Hot Lava Mama Posted September 5, 2013 Share Posted September 5, 2013 Ds's math book does not provide the answers for the even problems. I work out the "cumulative review" even problems in every chapter because it is good review for him (and me!). I must be too tired because I can't seem to figure these out. Can you help? Factor completely: 16 - (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum) Factor completely: 10x{the x is squared} + 13x - 3 Thanks for your help! Hot Lava Mama Quote Link to comment Share on other sites More sharing options...
Posted September 5, 2013 Share Posted September 5, 2013 If you are just lazy and want the answers, you can type them into wolfram alpha online, and have it solve them for you, with nice graphs and everything: http://www.wolframalpha.com/input/?i=16+-+%28x+-+11%29+%5E+2 http://www.wolframalpha.com/input/?i=10x%5E2++%2B+13x++-3 Quote Link to comment Share on other sites More sharing options...
ErinE Posted September 5, 2013 Share Posted September 5, 2013 Ds's math book does not provide the answers for the even problems. I work out the "cumulative review" even problems in every chapter because it is good review for him (and me!). I must be too tired because I can't seem to figure these out. Can you help? Factor completely: 16 - (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum) Factor completely: 10x{the x is squared} + 13x - 3 Thanks for your help! Hot Lava Mama A quick reply to the second10x^2 + 13 x - 3 Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3. Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is: (5x - 1)(2x +3) ETA: the first 16 - (x-11)^2 Find (x-11)^2 16 - (x^2 -22x + 121) Distribute the -1 16 - x^2 + 22x - 121 Combine like terms (16 and -121) -x^2 + 22x - 105 Factor out -1 -(x^2-22x+105) Factors of 105 that combine to 22 are 7 and 15 so solution is: -(x-15)(x-7) Quote Link to comment Share on other sites More sharing options...
Arcadia Posted September 5, 2013 Share Posted September 5, 2013 Factor completely: 16 - (x - 11) {x - 11 is squared, but I can't figure out how to write that in this forum) 16 - (x-11)^2 is the same as4^2 - (x-11)^2 a^2 - b^2 = (a-b )(a+b ) 4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11 = (15-x)(x-7) Quote Link to comment Share on other sites More sharing options...
8filltheheart Posted September 5, 2013 Share Posted September 5, 2013 16 - (x-11)^2 is the same as 4^2 - (x-11)^2 a^2 - b^2 = (a-b )(a+b ) 4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11 = (15-x)(x-7) x = 15, x = 7 This is a great explanation. The only thing I would not want the OP to misunderstand is that you should not solve for the variables. The original problem was not given as an equation but an expression. Quote Link to comment Share on other sites More sharing options...
Arcadia Posted September 5, 2013 Share Posted September 5, 2013 The only thing I would not want the OP to misunderstand is that you should not solve for the variables. The original problem was not given as an equation but an expression. Oops, I'll edit my post. Old habit from having to factorise, solve for x and plot the graph for full credit for a math question. Quote Link to comment Share on other sites More sharing options...
quark Posted September 5, 2013 Share Posted September 5, 2013 Oops, I'll edit my post. Old habit from having to factorise, solve for x and plot the graph for full credit for a math question. OT alert: I had a teacher who required it even if the question didn't ask for it. I liked your elegant steps! Quote Link to comment Share on other sites More sharing options...
Hot Lava Mama Posted September 5, 2013 Author Share Posted September 5, 2013 If you are just lazy and want the answers, you can type them into wolfram alpha online, and have it solve them for you, with nice graphs and everything: http://www.wolframalpha.com/input/?i=16+-+%28x+-+11%29+%5E+2 http://www.wolframalpha.com/input/?i=10x%5E2++%2B+13x++-3 :) Snicker! No, not lazy in this case, just stupid! :) Thanks for that link! I never heard of it. I am sure it will be useful during our coming high school days! (My oldest is just starting 9th, and I have 4 other kids behind him!) Thanks! Hot Lava Mama Quote Link to comment Share on other sites More sharing options...
Hot Lava Mama Posted September 5, 2013 Author Share Posted September 5, 2013 A quick reply to the second 10x^2 + 13 x - 3 Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3. Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is: (5x - 1)(2x +3) ETA: the first 16 - (x-11)^2 Find (x-11)^2 16 - (x^2 -22x + 121) Distribute the -1 16 - x^2 + 22x - 121 Combine like terms (16 and -121) -x^2 + 22x - 105 Factor out -1 -(x^2-22x+105) Factors of 105 that combine to 22 are 7 and 15 so solution is: -(x-15)(x-7) Thank you! My mind it on vacation right now! :) I appreciate the help! (Thank your brain for me, too!) :) Hot Lava Mama Quote Link to comment Share on other sites More sharing options...
Hot Lava Mama Posted September 5, 2013 Author Share Posted September 5, 2013 A quick reply to the second 10x^2 + 13 x - 3 Factors of 10 are 1, 2, 5, 10. Factors of -3 are -3, -1, 1, 3. Looking at the possible factor combinations, I know +15x (5x*3) -2x (2x*(-1)) equals 13x so the solution is: (5x - 1)(2x +3) ETA: the first 16 - (x-11)^2 Find (x-11)^2 16 - (x^2 -22x + 121) Distribute the -1 16 - x^2 + 22x - 121 Combine like terms (16 and -121) -x^2 + 22x - 105 Factor out -1 -(x^2-22x+105) Factors of 105 that combine to 22 are 7 and 15 so solution is: -(x-15)(x-7) Thanks! That -1 on the outside of (x-15) would have tripped me up. I was stumped with the middle piece. :) Hot Lava Mama Quote Link to comment Share on other sites More sharing options...
Cosmos Posted September 5, 2013 Share Posted September 5, 2013 OT alert: I had a teacher who required it even if the question didn't ask for it. I liked your elegant steps! So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation? I always wondered why so many of the students I tutored seemed not to understand the distinction between an expression and an equation (or between simplifying and expression and solving an equation). They were always trying to solve an equation when there wasn't one! Quote Link to comment Share on other sites More sharing options...
ErinE Posted September 5, 2013 Share Posted September 5, 2013 Thanks! That -1 on the outside of (x-15) would have tripped me up. I was stumped with the middle piece. :) Hot Lava Mama I factor out the -1 because otherwise there would be two possible combinations (15-x)(x-7) and (7-x)(x-15) Any of the three would be correct, but I would argue the most "factored" (if that's a real term) is: -(x-15)(x-7) Quote Link to comment Share on other sites More sharing options...
ErinE Posted September 5, 2013 Share Posted September 5, 2013 16 - (x-11)^2 is the same as 4^2 - (x-11)^2 a^2 - b^2 = (a-b )(a+b ) 4^2 - (x-11)^2 = (4-x+11)(4+x-11) where a = 4 and b = x - 11 = (15-x)(x-7) I like your solution as well. I always tell DS to be a thinker, not a computer then I find myself guilty of computing! Quote Link to comment Share on other sites More sharing options...
quark Posted September 5, 2013 Share Posted September 5, 2013 So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation? I always wondered why so many of the students I tutored seemed not to understand the distinction between an expression and an equation (or between simplifying and expression and solving an equation). They were always trying to solve an equation when there wasn't one! I'm trying to remember exactly what we did, what was expected and how it was phrased. I learned math in a different language and that is another reason why I stumble a lot with it in English although English is my more fluent language. I remember clearly having to show a lot of things not even asked for in the question and half of the time I was figuring out what exactly the teacher wanted and what the question was asking for. Not saying it was the right or wrong thing to do. Arcadia's remark brought back memories. :) One of the reasons why I always maintain that I don't teach my kid upper level math. I'm still re-learning math terms in English myself. Quote Link to comment Share on other sites More sharing options...
Arcadia Posted September 5, 2013 Share Posted September 5, 2013 So if the problem asked the student to factor an expression, the teacher expected the student also to imagine that the expression appeared in an equation and solve that imaginary equation? The problem is actually worded differently. Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml Quote Link to comment Share on other sites More sharing options...
quark Posted September 5, 2013 Share Posted September 5, 2013 The problem is actually worded differently. Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml Thanks for the link...does that bring back more memories! Yes, I remember having to complete that step where you find x when the two factors multiplied = 0. Still trying to find examples online. I'll update when I do. Quote Link to comment Share on other sites More sharing options...
8filltheheart Posted September 5, 2013 Share Posted September 5, 2013 The problems should be worded differently. Solving for the variable requires 2 expressions to be set equal to each other. An expression is not set equal to anything. In order to solve it, you would have to make the assumption that it equalled zero. The link Arcadia posted is an equation and can be solved. The expression equals zero. That site also has info on expressions. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/factorisinghirev1.shtml. They are factored, but not solved for the variable. Quote Link to comment Share on other sites More sharing options...
quark Posted September 5, 2013 Share Posted September 5, 2013 My apologies! I looked through an A Level maths book that I have at home that comes closest to the type of questions we did. I think I was confusing things. Cosmos, I was probably like one of your tutored students! :P Quote Link to comment Share on other sites More sharing options...
Cosmos Posted September 5, 2013 Share Posted September 5, 2013 The problem is actually worded differently. Below link is a typical workbook exercise question I had as a kid. Can't remember if NEM1 might have similar style questions. http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev1.shtml Sure. I think that type of problem is typical of any algebra program. I just think that a teacher should be cautious in his use of language to ensure that the students understand the difference between these two problems: Solve the equation x^2 - 5x + 4 = 0 and Simplify the expression x^2 - 5x + 4 The problem in the link clearly indicates the first type. I do not think it benefits students to give the impression that the simplifying of an expression is inevitably followed by setting that expression to zero and solving for roots, even though that is of course a very common procedure to follow. So it surprises me to hear that a teacher might say "simplify an expression" when he intends "solve an equation". Quote Link to comment Share on other sites More sharing options...
Cosmos Posted September 5, 2013 Share Posted September 5, 2013 I learned math in a different language and that is another reason why I stumble a lot with it in English although English is my more fluent language. Isn't that an interesting thing to think about? I wonder how much influence the language of instruction has on our math understanding. You hear people talk about the influence that numbering systems ("one-ten two" instead of "twelve", for example) has on children's understanding of place value, but I haven't heard that idea explored very fully into other areas of mathematics. A bit of a side track for this thread, though! Quote Link to comment Share on other sites More sharing options...
kiana Posted September 6, 2013 Share Posted September 6, 2013 I do not think it benefits students to give the impression that the simplifying of an expression is inevitably followed by setting that expression to zero and solving for roots, even though that is of course a very common procedure to follow. So it surprises me to hear that a teacher might say "simplify an expression" when he intends "solve an equation". +1 googolplex. I see people do this all the time in my classes (so it is a very common error), but if there is no equation, you cannot solve the expression. Or, as I tell my students, "If there is no 'equals' in the problem, why are you putting one in the answer?" This is compounded by the use of equals to mean "and my next step is" which is another extremely common error. Equals should only be used between things that are actually equal. This hinders students when they need to reason through chains of equalities using transitivity. Sorry about the continued sidetrack. Quote Link to comment Share on other sites More sharing options...
katilac Posted September 6, 2013 Share Posted September 6, 2013 Slight sidetrack: check your library and see if they offer free online tutoring. Ours does, in every subject! We just enter our library card number and 'wait in line' for a tutor. They ask if you need help with an entire lesson, if you are stuck on a certain problem, etc. We've never had to wait more than a few minutes. This saves us a LOT of time in math, lol. You and the tutor can type messages back and forth, and there is a whiteboard for solving problems. I haven't tried this part yet, but plan to: you can submit an essay/paper for feedback! Quote Link to comment Share on other sites More sharing options...
Dana Posted September 6, 2013 Share Posted September 6, 2013 Sure. I think that type of problem is typical of any algebra program. I just think that a teacher should be cautious in his use of language to ensure that the students understand the difference between these two problems: And that's why I think it's important that students also can explain why they're doing what they're doing (as per discussion on chat board about explanations in math). Getting the right answer doesn't mean much IMO if you can't also say why it's right. (And I'd also argue that this is a major problem with teachers who don't know their math well enough.) Quote Link to comment Share on other sites More sharing options...
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