致中小学生和他们的家长

最近看到一份数学5，6，年级的综合考题，很有感想。在燕京人一书中我已经写过我从小学习的体会，就是读懂一本自学的好书，学会思考，培养自学能力，对人一生都有帮助。我在五年级暑假1949年读了一本解放前升学数学指导，使我从糊涂的学习状态变得会思考了，使我后来的学习有突飞猛进的改变。看到后来不论在中国和外国的中国家长都愿意花大钱送子女上各种补习学校，使得孩子忙于应付，缺少独立思考，自学的能力。即使考高分，自己的能力并不强。在美国我没有读过美国的教材，不知道是否有很好的自学参考书。我现在给出这份考题，然后在另外两个博文中分别给出符合自学的中英文详解。便于学生和家长了解。希望有些帮助。也希望能有人多写自学丛书，用此来开发孩子们的智力。下面是这份考题。

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 ?