The same step every time
Linear patterns
The chess club
A chess club charges a joining fee of ₹200 and then ₹50 for every match a player plays. Anu pays ₹250 after her first match, ₹300 after her second, and so on.
Anu plays one more match each week. By how much does her total go up each time?
What this lesson covers
The idea
A characteristic feature of a linear polynomial is that the difference between its successive values at integers is constant; such patterns are called linear patterns.
The chess club
A chess club charges a joining fee of ₹200 and then ₹50 for every match a player plays. Anu pays ₹250 after her first match, ₹300 after her second, and so on.
Anu plays one more match each week. By how much does her total go up each time?
- Always by ₹50
- By more each time
- By less each time
Play the matches
Press the button to play one more match. Watch the change between one row and the next.
A constant difference
A characteristic feature of a linear polynomial is that the difference between its successive values at integers is constant. Such patterns are called linear patterns.
The amount paid is 200 + 50m, a linear polynomial in m. It goes up by ₹50 for every extra match. Compare it with m²: the values 1, 4, 9, 16, 25 change by 3, then 5, then 7, then 9. The change keeps changing, so m² is not linear.
Notes
A linear polynomial changes by the same amount each time the input goes up by 1. Such patterns are linear patterns.
Check yourself
A square has side x cm, so its perimeter is 4x. The side grows from 3 cm to 3.5 cm. By how many cm does the perimeter go up?
Answer: 2
The perimeter goes from 4 × 3 = 12 to 4 × 3.5 = 14, so it goes up by 2 cm. Every extra 0.5 cm of side adds 2 cm.
Which list of values shows a linear pattern?
The chess club charges 200 + 50m rupees for m matches. How much does a player pay for 10 matches?
Answer: 700
200 + 50 × 10 = 200 + 500 = 700 rupees.
Which of these changes by the same amount each time the number goes up by 1?
- 1, 4, 9, 16, …. The changes are 3, 5, 7: they keep changing.
- 2, 4, 8, 16, …. The changes are 2, 4, 8: they keep changing.
- 3, 7, 11, 15, … — correct. Yes. It goes up by 4 every time.
- 200 + 50m — correct. Yes. It goes up by 50 every time: a linear polynomial.
- m². m² goes 1, 4, 9, 16: the change keeps changing.
- m³. m³ goes 1, 8, 27, 64: the change keeps changing.