Signed numbers, place value and operation order · Higher
| English | 中文 | Pinyin |
|---|---|---|
| reciprocal/rɪˈsɪprəkl/ | 倒数 | dào shǔ |
Which way does the balance move?
- An account starts 12 yuan overdrawn. A payment of 35 yuan arrives, then a 9-yuan charge is taken. Which direction does each change move the balance?
- This lesson studies reciprocal 倒数: A number that multiplies a nonzero original number to give 1.
Choose the mathematical structure
- Larger numbers lie farther right on the number line. Adding a negative moves left; subtracting a negative moves right. Multiply or divide signs first, then magnitudes. Evaluate brackets, powers and roots before multiplication/division, then addition/subtraction, working left to right within equal priority.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines reciprocal?
A number that multiplies a nonzero original number to give 1.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
The balance is -12+35-9=14 yuan. Also -4-(-7)=3 and (-6)×(-3)=18. The reciprocal of -4 is -1/4, since their product is 1. For 18÷3×2, work left to right to obtain 12. In 3.047, the 4 means four hundredths and the 7 means seven thousandths. For decimal multiplication, first calculate 24×35: 24×30+24×5=720+120=840. The original 2.4×0.35 has three decimal places in total, giving 0.840. For 5.04÷0.12 multiply both numbers by 100 to get 504÷12; 12×40=480 leaves 24, so the answer is 42. To order negative fractions, -3/4=-0.75 and -4/5=-0.8, hence -4/5<-3/4. The symbols ≤ and ≥ include equality; ≠ means unequal.
Signed numbers, place value and operation order
Larger numbers lie farther right on the number line
Compare the model with the worked case and explain one change.
Find -12+35-9.
-12+35=23, then 23-9=14.
Test a tempting shortcut
- Subtraction is not commutative: 3-8 and 8-3 differ. A negative sign outside a square is not inside its base: -3²=-9, but (-3)²=9. Zero has no reciprocal.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Multiplication must always be performed before division, even when division is farther left. This claim is false. Explain which definition or assumption it violates.
Find -4-(-7).
Subtracting -7 adds 7: -4+7=3.
Multiplication must always be performed before division, even when division is farther left.
Subtraction is not commutative: 3-8 and 8-3 differ. A negative sign outside a square is not inside its base: -3²=-9, but (-3)²=9. Zero has no reciprocal.
Interpret a new situation
- Use a signed starting balance, a positive credit and a negative debit. A shop buying for 48 yuan and selling for 60 makes 12 yuan profit; reversing the prices gives a 12-yuan loss. Estimate the sign before calculating.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Evaluate 18÷3×2.
Equal priority: 18÷3=6, then 6×2=12.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.1. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A number that multiplies a nonzero original number to give 1. Choose the relationship, show the method, check its assumptions and interpret the result.