Estimation and Division
4A · Chapter 5 · Lesson 2 · MATH-4A-C05-L02
Learning goal. Estimate quotients with compatible numbers and use bounds to flag unreasonable answers.
Before you begin
Recall: Grade 3: single-digit division and remainder; 4A Ch4: multiplication estimates; Mental Math for Division. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Calculate 168 ÷ 6 by breaking apart the dividend.
Show answer & reasoning
Decompose 168 into convenient multiples of 6; the resulting quotients sum to 28.
Calculate 28 ÷ 2 by breaking apart the dividend.
Show answer & reasoning
Decompose 28 into convenient multiples of 2; the resulting quotients sum to 14.
Calculate 320 ÷ 8 by breaking apart the dividend.
Show answer & reasoning
Decompose 320 into convenient multiples of 8; the resulting quotients sum to 40.
Key vocabulary
- dividend
- number being divided
- divisor
- number of equal groups or group size
- quotient
- result of division
- remainder
- amount left after complete groups
Understand the concept
Estimate a quotient before applying a written division procedure. For 3,920 ÷ 8, use 4,000 ÷ 8 = 500 as a benchmark; the exact answer should be close. A sensible estimate lets you notice quotient digits written in the wrong place. Verify with multiplication afterward.
Model it: compatible-number benchmarks and number line. 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
Estimate 150 ÷ 5 to the nearest ten in the quotient.
The exact quotient has whole-number part 30; rounding that to the nearest ten gives 30.
Check your understanding 1
Estimate 2661 ÷ 9 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 295; rounding that to the nearest ten gives 300.
Worked example 2
Estimate 535 ÷ 2 to the nearest ten in the quotient.
The exact quotient has whole-number part 267; rounding that to the nearest ten gives 270.
Check your understanding 2
Estimate 757 ÷ 5 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 151; rounding that to the nearest ten gives 150.
Worked example 3
Estimate 635 ÷ 3 to the nearest ten in the quotient.
The exact quotient has whole-number part 211; rounding that to the nearest ten gives 210.
Check your understanding 3
Estimate 2823 ÷ 8 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 352; rounding that to the nearest ten gives 350.
Guided practice
Estimate 905 ÷ 3 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 301; rounding that to the nearest ten gives 300.
Estimate 4033 ÷ 9 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 448; rounding that to the nearest ten gives 450.
Estimate 344 ÷ 5 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 68; rounding that to the nearest ten gives 70.
Is 3019 ÷ 7 closer to 430 or 450?
Show answer & reasoning
Compare compatible products with 7; the quotient is near 431, which is closer to 430.
Is 1927 ÷ 8 closer to 240 or 260?
Show answer & reasoning
Compare compatible products with 8; the quotient is near 240, which is closer to 240.
Independent practice
Estimate 2050 ÷ 8 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 256; rounding that to the nearest ten gives 260.
Estimate 2097 ÷ 6 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 349; rounding that to the nearest ten gives 350.
Is 628 ÷ 4 closer to 160 or 180?
Show answer & reasoning
Compare compatible products with 4; the quotient is near 157, which is closer to 160.
Is 1040 ÷ 7 closer to 150 or 170?
Show answer & reasoning
Compare compatible products with 7; the quotient is near 148, which is closer to 150.
Is 373 ÷ 3 closer to 120 or 140?
Show answer & reasoning
Compare compatible products with 3; the quotient is near 124, which is closer to 120.
Estimate 578 ÷ 3 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 192; rounding that to the nearest ten gives 190.
Estimate 179 ÷ 3 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 59; rounding that to the nearest ten gives 60.
Estimate 427 ÷ 2 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 213; rounding that to the nearest ten gives 210.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
Estimate 2,462 ÷ 6 as about 400 or about 4,000. Explain the best estimate.
Show answer & reasoning
2,400 ÷ 6 = 400, a close compatible-number estimate for 2,462 ÷ 6.
Estimate 955 ÷ 6 to the nearest ten in the quotient.
Show answer & reasoning
The exact quotient has whole-number part 159; rounding that to the nearest ten gives 160.
Exit check
Is 1736 ÷ 5 closer to 350 or 370?
Show answer & reasoning
Compare compatible products with 5; the quotient is near 347, which is closer to 350.
Is 724 ÷ 3 closer to 240 or 260?
Show answer & reasoning
Compare compatible products with 3; the quotient is near 241, which is closer to 240.
Is 2410 ÷ 7 closer to 340 or 360?
Show answer & reasoning
Compare compatible products with 7; the quotient is near 344, which is closer to 340.