致中小學生和他們的家長

最近看到一份數學5，6，年級的綜合考題，很有感想。在燕京人一書中我已經寫過我從小學習的體會，就是讀懂一本自學的好書，學會思考，培養自學能力，對人一生都有幫助。我在五年級暑假1949年讀了一本解放前升學數學指導，使我從糊塗的學習狀態變得會思考了，使我後來的學習有突飛猛進的改變。看到後來不論在中國和外國的中國家長都願意花大錢送子女上各種補習學校，使得孩子忙於應付，缺少獨立思考，自學的能力。即使考高分，自己的能力並不強。在美國我沒有讀過美國的教材，不知道是否有很好的自學參考書。我現在給出這份考題，然後在另外兩個博文中分別給出符合自學的中英文詳解。便於學生和家長了解。希望有些幫助。也希望能有人多寫自學叢書，用此來開發孩子們的智力。下麵是這份考題。

1. (2 pts × 6 = 12 pts) Calculate mentally. Do not sketch. (a) 4357 + 9685 =   

(b) 4357 − 9685 =

(c) 57 × 9 =

(d) 92 ÷ 23 =

(e) 257 × 8 =

(f) 483 ÷ 7 =

2. (3 pts × 2 = 6 pts) Calculate with pencil. Show every single step of your calculation. For the division, round the result to the nearest hundredth.

9 8 × 7 3=

9 5/37=

3. (2 pts × 4 = 8 pts) Calculate with pencil. Simplify the results. Show every step of your calculation. 2 7 + 3 4 = (a) 8 7 − 55 44 = (b) Name: August 5, 2023 August 5, 2023Name: 28 27 × 9 35 = (c) 66 75 ÷ 11 25 = (d)

4. (2 pts × 4 = 8 pts) In each pair below, circle the greater number.

(a)92 /107 , 93/ 107

(b) 92/ 107 , 92 /106

(c) 92/ 107 , 46 /53 

(d) 29 /56  , 0.49

5. (2 pts × 4 = 8 pts) Calculate with pencil.

(a)15.8 + 2.371 =

(b) 630.02 − 34.176 =

(c) 1.4 × 5.05 =

(d) 7.38 ÷ 0.12 =

 6. (4 pts + 5 pts = 9 pts) Calculate with pencil. Show every step of your calculation.

(a)12 − 10 × (99 − 99 ÷ 33) + (87 × 2 − 174) =

(b) (3 × (19 + 6 × (7 − 9)) − 2) × (174 − (174 − 29) ÷ 29) ÷ 19 − 18 =

 7. (1.5 pts × 10 = 15 pts) Calculate with pencil. Sketching on another piece of paper is OK.

(a)123 − 123 × 11 =

(b) −123 + (−123) × 9 =

(c) 123 − 123 × (−9) =

(d) −123 − (−123) × (−9) =

(e) 123 − 123 ÷ (−123) =

(f) (−123) ÷ 123 − (−123) =

(g) (−123) ÷ (−123) + 123 =

(h) (−9) × 123 − 123 =

(i) (−123) ÷ (12.3 − (−12.3)) =

(j) 100 × 123 − 123 ÷ −123 50 =

(j) 8. (5 pts × 2 =

10 pts) Use the known results to complete the calculations. Show every step of your calculation.

(a)It is known that 123 + 234 + 345 + 456 + 567 + 678 + 789 + 890 + 901 = 4983. How much is 123 + 234 + 345 + 456 + 567 + 678 + 789 + 890?

(b) It is known that (−123)×234+ 345+ 456+ 567+ 678+ 789+ 890×901 = 775943.

How much is (−123) × 234 + 345 + 456 + 567 + 678 + 789 + 890 × 900?

 9. (8 pts × 3 = 24 pts) Word problems. (Show the details of your calculation.)

(a)A shopkeeper had 1080 apples and oranges altogether at the beginning. After selling 60 oranges and 20 apples, the number of apples he had was three times the number of oranges left. How many oranges did he have at first?

(b) Katy is 8 years younger than Tess. Nina is 13 years older than Tess. The sum of their ages is 80. How old is each of them?

(c) There are 260 sheep and chickens in total on a farm. The animals have 768 legs all together. How many wings do the animals have in total ?