Calculations and Place Value
4A · Chapter 1 · Lesson 7 · MATH-4A-C01-L07
Learning goal. Use powers-of-ten place shifts and decompositions to calculate without changing quantities.
Before you begin
Recall: Grade 3: four-digit place value, regrouping, comparison and rounding; Rounding 6-Digit Numbers. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Round 344,142 to the nearest 1,000.
Show answer & reasoning
The bounding multiples are 344,000 and 345,000. The midpoint is 344,500, so the nearer benchmark is 344,000.
What is the midpoint between 816,000 and 817,000?
Show answer & reasoning
Half of the interval of 1,000 is 500. Add it to the lower benchmark.
Explain why rounding 401,224 to the nearest 1,000 gives 401,000.
Show answer & reasoning
401,224 is between 401,000 and 402,000; the midpoint is 401,500.
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
Place value lets you compute with groups of tens, hundreds and thousands instead of treating long numerals as indivisible strings. Multiplying by 100 scales every unit one hundredfold; adding 3 thousands changes the number by exactly 3,000. Show the equivalent expanded forms and explain each exchange before relying on a written shortcut.
Model it: place-value arrows and expanded-form regrouping. 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
Calculate 5,862 × 10 using place value.
Every place-value unit becomes 10 times as large; 5,862 × 10 = 58,620.
Check your understanding 1
Calculate 4,343 × 10 using place value.
Show answer & reasoning
Every place-value unit becomes 10 times as large; 4,343 × 10 = 43,430.
Worked example 2
Calculate 5,325 × 100 using place value.
Every place-value unit becomes 100 times as large; 5,325 × 100 = 532,500.
Check your understanding 2
Calculate 3,233 × 10 using place value.
Show answer & reasoning
Every place-value unit becomes 10 times as large; 3,233 × 10 = 32,330.
Worked example 3
Calculate 180,560 ÷ 10 by grouping place-value units.
Dividing by 10 reduces each place-value unit by a factor of 10. The quotient is 18,056.
Check your understanding 3
Calculate 167,660 ÷ 10 by grouping place-value units.
Show answer & reasoning
Dividing by 10 reduces each place-value unit by a factor of 10. The quotient is 16,766.
Guided practice
Calculate 6,888 × 100 using place value.
Show answer & reasoning
Every place-value unit becomes 100 times as large; 6,888 × 100 = 688,800.
Calculate 8,252 × 100 using place value.
Show answer & reasoning
Every place-value unit becomes 100 times as large; 8,252 × 100 = 825,200.
Calculate 728,600 ÷ 100 by grouping place-value units.
Show answer & reasoning
Dividing by 100 reduces each place-value unit by a factor of 100. The quotient is 7,286.
Find 50,809 + 10,000. Explain which place-value units change.
Show answer & reasoning
Adding 10,000 increases the number by that place-value unit, with exchanges if needed: 60,809.
A teacher partitions 16,350 counters into equal groups of 10. How many counters are in each group?
Show answer & reasoning
Each group has 16,350 ÷ 10 = 1,635 counters.
Independent practice
Calculate 9,546 × 10 using place value.
Show answer & reasoning
Every place-value unit becomes 10 times as large; 9,546 × 10 = 95,460.
Calculate 340,100 ÷ 100 by grouping place-value units.
Show answer & reasoning
Dividing by 100 reduces each place-value unit by a factor of 100. The quotient is 3,401.
Find 29,000 + 1,000. Explain which place-value units change.
Show answer & reasoning
Adding 1,000 increases the number by that place-value unit, with exchanges if needed: 30,000.
A teacher partitions 45,150 counters into equal groups of 10. How many counters are in each group?
Show answer & reasoning
Each group has 45,150 ÷ 10 = 4,515 counters.
Calculate 48,795 + 4,142 and check by subtraction.
Show answer & reasoning
48,795 + 4,142 = 52,937; subtract 4,142 to recover 48,795.
Calculate 16,839 × 10 using place value.
Show answer & reasoning
Every place-value unit becomes 10 times as large; 16,839 × 10 = 168,390.
Calculate 14,357 × 10 using place value.
Show answer & reasoning
Every place-value unit becomes 10 times as large; 14,357 × 10 = 143,570.
Calculate 97,610 ÷ 10 by grouping place-value units.
Show answer & reasoning
Dividing by 10 reduces each place-value unit by a factor of 10. The quotient is 9,761.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A learner multiplies 5,500 by 100 and writes 55,000. Identify the scaling mistake and give the correct result.
Show answer & reasoning
Multiplying by 100 makes every place value one hundred times larger. The correct product is 550,000.
Calculate 46,668 + 3,456 and check by subtraction.
Show answer & reasoning
46,668 + 3,456 = 50,124; subtract 3,456 to recover 46,668.
Exit check
Find 76,193 + 1,000. Explain which place-value units change.
Show answer & reasoning
Adding 1,000 increases the number by that place-value unit, with exchanges if needed: 77,193.
A teacher partitions 68,520 counters into equal groups of 10. How many counters are in each group?
Show answer & reasoning
Each group has 68,520 ÷ 10 = 6,852 counters.
Calculate 36,044 + 2,582 and check by subtraction.
Show answer & reasoning
36,044 + 2,582 = 38,626; subtract 2,582 to recover 36,044.