OA4-13 Rounding on a Number Line
Pages 80–81
STANDARDS
3.NBT.A.1, 4.NBT.A.3
Goals
Students will round to the closest ten, except when the number is exactly
halfway between a multiple of ten.
PRIOR KNOWLEDGE REQUIRED
Number lines
Concept of closer
MATERIALS
number cards, string or tape, rope, a ring or hoop (see Activity)
The closest ten. Show a number line from 0 to 10 on the board:
012345678910
Circle the 2 and ask if the 2 is closer to the 0 or to the 10. When students
answer 0, draw an arrow from the 2 to the 0 to show the distance. Repeat
with several examples and then ASK: Which numbers are closer to 0?
Which numbers are closer to 10? Which number is a special case? Why
is it a special case?
Tell students you want to round numbers to the nearest ten. ASK: Would
you round 6 to 0 or to 10? (10) Why? (because it is closer to 10 than to 0)
Exercises:
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a)
b)
c)
d)
3 rounded to the nearest ten is
2 rounded to the nearest ten is
8 rounded to the nearest ten is
9 rounded to the nearest ten is
.
.
.
.
Give students a minute to answer these questions, then take up the answers
by having students signal rounding up to 10 with a thumbs up and signal
rounding down to 0 with a thumbs down.
Then draw an incorrect number line with numbers not equally spaced, so
that 4 appears closer to 10 than to 0. ASK: Is 4 closer to 0 or to 10 on
this number line? (closer to 10) Why? (because you drew the number line
incorrectly) Explain that when mathematicians say that 4 is closer to 0
than to 10, they mean 4 is closer to 0 than to 10 on any number line where
the numbers are equally spaced. The number line has to be drawn properly.
For the same reason, graph paper is printed with equally spaced lines, and
there is equal spacing between measurements on thermometers, measuring
cups, rulers, and many other instruments.
Operations and Algebraic Thinking 4-13
D-1
Connection
Real World
Ask students whether they’ve seen other number lines used to measure
different things (e.g., on thermometers, measuring cups, radio stations).
Ask students to identify cases where it’s acceptable to round a measurement
(e.g., indoor or outdoor temperature, percent of precipitation). Ask them
to identify cases where it is important not to round numbers from a number
line or measurement (e.g., body temperature, radio station, amount of
medication). Conclude by saying that rounding on a number line or a
measurement can be useful, but we should not do it in all situations.
Then draw a number line from 10 to 30, with 10, 20, and 30 a different color
than the other numbers.
10 11 12131415 1617 181920212223 24252627282930
Circle various numbers (not 15 or 25) and ask volunteers to draw an
arrow showing which number they would round to if they had to round
to the nearest ten.
Repeat with a number line from 50 to 70, again writing the multiples of 10
in a different color. Then repeat with number lines from 230 to 250 or 370
to 390, etc.
Ask students for a general rule to tell which ten a number is closest to. What
digit should they look at? (the ones digit) How can they tell from this digit
which multiple of ten a number is closest to?
Then ask students to determine the closest multiple of ten given two choices
instead of a number line. Example: Is 24 closer to 20 or 30? Is 276 closer to
270 or 280?
Write the numbers from 30 to 40 on cards. Use different colors to make
the numbers 30 and 40 stand out from the rest. Attach the 11 cards to
a rope using tape or by perforating the cards and attaching them with
string. Make sure that the cards are equally spaced by 10 cm. Also,
make sure that the entire line is centered on the rope. This is achieved
most easily if the card labeled 35 is attached to the center of the rope
and used as a reference for the placement of the rest of the cards. The
ends of the rope will be longer than is indicated in the diagram below.
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
40
Take a ring or hoop and pull the rope through it. A hoop used for
cross-stitching would be a good size.
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
40
D-2
Teacher’s Guide for AP Book 4.1
322
333
3
344
3
35
35
36
36
37
37
38
38
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ACTIVITY
30
31
32
33
34
35
36
37
38
39
40
Ask two volunteers to hold the number line taut. Ask a volunteer to find
the middle number between 30 and 40. ASK: How do you know that this
number is in the middle? What do you have to check? (the distance to
the ends
the rope—make
volunteer
do that)
volunteer stand
30 of 31
32
33
34 a 35
37
38 Let
39 a40
36
behind the line holding the middle number (35). Explain to students that
the volunteers and the number line make a rounding machine. The
machine will automatically round the number to the nearest ten. Put the
ring on 32. Ask the volunteer holding the middle of the line to pull up
from 35, so that the ring slides to 30.
35
30
31
32
33
34
36
37
38
39
40
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Try more numbers. Ask students to explain why the machine
works. You can repeat this activity with the multiples of 10 going up
to 100 or with any other numbers. Do not include numbers that
require rounding the digit 5.
Operations and Algebraic Thinking 4-13
D-3
OA4-14 R
ounding on a Number Line
Pages 82–83
(Hundreds and Thousands)
STANDARDS
4.NBT.A.4
Goals
Students will round to the closest hundred or thousand, except when
the number is halfway between a multiple of a hundred or a thousand.
PRIOR KNOWLEDGE REQUIRED
Number lines
Concept of closer
The closest hundred. Repeat the previous lesson with a number line from
0 to 100 that shows only the multiples of 10.
0102030405060708090100
At first, ask students whether numbers that are multiples of 10 (30, 70, 60,
and so on) are closer to 0 or 100. Example: Is 40 closer to 0 or 100? Draw
an arrow to show this. Repeat with several examples, then ASK: Which
multiples of 10 are closer to 0 and which multiples of 10 are closer to 100?
Which number is a special case? Why is it a special case?
Ask students for a general rule to tell which multiple of a hundred a number
is closest to. What digit should they look at? (the tens digit) How can they
tell from this digit which multiple of a hundred a number is closest to?
When is there a special case? Emphasize that the number is closer to the
higher multiple of 100 if its tens digit is 6, 7, 8, or 9, and it’s closer to the
lower multiple of 100 if its tens digit is 1, 2, 3, or 4. If the tens digit is 5, then
any ones digit except 0 will make it closer to the higher multiple. Only when
the tens digit is 5 and the ones digit is 0 do we have a special case where it
is not closer to either.
Repeat the lesson for thousands, emphasizing the importance of considering
the value of the hundreds digit.
Extensions
1. Rounding to the nearest ten, how many numbers round down to...
a) 20? (four: 21, 22, 23, 24)
b) 200? (four: 201, 202, 203, 204)
c) 2,000? (four: 2,001, 2,001, 2,003, 2,004)
D-4
Teacher’s Guide for AP Book 4.1
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Then ask students about numbers that are not multiples of 10. First ask
them where they would place the number 33 on the number line. Have a
volunteer show this. Then ask the rest of the class if 33 is closer to 0 or to
100. Repeat with several numbers. Repeat with a number line from 100 to
200 and another number line from 700 to 800.
2. a)Rounding to the nearest ten, how many numbers round down to 30?
(four: 31, 32, 33, 34)
b)Rounding to the nearest hundred, how many numbers round down
to 300? (forty-nine: 301, 302, 303, … , 349)
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c)Rounding to the nearest thousand, how many numbers round down
to 3,000? (four hundred ninety-nine: 3,001, 3,002, 3,003, … , 3,499)
Operations and Algebraic Thinking 4-14
D-5
OA4-15 Rounding
Pages 84–85
STANDARDS
3.NBT.A.1, 4.NBT.A.3
Goals
Students will round whole numbers to the nearest ten, hundred,
or thousand.
PRIOR KNOWLEDGE REQUIRED
Knowing which multiples of ten, a hundred, or a thousand a number
is between
Finding which multiple of ten, a hundred, or a thousand a given
number is closest to
Review rounding 2-digit numbers. Have students round 2-digit numbers
to the nearest ten. Do not at first include numbers that have ones digit 5.
Ask students how they know which multiples of ten the number is between.
ASK: How many tens are in 37? How many would be one more ten? So 37
is between 3 tens and 4 tens; that means it’s between 30 and 40. Which
multiple of ten is it closer to?
Connection
Real World
Rounding 2-digit numbers when the ones digit is 5. Tell students that
when the ones digit is 5, a number is not closer to either the smaller or
the larger ten, but we always round up. Give students many examples to
practice with: 25, 45, 95, 35, 15, 5, 85, 75, 55, 65. If some students find this
hard to remember, you could share the following analogy: I am trying to cross
the street, but there is a big truck coming, so when I am partway across I have
to decide whether to keep going or turn back. If I am less than halfway across,
it makes sense to turn back because I am less likely to get hit. If I am more
than halfway across, it makes sense to keep going because I am again less
likely to get hit. But if I am exactly halfway across, what should I do? Each
choice gives me the same chance of getting hit. Have students discuss what
they would do and why. Remind them that they are, after all, trying to cross
the street. So actually, it makes sense to keep going rather than to turn back.
That will get them where they want to be.
Here is another way to help students remember which way to round: Write
out all the 2-digit numbers that have the same tens digits, for example,
3 (30, 31, ... , 39). ASK: Which numbers should we round to 30 because
they’re closer to 30 than 40? Which numbers should we round to 40
because they’re closer to 40 than to 30? How many are in each list?
Where should we put 35 so that the lists are equal?
Rounding 3-digit numbers to the nearest ten. SAY: To round 3-digit
numbers to the nearest ten we look at the ones digit, the same way we
do when we round 2-digit numbers to the nearest ten. The difference is
that there is now a digit we need to ignore (the hundreds digit).
D-6
Teacher’s Guide for AP Book 4.1
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ASK: How many tens are in 94? How many would be one more ten? What
number is ten tens? So 94 is between what two multiples of 10? Which
multiple of ten is it closer to? Repeat this line of questioning for 97.
Write on the board:
372
SAY: The ones digit is less than 5, so we round down. Round 372 to 370.
Exercises: Round each number to the nearest ten.
a)
174b)
885c)
341d)
936
Bonus
e)3,456
f)28,712
Answers: a) 170, b) 890, c) 340, d) 940, e) 3,460, f) 28,710
Rounding 3-digit numbers to the nearest hundred. Write on the board:
240, 241, 242, 243, 244, 245, 246, 247, 248, 249
ASK: What do these numbers all have in common? (3 digits, hundreds digit
2, tens digit 4) Are they closer to 200 or 300? (200) How do you know?
PROMPT: On a number line, what number is halfway from 200 to 300? (250)
Point out that all these numbers are less than halfway from 200 to 300, so
they are closer to 200 than to 300. ASK: Did you need to look at the ones
digit to check that these numbers are closer to 200 than to 300? (no) What
digit did you look at? (the tens digit) Summarize by saying that, to round to
the nearest hundred, we need to look at the tens digit.
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Then tell students to look at these numbers: 250, 251, 252, 253, 254, 255,
256, 257, 258, 259. ASK: Which hundred are these numbers closest to?
Are they all closest to 300 or is there one that’s different? Why is that one
a special case? If you saw that the tens digit was 5 but you didn’t know
the ones digit, and you had to guess if the number was closer to 200 or 300,
what would your guess be? Would the number ever be closer to 200?
Tell students that when rounding a number to the nearest hundred, mathe­
maticians decided to make it easier and say that if the tens digit is a 5, you
always round up. It doesn’t make any more sense to round 250 to 200 than
to 300, so you might as well round it up to 300 like you do all the other
numbers that have hundreds digit 2 and tens digit 5.
Then ASK: When rounding a number to the nearest hundred, what digit do
we need to look at? (the tens digit) Write on the board:
Round to the nearest hundred: 234
547
651
850
973
Have a volunteer underline the hundreds digit because that is what they are
rounding to. Then ask another volunteer to draw an arrow to the digit that
determines whether to round up or round down, so the board now looks like:
2 3 4
5 4 7
6 5 1
8 5 0
973
ASK: Where is that digit compared to the underlined digit? (It is the next one
to the right.) How do you know when to round down and when to round
up? Have another volunteer decide in each case whether to round up or
down and write the correct rounded number.
Operations and Algebraic Thinking 4-15
D-7
Repeat with 4-digit numbers, having students again round to the nearest
hundred. Examples: 5,439, 2,964, 8,249. Then round 4-digit and 5-digit
numbers to the nearest thousand. Examples: 4,520, 73,618, 2,388,
28,103, 87,093, 9,843.
To help ensure that students round correctly, suggest that they always
underline the digit they are rounding to, then point their pencil tip at
the digit to the right of the one they underlined. This digit will tell them
whether to round up or down. If any students are having trouble with
rounding, teach them to round on a grid as shown on AP Book 4.1
p. 86–87 (Lesson OA4-6).
Extension
Another way to round a number to the nearest ten. First, add 5. Then,
replace the ones digit in the answer with 0. Example:
36 + 5 = 41
40 32 + 5 = 37
30
To round to the nearest hundred, add 50 instead of 5. The rounded number
will be the answer with the ones and tens digits replaced with 0. Example:
842 + 50 = 892
800
573 + 50 = 623
600
You can explain why this works as follows: The number 842 is between 800
and 900. Any number between 800 and 900 will round up to 900 if it is at
least halfway to 900. When you add half of 100 to a number that is less than
halfway to 900, you get a number still in the 800s; when you add half of 100
to a number that is at least halfway to 900, you get a number that is in the
900s. You can liken this to pouring liquid into a container that is half full. If
the amount you are pouring is at least half a container, you will reach the
top, and maybe overflow. If the amount you are pouring is less than half a
container, you will not reach the top.
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Challenge students to make up a rule for using this method to round to
the nearest thousand.
D-8
Teacher’s Guide for AP Book 4.1
OA4-16 Rounding on a Grid
Pages 86–87
STANDARDS
4.NBT.A.3
Goals
Students will round whole numbers to the nearest ten, hundred,
thousand, ten thousand, or hundred thousand.
PRIOR KNOWLEDGE REQUIRED
Knowing which multiples of ten, a hundred, or a thousand
a number is between
Finding which multiple of ten, a hundred, or a thousand a given
number is closest to.
Rounding on a grid (without regrouping). Show students how numbers
can be rounded on a grid. Follow the steps shown below to round 12,473 to
the nearest thousand.
Step 1: As before, underline the digit you are rounding to and put your pencil
on the digit to the right. If the digit under your pencil is 5, 6, 7, 8, or 9, you
will round up; write “round up” beside (or above) the grid. If the digit under
your pencil is 0, 1, 2, 3, or 4, you will round down; write “round down.” The
hundreds digit here is 4, so we write “round down.”
Round Down
1
2
4
7
3
Step 2: To round up, add 1 to the underlined digit; to round down, keep the
digit the same. In this case, we are rounding down, so we copy the 2.
1
2
4
7
3
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2
Isolate Step 2. If any students are struggling with Step 2, make up several
examples where the first step is done for them so that they can focus only
on rounding the underlined digit up or down. Sample questions:
Exercises: Round the underlined digit up or down as indicated:
Round Down
1
6
4
7
3
Round Up
2
0
7
5
2
Round Up
5
8
2
1
5
Once all students have mastered Step 2, move on.
Operations and Algebraic Thinking 4-16
D-9
Step 3: Change all numbers to the right of the rounded digit to zeros. Leave
all numbers to the left of the rounded digit as they were. The number 12,473
rounded to the nearest thousand is 12,000.
Round Down
1
2
4
7
3
1
2
0
0
0
Rounding with regrouping. When students have mastered rounding without
regrouping, give them several examples that demand regrouping as well.
Warn them that the digits to the left of the rounded digit might change now.
Round the 9 up to 10 and then regroup the 10 hundreds as 1 thousand and
add it to the 7 thousand.
Round Up
1
7
9
Regroup
7
8
1
7
9
7
8
10 0
0
1
8
0
0
0
Another way to do this rounding. Do an example together: Underline the
digit you want to round to and decide whether to round up or down, as
before, then change all numbers to the right of the rounded digit to zeros.
We obtain (for the same example as above):
Round Up
1
7
9
7
8
0
0
Then ASK: Which two hundreds is the number between? PROMPT: How
many hundreds are in 17,978? (179 hundreds are in 17,978, so the number
is between 179 hundreds and 180 hundreds) Remind students that we are
rounding up (point to the picture) because the tens digit is 7—the number
is closer to 180 hundreds than to 179 hundreds. Complete the rounding by
writing 180, not 179, as the number of hundreds.
1
7
9
7
8
1
8
0
0
0
Point out that both ways of rounding get the same answer.
Rounding larger numbers. Show grids with larger numbers, where
regrouping does not occur. Ask students to round 538,226 to the nearest
ten thousand. Here is the result they should obtain:
Round Up
D-10
5
3
8
2
2
6
5
4
0
0
0
0
Teacher’s Guide for AP Book 4.1
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Round Up
Repeat with numbers where there is regrouping. Ask students to round
745,391 to the nearest ten. Here is the result they should obtain:
Round Up
7
4
5
3
9
6
7
4
5
4
0
0
Round three more large numbers that require regrouping as a class:
a) round 439,889 to the nearest thousand
b) round 953,219 to the nearest hundred thousand
c) round 595,233 to the nearest ten thousand
Students can round more such numbers individually.
Exercises: Round each number to the given digit.
a)thousands
3
9
6
7
b)hundreds
3
1
2
9
7
1
c) ten thousands
1
2
9
9
3
d) hundreds
6
4
9
9
8
7
2
Extensions
1.
Regrouping twice when rounding. If you have an advanced class,
you can try teaching this to the whole class. SAY: Sometimes rounding
forces you to regroup two or more numbers. Let’s see what to do when
this happens. We want to round 3,999 to the nearest ten.
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First, we round 90 up to 100.
3
9
9
9
10
Then we regroup the 10 tens as 1 hundred and add it to the 9 hundreds.
This gives 10 hundreds (1,000).
3
9
9
9
10 0
Operations and Algebraic Thinking 4-16
D-11
Now we regroup the 10 hundreds as 1 thousand and add it to the
3 thousands.
3
9
9
4
0
0
9
To finish, we complete the rounded number by adding any
missing zeros.
3
9
9
9
4
0
0
0
SAY: Let’s do another example with a bigger number, such as 799,994.
We will round this number to the nearest hundred.
First, we round 900 up to 1,000.
7
9
9
9
9
4
10
Then we regroup the 10 hundreds as 1 thousand and add them to the
9 thousands in the original number. Now we have 10 thousands.
7
9
9
9
9
4
10 0
ASK: What’s the next step? (We regroup the 10 thousands as 1 ten
thousand and add it to the 9 ten thousands in the original number. Now
we have 10 ten thousands, or 1 hundred thousand (100,000).)
9
9
9
10 0
0
9
4
ASK: What’s the next step? (We add 1 to the hundred thousands.)
7
9
9
9
8
0
0
0
9
4
ASK: How do we finish the rounding? (We complete the rounded
number by adding any missing zeros.)
D-12
7
9
9
9
9
4
8
0
0
0
0
0
Teacher’s Guide for AP Book 4.1
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7
Have students round 3,997 to the nearest hundred and to the nearest
ten; 73,992 to the nearest hundred and to the nearest ten; and 39,997
to the nearest hundred and to the nearest thousand.
2. If 48,329 is rounded to 48,300, what digit has it been rounded to?
(MP.1)
3. a)Write down all numbers that round to 40 when rounded to the
nearest ten. How many such numbers are there?
b)Write down all numbers that round to 800 when rounded to the
nearest ten. How many such numbers are there?
c)What is the smallest number that rounds to 800 when rounded to
the nearest hundred? What is the largest number that rounds to
800 when rounded to the nearest hundred? How many numbers
round to 800 when rounded to the nearest hundred? Hint: If you
wrote down all the numbers from 1 to the largest number you
found, and took away all the numbers that don’t round to 800,
how many numbers would still be in the list?
Answers
a) 35, 36, 37, 38, 39, 40, 41, 42, 43, 44; there are 10 such numbers
b)795, 796, 797, 798, 799, 800, 801, 802, 803, 804; again, there are
10 such numbers
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c)750 is the smallest such number, and 849 is the largest.
There are 849 numbers from 1 to 849. We don’t want to include
all the numbers from 1 to 749. So there are 849 − 749 = 100
numbers in the list.
Operations and Algebraic Thinking 4-16
D-13
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
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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
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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
D-17