The expression \u201c 35 <\/sup>\u201d means \u201cmultiply 3 by itself 5 times\u201d: \u00a0\u00a035<\/sup>=3\u00d73\u00d73\u00d73\u00d73.\u00a0 I\u2019m not really worried about how to calculate this right now.\u00a0 I just want to agree that we call it \u201cthree to the fifth\u201d. \u00a0\u00a0And raising to the fifth power doesn\u2019t have any other special name.<\/p>\n And when we talk about x7<\/sup> , we say \u201cx to the seventh\u201d.\u00a0 Again, no special name.<\/p>\n But there are two powers that have additional names.\u00a0 When we talk about say, 52<\/sup>, we most often call it \u201c5 squared\u201d.\u00a0 \u00a0\u00a0Raising to the second power is called \u201csquaring\u201d.\u00a0\u00a0Raising to the third power is called \u201ccubing\u201d.\u00a0 And , of course, you knew that.\u00a0\u00a0 But it is easy to overlook why these (and only these)<\/i> powers have these additional names.\u00a0 The additional names have geometric meaning.<\/p>\n It is easy to overlook the geometric meanings of squaring and cubing.\u00a0 I know this because of how my students respond when we talk about FOIL-ing to square a binomial.\u00a0 The geometric meaning always seems to catch them by surprise.\u00a0 But it is usually a happy discovery.<\/p>\n \u00a0CURSES! FOILed\u00a0 AGAIN<\/strong><\/p>\n For those of you who haven\u2019t seen it in a while, I\u2019ll review.\u00a0 FOIL is a mnemonic device used to help students remember how to multiply\u00a0binomials.\u00a0 It is the result of distributing multiplication over addition multiple times.\u00a0 You can use it to square binomials.<\/p>\n You start with an expression such as: (x + y)2<\/sup><\/i><\/p>\n You re-write it:\u00a0 (x + y)(x + y)\u00a0\u00a0 <\/i><\/p>\n Now, think of that first term as the multiplier and think of the second term as the sum over which multiplication is distributed.\u00a0 You get:<\/p>\n (x + y)x + (x + y)y <\/i><\/p>\n Then you can distribute each of those as well.\u00a0 That leads you to: x2<\/sup> + xy + xy + y2<\/sup><\/i><\/p>\n Before we combine the middle two terms, note that there are 4 pairings.\u00a0 That\u2019s where the mnemonic FOIL comes in.\u00a0 You get the pairs by multiplying the FIRST terms, the OUTER terms, the INNER terms and the LAST terms.<\/p>\n I don\u2019t have a problem with this mnemonic itself.\u00a0 It\u2019s kind of robotic, but I understand that you have to make certain processes automatic so that you can get to the next step.\u00a0 I wouldn\u2019t want my students to have to go back to the distribution property every time they see (x + y)<\/em>2<\/sup>. \u00a0\u00a0In fact, I don\u2019t want them to have to foil it either.\u00a0 I want them to know that (x + y)2<\/sup> = x2<\/sup> + 2xy + y2<\/sup> <\/i>not because they have memorized it but rather because they can picture<\/i> it!<\/i><\/p>\n THE GEOMETRIC MEANING OF<\/strong> (x + y<\/em>)2<\/sup><\/strong><\/p>\n We call it \u201csquaring\u201d because it gives us the area of the square.<\/p>\n So (x + y)2<\/sup><\/i> is the area of the square with length (x + y)<\/i>.\u00a0 Now, let\u2019s draw some pictures.<\/p>\n Here is a line segment with a length of x + y<\/i>:<\/p>\n And here is\u00a0the square with a side that has\u00a0that same length:<\/p>\n The area of that square is certainly (x + y)2<\/sup><\/i>.<\/p>\n But what if we break that square into smaller regions?<\/p>\n The picture tells the story in a more intuitive and memorable way than foiling ever could.\u00a0 We are clearly looking at a big square (with area = x2<\/sup>), a little square (with area = y2<\/sup>), and two identical rectangles (each with area = xy).\u00a0 This is the diagram that goes with the rule for binomial squares:<\/p>\n (x + y)2<\/sup> = x2<\/sup> + 2xy + y2<\/sup><\/i><\/p>\n But it\u2019s better with pictures.<\/p>\n <\/p>\n <\/p>\n Next post — higher dimensions.<\/p>\n And then x2<\/sup> \u2013 y2<\/sup><\/i>. \u00a0\u00a0Spoiler: it really <\/i>is the difference of two squares.<\/i><\/p>\n","protected":false},"excerpt":{"rendered":" Let\u2019s start with a silly question about math vocabulary: Why do some powers have special names? The expression \u201c 35 \u201d means \u201cmultiply 3 by itself 5 times\u201d: \u00a0\u00a035=3\u00d73\u00d73\u00d73\u00d73.\u00a0 I\u2019m not really worried about how to calculate this right now.\u00a0 … Continue reading \n
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