Overview
This is a learning module for sixth grade students who are learning geometry. The learning module could also be used with students who are in 5th, 7th, or 8th grade. This learning module covers the following Common Core learning standards:
CCSS.MATH.CONTENT.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.
CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
CCSS.MATH.CONTENT.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
CCSS.MATH.CONTENT.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.
The prerequisite knowledge for this learning module includes:
- Knowing the names and properties of basic two-dimensional polygons
- Knowing how to find perimeter
This learning module is intended to be an online course, though it could be used to supplement a more traditional course as well. Therefore, students should have regular internet access at home. This course uses learning analytics to provide students with feedback on their progress throughout the course, and to calculate their final grades at the end of the learning module.
The students will also be regularly using Khan Academy as an assessment tool. Teachers can choose to set up a classroom on Khan Academy that automatically assigns and collects assessments, or students can simply email a screenshot of their scores to the teacher. The way the learning module is written assumes that teachers have set up a Khan Academy classroom and automatically have access to their students' progress.
Triangles and Rectangles
For the Student
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Learning Standard: CCSS.MATH.CONTENT.6.G.A.1 Goal: I can calculate the area of triangles and rectangles. |
Before beginning the learning module, let's identify your areas for growth with a pretest from Khan Academy. After you take the test, take a screenshot of your areas of growth. Here's an example:
You can click the topics you missed to learn more about them, though we will be covering these topics in this learning module as well.
This lesson goes with the "area of composite shapes" area of need from the pre-test. We need to learn how to find the area of individual shapes before we can start putting them together.
Now we're ready to begin learning! Start by watching this video lesson that will teach you what area is, how to write units for area problems, and how to find the area of rectangles and triangles.
Two of the key ideas shown in the video were the area formulas for rectangles and triangles. Here they are again:
Try this Shape Shoot game to practice finding the area of rectangles.
See if you can use your area and perimeter skills to design a zoo!
Now take on the computer or a friend in Area Blocks!
Practice finding the area of triangles here.
Can you win the trophy by finding the areas of all of these triangles?
Work for the week:
Assessment: Take the Khan Academy practice assessment on the area of triangles. You are welcome to review the lesson on Khan Academy before taking the assessment.
Comment: Which of your areas of growth sounds like it will be the easiest to master? Which sounds the most challenging? (If you had fewer than two areas of growth, you can choose from topics you answered correctly as well)
Update: A rectangle has an area of 48 square inches. What are the dimensions of this rectangle? Is it possible that there could be more than one answer -- if so, can you find others? If there are multiple answers, are the perimeters all the same?
Replies: Reply to three classmates. Check their math and coach them if you come across any errors.
For the Teacher
Purpose: In order for students to be able to solve the combined shapes problems, they will need to know how to find the area of triangles and rectangles. Many students come into 6th grade knowing how to find the area of rectangles, but it is less common for students to know how to find the area of triangles.
Extension: If students show that they are able to master this content, they can extend their learning by working on the Pythagorean theorem. From there, they can solve problems with a given perimeter and area and try to find the dimensions of a triangle that fits those parameters.
Common errors: Students struggle with the triangle formula. Many students will try to multiply all three side lengths and then divide it by 2. Instead, they need to identify the height of the triangle by looking for the right angle. The height of the triangle may or may not be a side length, rather it may be inside of the triangle. Once students learn the triangle formula, many will mistakenly divide the area of a rectangle by 2 as well.
Parallelograms and Trapezoids
For the Student
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Learning Standard: CCSS.MATH.CONTENT.6.G.A.1 Goal: I can find the area of parallelograms and trapezoids. |
This lesson goes with the "area of composite shapes" area for growth from the pre-test. We need to learn how to find the area of individual shapes before we can start putting them together.
Watch this video to learn what the formula for the area of a parallelogram is and why it works!
Did you know that two trapezoids make a parallelogram? Watch this video to see how that helps us find the area of a trapezoid.
You can also get the formula by turning a trapezoid into a triangle. Watch this video to see how:
Now you should know the area formulas for parallelograms and trapezoids. Here they are again:
Practice what you've learned about rectangles, parallelograms, and trapezoids by playing this Jeopardy! game. You can play with one team (yourself), or challenge a friend.
Try this activity to practice finding the area of triangles and trapezoids.
Work for the week:
Assessment: Take the Khan Academy practice assessments on the area of parallelograms and trapezoids. You are welcome to review the parallelogram or trapezoid lessons on Khan Academy before taking the assessment.
Comment: How do you know which length is the height on a shape?
Update: Find an example of a trapezoid and parallelogram in real life. This may be a bit more challenging than the triangles and rectangles from the last update, but you can do it! If you are able to measure the shapes and then find the areas, please do. If you are unable to measure your item, please create your own parallelogram andor trapezoid and find the area(s). Here are some examples of how you can get creative:
Replies: Reply to three of your classmates. Check their math and coach them if you come across any errors.
For the Teacher
Purpose: Similarly to the previous update, students spend time learning the area formulas for parallelograms and trapezoids so that they can use those shapes to find the area of combined shapes. By learning the derivations for the formulas, students also see how parallelograms and trapezoids are actually made up of triangles and rectangles.
Extension: Students can begin working with combined shapes here. If they have the formulas for parallelograms and trapezoids down, students can practice finding the areas of each by breaking them up into triangles and rectangles. They can show how this answer is the same as the answer from the formula. Students can also work with tangrams to create their own combined shapes, then find the sum of the areas.
Common errors: Just like with triangles, students have trouble identifying the height. Reinforce that the height of a shape must make a right angle with the base. Students may struggle with the order of operations for the trapezoid formula. They must add up the two bases first. It doesn't matter whether they divide by 2 or multiply by the height next, since multiplication is commutative. Some students may like to multiply by 1/2, while others may like to divide by 2, and these are also the same thing so it doesn't matter which they use.
Circles
For the Student
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Learning Standard: CCSS.MATH.CONTENT.7.G.B.4 Goal: I can find the area and the circumference of a circle. |
This lesson goes with the "circumference of a circle," "area of a circle," and "radius and diameter" areas for growth from the pre-test.
Let's start with some vocab. The radius of a circle is the distance from the center of a circle to its edge. The diameter of a circle is the distance from edge-to-edge through the center of a circle. The circumference is a special name for the perimeter of a circle. Though not shown in the image below, the area of a circle is the space inside of it, just like the area of the other shapes we've learned about.
When we find the area and circumference of a circle, we will need to use a special ratio called π (pi). This video introduces π and shows how it relates to circles.
Now watch this video to learn about the area and circumference formulas for circles.
Now that you've learned the area and circumference formulas, here is a fun way to remember them:
Practice what you've learned with this activity about circles.
Here's a Jeopardy! style game so you can challenge a friend or relative!
Try the problems in this file:
Work for the week:
Assessment: Take the Khan Academy practice assessment on radius and diameter, area, and circumference. You are welcome to review the lessons on Khan Academy on radius and diameter, area, and circumference, before taking the assessment.
Comment: Give one example of when you would need to know circumference in real life.
Update: Choose one of the problems from the packet linked above. Film a video lesson showing how to solve it. Also, include a written explanation.
Replies: Reply to three of your classmates. Coach your peers if necessary on the math, and give them feedback on the video lesson (volume, clarity, visuals, pace, etc).
For the Teacher
Purpose: This is the last area formula students will need to know in order to find the area of combined shapes. They will also need to know the circle formulas in order to find the surface area and volume of a cylinder and cone.
Extension: If students master circles early on, they can find the area given the circumference and vice versa. In order to solve that, students will need to know how to solve basic equations and find square roots. They can also do problems with half circles, quarter circles, and percentages of circles. For the perimeter half circles, remember that that we can't just take half of the circumference. We also need to add in the diameter that represents the length of the flat/cut-off part of the half circle. Similarly with quarter circles and percentages of circles, we need to add back in two radii (or a diameter).
Common errors: Order of operations is important in the area formula, as many students will multiply π by r, then square the result. Instead, it should be r squared, then multiplied by π. Some students may prefer to use the circumference formula 2πr, while others prefer 2d, and either is fine.
Combined Shapes
For the Student
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Learning Standard: CCSS.MATH.CONTENT.6.G.A.1 Goal: I can find the area of combined shapes. |
This lesson goes with the "area of composite shapes" area for growth from the pre-test. Now that we know how to find the areas of individual shapes, we can put them together into combined shapes. Composite shapes, combined shapes, complex shapes, compound shapes, and irregular shapes are all the same thing.
Watch this video to learn how to find the area and perimeter of combined shapes.
This video has a few more examples.
Practice breaking down combined shapes into more familiar figures in this activity.
This activity will allow you to practice with perimeter of combined shapes.
Try the problems in this file:
Work for the week:
Assessment: Take the Khan Academy practice assessment on the area of composite shapes and shaded areas. You are welcome to review the lesson on composite shapes on Khan Academy before taking the assessment.
Comment: Where do you see combined shapes out in the real world? What familiar shapes make up your example?
Update: Choose three of the real-world problems in the file below. Solve them, showing all work to clearly communicate your thinking, and share your work in the update. Be sure to include a screenshot or drawing of the original problem so we know what you're solving.
Replies: Reply to three of your classmates. Check their math and coach them if you come across any errors. Also, give them feedback on whether you feel they showed enough work to clearly communicate their thinking.
For the Teacher
Purpose: Combined shapes are seen throughout math and in real life. In math, learning about combined shapes allows students to solve complex geometry problems, and also lends itself to finding surface area using nets. In real life, combined shapes are seen in the floor plans of buildings and the layouts of other architectural designs.
Extension: Challenge students to find the answer using multiple strategies. For example, many problems can be solved by finding the sum of smaller figures, and also by finding the difference of a bigger "whole" and smaller "parts." Also, problems where it is less obvious how to decompose the shape into smaller parts are always more challenging.
Common errors: Students are likely to confuse formulas while they work with multiple shapes at once. For example, if the combined shape has both a rectangle and a triangle, some students will take half of both areas, instead of just taking half of the triangle. Keep in mind that there are often multiple ways to decompose a shape, so as long as the answer is correct and the work is logical, it is okay if students use different methods.
3D Shapes
For the Student
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Learning Standard: CCSS.MATH.CONTENT.6.G.A.4 Goal: I can name the different 3D shapes and their properties. |
This lesson goes with the "surface area" area for growth from the pre-test. We need to learn what 3D shapes are before we can find the surface area.
Begin by watching this video to preview the shapes you are about to learn about. You'll see some real-life examples of each of the shapes in the video.
Go to this website and answer the following questions on a piece of paper. You can either print these questions or simply copy down the answers on plain paper:
1. Define prism.
2. What is the difference between a “base” and a “lateral face”?
3. What shape are the lateral faces of prisms?
4. How is a prism described or named?
5. What would you call a prism with hexagon-shaped bases?
6. Play with the activity at the bottom of the website and fill in the table below.
7. Based on the activity, what does it seem a “net” is?
8. Based on the activity, create a definition for the following words.
- Face:
- Vertex:
- Edge:
9. Click on the link for pyramids. Define pyramid.
10. What shape are the lateral faces of pyramids?
12. How is a pyramid described or named?
13. What would you call a pyramid with a decagon-shaped base?
14. Play with the activity at the bottom of the website and fill in the table below.
This video will help you check over some of your answers, as well as introduce you to some new shapes.
Play this game to practice what you've learned!
Work for the week:
Assessment: Take the Khan Academy practice assessment on parts of 3D shapes and on 3D shapes. You are welcome to review the lesson on parts of 3D shapes and on 3D shapes on Khan Academy before taking the assessment.
Comment: Name a common household item that is one of the shapes mentioned in this update. What shape is it?
Update: Look online and find examples of buildings or structures with the following shapes: (1) prism, (2) pyramid, (3) cylinder, (4) cone, (5) sphere. Include a picture of each example and tell us how many faces, edges, and vertices each one has. For the prism and pyramid, specify what type of prism and what type of pyramid it is.
Replies: Reply to three of your classmates.
For the Teacher
Purpose: This lesson is all about building a common vocabulary. While students have been talking about rectangles, circles, and triangles since they were very young, the technical definitions of prisms and pyramids are probably new to them. Establishing common vocabulary helps make the rest of the learning module go smoothly.
Extension: Students could look at examples of combined 3D shapes and identify the shapes that compose it, as well as count the faces, edges, and vertices.
Another extension would be to come up with a formula, pattern, or rule for finding the faces, edges, and vertices of prisms and pyramids (separately) based on how many sides the base of the shape has.
| Prism | Pyramid | |
| Faces | n+2 | n+1 |
| Edges | 3n | 2n |
| Vertices | 2n | n+1 |
(n is the number of sides the base has)
Common errors: Students often forget to count what they can't see, such as faces and edges that are in the "back" of the image. Sometimes it helps if students highlight or color in the back face before they start counting.
Nets
For the Student
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Learning Standard: CCSS.MATH.CONTENT.6.G.A.4 Goal: I can draw and recognize nets of 3D figures. |
This lesson goes with the "surface area" area for growth from the pre-test.
Watch this video to learn what nets are and how they can be used to find surface area.
Nets are an "unfolded" representation of a 3D shape. Once you have the net, you can use what you know about combined shapes to find the total area, or surface area.
Print this document single-sided, cut out the nets, and fold them to make 3D solids. If you don't have access to a printer, you can also draw the nets on paper yourself
Here are some more examples of nets:
This image shows what several different nets look like:
Practice identifying nets with this activity.
See how quickly you can find all the nets that will make a cube in this game.
Work for the week:
Assessment: Take the Khan Academy practice assessment on nets and surface area using nets. You are welcome to review the lesson on nets and surface area using nets on Khan Academy before taking the assessment.
Comment: How are nets used to find surface area? How do nets relate to combined shapes?
Update: "Unfold" the net of a 3D solid in your house, such as a prism cereal box or cylindrical oatmeal box. Take a picture before and after the net has been unfolded. Use the net to find the surface area of your item. Post your pictures and math work.
Replies: Reply to three of your classmates. Check their math and coach them if you come across any errors.
For the Teacher
Purpose: Surface area is fairly difficult for some students to comprehend, so by breaking it down into a 2D net, students are able to comprehend it better and relate it to what they already know about combined shapes.
Extension: Students who understand nets can try creating their own cut outs. They can use a given surface area and create a net that would fit the parameter. Perhaps they can even find the net of different shapes.
Common errors: Again, students often leave out a side that they can't see. It may be helpful to have students first identify the number of faces the shape should have, and then draw the net. Many students also mislabel the sides of the nets based on the given measurements, so it can help to have students label the shapes in the nets with labels such as "top," "bottom," "left," "front," etc. Then, students can double check that the numbers match up with the side.
Surface Area
For the Student
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Learning Standard: CCSS.MATH.CONTENT.7.G.B.6 Goal: I can find the surface area of 3D shapes. |
This lesson goes with the "surface area" area for growth from the pre-test.
Go through this interactive lesson to learn about surface area.
This manipulative lets you see how the net changes as you change the dimensions of the 3D shape.
This video goes into detail about finding surface area using nets. This is one method for finding surface area.
It's possible to find surface area without drawing the net first. This video shows you how to do that.
This video shows how to do surface area word problems.
Try out some more word problems with surface area with this activity.
Try these surface area problems using nets:
Work for the week:
Assessment: Take the Khan Academy practice assessment on the surface area. You are welcome to review the lesson on Khan Academy before taking the assessment.
Comment: Which method do you prefer: finding surface area by drawing a net, or finding surface area by considering the sides? Why?
Update: Choose one of the problems from this activity and create a video lesson teaching how to solve it.
Replies: Reply to three of your classmates. Coach your peers if necessary on the math, and give them feedback on the video lesson (volume, clarity, visuals, pace, etc).
For the Teacher
Purpose: Surface area has a lot of relevance to real-life situations. For example, surface area helps us figure out how much paint we need to paint a room, how much wrapping paper we need to wrap a gift, how big of a box we need to package something, etc.
Extension: Students who are able to find the surface area of single shapes can be challenged by finding the surface area of combined shapes. They can also create their own 3D shapes (either with a model or on paper) and find the surface area of those.
This document has some very challenging problems with combined shapes.
Common errors: At the risk of sounding like a broken record, students often forget to include sides they can't see. It really helps when students label their scratch work "front," "back," "left," "right," "top," "bottom," or whatever labels are appropriate for the figure.
With rectangular prisms, the front/back, left/right, and top/bottom all have the same area. This is not the case for every triangular prism. While the triangular bases will have the same area, the rectangular lateral faces may be the same, may be all different, or may have only two that are the same. It's important that the students don't make assumptions and actually look at the measurements.
Volume
For the Student
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Learning Standard: CCSS.MATH.CONTENT.7.G.B.6 Goal: I can find the volume of 3D shapes. |
The first pretest we took didn't cover volume, so begin by identifying your areas for growth by taking pretest 1 and pretest 2.
Learn about what volume is and how to find it by watching this video.
This video shows you how to tackle a more complex volume word problem.
Try some volume word problems with this activity.
Use this manipulative to see how volume and surface area change as the dimensions of the shape change.
Play this Jeopardy! game with both volume and surface area.
Work for the week:
Assessment: Take the Khan Academy quiz on volume of prisms and volume of other shapes. You are welcome to review the lesson on prisms and other shapes Khan Academy before taking the assessment.
Comment: What is the difference between volume and surfaced area?
Update: Try all of the problems below. Choose one problem to create an update about, describing how to solve it in words. In your update, include where you think students will make errors, and how to avoid those errors.
Replies: Reply to three of your classmates. Check their math and coach them if you come across any errors. Also, give them feedback on whether you feel they showed enough work to clearly communicate their thinking.
For the Teacher
Purpose: Volume is also relevant in real-life situations. Any time we need to find the capacity of something, or pack an item with other items, we will need to find the volume. Volume is often used in science for measurements and experiments.
Extension: Just like with surface area, a great extension is to find the volume of combined shapes. The document below has some very challenging problems.
Another extension option is for students to try to find a shape with the greatest volume given a particular surface area. Students could also make their own shapes and find the volume of those.
One last idea is having students try some volume problems using the Pythagorean theorem. In the problem below, students would need to find the height of the triangle in order to find the height of the pyramid.
Common errors: By the time students get here, they usually find volume to be a piece of cake. It is significantly easier than surface area and lends itself to fewer mistakes. Students do struggle with units and tend to mix up cubic units (3D volume) with square units (2D surface area). The most common mistake would be students who use the slant height instead of the altitude. Emphasize that the height of the 3D shape must make a right angle with its base, just like with 2D shapes.
Peer-Reviewed Project
For the Student
You will be designing four mini golf holes. There will be four components to this project: combined area, surface area, volume, and presentation.
Note: After reading the instructions, if you'd like to change the project to something other than mini golf, you may do so as long as you still meet the requirements. Please message the instructor for approval before you begin. Other options include designing a playground, sports arena, house, etc.
Combined Area:
The bird’s-eye layout of each of your mini golf holes must be a combined area shape. You must use at least two rectangles, two triangles, two circles, one parallelogram, and one trapezoid. After those are used, you can use any additional shapes you’d like to use. You will include calculations showing how to find the total area of each of the mini golf holes. The calculations can look like traditional math "numbers" work, or can be something different, such as a video or written explanation.
Surface Area and Volume:
There must be at least one structure on each mini golf hole. You must use at least one prism, one pyramid, and one cylinder. After those are used, you can use any additional shapes you’d like to use. You will include calculations showing how to find the surface area and volume of each of the structures. As with the combined area, the calculations can look like traditional math "numbers" work, or can be something different, such as a video or written explanation.
Presentation:
You must include drawings/images of each of your mini golf holes and structures. You can draw them by hand or you can use pictures from Google images. The drawings/images should include labeled measurements, even if the picture is not to scale.
For the Teacher
This project can be distributed around week 6 of the learning module.
Information Survey
For the Student
Please take this survey.
For the Teacher
This survey can be used at the beginning of the learning module and again at the end. You can collect data on what topics the students are stronger and weaker in, based on self-assessment. You can also collect data on how the students prefer to learn online. Comparing the beginning and end results of the survey will give feedback on what updates were most effective.
Once you have the data, you can modify the updates so they are more tailored toward common mistakes and problem areas. If many students struggled with the same topic, it could be a sign that the update on that topic needs to be heavily revised. If many students feel they are successful with the same topic, perhaps that updates could be modified to include more challenging problems to extend the students' learning.