Quod Erat Demonstrandum

2008/11/16

F.2 Mathematics: factorization by cross method

Filed under: Junior Form Mathematics — johnmayhk @ 6:20 下午
Tags: ,

Factorize

x^2 - 5x - 6.

By using the cross method, students may give the following two ‘possible answers’.

A. (x - 2)(x - 3)
B. (x + 1)(x - 6)

The correct answer is B. Some students may opt for A because the expression in A could give (so-called) the middle term -5x. However, the constant term in A is +6 (not -6, the correct one), hence A is not the answer.

Setting this type of question may remind students of the importance of checking the constant term.

Other examples like

x^2 - 13x - 30 =

A. (x - 3)(x - 10)
B. (x + 2)(x - 15)

x^2 - 15x - 54 =

A. (x - 6)(x - 9)
B. (x + 3)(x - 18)

Correct answers to the above are B. But some students may get it wrongly.

To set up such kind of questions, just consider two expressions:

(x - a)(x + b)
(x - c)(x - d)

All we need is setting

ab = cd and a - b = c + d

Yield

(c - b)(d - b) = 2b^2 – – – – – – (*)

Then, we may put different positive integral value of b and evaluate c, d and finally a.

Just give an example.

Put b = 7 (say), by (*),

(c - 7)(d - 7)  = 2\times 7\times 7

Just take c - 7 = 7 and d - 7 = 2 \times 7, thus

c = 14, d = 21 and hence a = \frac{14 \times 21}{7} = 42

Hence we have two expressions

(x - 14)(x - 21) \equiv x^2 - 35x + 294
(x + 7)(x - 42) \equiv x^2 - 35x - 294

having the same ‘middle terms’ with constant terms differing in sign.

Of course, this article is rubblish once we have set program in calculators.

14 則迴響 »

  1. 宜家覺得最恐怖係
    聽見小學都用計數機教乘數
    唔使諗乘數表

    迴響 由 溟天凱 — 2008/11/16 @ 6:29 下午 | 回覆

  2. 唔係 fai….
    想當年….. 媽媽指出九因歌是中國的國粹之一
    之後在籐條輔助下,乘數表我當年背背下就識了 =_="

    幾年前已有報導指外國的小朋友乘數根底薄弱…
    仲唔到毒瘤像金融海嘯般殺到香港… = =

    假若未來的主人翁 foundation 禁差….
    佢地幾時先可以成為 “真正" 的主人翁?不解。

    迴響 由 Ricky — 2008/11/17 @ 3:18 上午 | 回覆

  3. 我幫就讀小學五年級的侄兒看數學功課時,知道他是用心算的。

    「理解」是重要,但相信「背誦」也有一定的正面作用。背誦國寶級文物「九因歌」,除了有助學習乘數,相信在小朋友的腦袋中,或多或少會產生某些網絡連繫(嗯,吹水的,手上沒有科學實證),腦部得以成長,對學習新事物或有所裨益(我用「或」,因為那是憑空猜測的。)

    用一月、二月、三月等等,在某程度上比 January,February,March 更易於讓小朋友掌握。以英語來學乘法,可能較用中國人的「九因歌」困難。這個「國技」,絕對有保留的價值。

    迴響 由 johnmayhk — 2008/11/17 @ 1:10 下午 | 回覆

  4. “或多或少會產生某些網絡連繫.."
    從認知心理學角度上 (記憶),這是有可能的。

    至於是如何,小弟只知皮毛…
    不敢在各高人前亂拋書包。
    還望高手指點。

    迴響 由 Ricky — 2008/11/17 @ 7:33 下午 | 回覆

  5. 真是有一間中學的數學老師,教學生用 program 計算 factorization 的題目,說用 cross method 浪費時間云云,留待中四時才學好了!

    迴響 由 — 2008/11/22 @ 6:17 下午 | 回覆

  6. 中二學 cross method?

    迴響 由 W — 2009/03/25 @ 5:27 下午 | 回覆

  7. Yes, nearly all methods and identities (including a^3 + b^3 , a^3 – b^3) about factorization are taught in F.2 in my school.

    迴響 由 johnmayhk — 2009/03/25 @ 5:59 下午 | 回覆

  8. 唔知可唔可以在cross-method中用分數or小數點?

    迴響 由 Lam Ho Hung — 2009/10/28 @ 5:29 下午 | 回覆

    • Cross method 的目的是因式分解。

      比如要因式分解

      x^2 - 2.7x + 0.4301

      你又厲害到,在沒有二次公式或/和計算機的幫助下,可以想到

      0.4301 = 0.17 \times 2.53

      從而得到

      x^2 - 2.7x + 0.4301 \equiv (x - 0.17)(x - 2.53)

      那麼 cross method 為何不能出現小數?

      但如果同學用計算機的程式,進行以下的因式分解:

      10000x^2 - 27000x + 4301

      計算機顯示了

      0.172.53

      從而同學誤以為

      10000x^2 - 27000x + 4301 \equiv (x - 0.17)(x - 2.53)

      的話,那當然是錯。

      (注:正確是 10000x^2 - 27000x + 4301 \equiv 10000(x - 0.17)(x - 2.53) \equiv (100x - 17)(100x - 253)

      又例如,你可以 cross method 分解

      x^2 -2x - 1

      如果你厲害到,在沒有二次公式或/和計算機的幫助下,可以想到

      -1 = (1 + \sqrt{2})(1 - \sqrt{2})

      從而得到

      x^2 -2x - 1 \equiv (x - 1 + \sqrt{2})(x - 1 - \sqrt{2})

      那麼 cross method 為何不能出現無理數?

      注:題目應要說清楚容不容許 \sqrt{2} 在答案中出現。

      迴響 由 johnmayhk — 2009/10/28 @ 8:42 下午 | 回覆

  9. 我而+中二教identity and fatorization 都冇教cross method-.-我想學ah-.-

    迴響 由 warren — 2009/11/01 @ 4:06 下午 | 回覆

    • 我上年中二冇教cross method, 但中三有教, 所以我唸你中三都有得學嫁啦

      迴響 由 winnie — 2011/09/23 @ 6:32 下午 | 回覆

  10. 我唔明hence 之後要點訐 比左個hence係唔係無關係 ? 只係叫你計埋佢?

    迴響 由 zero — 2011/12/01 @ 10:39 下午 | 回覆

  11. No . “Hence" means you have to use the result of the last question

    迴響 由 Tony — 2012/03/31 @ 4:02 下午 | 回覆


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