Patterns describe change
Lists can change by adding, multiplying or following another rule. Arithmetic progressions are the constant-addition family.
Learning goals
- Distinguish additive, multiplicative and other patterns.
- Generate terms from a contextual rule.
- Justify whether a sequence is arithmetic.
Concept lesson
Describe how terms change
Compare each term with the one immediately before it. Record whether the operation is add, subtract, multiply or another rule.
Arithmetic means constant addition
A sequence is arithmetic only when the same number is added to obtain every next term. Repeated multiplication, squares and irregular changes are not APs.
Evidence from every available gap
Calculate each consecutive difference. One matching pair is insufficient; all displayed differences must agree.
Generate ladder-rung lengths
A bottom rung is 45 cm and each higher rung is 2 cm shorter. The lengths are 45,43,41,39,... because 2 is subtracted each time.
Compare additive and multiplicative patterns
5,9,13,17 adds 4 each time and is an AP. 1,2,4,8 multiplies by 2; its differences 1,2,4 are unequal, so it is not an AP.
Test every gap
Classify 3,7,11,15; 1,4,9,16; and 4,9,14,20. List every difference before deciding.
Context rule or AP?
Compare a savings amount increasing by 50 rupees yearly with an amount multiplied by 5/3 every three years. Generate terms and decide which is an AP.
One-mark recognition pattern
Check consecutive differences rather than visual regularity. In a complete answer, state the selected sequence and its constant difference.
Exit ticket
Generate five terms for a quantity starting at 70 and decreasing by 6. Decide whether it is an AP. Then test 2,6,10,15 and explain the failure.