OA4-17 Estimating Sums and Differences
Pages 88–89
STANDARDS
4.OA.A.3
Vocabulary
estimating
the approximately equal
to sign ( ≈ )
Goals
Students will estimate sums and differences by rounding each addend to
the nearest ten, hundred, thousand, ten thousand, or hundred thousand.
PRIOR KNOWLEDGE REQUIRED
Rounding to the nearest ten, hundred, thousand, ten thousand,
or hundred thousand
Estimations in calculations. Show students how to estimate 52 + 34 by
rounding each number to the nearest ten: 50 + 30 = 80. SAY: Since 52 is
close to 50 and 34 is close to 30, 52 + 34 will be close to, or approximately,
50 + 30. Mathematicians have invented a sign to mean “approximately
equal to.” It’s a squiggly equal sign: “≈.” So we can write 52 + 34 ≈ 80. It
would not be right to put 52 + 34 = 80 because they are not actually equal;
they are just close to, or approximately equal.
Connection
Real World
Tell students that when they round up or down before adding, they aren’t
finding the exact answer, they are just estimating. They are finding an
answer that is close to the exact answer. ASK: When do you think it
might be useful to estimate answers? Sample answer: in a grocery store,
estimating total price or change expected.
Have students estimate the sums of 2-digit numbers by rounding each
to the nearest ten. Remind them to use the approximately equal to sign.
Exercises:
a)41 + 38
d)84 + 13
93 − 21 ≈ 90 − 20
= 70
b) 52 + 11
e) 92 + 37
c) 73 + 19
f) 83 + 24
Then ASK: How would you estimate 93 − 21? Write the estimated
difference on the board (see margin).
a)53 − 21
d)48 − 17
b) 72 − 29
e) 63 − 12
c) 68 − 53
f) 74 − 37
Then have students practice estimating the sums and differences of
• 3-digit numbers by rounding to the nearest ten.
(Examples: 421 + 159, 904 − 219)
• 3- and 4-digit numbers by rounding to the nearest hundred.
(Examples: 498 + 123, 4,501 − 1,511)
• 4- and 5-digit numbers by rounding to the nearest thousand.
(Examples: 7,980 + 1,278, 13,891 − 11,990, 3,100 + 4,984)
• 5- and 6-digit numbers by rounding to the nearest ten thousand.
(Examples: 54,392 + 38,447, 679,029 − 626,928)
D-14
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Have students estimate the differences of 2-digit numbers by again
rounding each to the nearest ten. Exercises:
• 6- digit numbers by rounding to the nearest hundred thousand.
(Examples: 928,283 − 244,219, 467,835 + 384,234)
Is the estimate too high or too low? Write on the board:
3330
+ 41
+ 40
70
SAY: I estimated 33 + 41 to be 70. Do you think this is higher than the
actual answer or lower? (lower) Why? PROMPT: Is 30 more or less than
33? (less) Is 40 more or less than 41? (less) SAY: I rounded both numbers
down, so the sum I get will be less than the actual sum. Have students
verify this by calculating the actual sum. (74; indeed, 70 is less than 74)
(MP.8)
Exercises: Calculate both the actual sums and the rounded sums. Circle
the larger sum.
a)
3230 b)
2320 c)
4240
+ 41 + 40
+ 64 + 60
+ 73 + 70
Answers: a) 73, 70, b) 87, 80, c) 115, 110. The actual sum should be
circled in all cases.
ASK: Which sum is larger, the actual sum or the rounded sum? (always the
actual sum) Why was the actual sum always larger? (because the rounded
numbers were smaller than the actual numbers; we always rounded down)
Exercises: Calculate both the actual sums and the rounded sums. Circle
the larger sum.
a)
3640 b)
2930 c)
3740
+ 48 + 50
+ 86 + 90
+ 56 + 60
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 84, 90, b) 115, 120, c) 93, 100. The rounded sum should be
circled in all cases.
ASK: Which sum is larger, the actual sum or the rounded sum? (always
the rounded sum) Why was the rounded sum always larger? (because the
rounded numbers were larger than the actual numbers) Point out that when
both numbers are rounded up, the rounded sum is larger, and when both
numbers are rounded down, the rounded sum is smaller.
Exercises: Predict whether Ahmed’s estimate is too high or too low, then
check your prediction by calculating the actual sum.
a)
b)
c)
d)
e)
Ahmed estimates 63 + 71 as 60 + 70 = 130.
Ahmed estimates 752 + 689 as 800 + 700 = 1,500.
Ahmed estimates 432 + 514 as 430 + 510 = 940.
Ahmed estimates 23,912 + 14,706 as 20,000 + 10,000 = 30,000.
Ahmed estimates 65,532 + 23,964 as 66,000 + 24,000 = 90,000.
Answers
a)Too low because 60 is less than 63 and 70 is less than 71, so 60 + 70
will be less than 63 + 71. Indeed, 63 + 71 = 134 is more than 130.
Operations and Algebraic Thinking 4-17
D-15
b)Too high because 800 is more than 752 and 700 is more than 689, so
800 + 700 will be more than 752 + 689. Indeed, 752 + 689 = 1,441.
c) Too low. Indeed, 432 + 514 = 946.
d) Too low. Indeed, 23,912 + 14,706 = 38,618.
e) Too high. Indeed, 65,532 + 23,964 = 89,496.
(MP.2)
Recognizing when an answer is reasonable or not. For example, Daniel
added 273 and 385, and got the answer 958. Does this answer seem
reasonable? Students should see that even rounding both numbers up
gives a sum less than 900, so this answer can’t be correct.
Do the following answers seem reasonable? Invite students to explain
using estimates and perform the actual calculation to check their answers.
a) Xian added 444 and 222 and got 888.
b) Melissa added 196 and 493 and got 709.
c) Enrico added 417 and 634 and got 951.
(MP.8)
Rounding to smaller place values is more accurate. SAY: Let’s try
estimating the sum 353 + 828 by rounding to the tens and then to the
hundreds. ASK: Which way do you think will give an answer closer to
the actual sum? Write the sum on the board and get students to help you
round the numbers to the given place value and then do the calculation.
nearest ten:
nearest hundred:
353
353
≈350
353≈400
+ 828
+ 828 ≈ 830
+ 828 ≈ 800
1,180
1,200
ASK: The sum is closest to which answer, the one obtained by rounding
the tens or the hundreds? (the tens) Explain that the lower the place value
we round to in our estimation, the closer we get to the actual sum. Discuss
how this is similar to measuring. Measuring to the nearest millimeter is
more accurate and gives more information than measuring to the nearest
centimeter because millimeters are smaller than centimeters.
Do the same type of exercise with two 4-digit numbers: 5,938 + 8,213.
Round to the … tens: hundreds:thousands:
5,938
≈5,940
5,938≈5,900
5,938≈6,000
+ 8,213 ≈ 8,210
+ 8,213 ≈8,200
+ 8,213 ≈8,000
14,15014,10014,000
The actual sum is 14,151, so again rounding to the closest ten is the
most accurate.
Now estimate the sum 2,356 + 1,432 by rounding each number
to the nearest:
a)ten
b)hundred
c)thousand
d)ten thousand
D-16
Teacher’s Guide for AP Book 4.1
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Then calculate the sum of the two numbers and compare it with the two
values we just obtained by estimating. (353 + 828 = 1,181)
Have students put their answers in order from closest to the actual answer
to furthest from the actual answer. What do students notice? (rounding to
smaller place values is more accurate)
Point out that the answer to part d) above is 0 + 0 = 0. Emphasize that
rounding to too big a place value can become absurd. SAY: It would be like
rounding the distance from my desk to your desk to the nearest mile.
(MP.5)
Choosing between speed and accuracy. ASK: Was adding more accurate
when we rounded to the nearest tens, hundreds, or thousands? (tens)
ASK: Was adding faster when we rounded to the nearest tens, hundreds,
or thousands? (thousands) Why? (Adding 6,000 and 8,000 is as easy as
adding 6 and 8, two 1-digit numbers, but adding 5,900 and 8,200 is like
adding 59 and 82, two 2-digit numbers; 1-digit numbers are easier to add
than 2-digit numbers.) Point out that we often need to choose between
being fast and being more accurate. Sometimes we need more accuracy,
and sometimes we need to be faster.
Extensions
1. a)Estimate 427 + 516 by rounding both numbers to the nearest
hundred. Is your estimate higher or lower than the actual answer?
b) Estimate 427 + 516 by rounding both numbers to the nearest ten.
Is your estimate higher or lower than the actual answer?
Bonus: Make up another question where rounding to the nearest
hundred is lower than the actual answer, but rounding to the nearest
ten is higher than the actual answer.
(MP.3)
2.Have students investigate when rounding one number up and one
number down is better than rounding each to the nearest hundred by
completing the following chart and circling the estimate that is closest
to the actual answer:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
763
+751
Actual
Answer
796
+389
648
+ 639
602
+ 312
329
+ 736
1,514
Round to 800 + 800
the Nearest
= 1,600
Hundred
Round One
700 + 800
Up and
Round One = 1,500
Down
Operations and Algebraic Thinking 4-17
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