3A · Chapter 1 · Lesson 06
Number Patterns
Identify constant additive and subtractive steps., Continue a pattern and find missing terms., Explain and verify a candidate pattern using more than one transition.
Your starting challenge
What comes next after 2,150, 2,300, 2,450? How can you prove your rule from all three terms?
By the end, you can…
- Identify constant additive and subtractive steps.
- Continue a pattern and find missing terms.
- Explain and verify a candidate pattern using more than one transition.
Before starting, you should know…
- Add and subtract hundreds and tens mentally.
- Read number-line scales.
Entry · 2 questions
Quick prerequisite check
Try these two before the lesson. If either is confusing, review the listed prerequisites first.
- E1 A number line begins at 1,000 and each tick increases by 100. What is the value at tick 3 after the start?
Show answer and reasoning
Answer: 1,300
Add 3 intervals of 100: 1,000 + 3 × 100 = 1,300.
- E2 What is 2,200 + 200?
Show answer and reasoning
Answer: 2,400
Add two hundreds to 22 hundreds to get 24 hundreds.
Vocabulary to know
| term | One number in a sequence. |
|---|---|
| rule | A relationship used to generate or describe terms. |
| increasing | Becoming larger from term to term. |
| decreasing | Becoming smaller from term to term. |
Learn the concept
Find a common step
Subtract neighboring terms. If the differences match, the pattern may have a constant step. Check all shown gaps instead of guessing from the first two numbers.
Think in place-value units
A step of +200 is “two more hundreds.” When crossing a thousand boundary, the rule remains +200 even though the thousands digit changes.
Distinguish a rule from a guess
A few listed terms can fit more than one pattern. For our core problems, we state or use a constant additive step; harder challenges may ask you to justify other possibilities.
Work through examples
Study the explanation; then try each check before opening its answer.
Worked example 1
Find the constant change
Continue 2,150; 2,300; 2,450.
- 2,300 − 2,150 = 150.
- 2,450 − 2,300 = 150.
- Add 150 again.
Result: 2,600
Now you try · Understanding check 1
Find the next two numbers: 3,400, 3,525, 3,650, 3,775, ___, ___.Reveal answer and explanation
3,900, 4,025 — The same signed step, +125, connects consecutive terms. Continue twice.
Worked example 2
Work backward
Find the missing number: 5,600; ___; 5,000; 4,700.
- The last visible step is −300.
- The missing term is 5,600 − 300 = 5,300.
- Check: 5,300 − 300 = 5,000.
Result: 5,300
Now you try · Understanding check 2
Complete the pattern: 4,200, ___, 3,800, 3,600.Reveal answer and explanation
4,000 — The first and third terms span two equal steps: (3,800 − 4,200) ÷ 2 = -200 per step.
Worked example 3
Cross a place-value boundary
Continue 9,700; 9,800; 9,900.
- The common step is +100.
- 9,900 + 100 = 10,000.
Result: 10,000
Now you try · Understanding check 3
Find the next two numbers: 9,600, 9,700, 9,800, ___, ___.Reveal answer and explanation
9,900, 10,000 — The same signed step, +100, connects consecutive terms. Continue twice.
Guided · 4 questions
Guided practice
Try each problem; return to a visual model if you need one.
- G1 Find the next two numbers: 2,200, 2,350, 2,500, 2,650, ___, ___.
Show answer and reasoning
Answer: 2,800, 2,950
The same signed step, +150, connects consecutive terms. Continue twice.
- G2 Find the next two numbers: 7,850, 7,650, 7,450, 7,250, ___, ___.
Show answer and reasoning
Answer: 7,050, 6,850
The same signed step, -200, connects consecutive terms. Continue twice.
- G3 Complete the pattern: 1,300, ___, 1,650, 1,825.
Show answer and reasoning
Answer: 1,475
The first and third terms span two equal steps: (1,650 − 1,300) ÷ 2 = 175 per step.
- G4 Find the next two numbers: 9,000, 8,875, 8,750, 8,625, ___, ___.
Show answer and reasoning
Answer: 8,500, 8,375
The same signed step, -125, connects consecutive terms. Continue twice.
Independent · 8 questions
Independent practice
Work without looking at examples first. Show your thinking on paper.
- I1 Find the next two numbers: 4,050, 4,175, 4,300, 4,425, ___, ___.
Show answer and reasoning
Answer: 4,550, 4,675
The same signed step, +125, connects consecutive terms. Continue twice.
- I2 Find the next two numbers: 8,100, 7,850, 7,600, 7,350, ___, ___.
Show answer and reasoning
Answer: 7,100, 6,850
The same signed step, -250, connects consecutive terms. Continue twice.
- I3 Complete the pattern: 3,200, ___, 3,800, 4,100.
Show answer and reasoning
Answer: 3,500
The first and third terms span two equal steps: (3,800 − 3,200) ÷ 2 = 300 per step.
- I4 Find the next two numbers: 1,400, 1,850, 2,300, 2,750, ___, ___.
Show answer and reasoning
Answer: 3,200, 3,650
The same signed step, +450, connects consecutive terms. Continue twice.
- I5 Find the next two numbers: 9,500, 9,325, 9,150, 8,975, ___, ___.
Show answer and reasoning
Answer: 8,800, 8,625
The same signed step, -175, connects consecutive terms. Continue twice.
- I6 Complete the pattern: 7,200, ___, 6,950, 6,825.
Show answer and reasoning
Answer: 7,075
The first and third terms span two equal steps: (6,950 − 7,200) ÷ 2 = -125 per step.
- I7 Find the next two numbers: 6,500, 6,725, 6,950, 7,175, ___, ___.
Show answer and reasoning
Answer: 7,400, 7,625
The same signed step, +225, connects consecutive terms. Continue twice.
- I8 Find the next two numbers: 8,975, 8,850, 8,725, 8,600, ___, ___.
Show answer and reasoning
Answer: 8,475, 8,350
The same signed step, -125, connects consecutive terms. Continue twice.
Challenge · 2 questions
Challenge and transfer
These nonroutine problems require you to explain or adapt what you learned.
- C1 A pattern starts at 1,250 and increases by 175 each time. What is the sixth term, counting 1,250 as the first?
Show answer and reasoning
Answer: 2,125
Sixth term is five steps after first: 1,250 + 5 × 175 = 2,125.
- C2 A pattern shows 2,000; 2,100; 2,300; 2,600. Find the next term if the increases themselves grow by 100 each time.
Show answer and reasoning
Answer: 3,000
Changes are +100,+200,+300, so next change is +400: 2,600 + 400 = 3,000. This is an extension, not a constant-step pattern.
Exit · 3 questions
Exit check
Use these fresh problems to decide whether to proceed. Answers should be independent.
- E1 Find the next two numbers: 3,500, 3,625, 3,750, ___, ___.
Show answer and reasoning
Answer: 3,875, 4,000
The same signed step, +125, connects consecutive terms. Continue twice.
- E2 Complete the pattern: 6,500, ___, 6,200, 6,050.
Show answer and reasoning
Answer: 6,350
The first and third terms span two equal steps: (6,200 − 6,500) ÷ 2 = -150 per step.
- E3 What is the constant step in 1,250; 1,500; 1,750?
Show answer and reasoning
Answer: +250
1,500 − 1,250 = 250, and 1,750 − 1,500 = 250.