What picture would prove what (a + b)^2 is equal to? - September 18-22, 1995
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How about (a + b)(c + d)? (Can you think of another?)Proofs Without Words are drawings that 'prove' mathematical statements. For example, you could 'prove' that a(a + b) = a^2 + ab by drawing the picture
______________________ | | | | | | | | | | | | a | | | | | | | | | | | | |_______________|______| a bCan you see how that makes sense? (What property does that illustrate?) What picture would prove what (a + b)^2 is equal to? How about(a + b)(c + d)? Can you think of another?(When drawing pictures for your answer, use a monospaced font and spaces instead of tabs - tabs have different sizes on different computers, so your drawing might come out looking really weird.)
Annie says: Solutions
Proofs Without Words are really cool. As you can see from the answers to the first part [illustrate (a+b)^2] it makes all that FOIL stuff make sense!
Once upon a time, perhaps in elementary school, we were taught to look at area and such and actually draw rectangles and squares to 'prove' how things worked. There's no reason you can't do that for many more concepts in math. The ones illustrated here are just a few examples - and the distributive property is a natural for this sort of thing.
Nice job this week - some of the pictures came through a bit muddled (it's tough, and you have to make sure you don't use any tabs), but I got almost all of them straightened out, I think.
Following are highlights, and the full list of names and solutions is also available.
Denise Lin, Kate Longhurst, Michelle Ross Grade: Year 9 School: St Hilda's Anglican School for Girls Answer: (a+b)^2=a^2+2ab+b^2 a b ______________ | | | | | | a | a^2 | ab | | | | |________|_____| | | | b | ab | b^2 | |________|_____| 2(a+b) 2 __________ | | | | b | 2b | | | |__________| | | a | 2a | |__________| The Associative Law: a+(b+c)=(a+b)+c ______ _________ ___________ a b c _______________ ___________ a+b c ______ ____________________ a b+c
Erin Phillips Grade: 9 School: Summit Answer: It illustrates the distributive property. (a+b)^2 _______________________ | | | b| | | |_________________|_____| | | | | | | | | | | | | a| | | | | | | | | | | | | | | |_________________|_____| a b (a+b)(c+d) _______________________ | | | | | | c| | | | | | |_________________|_____| | | | | | | | | | d| | | | | | | | | |_________________|_____| a b (a+b+c)(d+e+f) _________________________________ | | | | | | | | d| | | | |_________________|_____|_________| | | | | e|_________________|_____|_________| | | | | | | | | | | | | f| | | | | | | | | | | | |_________________|_____|_________| a b c
Christopher Roth School: Lincoln-Sudbury Regional High School. I have got it! The answer is: (a+b)^2 _________________ | | | b | ab | b^2 | |_________________| | | | | | | = 2ab + a^2 + b^2 a | a^2 | ab | | | | | | | |_____a____|___b__| (a+b)(c+d) ____________ | | | d | ad | bd | | | | = ac + ad + bc + bd |____________| | | | c | ac | bc | |_____|______| a b (a+b+c)^2 ______________ | | | | c | ac | bc | c^2| |____|____|____| | | | | = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc b | ab | b^2| bc | |____|____|____| a | a^2| ab | ac | |____|____|____| a b c
Eric Jankowski Grade: 8 School: Center for the Arts and Sciences Saginaw, MI Answer: The example demonstrated the distributive property. (a+b)^2 = a^2+b^2+2ab (a+b)(c+d)= ac + ad + bc + bd ____a______b_ c d b | ab |b^2| ______________________ |________|___| b| | | | | | | bc | bd | a | a^2 |ab | |---------------------| | | | | | | | | | a| | | -------------- | ac | ad | | | | --------------------- (a+b+c)^2 = a^2 + b^2 + c^2 +2ab + 2ac + 2bc __a____b_______c____ | | | | a|a^2| ab | ac | |------------------| |ab | b^2| bc | b| | | | | | | | |------------------| | | | | c|ac | bc | c^2 | | | | | | | | | --------------------
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