Number Patterns
4A · Chapter 1 · Lesson 3 · MATH-4A-C01-L03
Learning goal. Infer, extend and justify integer sequences based on consistent increments, decrements and place-value changes.
Before you begin
Recall: Grade 3: four-digit place value, regrouping, comparison and rounding; Numbers to 1,000,000. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
From 327,627 add 3 groups of 1,000.
Show answer & reasoning
3 groups of 1,000 are 3,000; then add to the starting value.
In 597,513, what is the value of the digit in the hundred-thousands place?
Show answer & reasoning
The digit 5 represents 5 groups of 100,000, or 500,000.
Write 172,606 as a number of whole thousands and remaining ones.
Show answer & reasoning
172 × 1,000 + 606 = 172,606. This grouping preserves the original value.
Key vocabulary
- place value
- value determined by a digit position
- expanded form
- sum of each nonzero place value
- benchmark
- nearby convenient multiple
- midpoint
- number exactly halfway between two bounds
Understand the concept
Number patterns depend on a rule that fits every neighboring pair. A sequence such as 120,000; 125,000; 130,000 is growing by 5,000 each time. Test a proposed rule against every step, not merely the first. Some patterns alternate rules, but such a pattern must say or show that structure clearly.
Model it: open number line and uniform jumps. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Extend this pattern: 438,594, 438,694, 438,794, ___.
Each term increases by 100; add that step once more.
Check your understanding 1
Extend this pattern: 212,432, 212,682, 212,932, ___.
Show answer & reasoning
Each term increases by 250; add that step once more.
Worked example 2
Extend this pattern: 953,795, 953,845, 953,895, ___.
Each term increases by 50; add that step once more.
Check your understanding 2
Extend this pattern: 246,697, 251,697, 256,697, ___.
Show answer & reasoning
Each term increases by 5,000; add that step once more.
Worked example 3
Extend this pattern: 305,969, 310,969, 315,969, ___.
Each term increases by 5,000; add that step once more.
Check your understanding 3
Extend this pattern: 809,303, 809,553, 809,803, ___.
Show answer & reasoning
Each term increases by 250; add that step once more.
Guided practice
Extend this pattern: 547,934, 548,934, 549,934, ___.
Show answer & reasoning
Each term increases by 1,000; add that step once more.
Extend this pattern: 139,609, 140,609, 141,609, ___.
Show answer & reasoning
Each term increases by 1,000; add that step once more.
Extend this pattern: 709,996, 710,996, 711,996, ___.
Show answer & reasoning
Each term increases by 1,000; add that step once more.
Identify the rule for 538,232, 538,182, 538,132.
Show answer & reasoning
The difference between adjacent terms is consistently −50.
A sequence begins at 103,763 and increases by 50 each time. What is its fifth term?
Show answer & reasoning
The fifth term is four equal steps after the first: 103,763 + 4 × 50.
Independent practice
Extend this pattern: 451,533, 456,533, 461,533, ___.
Show answer & reasoning
Each term increases by 5,000; add that step once more.
Extend this pattern: 759,019, 759,269, 759,519, ___.
Show answer & reasoning
Each term increases by 250; add that step once more.
Identify the rule for 970,122, 965,122, 960,122.
Show answer & reasoning
The difference between adjacent terms is consistently −5,000.
A sequence begins at 710,753 and increases by 250 each time. What is its fifth term?
Show answer & reasoning
The fifth term is four equal steps after the first: 710,753 + 4 × 250.
Maya says a pattern 119,716, 119,816, 120,016 grows by the same amount every time. Is she right?
Show answer & reasoning
The differences are 100 and 200, so the step is not constant.
Extend this pattern: 391,945, 391,995, 392,045, ___.
Show answer & reasoning
Each term increases by 50; add that step once more.
Extend this pattern: 852,099, 853,099, 854,099, ___.
Show answer & reasoning
Each term increases by 1,000; add that step once more.
Extend this pattern: 134,487, 134,737, 134,987, ___.
Show answer & reasoning
Each term increases by 250; add that step once more.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student predicts 7,776 after the pattern 6,776, 7,026, 7,276. Explain and give the correct next term.
Show answer & reasoning
Adjacent terms each differ by 250; add it once to 7,276.
Extend this pattern: 305,512, 305,612, 305,712, ___.
Show answer & reasoning
Each term increases by 100; add that step once more.
Exit check
Identify the rule for 687,836, 687,786, 687,736.
Show answer & reasoning
The difference between adjacent terms is consistently −50.
A sequence begins at 404,794 and increases by 1,000 each time. What is its fifth term?
Show answer & reasoning
The fifth term is four equal steps after the first: 404,794 + 4 × 1,000.
Maya says a pattern 478,406, 483,406, 493,406 grows by the same amount every time. Is she right?
Show answer & reasoning
The differences are 5,000 and 10,000, so the step is not constant.