Signed numbers, place value and operation order · Foundation
הוצג| English | משמעות |
|---|---|
| reciprocal/rɪˈsɪprəkl/ | הופך |
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.
The method
- 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.
Why is -4-(-7) equal to 3?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Subtracting a negative is adding its opposite. On the number line, -4 plus 7 lands on 3; treating the signs as "-- makes negative" loses exactly that move.
A worked case
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
-12+35=23, then 23-9=14.
Where it goes wrong
- 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.
Not true: Multiplication must be performed before division, even when division is farther left. Multiplication and division have equal priority and are done from left to right: 12÷3×2=8.
Find -4-(-7).
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Subtracting -7 adds 7: -4+7=3.
Multiplication and division have equal priority and are done from left to right.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
True. Multiplication and division have equal priority and are done from left to right: 12÷3×2=8.
Applying it
- 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.
Evaluate 18÷3×2.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Equal priority: 18÷3=6, then 6×2=12.
Match each operation rule to its example.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Sign rules act on the product or quotient as a whole, and equal-priority operations run left to right.
In your course
- AQA GCSE Mathematics (8300), Foundation tier, specification section 3.1.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Reciprocal: A number that multiplies a nonzero original number to give 1.
- Larger numbers lie farther right on the number line.
- Watch for: subtraction is not commutative: 3-8 and 8-3 differ.