Numbers to 100,000
4A · Chapter 1 · Lesson 1 · MATH-4A-C01-L01
Learning goal. Represent and explain five-digit quantities using place-value exchanges, expanded forms and number names.
Before you begin
Recall: Grade 3: four-digit place value, regrouping, comparison and rounding. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
What is the value of the 7 in 7,436?
Show answer & reasoning
Seven thousands is 7,000.
Write 4,205 in expanded form.
Show answer & reasoning
The tens place contains zero tens.
Round 6,382 to the nearest hundred.
Show answer & reasoning
6,382 lies above the midpoint 6,350.
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
A five-digit number represents whole groups of ten-thousands, thousands, hundreds, tens and ones. For 42,306, the 4 does not mean four ones: it represents four ten-thousands. Ten thousands exchange for one ten-thousand without changing the quantity. Zero can hold an empty place, so 42,306 and 4,236 are not the same number.
Model it: place-value chart, bundles of 10,000. 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
In 13,106, what is the value of the digit in the ten-thousands place?
The digit 1 represents 1 groups of 10,000, or 10,000.
Check your understanding 1
In 34,108, what is the value of the digit in the ten-thousands place?
Show answer & reasoning
The digit 3 represents 3 groups of 10,000, or 30,000.
Worked example 2
Write 67,382 as a number of whole thousands and remaining ones.
67 × 1,000 + 382 = 67,382. This grouping preserves the original value.
Check your understanding 2
Write 40,370 as a number of whole thousands and remaining ones.
Show answer & reasoning
40 × 1,000 + 370 = 40,370. This grouping preserves the original value.
Worked example 3
Find 76,945 − 75,945 and explain the place-value change.
Only the increment changes: 76,945 − 75,945 = 1,000.
Check your understanding 3
Find 163,477 − 63,477 and explain the place-value change.
Show answer & reasoning
Only the increment changes: 163,477 − 63,477 = 100,000.
Guided practice
In 66,051, what is the value of the digit in the ten-thousands place?
Show answer & reasoning
The digit 6 represents 6 groups of 10,000, or 60,000.
Write 94,723 as a number of whole thousands and remaining ones.
Show answer & reasoning
94 × 1,000 + 723 = 94,723. This grouping preserves the original value.
Find 178,384 − 78,384 and explain the place-value change.
Show answer & reasoning
Only the increment changes: 178,384 − 78,384 = 100,000.
Multiply 62,972 by 10.
Show answer & reasoning
Scaling each place value by 10 gives 629,720.
How many complete groups of 10 can be made from 52,702 objects, and how many remain?
Show answer & reasoning
52,702 = 5270 × 10 + 2.
Independent practice
Write 37,451 as a number of whole thousands and remaining ones.
Show answer & reasoning
37 × 1,000 + 451 = 37,451. This grouping preserves the original value.
Find 98,567 − 88,567 and explain the place-value change.
Show answer & reasoning
Only the increment changes: 98,567 − 88,567 = 10,000.
Multiply 1,000 by 1000.
Show answer & reasoning
Scaling each place value by 1000 gives 1,000,000.
How many complete groups of 10 can be made from 53,658 objects, and how many remain?
Show answer & reasoning
53,658 = 5365 × 10 + 8.
From 35,898 add 3 groups of 100.
Show answer & reasoning
3 groups of 100 are 300; then add to the starting value.
In 36,181, what is the value of the digit in the ten-thousands place?
Show answer & reasoning
The digit 3 represents 3 groups of 10,000, or 30,000.
Write 34,690 as a number of whole thousands and remaining ones.
Show answer & reasoning
34 × 1,000 + 690 = 34,690. This grouping preserves the original value.
Find 87,888 − 86,888 and explain the place-value change.
Show answer & reasoning
Only the increment changes: 87,888 − 86,888 = 1,000.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student says the 5 in 54,105 means 5 ones. Explain its actual value.
Show answer & reasoning
The 5 occupies the ten-thousands position, representing 5 groups of 10,000.
From 92,982 add 1 groups of 100.
Show answer & reasoning
1 groups of 100 are 100; then add to the starting value.
Exit check
Multiply 16,064 by 10.
Show answer & reasoning
Scaling each place value by 10 gives 160,640.
How many complete groups of 100 can be made from 24,286 objects, and how many remain?
Show answer & reasoning
24,286 = 242 × 100 + 86.
From 45,982 add 3 groups of 1,000.
Show answer & reasoning
3 groups of 1,000 are 3,000; then add to the starting value.