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Lesson 1.2
Round to the Nearest Ten or
Hundred
• Between which two tens is 32? Which ten is
it closer to? Explain. 30 and 40. 32 is closer to 30.
Common Core Standard CC.3.NBT.1
Use Math Talk to focus students’ thinking on
the fact that more than one number would
round to 30.
• What makes a number able to be rounded
to 30? It must be 25 or greater, or less than 35.
• In which direction would you round 28 to
get to 30? up
• In which direction would you round 34 to
get to 30? down
You might draw a number line on the board
from 20 to 40 so students can see that the
numbers 25–29 and 31–24 would round to 30.
• Look at the second number line. How is
rounding to the nearest hundred similar to
rounding to the nearest ten? Possible answer:
It is only 2 numbers away from 30 but 8 numbers away
from 40.
Use place value understanding to round whole numbers to the
nearest 10 or 100.
Lesson Objective Round 2- and 3-digit numbers to
the nearest ten or hundred.
Essential Question How can you round numbers?
Vocabulary round
1 ENGAGE
Access Prior Knowledge Discuss with students
situations where you do not need to know an
exact number, but knowing about how much
or about how many is sufficient. For example,
the length of a car is about 10 feet. There are
about 100 seats in the auditorium. The height
of a house is about 20 feet.
• How are these numbers alike? They all have
zeros. They all tell about how many.
2 TEACH and TALK
Unlock the Problem
GO
Online
Animated
Math Models
when I round to the nearest ten, I look at the number
line to see which ten the number is closer to. When I
round to the nearest hundred, I look at the number line
to see which hundred the number is closer to.
• What is 144 rounded to the nearest
hundred? 100
MATHEMATICAL
PRACTICES
When would you round a number?
Discuss the problem. Be sure that students
understand that 32 is an exact number and
they need to round 32 to the nearest ten.
• What is an example of a rounded number?
© Houghton Mifflin Harcourt Publishing Company
Numbers with a zero at the end can be examples of
rounded numbers, such as 10, 20, 30, and so on.
• What are the tens that are closest to the
number 32? 30 and 40
c
One Way
• Why is a number line a good way to think
about which numbers should be rounded?
A number line can show how far apart numbers are
from each other so they can be compared easily.
• Why does the first number line include
tens and not hundreds? Possible answer: because
32 is a 2-digit number and I am rounding 32 to the
nearest 10
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Lesson 1.2
Name
Round to the Nearest Ten or Hundred
Essential Question How can you round numbers?
COMMON CORE STANDARD CC.3.NBT.1
Use place value understanding and properties of
operations to perform multi-digit arithmetic.
UNLOCK the Problem
When you round a number, you find a number
that tells you about how much or about how many.
Mia’s baseball bat is 32 inches long. What is its
length rounded to the nearest ten inches?
One Way Use a number line to round.
A Round 32 to the nearest ten.
32
0
10
30
20
40
Find which tens the number is between.
30
40 .
32 is between _
and _
30 than it is to _
40 .
32 is closer to _
MATHEMATICAL PRACTICES
Name three
other numbers that round
to 30 when rounded to
the nearest ten. Explain.
30 .
32 rounded to the nearest ten is _
So, the length of Mia’s bat rounded to the
© Houghton Mifflin Harcourt Publishing Company
B Round 174 to the nearest hundred.
174
0
100
200
© Houghton Mifflin Harcourt Publishing Company
Possible answer: 28, 31, 34;
possible explanation: 28 is
between 20 and 30 but it is
closer to 30; 31 and 34 are
between 30 and 40 but they
are closer to 30.
30
inches.
nearest ten inches is _
300
Find which hundreds the number is between.
100 and _
200 .
174 is between _
200 than it is to _
100 .
174 is closer to _
200 .
So, 174 rounded to the nearest hundred is _
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Try This!
Discuss with students that there are different
ways to round a number. Then, have students
complete Parts A and B.
• How is rounding a 3-digit number, like 718,
to the nearest ten similar to rounding the
number to the nearest hundred? Possible
answer: in each case, I find the two tens or hundreds
the number is between and then I decide which is
closer.
• How is rounding a 3-digit number to the
nearest ten different than rounding to the
nearest hundred? Possible answers: the number
lines I use are different. To round to the nearest ten, I
need to find the two tens that the number is between.
To round to the nearest hundred, I need to find the two
hundreds that the number is between. I write a zero for
the digit in the ones place when rounding to ten. I write
a zero for the digits in the tens and ones places when
rounding to hundred.
c
Another Way
Students should recognize that the result of
rounding using place value is the same as
rounding on a number line.
• How might rounding using place value be
quicker? Possible answer: I don’t have to draw a
DO NOT
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• Which place value digit should you look at
to decide if 168 should be rounded to 200?
Look at the tens place value.
• In which place value digit should you look
at to decide if 81 should be rounded to
100? Look at the tens place value.
• In which place value digit should you look
at to decide if 81 should be rounded to 80?
Look at the ones place value.
COMMON ERRORS
COMMON ERRORS
Error Students may round numbers
incorrectly because they do not look at
the place to the immediate right.
Example To round 718 to the nearest
hundred, students may look at the 8 and
round to 800.
Springboard to Learning Have students circle
the place to which they are rounding and
underline the number to the immediate right
before rounding.
number line to see the numbers.
• How is rounding to the nearest ten and
rounding to the nearest hundred using
place value the same? In each case, I look at
the digit to the right of the place I am rounding to. If
the digit is less than 5, the digit in the rounding place
stays the same. If the digit is 5 or greater, the digit in
the rounding place increases by 1. I write zeros for the
digits to the right of the rounding place.
© Houghton Mifflin Harcourt Publishing Company
Use Math Talk to show students how using
place value is similar to using a number line.
• How can you look at the ones place to tell
if 54 should be rounded up or down? If the
number in the ones place is 1, 2, 3, or 4, it should be
rounded down. If the number in the ones place is 5, 6,
7, 8, or 9, it should be rounded up.
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Try This! Round 718 to the nearest ten and hundred.
Locate and label 718 on the number lines.
A Nearest Ten
B Nearest Hundred
718
700
Check students’ work.
710
718
700
720
750
800
720 than it is
718 is closer to _
700 than it is
718 is closer to _
710 .
to _
800 .
to _
720 .
So, 718 rounds to _
700 .
So, 718 rounds to _
Another Way
Use place value.
A Round 63 to the nearest ten.
Think: The digit in the ones place tells if
the number is closer to 60 or 70.
63
l
3 , 5
So, the tens digit stays the same. Write 6 as the
tens digit.
• Find the place to which you want
to round.
• Look at the digit to the right.
• If the digit is less than 5, the digit
in the rounding place stays the same.
• If the digit is 5 or greater, the digit
in the rounding place increases by one.
• Write zeros for the digits to the
right of the rounding place.
Write zero as the ones digit.
So, 63 rounded to the nearest ten
B Round 457 to the nearest hundred.
Think: The digit in the tens place tells if
the number is closer to 400 or 500.
457
l
5 5 5
So, the hundreds digit increases by one.
Write 5 as the hundreds digit.
Write zeros as the tens and ones digits.
So, 457 rounded to the nearest hundred
500 .
is _
Math Talk: Possible explanation:
when you use place value, you
look at the digit to the right of the
rounding place to see to which ten
or hundred the digit is closer. If the
next digit is 5 or greater, it will be
closer to the next ten or hundred.
When you use a number line, you
can see to which ten or hundred
the number is closer.
MATHEMATICAL PRACTICES
Explain how using place
value is similar to using a number line.
© Houghton Mifflin Harcourt Publishing Company
60 .
is _
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• Why are you being asked to find which
hundred 548 is closer to? That is how you figure
3 PRACTICE
out how to round the number.
Share and Show •
Guided Practice
The first problem connects to the learning
model. Have students use the MathBoard to
explain their thinking.
Use Math Talk to focus on students’
understanding of rounding. Encourage
students to explain their thinking.
Use Exercises 6 and 7 for Quick Check. Students
should show their answers for the Quick Check
on the Math Board.
Quick Check
Quick Check
c
3
2
3
1
2
1
Rt I
R
Rt I
If
If
a student misses Exercises 6 and 7
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 9B
• Reteach 1.2
Soar to Success Math 15.15, 15.17
On Your Own •
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• In Exercises 13–15, is it possible to have the
same answer when rounding to the nearest
ten and hundred? Explain. Yes, when a number
is rounded to the closest ten, it is possible that the
number rounds to a hundred, such as Exercise 14. The
closest ten to 298 is 300. The closest hundred to 298 is
also 300.
Go Deeper
MATHEMATICAL
PRACTICES
To extend their thinking, ask students to
find the greatest number that rounds to 500.
Remind them that they are rounding to the
hundreds place, not the tens place.
• Would you be rounding up or down to find
the greatest number that rounds to 500?
down
• What is that number? 549
Independent
Practice
If students complete Exercises 6 and 7 correctly,
they may continue with Independent Practice.
Encourage students to complete the On Your
Own section independently, but provide
guidance as necessary. Ask questions to make
sure students know what they need to find
out.
• In Exercises 10–12, why is 550 not a
possible answer? I am asked to round to the
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
nearest hundred, and 550 is not a hundred.
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Name
Share and Show N
Locate and label 46 on the number line.
Round to the nearest ten. Check students’ work.
MATHEMATICAL PRACTICES
What is the greatest
number that rounds to 50 when
rounded to the nearest ten? What
is the least number? Explain.
46
30
40
60
50
1.
40
50 .
46 is between _
and _
2.
40
50
46 is closer to _
than it is to _
.
3.
50
46 rounded to the nearest ten is _
.
Round to the nearest ten.
4.
20
19 _
5.
greatest 54; least 45; Possible
explanation: since this group of
numbers rounds to 50 (45, 46, 47,
48, 49, 50, 51, 52, 53, 54), the
greatest number is 54 and the
least number is 45.
70
66 _
6.
50
51 _
9.
700
658 _
15.
840
844 _
Round to the nearest hundred.
7.
500
463 _
8.
200
202 _
On Your OwnN
Locate and label 548 on the number line.
Round to the nearest hundred. Check students’ work.
548
600
500
10.
600 .
500
548 is between _
and _
11.
500
600 .
548 is closer to _
than it is to _
12.
500 .
548 rounded to the nearest hundred is _
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
400
Round to the nearest ten and hundred.
13.
580
576 _
600
_
14.
300
298 _
300
_
800
_
Chapter 1 • Lesson 2
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Problem Solving
MATHEMATICAL
PRACTICES
For Exercises 16–18, students use information
from a table and rounding to the nearest ten
or hundred to solve problems. In Exercise 18,
encourage students to discuss strategies they
can use to determine which numbers round
to 800.
Problem
To solve Exercise 19, students will have to use
higher order thinking skills. Remind them of
when a number is rounded up to the nearest
ten and when a number is rounded down to
the nearest ten.
• Does 351 round to 360 when rounding to
the nearest ten? Explain. No, 351 is closer to 350
4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How can you round numbers? I can use a number
line or place value.
Math Journal
Describe how to round 678 to the nearest
hundred.
than it is to 360, so it rounds to 350 instead of 360.
• Does 357 round to 360 when rounding to
the nearest ten? Explain. Yes, the closest ten to
357 is 360.
• How can you use this thinking to find other
numbers that round to 360? I can think of
which numbers are closest to 360 without rounding to
another ten. For example, 356, 359, 361, and 364 all
round to 360.
Test Prep Coach
© Houghton Mifflin Harcourt Publishing Company
Test Prep Coach helps teachers to identify
common errors that students can make.
In Exercise 21, if students selected:
• A or C, they rounded to the wrong ten.
• D, they rounded to the nearest hundred.
11
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MATHEMATICAL
PRACTICES
Model • Reason • Make Sense
Problem Solving
Use the table for 16–18.
16.
On which day did about 900 visitors
come to the giraffe exhibit?
Visitors to the Giraffe
Exhibit
Sunday
17.
Day
To the nearest ten, how many visitors
came to the giraffe exhibit on Sunday?
890 visitors
18.
On which two days did about 800 visitors
come to the giraffe exhibit each day?
Number of Visitors
Sunday
894
Monday
793
Tuesday
438
Wednesday
362
Thursday
839
Friday
725
Saturday
598
Monday and Thursday
19.
Write five numbers that round to
360 when rounded to the nearest ten.
Possible answer: 356, 357, 359, 361, 364
20.
What’s the Error? Cole
said that 555 rounded to the nearest ten
is 600. What is Cole’s error? Explain.
The answer should be 560. Possible explanation:
Cole rounded to the nearest hundred instead of
the nearest ten.
12
Test Prep What is 438 rounded to the
nearest ten?
A
450
B
440
C
430
D
400
FOR MORE PRACTICE:
Standards Practice Book, pp. P5–P6
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21.
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Lesson 6.6
Investigate • Model with Arrays
Common Core Standard CC.3.OA.3
Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem.
Also CC.3.OA.2
Lesson Objective Model division by using arrays.
Essential Question How can you use arrays to
solve division problems?
• What should you do if you put the array
together and do not have an equal number
of tiles in each row? I must count again to see if
I made an error, because an array must have an equal
number of tiles in each row.
c Draw Conclusions
• What number did you divide? Explain.
30; possible answer: I started with 30 tiles.
• What number did you divide by? Explain.
5; possible answer: I made rows of 5 tiles each.
Problem
1 ENGAGE
Materials square tiles
Access Prior Knowledge Have students use
square tiles to review making arrays for
multiplication. Remind students that an array
is a set of objects arranged in rows. There is
the same number of tiles in each row.
• What real-world examples can arrays
represent? Possible answers: marching bands or
chairs set up in equal rows
Have students model 4 3 5 5 ______.
• Why did you make 4 rows of 5? Possible
answer: 4 x 5 means 4 groups, or rows, of 5
• How did you find the product? Possible
Exercise 3 requires students to generalize from
the 5 by 6 array to finding a new array with 6
tiles in each row.
• Use the same array to describe a strategy
for finding the number of rows of 6 tiles
that are in 30. Possible answer: I knew that
30 4 5 5 6, so 30 4 6 5 5. Count the number of tiles
in a row in the first array; Think: What number times
6 equals 30?
• How is this array different from the one
you put together in the Investigate section
that shows how many rows of 5 there are
in 30? The final number, 30, is the same for both
arrays. This one has 6 tiles in one row, while the first
array had 5 tiles in one row.
answer: I skip counted by 5 four times to get 20.
2 TEACH and TALK
© Houghton Mifflin Harcourt Publishing Company
c
Investigate
GO
iTools
Online
MATHEMATICAL
PRACTICES
Work together with students to complete the
steps of the activity. Be sure students make
equal rows of 5. Explain that making equal
rows is necessary so that the array is easy to
follow and analyze. Remind students of the
difference between a row and the number of
tiles in the row.
• In which direction must you work to make
a row of tiles? across from left to right
• In which direction must you count to find
how many rows are in your completed
array? from top to bottom
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Lesson 6.6
Name
Model with Arrays
COMMON CORE STANDARD CC.3.OA.3
Represent and solve problems involving
multiplication and division.
Essential Question How can you use arrays to solve
division problems?
Investigate
Materials ■ square tiles
You can use arrays to model division and find
equal groups.
A. Count out 30 tiles. Make an array to find
how many rows of 5 are in 30.
B. Make a row of 5 tiles.
C. Continue to make as many rows of 5 tiles
as you can.
6 rows
How many rows of 5 did you make? _________
Draw ConclusionsN
1.
Explain how you used the tiles to find the number of
rows of 5 in 30.
Possible explanation: I placed the 30 tiles in rows of 5 tiles. Then I counted
the number of rows I made. There were 6 rows. So, there are 6 rows of 5 in 30.
What multiplication equation could you write for the
array? Explain.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
2.
6 3 5 5 30; Possible explanation: there are 6 rows of 5 tiles each.
6 groups of 5 are 30.
3.
Apply Tell how to use an array to find how
many rows of 6 are in 30.
Possible answer: make rows of 6 until all 30 tiles are used. There are 5 rows of 6 in 30.
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c Make Connections
Help students connect the array to a division
equation.
• Why is 30 the dividend? because that is the
number of tiles being divided, or separated into equal
groups
COMMON ERRORS
Error Students may make an incorrect array
for a division problem.
Example How many rows of 3 are in 18?
• What represents the quotient in this array?
The number of rows is the quotient.
© Houghton Mifflin Harcourt Publishing Company
Try This!
After students find the quotient, have them
share how they wrote the division equation
that the array represents. Then check their
quotients.
• To divide, you have used drawing equal
groups or circling equal groups, repeated
subtraction, counting back on a number
line, and arrays. Which method do you
think is the easiest? Explain.
Have several students explain their choices.
Students’ explanations may include the
following:
• Drawing or circling the groups—it’s easier
when you can see the problem in a picture.
• Repeated subtraction—its faster to keep
subtracting than to make a drawing.
• Number line—it’s easy to count the jumps
to get the quotient.
• Array—it’s easy to put the tiles in equal
rows and then count the number of rows.
Use Math Talk to focus on students’
understanding of using an array to divide.
Explain that students can count a whole
row of an array as one number in a division
problem. So, if an array has 4 rows, students
can think: 24 4 4. The answer to that problem
should be found by looking at the number of
tiles in a row in the array, which is 6.
Similarly, if there are 6 tiles in a row in the
array, students can think: 24 4 6. They can
find the answer by looking at the number of
rows, which is 4.
• Why is it important to make neat rows
when making an array? If neat rows are not
COMMON ERRORS
Springboard to Learning Review with
students that if the problem asks for rows
of 3, you put that number in each row
and count the number of rows to get the
answer. If the problem gives the number
of rows, you start by making that many
rows with one tile in each row. You keep
adding one tile to each row until all
the tiles are used. Then you count the
numbers of tiles in each row.
made, it is easy to make mistakes counting the rows
and number of titles in a row, and therefore get the
wrong answer.
1
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Make ConnectionsN
You can write a division equation to show how
many rows of 5 are in 30. Show the array you made
in Investigate by completing the drawing below.
You can divide to find
the number of equal
rows or to find the
number in each row.
30 4 5 5 j
6
There are _
rows of 5 tiles in 30.
6 .
So, 30 4 5 5 _
Try This!
Possible explanation: it
helps me solve the division
problem by showing how
many tiles are in each row.
MATHEMATICAL PRACTICES
Explain how making
an array helps you divide.
•
•
6 tiles
How many tiles are in each row? ___
24 4 4 5 6 or 24 4 6 5 4
What division equation can you write for your array? _____
© Houghton Mifflin Harcourt Publishing Company
Count out 24 tiles. Make an array with the same number of tiles in
4 rows. Place 1 tile in each of the 4 rows. Then continue placing
1 tile in each row until you use all the tiles. Draw your array below.
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3 PRACTICE
Share and Show •
Quick Check
Guided Practice
Exercises 5–8 are examples of partitive
division. Before students complete the page,
ask a volunteer to explain how he or she will
find the answer to Exercise 5. Separate 25 tiles
into 5 groups by placing one tile in each of 5 rows. Place
one tile at a time in each row until all tiles are used. Count
the number of tiles in each row to find the quotient.
3
2
1
Rt I
R
If
a student misses Exercises 2 and 6
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 231B
• Reteach 6.6
Soar to Success Math 13.17
Remind students to write the division
equations their arrays represent.
Use Exercises 2 and 6 for Quick Check. Students
should show their answers for the Quick Check
on the MathBoard.
Use Math Talk to focus on students’
understanding of using arrays to model
division.
• How are the number of rows in an array
and the number of tiles in a row related?
You must find both to know how to divide using an
array. If you know the total and one of the numbers,
you can find the other number.
Give students an extra example to make sure
they understand the concept.
• Look at Exercise 9 again. What does the
number 3 in the problem tell you? how many
are in each row
• What does the number 5 in the problem tell
you? the number of rows in the array
• What division problem can you write if you
know that there is a total of 15 tiles and
there are 5 tiles in a row? 15 4 5 5 3
• What does the 3 tell you about the array?
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
It tells the number of rows.
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Share and Show
Use square tiles to make an array. Solve. Check students’ arrays.
1.
How many rows of 3 are in 18?
2.
6 rows
_____
3.
How many rows of 6 are in 12?
2 rows
_____
How many rows of 7 are in 21?
4.
3 rows
_____
How many rows of 8 are in 32?
4 rows
_____
Make an array. Then write a division equation. Check students’ arrays.
5. 25 tiles in 5 rows
6. 14 tiles in 2 rows
25 4 5 5 5
_____
7.
14 4 2 5 7
_____
28 tiles in 4 rows
8.
28 4 4 5 7
_____
9.
27 tiles in 9 rows
27 4 9 5 3
_____
How many rows of 3 are in 15?
10.
15 4 3 5 5
_____
How many rows of 8 are in 24?
24 4 8 5 3
_____
MATHEMATICAL PRACTICES
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Possible explanation: you count the number
of rows when you know the number in each
row. You count the number of tiles in each
row when you know the number of rows.
11.
Explain when you count the
number of rows to find the answer
and when you count the number of
tiles in each row to find the answer.
Show two ways you
could make an array with tiles for
18 4 6. Shade squares on the grid
to record the arrays.
Check students’ drawings.
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Unlock the Problem
MATHEMATICAL
PRACTICES
In Exercise 12, students draw an array to solve
a problem in context. In Step d, have students
share other strategies they could use to solve
the problem.
• Do you find repeated subtraction to be
easier or harder than making arrays?
Explain. Answers will vary.
Encourage students to express their thoughts
about making arrays and why they find them
easier or harder than repeated subtraction. As
students express their thoughts about arrays
and what they find difficult or easy about
them, address their apprehensions. You may
find that students are unsure about whether
they have made the correct number of rows or
put the correct number of objects in each row.
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4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How can you use arrays to solve division
problems? Possible answer: I can find how many equal
groups by placing that number of tiles in each row of
an array until all tiles are used. The number of rows is
the answer. I can divide the tiles into a number of rows,
placing 1 tile at a time in each row, until all the tiles are
used. The number of tiles in each row is the answer.
Math Journal
Draw an array to show how to arrange
20 chairs into 5 equal rows. Explain what
each part of the array represents.
Remind them that a row is the distance across
from left to right. If they are showing that
Thomas planted 4 seedlings in a row, they
must draw 4 objects in one single row. Then
they can find the number of rows it takes to
reach 28.
Test Prep Coach
Test Prep Coach helps teachers to identify
common errors that students can make.
For Exercise 13, if students selected:
A They added 36 and 6.
B They subtracted 6 from 36.
© Houghton Mifflin Harcourt Publishing Company
C They incorrectly divided by 6.
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MATHEMATICAL
PRACTICES
Model • Reason • Make Sense
TEST
PREP
UNLOCK the Problem
12.
Thomas has 28 tomato seedlings to plant in his garden.
He wants to plant 4 seedlings in each row. How many
rows of tomato seedlings will Thomas plant?
5
A
B
6
C
7
D
8
a. What do you need to find? how many rows of tomato seedlings
Thomas will plant
b. What operation could you use to solve the problem? Possible answer:
division
c. Draw an array to find the number of
rows of tomato seedlings.
d. What is another way you could
have solved the problem?
Check students’ drawings.
There should be an array with
7 rows of 4 tiles.
Possible answer: I could have used
repeated subtraction.
e. Complete the sentences.
28
tomato
Thomas has __
seedlings.
4
He wants to plant __
row
.
seedlings in each __
f. Fill in the bubble for the correct
answer choice above.
13.
Faith plants 36 flowers in 6 equal
rows. How many flowers are in
each row?
A
234
42
B
30
C
7
D
FOR MORE PRACTICE:
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14.
There were 20 plants sold at a store
on Saturday. Customers bought
5 plants each. How many customers
bought the plants?
A
3
B
4
C
5
D
6
© Houghton Mifflin Harcourt Publishing Company
7
So, Thomas will plant __
rows of tomato seedlings.
FOR EXTRA PRACTICE:
Standards Practice Book, p. P124
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Lesson 1.12
Problem Solving •
Model Addition and Subtraction
Common Core Standard CC.3.OA.8
Solve two-step word problems using the four operations. Represent
these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using
mental computation and estimation strategies including rounding.
Lesson Objective Solve addition and subtraction
problems by using the strategy draw a diagram.
Essential Question How can you use the strategy
draw a diagram to solve one- and two-step addition and
subtraction problems?
1 ENGAGE
Access Prior Knowledge Introduce the lesson
by asking students:
• Have you ever played a computer game
with another person? What kind of game
was it? Did you keep score? How did you
know who won?
2 TEACH and TALK
c
Unlock the Problem
• How did you use the information from the
problem to label the first bar model? I labeled
the longer bar 84 to show the points Sami scored in
the first round. I labeled the shorter bar 21 to show
how many more points he scored in the second round.
The number under the bars shows the total number of
points Sami scored in the second round.
• What does the second bar model show? The
second bar model shows Sami’s total score for the two
rounds.
• How are the bar models related to the
addition sentences that represent them?
Possible answer: the addends are in the bars, and the
sum is represented by the number under the bars.
Students may have difficulty with the fact that
this is a two-step problem. They may neglect
to complete the second step of the problem
and think that the answer is 105 points. Tell
students that they should reread the problem
carefully after they find their answer to be
sure that they have answered the question
that is asked.
• Why is it important to check the original
problem when you are finished to make
sure you answered the right question? It is
possible to get an answer that is mathematically correct,
but answers the wrong question.
MATHEMATICAL
PRACTICES
Have students read the problem. Point out
that a bar model is a diagram that can help
them decide what operation to use to solve
a problem.
Guide students to read each question in the
graphic organizer and answer it before solving
the problem.
• What question are you trying to answer?
© Houghton Mifflin Harcourt Publishing Company
What was Sami’s total score?
• What information do you know? He scored
84 points in the first round and 21 more points in the
second round.
• Did Sami score more points in the first
round of the game or the second round?
He scored more points in the second round.
• How did you use the first bar model? I used
the first bar model to figure out how many points Sami
scored in the second round.
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PROBLEM SOLVING
Name
Lesson 1.12
Problem Solving •
Model Addition and Subtraction
COMMON CORE STANDARD CC.3.OA.8
Solve problems involving the four operations, and
identify and explain patterns in arithmetic.
Essential Question How can you use the strategy draw a diagram to
solve one- and two-step addition and subtraction problems?
UNLOCK the Problem
Sami scored 84 points in the first round of a new
computer game. He scored 21 more points in the
second round than in the first round. What was
Sami’s total score?
You can use a bar model to solve the problem.
Read the Problem
What do I need to
find?
What information do
I need to use?
How will I use the
information?
I need to find
84
Sami scored _
points in the first round.
I will draw a bar model to
show the number of points
Sami scored in each
round. Then I will use the
bar model to decide which
operation to use.
Sami’s
total score
____
.
21 more
He scored _
points than that in the
second round.
Solve the Problem
© Houghton Mifflin Harcourt Publishing Company
•
Complete the bar model to show the
number of points Sami scored in the
second round.
84
points
21
•
Complete another bar model to
show Sami’s total score.
points
84
points
points
84
21
__
1 __
5n
84
__
1
105
points
105
__
points
5s
189
__
5s
105
__
5n
105 points
1. How many points did Sami score in the second round? __
189 points
2. What was Sami’s total score? __
Chapter 1
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c
Try Another Problem
Have students read the problem and then
answer the questions in the graphic organizer
and solve the problem. Invite students to
share their diagrams and explanations.
• Who scored more points? How does the bar
model show this? Anna; the bar for Anna’s points
is longer.
• How does the bar model help you solve
the problem? Possible answers: it shows that the
unknown part is the difference between Anna’s and
Greg’s scores.
• What numbers do you need to subtract to
solve the problem? I need to subtract Greg’s score
from Anna’s score.
• How is the bar model related to the
subtraction sentence that represents it?
Possible answer: the number in the shorter bar is
subtracted from the number in the longer bar to find
the difference, which is represented by the unknown
quantity to the right of the shorter bar.
DO NOT
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You may suggest that students place
completed Try Another Problem graphic
organizers in their portfolios.
COMMON
COMMON ERRORS
ERRORS
Error Students may have difficulty determin-
ing how to label the bars in the bar model.
Example In Try Another Problem, students
may draw a shorter bar for Anna’s points
than for Greg’s points and, therefore, label it
incorrectly.
Springboard to Learning Remind students
that although the bars in a bar model do not
have to be in exact proportion, longer bars
should represent greater numbers. Have students first determine whose score is greater
before drawing their bar models.
In problem 4, there are several ways to check
for reasonableness, including estimation.
Invite students to share their answers and
discuss the different ways to estimate, such
as rounding or using compatible numbers, to
check for reasonableness.
s
Use Math Talk to focus on students’
understanding of how to use bar models to
solve a problem. Ask students to look at the
bar models again and compare the length of
each bar model compared to Anna and Greg’s
scores.
• What would the bar model look like if Greg
scored more points than Anna? Greg’s bar
h
model would be longer than Anna’s.
• Whose bar model would show the
unknown part with the gray box?
© Houghton Mifflin Harcourt Publishing Company
Greg’s bar model
Lesson 1.12
1
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Try Another Problem
Anna scored 265 points in a computer game. Greg scored
142 points. How many more points did Anna score than Greg?
You can use a bar model to solve the problem.
Read the Problem
What do I need to
find?
What information do
I need to use?
I need to find how many
more points Anna scored
than Greg.
Anna scored 265 points.
Greg scored 142 points.
How will I use the
information?
I will use a bar model to
show the points each
person scored. Then I will
use the bar model to decide
which operation to use.
Solve the Problem
Record the steps you used to solve the problem.
I need to complete the bar model to show
each person’s score.
265 2 142 5
Anna
Greg
I need to subtract to find the unknown
part.
265
142
points
123
n
5n
points
points
123 points
4. How do you know your answer is reasonable?
Possible answer: I can use friendly numbers to
subtract mentally. I subtract 2 from 142 to get 140. Then
I subtract 2 from 265 to get 263; 263 2 140 5 123.
5. How did your drawing help you solve the problem?
Possible answer: the bar model helped me see that I had
to subtract and what numbers I needed to subtract.
MATHEMATICAL PRACTICES
Explain how the
length of each bar in the
model would change if
Greg scored more points
than Anna but the totals
remained the same.
Anna’s bar would be
the shorter bar and
Greg’s bar would be
the longer bar.
© Houghton Mifflin Harcourt Publishing Company
3. How many more points did Anna score than Greg?
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Go Deeper
3 PRACTICE
Share and Show •
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Guided Practice
Be sure to point out to students that Exercise
1 is a two-step problem. Ask students to
determine what they need to find, what
information they need to use, and how they
can use the information.
Problem
Exercise 2 requires that students use higher
order thinking skills. The problem varies the
scenario presented in Exercise 1. Students
should connect the number and length of the
bars to the numeric label of each bar.
• What would be used to represent the third
student? A third bar model
• Is there just one answer for the number
of votes that each student gets? Explain.
MATHEMATICAL
PRACTICES
To extend their thinking, have students write
a problem and draw their own bar model
to solve it. Explain that the problem can be
about anything they wish, and can have two
or more bar models. Remind students that
they would need at least two bar models so
that two quantities can be compared.
• What kind of problem can you write that
would use a bar model to help solve it? The
problem could be about comparing scores, lengths
or weights of items, or any comparison in which one
quantity is known, part of another quantity is known,
and the total is unknown.
No; There are many possibilities for the number of
votes each student gets. The numbers can be any three
numbers that add up to 121 votes.
Use Exercises 3 and 4 for Quick Check. Students
should show their answers for the Quick Check
on the MathBoard.
Quick Check
3
2
1
Rt I
R
a student misses Exercises 3 and 4
Then
Differentiate Instruction with
• RtI Tier 1 Activity, p. 51B
• Reteach 1.2
Soar to Success Math 70.04
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
If
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UNLOCK the Problem
Name
√ Use the problem solving
Share and ShowN
1.
Tips
MathBoard.
Sara received 73 votes in the school
election. Ben received 25 fewer votes than
Sara. How many students voted in all?
First, find how many students voted for Ben.
Think: 73 2 25 5 n
√ Choose a strategy you know.
Sara
votes
73
Ben
votes
25
Write the numbers in the bars.
votes
48 votes.
So, Ben received _
Next, find the total number of votes.
Think: 73 1 48 5 s
48
5
73
Write the numbers in the bars.
votes
votes
48
votes
votes
121
students voted in all.
So, __
5 121 votes
2.
What if there were 3 students
in another election and the total number
of votes was the same? What would the
bar model for the total number of votes
look like? How many votes might each
student get?
Possible answer: there would be 3 bars. Possible
number of votes: 55, 30, and 36
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
3.
Pose a Problem Use the bar model at the
right. Write a problem to match it.
Possible problem: Russ and Juan collect stamps.
89
157
Together they collected 157 stamps. Russ collected
89 stamps. How many stamps did Juan collect?
4.
Solve your problem. Will you add or subtract?
Possible answer: subtract; 157 2 89 5 68.
Chapter 1 • Lesson 12
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c
On Your Own •
Independent
Practice
If students complete Exercises 3 and 4 correctly,
they may continue with Independent Practice.
Encourage students to solve the problems
independently, but provide assistance as
needed.
If students struggle with the On Your Own
problems, ask them what they are trying to
find out, and what they already know about
the problem. Encourage them to use the Show
Your Work area of the page and then show it
to you to see if their thinking and calculations
are correct. Circle parts of their work that
contains errors, and guide them through the
process of making corrections.
es
Problem
Exercise 8 requires that students use higher
order thinking skills to solve the problem.
• What can you do to find out the greatest
number that could be rounded to 400?
s
I must think of the greatest number that can be
rounded down to 400.
Test Prep Coach
Test Prep Coach helps teachers to identify
common errors that students can make.
In Exercise 9, if students selected:
A They subtracted instead of added.
B They did not add the regrouped ten.
D They added incorrectly.
4 SUMMARIZE
MATHEMATICAL
PRACTICES
Essential Question
How can you use the strategy draw a diagram
to solve one- and two-step addition and
subtraction problems? Possible answer: I can draw a
bar model to see if I need to add or subtract.
Math Journal
Write an addition or subtraction problem and
draw a diagram to solve it.
• What number would be in the hundreds
place? Explain. 4; I know I will be rounding down
because I am looking for the greatest number that
rounds to 400.
• What number would be in the tens place?
Explain. 4; I know that any number that rounds to
400 must be less than 450.
• What number would be in the ones place?
Explain. 9; If I have a 4 in the tens place, the greatest
© Houghton Mifflin Harcourt Publishing Company
number that can be in the ones place and still round
down to 400 would be 9. The answer is 449.
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MATHEMATICAL
PRACTICES
On Your Own
5.
Model • Reason • Make Sense
Tony’s Tech Store is having a sale. The store
had 142 computers in stock. During the sale,
91 computers were sold. How many computers
were not sold?
51 computers
6.
The number of computer games sold during
the sale was 257. This is 162 more than the
number sold the week before the sale. How
many computer games were sold the week
before the sale?
95 computer games
7.
In one week, 128 cell phones were sold. The
following week, 37 more cell phones were sold
than the week before. How many cell phones
were sold in those two weeks?
293 cell phones
8.
On Monday, the number of
customers in the store, rounded to the nearest
hundred, was 400. What is the greatest number
of customers that could have been in the
store? Explain.
449 customers; Possible explanation: to round to
400, the greatest number must have a 4 in the tens
© Houghton Mifflin Harcourt Publishing Company
place so that the hundreds place stays the same.
The greatest number in the ones place is 9.
9.
54
Test Prep The number of laptop computers
sold in one day was 42. That is 18 fewer than
the total number of desktop computers sold.
How many desktop computers were sold?
A
24
C
60
B
50
D
61
FOR MORE PRACTICE:
Standards Practice Book, pp. P25–P26
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