s). This rule applies to any factor, including those that are less than 1.
Advances: Listening, Speaking
Anticipated Misconceptions
Some students may come up with one of these arrangements for the first question, because they assume the answer will take 2 blocks to build:
You could use one pattern block to demonstrate measuring the lengths of the sides of their shape, to show them which side they have not doubled.
Students may also come up with:
for tripling the trapezoid, because they triple the height of the scaled copy but they do not triple the length. You could use the process described above to show that not all side lengths have tripled.
Standards
- 7.G.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.6·Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
- 7.G.A.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.B.6·Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
- 7.RP.2.a·Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
- 7.RP.A.2.a·Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Scaling and Area
10 min
Teacher Prep
Setup
Required Preparation
Prepare to distribute the pattern blocks, at least 18 blue rhombuses, 18 green triangles, and 10 red trapezoids per group of 3–4 students. For the Are You Ready for More? prepare to distribute 7 yellow hexagons per group.
If there are not enough pattern blocks for each group to have a full set, consider rotating the blocks of each color through the groups. Using real pattern blocks is preferred, but the Digital Activity can be used if physical manipulatives are unavailable.
For the digital version of the activity, acquire devices that can run the applet.
Narrative
By now, students understand that lengths in a scaled copy are related to the original lengths by the scale factor. Here they see that the area of a scaled copy is related to the original area by the square of the scale factor.
Students build scaled copies of a single pattern block, using blocks of the same shape to do so. They determine how many blocks are needed to create a copy at each specified scale factor. Each pattern block serves as an informal unit of area. Because each original shape has an area of 1 block, the (scale factor)2 pattern for the area of a scaled copy is easier to recognize.
Students use the same set of scale factors to build copies of three different shapes (a rhombus, a triangle, and a hexagon). They notice regularity in their repeated reasoning and use their observations to predict the number of blocks needed to build other scaled copies (MP8).
If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use several applets to build scaled copies of a single pattern block. The applets organize the placement of the blocks as they are added. The digital version may help students build the shapes quickly and accurately so they can focus more on the number of pattern blocks it takes to make the scaled copy rather than on how the blocks are arranged.
Launch
Arrange students in groups of 3–4. Distribute pattern blocks and ask students to use them to build scaled copies of each shape as described in the task. Each group would need at most 16 blocks each of the green triangle, the blue rhombus, and the red trapezoid. If there are not enough for each group to have a full set with 16 each of the green, blue, and red blocks, consider rotating the blocks of each color through the groups, or having students start with 10 blocks of each and ask for more as needed.
Give students 6–7 minutes to collaborate on the task and follow with a whole-class discussion. Make sure all students understand that “twice as long” means “2 times as long."
Student Task
Your teacher will give you some pattern blocks. Work with your group to build the scaled copies described in each question.
-
How many blue rhombus blocks does it take to build a scaled copy of Figure A:
-
Where each side is twice as long?
-
Where each side is 3 times as long?
-
Where each side is 4 times as long?
-
-
How many green triangle blocks does it take to build a scaled copy of Figure B:
-
Where each side is twice as long?
-
Where each side is 3 times as long?
-
Using a scale factor of 4?
-
-
How many red trapezoid blocks does it take to build a scaled copy of Figure C:
-
Using a scale factor of 2?
-
Using a scale factor of 3?
-
Using a scale factor of 4?
-
Sample Response
-
- Three scaled copies of an equilateral triangle. First labeled 4 blocks. Second labeled 9 blocks. Third labeled 16 blocks.
- Three scaled copies of a red trapezoid. First figure labeled 4 blocks. Second figure labeled 9 blocks. Third figure labeled 16 blocks.
Activity Synthesis (Teacher Notes)
Display a table with only the column headings filled in. For the first four rows, ask different students to share how many blocks it took them to build each shape and record their answers in the table.
| scale factor | number of blocks to build Figure A | number of blocks to build Figure B | number of blocks to build Figure C |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 10 | |||
| s | |||
| 21 |
To help students notice, extend, and generalize the pattern in the table, guide a discussion using questions such as these:
- “In the table, how is the number of blocks related to the scale factor? Is there a pattern?” (The number of blocks is equal to the scale factor times itself.)
- “How many blocks are needed to build scaled copies using scale factors of 5 or 10? How do you know?” (25; 100; 5⋅5=25, 10⋅10=100)
- “How many blocks are needed to build a scaled copy using any scale factor s?” (s2)
- “If we want a scaled copy where each side is half as long, how much of a block would it take? How do you know? Does the same rule still apply?” (41 because 21⋅21=41)
If not brought up by students, highlight the fact that the number of blocks it took to build each scaled shape equals the scale factor times itself, regardless of the shape (look at the table row for s). This rule applies to any factor, including those that are less than 1.
Advances: Listening, Speaking
Anticipated Misconceptions
Some students may come up with one of these arrangements for the first question, because they assume the answer will take 2 blocks to build:
You could use one pattern block to demonstrate measuring the lengths of the sides of their shape, to show them which side they have not doubled.
Students may also come up with:
for tripling the trapezoid, because they triple the height of the scaled copy but they do not triple the length. You could use the process described above to show that not all side lengths have tripled.
Standards
- 7.G.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.6·Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
- 7.G.A.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.B.6·Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
- 7.RP.2.a·Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
- 7.RP.A.2.a·Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.