Altersgruppe: Grade 6 Virginia - Mathematics Standards of Learning (2009): 3.14, 5.8b,

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1 1 U n t er r ich t splan Nets of Rectangular Prisms Altersgruppe: Grade 6 Virginia - Mathematics Standards of Learning (2009): 3.14, 5.8b, 5.8c, 6.10d Virginia - Mathematics Standards of Learning (2016): 5.8.a, 5.8.b, 5.9.b Fairfax County Public Schools Program of Studies: 3.14.a.3, 3.14.a.4, 5.8.b.1, 5.8.b.2, 5.8.b.3, 5.8.c.3, 6.10.d.1 Online-Ressourcen: Cast yo ur N e t Wi de, P ush t he E nvel o pe Opening Teacher present s Students pract ice Class discussion Teacher present s Students pract ice Closing M at h Obj ect ives E xpe ri e nc e nets as a 2-dimensional representation of rectangular prisms P rac t i c e constructing a rectangular prism from a net, and vice versa Learn to connect nets (2-dimensions) with rectangular prisms (3-dimensions)

2 2 De vel o p strategies for correctly coloring the faces of a net or solid Ope ni ng 8 Prepare for the opening with two or more identical objects that are rectangular prisms, that can be unfolded into nets. The example that follows will use cereal boxes. Show a cereal box in its 3-dimensional prism form. What is the name of this shape? While many students will likely say box, encourage them to find another, more mathematical name. Ideally, they reach the conclusion of right rectangular prism, though it may require some guidance. Say: Each side of the cereal box has unique features to it. Some sides have pictures, one side has nutritional information, and other sides are mostly blank. There are other features too. For example, the front and the back faces do not share any edges. All of these ideas will be important. Show a cereal box in its 2-dimensional net form. What does it look like if we cut along some of the edges to flatten out the rectangular prism--the box--so that we can see all of the unique faces? Your students may say it looks like a bunch of rectangles. While this is true, dive a little deeper into why. Namely, each face of the cereal box is a rectangle (2- dimensional), so flattening the box should not change that.

3 3 If you have another cereal box, consider showing a different version of how that net could be created. A generic example of two nets for the same rectangular prism is shown below. Flip one of the nets over so that only the brown cardboard is showing. Write on a couple of the faces to denote which face of the box they correspond to, such as front and nutritional information. Note that your choice of faces may require you to include directional information like up to avoid the prospect of multiple correct solutions (since you will eventually flip the net back over to see if the faces are correctly labeled). One such example is using front and back. Use the box to help detere how the remaining faces should be labeled. Invite several students to offer up suggestions, while also explaining why. Encourage arguments in support or against the choice of labels. Write down the consensus answers, even if you know they will be incorrect. Looking back and discussing why the faces were incorrectly labeled can be just as powerful as getting them right in the first place.

4 4 T e ac he r prese nt s M at h game : Cast yo ur N e t Wi de - Cubo i ds 8 Present Matific s episode C a s t y o ur N e t Wide - C ubo ids to the class, using the projector. The examples shown in this section result from the episode being in Preset mode. The goal of this episode is to connect the 2-dimensional net with the 3- dimensional rectangular prism. Each screen requires your students to select the correct colors for each face of the net, so that they align with the prism shown. The first screen uses a cube, as shown below. The cube on the right is rotating, but clicking on the cube will pause the rotation. Additionally, clicking and dragging will rotate the shape in any desirable direction. Below are two rotations of the same cube from the above screen. The net can be folded back into the prism (and vice versa), as shown below. This allows your students to compare the coloring of the net with the intended coloring of the prism as they progress.

5 5 For some nets, there may be more than one correct coloring. For the cube above, the red faces are opposite each other, meaning any two faces of the net that end up opposite each other could be red (provided red is colored first). The light and dark blue have the same property, but once red is selected, there will be a fixed relationship to which faces must be light or dark blue. Continue on to one or two more examples, as you see fit. Otherwise, you may opt to proceed more quickly to the practice portion, where your students can explore strategies individually. St ude nt s prac t i c e M at h game : Cast yo ur N e t Wi de - Cubo i ds 10 Have the students play C a s t y o ur N e t Wide - C ubo ids on their personal devices. Encourage your students to spend some time exploring the relationship between the nets and the prisms by folding and unfolding the net. As they progress through the screens, circulate and ask about what strategies they are using to find the correct coloring for each net.

6 6 Cl ass di sc ussi o n 5 What was challenging about coloring the nets? The relationship between 2-dimensional objects and 3- dimensional objects is often challenging at first, so it is important to address that challenge. How did you overcome those challenges? What strategies did you try? As with nearly all obstacles in math, there are multiple paths to success. Sharing the strategies developed by a few students will help other students realize the multitude of possible solutions. In turn, this will hopefully result in more resilient problem solving in the future. Consider also addressing some strategies that did not pan out. See if those strategies can be revised in order to make them more fruitful. T e ac he r prese nt s M at h game : P ush t he E nvel o pe - Cubo i ds 6 Present Matific s episode Pus h t he Envelo pe - C ubo ids to the class, using the projector. The examples shown in this section result from the episode being in Preset mode. Once again, the goal of this episode is to connect the 2-dimensional net with the 3-dimensional prism. This time, the colored net is provided. Each screen requires your students to select the correct color for only one face of the rectangular prism, while the other faces are already colored.

7 7 At the start of the first screen only, the colored net at right will fold into a rectangular prism, then unfold back into the net. Subsequent screens will only provide the static net. Notice that, in the above screen, orange is a tempting color to use. Encourage your students to pause and think about the color for 10 seconds before responding. Ask your students to raise their hands to show which color they think is correct: How many of you think we should color the face red? How many of you think we should color the face purple? How many of you think we should color the face orange? Ask a few students to explain why they have chosen that color. In particular, the discussion about using orange or purple is an opportunity to discuss how rotations of a 3-dimensional figure affect the coloring of its net. Continue on to subsequent screens, inviting the same kinds of explanations and reasoning as discussed above.

8 8 St ude nt s prac t i c e M at h game : P ush t he E nvel o pe - Cubo i ds 6 Have the students play Pus h t he Envelo pe - C ubo ids on their personal devices. While it is possible to simply guess colors until finding the correct answer, encourage your students to develop strategies for finding the right color. You can always turn this into a game by asking: who can find the right color in the fewest number of tries? This encourages thoughtful reflection on the color of choice, while discouraging guessing. Cl o si ng 5 We saw two different types of scenarios today. First, we were given the rectangular prism and had to color the net. Then, we were given the net and had to color the prism. Which one was more challenging? Why? Take a ute or two to discuss where your students struggled and why. What helped you overcome those challenges? Revisit some of the useful strategies your students devised. Red your students that converting information between the 2- dimensional net and the 3-dimensional prism can take some time to get accustomed to, but that these challenges provide opportunities for growth.