Count through place boundaries and mentally change tens or hundreds

Learning goal: Count forward and backward within 1,000 by ones, tens and hundreds from varied starting numbers, skip-count by fives from multiples of five, and mentally add or subtract ten or one hundred from numbers in the specified 100–900 starting range. Explain what changes in value, including when a place boundary is crossed.

Counting by ones can cross a ten or hundred boundary. From 198, continue 199, 200, 201. Backward from 302, say 301, 300, 299. The numeral changes in more than one place at a boundary, but each step still changes the value by exactly one.

Counting by tens does not have to begin at a multiple of ten. Starting at 237 gives 247, 257, 267 and so on. The ones stay seven because each jump adds a complete ten. Backward by tens from 312 gives 302, 292, 282. Crossing a hundred boundary does not change the step size.

Counting by hundreds also works from a varied start. From 146, count 246, 346, 446. The tens and ones remain forty-six while the hundreds increase. Stay within the lesson's range when choosing the starting point and number of steps. Counting backward from 346 by hundreds gives 246, 146, 46, all within 0–1,000.

For skip-counting by fives, start at a multiple of five such as 35 or 120. From 35: 40, 45, 50, 55. From 120 backward: 115, 110, 105, 100. The sequence alternates a ones digit of zero and five for these whole-number multiples. Do not apply that ending pattern to a sequence started at 37, which is not the specified starting kind.

Mental addition and subtraction use the same place-value relationships. Adding ten to 190 yields 200; it is not enough to replace the tens digit nine with ten inside a numeral. Compose the ten tens into one hundred. Subtracting ten from 300 gives 290, which uses two hundreds and nine tens in its usual form.

Adding one hundred to 506 gives 606, while subtracting one hundred gives 406. The tens and ones retain their values in these examples. From the starting number 900, adding one hundred gives 1,000. The starting-number restriction describes where the problem begins; the result can reach the boundary 1,000.

The diagram separates a +10 sequence from a +100 sequence. Each arrow has a stated step size. Equal arrow lengths on the page organize the examples; they are not a number-line scale making ten and one hundred equal distances. Read the operation label before interpreting the next value.

Worked example 1: Cross a boundary by ten. For 190 + 10, combine nineteen tens and one ten to make twenty tens, or 200. Explain the regrouping instead of writing an impossible two-digit entry in the tens place.

Worked example 2: Count backward carefully. From 312, subtract ten repeatedly: 302, 292, 282. Check each change by adding ten back. The sequence does not jump from 302 to 202, which would remove one hundred.

Worked example 3: Distinguish step sizes. From 237, +10 gives 247 and +100 gives 337. Both additions preserve the seven ones, but they add different amounts. A similarity in the ones digit does not make the two operations equivalent.

Guided practice: Choose starting numbers near ten and hundred boundaries and count aloud in both directions. Use base-ten models when a boundary is confusing, then return to mental reasoning. Mix +10, −10, +100 and −100 problems from suitable starting values.

Independent tasks: Continue unfamiliar one-, ten- and hundred-step sequences, skip-count by five from a multiple of five and explain a mental place-value change. Include forward and backward examples; success on a memorized forward sequence alone is insufficient.

Feedback: If a learner changes the wrong place, name the value of the step and model it. If a sequence leaves the 0–1,000 range, revise the task's number of steps rather than silently changing its step size. Require a check of at least one boundary transition.