代數(shù)——美國高中畢業(yè)文憑的攔路虎

一位中國訪問學者曾講過他在美國大學的一段親身經(jīng)歷。

一天，一位愁眉苦臉的大學生來問他一個“十分困難”的問題，他看后感到很驚訝。原來這個“十分困難”的問題，只是一個一元一次方程式的求解問題。在中國，初中生就會做了，但這位大學生似乎對此毫無頭緒。

其實這不能怪學生。在美國許多州，例如加州就沒有把代數(shù)這門課列為中學生的必修課。換句話說，不會解一元一次方程，不會因式分解，照樣可以拿到高中文憑。這點對于亞洲的學生來說有點不可思議。但是近年來，美國一些州陸續(xù)采取了相應措施，要求高中生必須通過初等代數(shù)的考試才能獲得畢業(yè)文憑。

既然代數(shù)這么重要，那就讓我們一起來看看國外的學生學習代數(shù)的小竅門吧。

Many people have difficulties in the area of algebra mathematics. Many attempts to read books， attend lessons and research the web to find information and attend interactive lessons that could help them improve their algebra skills. Interactive learning is a lot more fun than reading algebra books but is that enough to make someone good at algebra? The concept of algebra has to be fully understood before one can have full knowledge of mathematics. Algebra is only going to become more advanced as the years of school pass and if one has not gained full knowledge of it at its most basic level， it will be impossible to venture on. There are thousands of algebra books out there that break down algebra to its easiest components. Many learning styles in these books are different so take the time to go through them and find one that looks like you might understand or one that accommodates your learning style. Practice is really important in algebra and you can’t expect to learn everything from online courses or tutorials. You need to go through the concepts and the principles of algebra again and again in order to learn them.

Since practice makes perfect， one effective method is to use your text book and write on paper all of the concepts along with equations that represent that concept. Keep each concept on a different sheet. Take notes and write down your opinion for every equation or algebra concept. You may not like it at first， but the more you practice and use this personal algebra notebook the better for you. You must be patient because at first it might seem like you are never going to be able to figure out how all of these numbers plug into one and other， so don’t rush， remember that learning algebra takes time.

In fact， if you successfully learn a few algebra concepts then you’ll want to have more. You have to algebra one step at a time. If you do these steps over and over， your brain will start to adjust and you will like it.

很多人都感到學習代數(shù)很困難。他們試圖通過閱讀、上課、上網(wǎng)搜集資料和參加互動課堂來提高代數(shù)成績。的確， 互動式的學習比讀代數(shù)書有趣多了，但是這真的能讓人學好它嗎？其實，只有理解了代數(shù)定理，才能夠全面地掌握數(shù)學知識。隨著年級逐漸升高，代數(shù)知識的難度只會不斷加深。如果你不能完全掌握代數(shù)的基礎知識，那么將來要想提高成績是不太可能的。市面上有許多的代數(shù)書都達不到深入淺出的程度，而且這些書里所介紹的學習方法各異。你需要花時間仔細讀一讀，找一本你覺得容易理解的或者是一本能夠給你推薦適合的學習方式的書來讀。學習代數(shù)的時候多做練習也很重要，因為你不能指望從在線課程或者家教那里學到所有的東西。你得充分理解代數(shù)中的那些概念和定理，這樣才能學好代數(shù)。

常言道“熟能生巧”，因此，一個有效的學習方法就是充分利用你的教材來做練習，并且在卡片上寫下所有能指代概念的數(shù)學式，每個概念都分開用不同的卡片寫。在數(shù)學式或者代數(shù)概念旁邊做好筆記，并且寫下你對它的理解。你可能一開始并不喜歡這樣做，但是越頻繁地練習并使用這種個人代數(shù)筆記，你的收獲就會越多。當然，你還需要有足夠的耐心，因為一開始你是絕對不可能計算出所有結果的。不過，別著急，學好代數(shù)是需要時間的。

實際上，只要掌握了一部分代數(shù)知識，你就會激發(fā)出更大的求知欲。在某一個時期內，你的水平可能只停留在一個階段，但是如果你一步一個腳印，腳踏實地地跟上，你的腦筋將會越來越靈活，你也會愛上代數(shù)。

一、選擇題

1. Correct Answer: D

2. Correct Answer: A

Explanation

Since the coordinates of two points on line l are given， the slope of l is =2. Line m， which is perpendicular to l will have a slope of -， since slopes of perpendicular lines are negative reciprocals of each other. The equation of m can be written as y=-x+b. Since line m also contains point （0， 0）， it follows that b=0.

Therefore， an equation of line m is y=-x.

3. Correct Answer: B

4. Correct Answer: E

Explanation

This is a scatterplot in which each point represents a college. The x－coordinate is the enrollment in 1980， and the y－coordinate is the enrollment in 2000. Point E has approximate coordinates （7 500， 7 700）， so the college represented by this point had an enrollment that increased only ≈0.027＝2.7％ between 1980 and 2000. All the other labeled points represent colleges with greater percent increases in enrollment.

二、填空題

5. Explanation

Since

4x-7

=5the value of 4x-7 is either 5 or －5.

4x-7=5 4x-7=-5

4x=124x=2

x=3or x=

The two values of x that satisfy the first equation are 3 and .

Since

3-8x

the value of 3 － 8x is either 1 or －1.

3-8x=1 3-8x=-1

8x=28x=4

x= orx=

The two values of x that satisfy the second equation areand .

You are asked to find the value of x that satisfies both equations. That value is .

6. Explanation

Drawing the figure described in the problem will help you visualize the correct solution to the problem. The figure below shows three parallel lines intersected by a fourth line. The acute angle is labeled 28°， as shown below.

[28°]

圖1

Using the fact that vertical angles and alternate interior angles are equal， you can put a check showing the other angles in the figure that also measure 28°， as shown below.

[28°][1][2][3][4][5]

圖2

There are 5 other angles that measure 28°. Therefore， the correct answer to this problem is 5. The number 5 can be gridded in any of the four columns on the answer grid.

7. Correct Answer: 3

1. A box containing 3 oranges， 2 apples， and one banana weighs 15 units. Another box containing 5 oranges， 7 apples， and 2 bananas weighs 44 units. A third box containing 1 orange， 3 apples， and 5 bananas weighs 26 units. How much does each fruit weigh?

2. The expression x5+y5 can be written as the product of x+y and another factor. Find that other factor.

3. If x=， then what is ?

4. Find all values of z such that z4+4z2+3=0.

5. Find the radius and center of the circle that is the graph of the equation

4x2+4y2+4x-16y=7.

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