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also took advantage of the opportu- nity to reinforce understanding of the math and computer science concepts involved. Computer Science Before starting our project, students investigated information processing to better understand how computers assist us in art-making. We discussed how computer programs send instruc- tions to a computer, which carries out the desired operation. As students watched my screen on their displays, I navigated Photoshop, demonstrating that every tool sends a different set of instructions about how our images should change. We also talked about how Photoshop is a "raster-based" image editor, which means that the computer has to sort through directions about the color of each and every pixel in an image. This would be an important consideration as we discussed the mathematics of creating an image average. Image Averages After learning about how the software works, I encouraged students to share ideas about how to make multiple images visible simultaneously. They identified that changing the layer- opacity setting creates semitrans- parent images. We discovered that changing the percent opacity allows multiple images to share the same canvas. However, if two images are both set to 50% opacity, the result- ing image is still translucent. This is because the top layer takes 50% while the bottom layer takes up half of the remaining 50%, resulting in an image that is 75% opaque. To get an image that is equal parts of two images, we needed to set the top layer to 50% opacity and the bottom layer to 100%. The resulting image was an average of the two. Integrating Art and Math I invited students to connect their new knowledge of image averages to their knowledge of mathematical aver- ages. We discussed, for example, that the number three is neither two nor four, but equal parts of both. Since each pixel's color is assigned a number in Photoshop, we discussed how the program makes similar calculations when we adjust the opacity. I informed students that we would be creating aver- ages from two images at a time because opacity settings cannot be adjusted to fractions of a per- centage. By combining two at a time, we could reduce the number of layers by half. We could continue to combine image averages, cutting our number of layers in half, until we ended up with a single image. To do this, however, we would need to make use of a geo- metric sequence. Geometric Sequences A geometric sequence is a series of numbers in which each number is the product of the one before it, multiplied by a common ratio. For example, the numbers 2, 6, 18, and 54 are part of a geometric sequence with a common ratio of three. For our purposes, we would need to find a geometric sequence with a common ratio of two, since the number of lay - ers would be divided by two every time we averaged the paired layers. The sequence we needed was 1, 2, 4, 8, 16, 32. By starting with thirty-two images, we con - cluded that we could reduce our number of layers by half, five times, to come up with a single image average. Art-Making Now that students understood the process, it was time to find images to combine. I asked students to each select a single theme, based on a hobby or interest, to use in an image search. Students searched for medium-sized images on Google. They were instructed to find images that were cropped in a similar way. For example, if they were searching for pictures of lions, they could not use both pictures of a standing lion and pictures of a close-up lion face. I invited students to connect their new knowledge of image averages to their knowledge of mathematical averages. Austin, grade seven. 20 April 2014 SchoolArts A_pages_4_14.indd 20 2/20/14 3:04 PM

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