1. science
2. mathematics
3. arithmetic

Unit 2 Number and Operations in Base Ten:
Place Value, Addition, and Subtraction
Introduction
In this unit, students will review the place value system for reading and writing numbers in base
ten. Students will distinguish between the place values of a digit in 1-, 2-, and 3-digit numbers,
and they will understand the value of each digit. Students will read and write 3-digit numbers
using words and numerals, and understand the connection between number words, expanded
form, and base ten block representations.
Students will also compare 3-digit numbers using the symbols <, >, and = with base ten blocks
and using place value. Students will add and subtract 3-digit numbers using base ten blocks
and using the standard algorithm, with and without regrouping.
NOTE: BLM Place Value Bingo has two “spinner circles,” which look like pie charts. A pencil
and paper clip can be used to operate these spinners. With the piece of paper resting on a
desk, hold the pencil vertically so that the sharp end faces down, goes through one end of the
paper clip, and presses against the center of the spinner circle. The paper clip rests on top of
the piece of paper, with one end anchored at the center of the spinner circle by the pencil. Flick
the other end of the paper clip. The paper clip should spin around the pencil, and eventually
stop on one of the sectors of the spinner circle.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-1
NBT3-1 Place Value—Ones, Tens, and Hundreds
Pages 15–16
Standards: preparation for 3.NBT.A.2
Goals:
Students will identify the place value of digits in 2- and 3-digit numbers, and understand the
value of the digits.
Prior Knowledge Required:
Knows the number words one, ten, hundred, and their corresponding numerals (1, 10, 100)
Vocabulary: digit, hundreds, hundreds digit, hundreds place, ones, ones digit, ones
place, place value, tens, tens digit, tens place
Materials:
BLM Place Value Cards (p. C-78)
BLM Place Value Bingo (p. C-79)
bingo chips, paper clips, pencils
(MP.6) Review of Place Value. Photocopy BLM Place Value Cards and cut out the three
cards. Write the number 347 on the board, leaving extra space between all the digits, and hold
the ones card under the 3. ASK: Did I put the card in the right place? (no) Is 3 the ones digit?
(no) Have a volunteer put the card below the correct digit. (below the 7) Invite volunteers to
position the other cards correctly. Cards can be affixed to the board temporarily using tape or
sticky tack.
Erase the 3 and take away the hundreds card. ASK: Are these cards still in the right place?
(yes) Write the 3 back in, put the hundreds card back beneath the 3, erase the 7, and remove
the ones card. ASK: Are these cards still in the right place? (no) Have a volunteer reposition the
cards correctly. (tens card under 3, ones card under 4). Repeat this process with the 3 and 7
(i.e., erase the 4).
Write 989 on the board and ask students to identify the place value of the underlined digit.
(hundreds) NOTE: If you give each student a copy of BLM Place Value Cards, individuals can
hold up their answers. Have students cut out the cards before you begin the lesson. Repeat with
several 2- and 3-digit numbers that have an underlined digit.
Vary the question slightly by asking students to find the place value of a particular digit without
underlining it. Example: Find the place value of the digit 4 in these numbers: 401, 124, 847.
(hundreds, ones, tens) Continue until students can identify place value correctly and confidently.
Include examples where you ask for the place value of the digit 0. Example: 108, 970, 302 (tens,
ones, tens)
C-2
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Place Value Chart. Introduce the place value chart and have a volunteer write the digits from
the number 231 in the correct columns (answers are in italics):
Hundreds
Tens
Ones
2
3
1 231
Use the words ones place, tens place, and hundreds place—or use the word column if students
have learned it. Repeat with the following examples, this time having students signal the correct
digit for each place value: 458, 703, 241, 89, 5, 333.
Exercises: Write the digits from the number in the place value chart.
Hundreds
Tens
Ones
a) 932
b) 426
c) 502
d) 47
Answers: a) Hundreds: 9, Tens: 3, Ones: 2; b) Hundreds: 4, Tens: 2, Ones: 6; c) Hundreds: 5,
Tens: 0; Ones: 2; d) Hundreds: 0, Tens: 4, Ones: 7
Write 836 on the board. SAY: The number 836 is a 3-digit number. What is the place value of
the digit 8? (hundreds) If necessary, point to each digit as you count aloud from the right: ones,
tens, hundreds. SAY: The 8 is in the hundreds place, so it stands for 800. What does the digit 3
stand for? (30) The 6? (6)
Exercises: Fill in the blank.
a) In the number 427, the digit ____ is in the hundreds place.
b) In the number 385, the digit ____ is in the tens place.
c) In the number 102, the digit ____ is in the ones place.
d) In the number 347, the digit 4 stands for ____.
e) In the number 598, the digit 5 stands for ____.
f) In the number 273, the digit 3 stands for ____.
Answers: a) 4, b) 8, c) 2, d) 40, e) 500, f) 3
ASK: What does the digit 6 stand for in 608? (600) In 306? (6) In 762? (60) In 506? (6)
Exercises: What does the digit 4 stand for?
a) 345 ______
b) 408 _______
c) 514 ______
d) 647 _______
Answers: a) 40, b) 400, c) 4, d), 40
ASK: In the number 831, what does the digit 3 stand for? (30) The 1? (1) The 8? (800)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-3
Exercises: Fill in the blanks.
a) In 982, the digit 9 stands for ____, the digit 8 stands for ____, and the digit 2 stands for ___.
b) In 173, the digit 1 stands for ____, the digit 7 stands for ____, and the digit 3 stands for ___.
c) In 547, the digit 5 stands for ____, the digit 4 stands for ____, and the digit 7 stands for ___.
d) In 628, the digit 6 stands for ____, the digit 2 stands for ____, and the digit 8 stands for ___.
Answers: a) 900, 80, 2; b) 100, 70, 3; c) 500, 40, 7; d) 600, 20, 8
ASK: What does 0 stand for in 340? (0) In 403? (0) In 809? (0) Emphasize that 0 always stands
for 0, no matter what position it is in. Finally, ASK: In the number 856, what is the tens digit? (5)
The ones digit? (6) The hundreds digit? (8). Repeat, having students signal their answers, for
350, 503, 455, 770, and 820.
Activities 1–2
1. Collaborative Place Value Bingo. Draw a large 3 × 3 table (the bingo card) on the board
and fill each cell with a 3-digit number. Example:
354
206
781
923
132
428
615
590
847
Ahead of time, write the word “hundreds” on a small piece of paper, and fold it twice so that the
word cannot be seen; do the same with the words “tens” and “ones.” Place these 3 folded
pieces of papers in a container. Similarly, write each of the digits from 0 to 9 on a separate small
piece of paper, and put these ten folded pieces of paper in another container. Have a volunteer
draw (without peeking) a folded piece of paper from the “digits container” and a folded piece of
paper from the “place value” container. The student then calls out what was drawn: for example,
if “3” and “tens” are drawn, the student says, “3 in the tens place.” Have all students scan the
bingo card to see if they can find a number with a 3 in the tens place. If there is such a number
on the bingo card (in this case, 132) they circle the number. Otherwise, no number gets circled.
Fold and return the pieces of paper to their respective containers, and repeat with a new
volunteer. The whole class wins if three numbers are circled in a row. (For a faster game, allow
rows, columns, and diagonals for the class to win, or declare a win if students circle four
numbers anywhere on the bingo card.)
2. Students who finish Activity 1 early might enjoy playing bingo individually, using the spinners
provided on BLM Place Value Bingo to determine a digit and a place value. Students will need
9 bingo chips, 1 paper clip, and 1 pencil. (See Unit Introduction, p. C-1, for instructions on using
these spinners with a pencil and paper clip.) You can fill in the numbers on the bingo card
yourself, or have students strategically choose numbers before their first spin. Now, if the digits
spinner turns up “7” and the place value spinner turns up “tens,” the student has to scan the
nine numbers in the chart to see if any of them have a 7 in the tens place. If so, the student
places a bingo chip on the number. Play continues until there are 3 bingo chips in a row or
column (or until there are 4 bingo chips anywhere on the card).
(end of activity)
C-4
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Extensions
(MP.6) 1. In the number 467, the digit 4 stands for 400. Teach students alternatives to the
phrase “stands for.” For example: the digit 4 “means” 400, the digit 4 “represents” 400, the digit
4 “is short for” 400, the digit 4 “has a value of” 400. If you choose to use the latter phrase, make
sure students understand the difference between the value of a digit and place value. Have
students practice these phrases in oral and written form for a variety of numbers.
2. Write the following numbers on the board: 350, 503, 435, 537, 325, and 753. Ask students to
identify which digit, the 5 or the 3, is worth more in each number. Students should be using the
phrases “stands for,” “has a value of,” “is short for,” and so on.
Sample answer: In 350, the 5 stands for 50 and the 3 stands for 300, so the digit 3 is worth
more.
(MP.7) 3. Teach students the Egyptian system for writing numerals to help them appreciate the
utility of place value.
1=
(stroke)
10 =
(arch)
100 =
(coiled rope)
Write the following numbers using both the Egyptian and the Arabic systems:
234
848
423
Invite students to study the numbers for a moment. ASK: What is different about the Egyptian
system for writing numbers? (it uses symbols instead of digits) You have to show the number of
ones, tens, and so on individually. If you have 7 ones, you have to draw 7 strokes. In the Arabic
system, a single digit (7) tells you how many ones there are. Review the ancient Egyptian
symbols for 1, 10, and 100 and ask students to write a few numbers the Egyptian way and to
translate those Egyptian numbers into regular numbers (using Arabic numerals). Emphasize
that the order in which you write the symbols doesn’t matter:
234 =
=
ASK: Does the order in which you write regular digits matter? (yes) Is 234 the same as 342?
(no) In the Egyptian way, does the value of a symbol depend on its place? (no) In our way, does
the value of a digit depend on its place? (yes) Are the ones, tens, and so on always in the same
place in our system? (yes) In the Egyptian system? (no) Why is our way called a place value
system? Have students write a number that is really long to write the Egyptian way (e.g., 798).
ASK: How is our system more convenient? Why is it helpful to have a place value system—i.e.,
a system where the ones, tens, and so on are always in the same place? Explain that having a
place value system allows you to use the same symbol to mean many different values. The digit
7, for example, can mean 7 ones, 7 tens or 7 hundreds depending on where it is in the number.
Students might want to invent their own number system using the Egyptian system as a model.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-5
4. Have students identify and write numbers according to specific criteria. NOTE: Some of the
examples use the words “even,” “odd,” and “sum”, which are taught formally later. Examples:
a) Write a number between 30 and 40.
b) Write an even number with a 6 in the tens place.
c) Write a number that ends with a zero.
d) Write a 2-digit number.
e) Write an odd number greater than 70.
f) Write a number with a tens digit one more than its ones digit.
g) Write a 2-digit number where the digits are the same.
h) Write a number between 50 and 60 where the digits are the same.
i) Find the sum of the digits in each of these numbers: 37, 48, 531, 225, 444, 372.
j) Write a 2-digit number where the sum of the digits is 11.
k) Write a 2-digit number where the digits are the same and the sum of the digits is 14.
l) Write a 3-digit number where the digits are the same and the sum of the digits is 15. Is there
a 2-digit number where the digits are the same and the sum of the digits is 15?
Make up more such questions, or have students make up their own.
Bonus:
m) Which number has a tens digit one less than its ones digit: 34, 47, 88, 90?
n) Write a 2-digit number with a tens digit eight less than its ones digit.
o) Write a 3-digit number where all three digits are odd.
p) Write a 3-digit number where the ones digit is equal to the sum of the hundreds digit and the
tens digit.
Sample answers: a) 37; b) 364; c) 170; d) 13; e) 73; f) 65; g) 88; h) 55; i) 10, 12, 9, 9, 12, 12;
j) 47; k) 77; l) 555, no; Bonus: m) 34; n) 19; o) 375; p) 459
C-6
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-2 Base Ten Blocks
Pages 17–19
Standards: preparation for 3.NBT.A.2
Goals:
Students will represent numbers with base ten materials.
Prior Knowledge Required:
Understands place value
Vocabulary: base ten blocks, digit, hundreds block, hundreds digit, ones block, ones digit,
tens block, tens digit,
Materials:
transparency and overhead projector (if available)
base ten blocks (of two different colors, if available)
BLM Hundreds Chart and Base Ten Materials (p. C-80)
BLM Hundreds Charts up to 200 (p. C-81)
Base ten models for 1- and 2-digit numbers. Photocopy BLM Hundreds Chart and Base
Ten Materials onto a transparency, if available. Demonstrate how to find 3 + 4 by taking 3 ones
blocks and then another 4 ones blocks and placing them on the chart in order, so that the last
block is on square 7. ASK: How can I find 13 + 5 by using ones blocks and the hundreds chart?
How is the counting already done for you when you put the ones blocks on the chart in order?
Emphasize that students can see the answer by looking under the last ones block. Tell your
students that instead of using 10 ones blocks to cover a row, you find it easier to use one bigger
block. Show them a tens block and ask if anyone remembers what the block is called. Show
them how 10 ones blocks can be joined together to make 1 tens block.
Provide your students with BLM Hundreds Charts up to 200, as well as 10 tens blocks and 9
ones blocks each. Have students use 3 tens blocks and 5 ones blocks and cover the squares
from the top chart in order. (The hundreds charts are 10 cm x 10 cm so that a ones block will
cover a grid square exactly. If the ones blocks you are using are larger than 1 cubic centimeter,
you may need to enlarge BLM Hundreds Charts up to 200 when printing.) ASK: How many
squares are covered? (35) PROMPT: Which number do you see under the last ones block? (35)
Repeat for several examples. (41, 23, 59, 74, 99) ASK: What number do you get if you use 2
tens blocks and 0 ones blocks? (20) What number do you get if you use 5 tens blocks? (50) 7
tens blocks? (70) 10 tens blocks? (100)
Base ten models for 3-digit numbers. Tell your students that we used a tens block instead of
10 separate ones blocks. ASK: What can we use instead of 10 tens blocks? (a hundreds block)
Show them how 10 tens blocks can be put side by side to make 1 hundreds block. Place the 10
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-7
tens blocks on top of the hundreds block. (If possible, use one color for the tens blocks and a
different color for the hundreds block.) Give your students 2 hundreds blocks to add to their 10
tens blocks and 9 ones blocks. ASK: What number do you get if you place a hundreds block on
the first hundreds chart and then 3 tens blocks and 7 ones blocks in order on the next hundreds
chart? (137) Repeat with:
a) 1 hundreds block, 5 tens blocks, 4 ones blocks (154)
b) 1 hundreds block, 6 tens blocks, 2 ones blocks (162)
c) 1 hundreds block, 7 tens blocks, 5 ones blocks (175)
d) 1 hundreds block, 3 tens blocks (130)
e) 1 hundreds block, 10 tens blocks (200)
f) 2 hundreds blocks (200)
(MP.2) Show models of base ten blocks without using the hundreds chart and have students tell
you what number is represented. Example: 3 hundreds blocks, 0 tens blocks, 5 ones blocks.
(305) Show base ten models of the following numbers, and have students write what number is
represented: 412, 352, 160, 27, 518, 231.
Drawing base ten models. Demonstrate drawing a base ten model for 145 on grid paper:
Shade the blocks and ASK: How many little squares are shaded altogether? (145) Have
students draw base ten models for other 2- and 3-digit numbers. Students can use grid paper to
draw models as shown in the above diagram; however, some students might have difficulty
drawing such base ten models. Teach students a faster way to sketch base ten models on
paper without grid markings:
Hundreds Block
Tens Block
Ones Block
Some students may prefer to simply draw straight lines for tens, and dots for ones:
Tens Block
C-8
Ones Block
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Let students choose their method for sketching base ten models, and have them practice with
many examples: 45, 60, 74, 104, 251, 300, 260.
(MP.1) Activity
Give your students ones, tens, and hundreds blocks. Students could work in groups. Have
students sketch their answers, so you can verify that they have successfully completed the
work.
a) Show 17, 31, 252, and 346 with base ten blocks.
b) Show 22 using exactly 13 blocks.
c) Show 31 using 13 blocks.
d) Show 315 using exactly 36 blocks.
HINT: For b), c), and d), start with a standard model and trade for blocks of equal value.
Answers: b) 1 tens blocks and 12 ones blocks; c) 2 tens blocks and 11 ones blocks, d) 31 tens
blocks and 5 ones blocks
(end of activity)
Extensions
(MP.7) 1. Draw sketches of base ten models where the hundreds, tens, and ones blocks are out
of order. Have students identify the number being modeled. Examples:
a)
b)
Answers: a) 235, b) 123
(MP.7) 2. Write equations on the board in which one side is a base ten model sketch and the
other side is a number. Examples:
a)
=
214
b)
=
125
Ask the students whether the equations are true or false. Vary the examples so that some are
true and some are false. For those that are false, have the students draw missing base ten
blocks or cross out extra base ten blocks to make the equations true.
Answers: a) true, b) false
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-9
NBT3-3 Expanded Form
Pages 20–21
Standards: preparation for 3.NBT.A.2
Goals: Students will understand how to represent numbers in expanded form, and how to
recognize a number written in expanded form.
Prior Knowledge Required:
Understands place value
Vocabulary: digit, expanded form, hundreds, numeral, ones, tens
Materials:
BLM Make Up Your Own Cards (p. C-82)
Write the expanded form when given the numeral. Show the base ten model drawing for 145
again:
ASK: How many little squares are colored? (145) Point to the hundreds block, the tens blocks,
and the ones blocks, and ask in turn how many little squares are colored from each type of
block. Then write on the board: 145 = 100 + 40 + 5. Repeat this exercise with the following
numbers: 361, 218, 427, 659. Explain that this is called expanded form with numerals. Tell the
students that we use the word numeral to denote a number written as a symbol, as in “10”
rather than the word “ten”.
Ask a volunteer to write the expanded form for 145 using the words “hundreds,” “tens,” and
“ones”. (145 = 1 hundred + 4 tens + 5 ones; point out that the “s” on the word “hundreds” is
dropped here since 145 has only 1 hundred). Explain that this is called expanded form with
numerals and words. Point out that in the expanded form using numerals and words, you are
counting how many of each type of block you need to make a number. In the expanded form
using numerals only, you are counting how many ones in each type of block are used to make
the number.
C-10
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises:
1. Fill in the blanks.
a) 325 = ___ hundreds + ___ tens + ___ ones b) 413 = ___ hundreds + ___ ten + ___ ones
c) 937 = ___ hundreds + ___ tens + ___ ones d) 108 = ___ hundred + ___ tens + ___ ones
Answers: a) 3, 2, 5; b) 4, 1, 3; c) 9, 3, 7; d) 1, 0, 8
2. Fill in the blanks.
a) 479 = 4 ________ + 7 ______ + 9 ______
b) 218 = 2 ________ + 1 ______ + 8 ______
c) 173 = 1 ________ + 7 ______ + 3 ______
d) 546 = 5 ________ + 4 ______ + 6 ______
Bonus:
e) 791 = 9 _________ + 1 _______ + 7 _______
f) 316 = 6 _________ + 3 _______ + 1 _______
Answers: a) hundreds, tens, ones; b) hundreds, tens, ones; c) hundreds, tens, ones;
d) hundreds, tens, ones; Bonus: e) tens, one, hundreds; f) ones, hundreds, tens
(MP.7) Have students draw base ten models for several numbers and record the expanded form
in two different ways (using numerals and words or numerals only). Examples: 135, 241, 129,
302. Have students expand several numbers using numerals only. Examples: 348, 640, 301.
(348 = 300 + 40 + 8, 640 = 600 + 40, 301 = 300 + 1). Ensure students understand that when we
have a 0 digit, we do not need to include 0 in the expanded sum. For example, we write
405 = 400 + 5, not 400 + 0 + 5. Similarly, we would normally write 405 = 4 hundreds + 5 ones,
even though it is important to understand that there are 0 tens.
Write the numeral for the expanded form. Have students write the numeral for several sums
written in expanded form. Write on the board:
200 + 70 + 6 = ___ ___ ___
Ask students what should be written in the blanks. PROMPT: What is the hundreds digit? (2)
The tens digit? (7) The ones digit? (6)
Exercises: Write the number for the expanded form.
a) 400 + 30 + 6 = _____
b) 700 + 60 + 5 = _____
c) 200 + 80 + 8 = _____
d) 800 + 10 + 3 = _____
Answers: a) 436, b) 765, c) 288, d) 813
Write on the board:
400 + 70 = ___ ___ ___
400 + 7 = ___ ___ ___
Ask students what should be written in the blanks for each question. PROMPT: What is the
hundreds digit? The tens digit? The ones digit? ASK: How are these two examples different?
(the first has 0 in the ones place, the second has 0 in the tens place)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-11
Exercises: Write the number for the expanded form.
a) 500 + 30 = _____
b) 700 + 5 = _____ c) 200 + 8 = _____
Answers: a) 530, b) 705, c) 208, d) 380
d) 300 + 80 = _____
(MP.1) Write on the board:
600 + 20 + ___ = 625
400 + ___ + 3 = 473
___ + 90 + 2 = 892
Challenge students to say how to fill in the blanks. For each question, PROMPT: What is
missing, the ones, tens, or hundreds? (ones, 5; tens, 70; hundreds, 800)
Exercises: Fill in the blank.
a) 500 + 30 + ____ = 534
b) 641 = 600 + ____ + 1
d) 700 + ___ + 2 = 742
e) 400 + ___ = 420
Answers: a) 4, b) 40, c) 800, d) 40, e) 20, f) 2
c) 812 = ____ + 10 + 2
f) 400 + ____ = 402
Expanded form and base ten sketches. Remind students how to draw rough sketches of
base ten models (see Lesson NBT3-2, p. C-7). Have students write various numbers in
expanded form and then draw a rough sketch of a base ten model. Examples: 732, 456, 57.
(MP.4) Tell your students that you read one book with 300 pages and another book with 70
pages. ASK: How many pages did I read altogether? Have a volunteer write the corresponding
addition sentence (300 + 70 = 370).
Exercises: Use expanded form to answer the question.
a) A store has 100 red bikes, 40 blue bikes, and 6 green bikes.
How many bikes does the store have altogether?
b) On a class field trip, there were 200 children, 10 parent volunteers, and 7 teachers.
How many people went on the field trip?
(MP.1) Bonus: In a school in Buffalo with 498 children, 400 students are from the United
States, and 90 students are from Canada. How many students are from neither the United
States nor Canada? (498 = 400 + 90 + ______)
Solutions: a) 100 + 40 + 6 = 146, b) 200 + 10 + 7 = 217, Bonus: 498 = 400 + 90 + ___, so
there are 8 students who are from neither the United States nor Canada.
Activity
I have ___, Who has ___? Using BLM Make Up Your Own Cards, make enough cards so that
everyone in the class can have one. Use expanded form and base ten sketches to make the
cards. For example, the student has the following card:
I have
300 + 40 + 8
Who has
?
C-12
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten The student says, “I have 348, who has 213?” and the person with 200 + 10 + 3 then says, “I
have 213, who has …” depending on which base ten model is on the bottom of that card. Play
continues until everyone gets a turn. You might arrange it so that the bottom of the last card
matches with the top of the first card, so that students know when they get back to the first card.
(end of activity)
Extensions
(MP.6) 1. Have students determine numbers written in expanded form when the hundreds, tens,
and ones are out of order. For example:
a) 30 + 5 + 400 = ____
b) 50 + 600 = ____ c) 5 tens + 6 ones + 7 hundreds = ____
Answers: a) 435, b) 650, c) 756
(MP.6) 2. Have students fill in the blanks for numbers in expanded form where the blanks are
out of order. For example:
a) 40 + ___ + 7 = 547
b) ___ + 60 = 69
c) ___ ones + ___ tens + 1 hundred = 198
Answers: a) 500; b) 9; c) 8, 9
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-13
NBT3-4 Writing and Reading Number Words
Pages 22–24
Standards: preparation for 3.NBT.A.2
Goals:
Students will read and write number words for any number between 0 and 999.
Prior Knowledge Required:
Understands place value
Vocabulary: digit, expanded form, number word, numeral
Materials:
BLM Blank Checks (p. C-83)
BLM Number Word Search (p. C-84, see Extension 1)
BLM Number Word Crossword Puzzle (p. C-85, see Extension 2)
BLM Crossword without Clues (p. C-86, see Extension 3)
(MP.6) Number words from 0 to 10. Write the numerals 0 to 10 vertically on the board. ASK:
Are these numbers written as numerals or as number words? (numerals) What are the number
words for these numerals? Have the students help you write the number words “zero, one, two,”
and so on beside the corresponding numerals. Ensure students understand the difference
between numerals (numbers written with symbols, as in “3”) and number words (as in “three”).
Number words from 11 to 19. Write the following words on the board, all in a row:
fourteen
thirteen
seventeen
sixteen
nineteen
fifteen
Ask the class to read the words out loud together. Ask volunteers to write the corresponding
numerals under the words. (14, 13, 17, 16, 19, 15)
ASK: What number does the word “teen” remind you of? (ten) Guide them by asking them to
look at the letters—is “teen” spelled almost the same as a number they know? Tell them that
fourteen is 4 + 10 = 14. ASK: Where can you see “four” in fourteen? Where can you see a word
that looks like “ten” in the word fourteen? Have volunteers demonstrate the first two parts of
both exercises below before having all students complete the rest.
(MP.8) Exercises:
1. Fill in the blanks with the correct number words.
a) fourteen = ______ + ten
b) seventeen = seven + __________
c) nineteen = ________ + ________
d) ________ = six + ten
Answers: a) four; b) ten; c) nine, ten; d) sixteen
C-14
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten 2. Circle the beginning letters that are the same in both number words.
a) six
sixteen
b) five
fifteen
c) nine
nineteen
d) four
fourteen
e) three
thirteen
f) two
twelve
g) seven
seventeen
h) eight
eighteen
Answers: a) six, b) fi, c) nine, d) four, e) th, f) tw, g) seven, h) eight
ASK: Which of these pairs have the entire 1-digit number in the word for the 2-digit number?
(four and fourteen, six and sixteen, seven and seventeen, eight and eighteen, nine and
nineteen) ASK: Why is the pair “eight and eighteen” a bit different from these other pairs?
(because “eight” ends with “t”, and there is only one “t” in “eighteen”)
Exercises: Circle the digits that are the same in both numbers.
a) 6
16
b) 5
15
c) 9
19
e) 3
13
f) 2
12
g) 7
17
Answer: a) 6, b) 5, c) 9, d) 4, e) 3, f) 2, g) 7, h) 8
d) 4
h) 8
14
18
Point out that the number word “eleven” is a special case, because even though eleven = one +
ten, the beginning letters of “one” do not match the beginning letters of “eleven.”
(MP.8) Exercises: Fill in the blank.
a) sixteen = ______ + ten
b) seventeen = ________ + ten
c) nineteen = nine + _______
d) thirteen = ________ + ten
e) fourteen = ______ + ten
f) fifteen = ________ + ten
g) twelve = _______ + ten
h) eleven = ________ + ten
i) eighteen = eight + _____
Answers: a) six, b) seven, c) ten, d) three, e) four, f) five, g) two, h) one, i) ten
Number words for multiples of 10 from 20 to 90. Write on the board:
twenty = 20
two = 2
ASK: What two beginning letters do these number words have in common? (“tw”) What digit is
in both numbers? (2)
Write on the board:
thirty
ASK: Can anyone think of a word for a 1-digit number that also starts with “th”? (three) Write:
thirty = _ 0
three = 3
Have a volunteer fill in the blank. (3) Write on the board:
forty = _ 0
fifty = _ 0
seventy =_ 0
sixty =_ 0
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-15
Have volunteers fill in the blanks by looking carefully at the beginning letters and asking
themselves what 1-digit number those letters remind them of.
ASK: Which ones digit do these numbers all have? (0) Which letters do the words all end with?
(“ty”) Tell students that any number word ending with “ty” will always mean a number having a
ones digit 0. Ask volunteers to guess how the following number words are written as numbers:
eighty, ninety. (80, 90) Challenge them to find a 2-digit number having the ones digit 0 whose
number word doesn’t end with “ty.”(10)
Exercises: Write the numerals for each number word.
a) thirty
thirteen
three
b) twenty
two
twelve
c) four
fourteen
forty
d) eighteen
eighty
eight
e) seven
ninety
thirteen
eighty
nine
fourteen
f) nineteen
sixty
forty
fifteen
twelve
eight
Answers: a) 30, 13, 3; b) 20, 2, 12; c) 4, 14, 40; d) 18, 80, 8; e) 7, 90, 13, 80, 9, 14; f) 19, 60,
40, 15, 12, 8
Exercises: Write the number word ending.
a) 30 = thir_____
b) 20 = twen ______
c) 13 = thir_____
d) 17 = seven____
e) 40 = for____
f) 80 = eigh____
g) 18 = eigh____
h) 19 = nine_____
i) 90 = nine____
Answers: a) ty, b) ty, c) teen, d) teen, e) ty, f) ty, g) teen, h) teen, i) ty
Finally, have students write the full number words.
Exercises: Write the number word.
a) 20 = _______
b) 19 = _______
c) 90 = ________
d) 17 = __________
e) 13 = _______
f) 80 = _______
g) 50 = ________
h) 15 = __________
i) 11 = ________
j) 18 = ________
k) 12 = ________
l) 70 = __________
Answers: a) twenty, b) nineteen, c) ninety, d) seventeen, e) thirteen, f) eighty, g) fifty, h) fifteen,
i) eleven, j) eighteen, k) twelve, l) seventy
Reading number words from 0 to 99. Write “twenty” on the board and ask a volunteer to write
the corresponding numeral. (20) Ask the student to state what the number word “twenty-three”
means. Can the student think of an addition sentence from this word? (20 + 3 = 23) Remind
students that this is the expanded form. Repeat for twenty-seven and twenty-one.
Exercise: Write the numeral for the given number word.
a) twenty-two
b) twenty-five
c) twenty-nine
d) twenty-six
Answers: a) 22, b) 25, c) 29, d) 26
C-16
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Write on the board:
thirty-six
Underneath the word “thirty,” write the numeral “30”; underneath the “six,” write “6.” You
might draw arrows connecting “thirty” to “30” and “six” to “6.” ASK: What numeral do you
think the number word “thirty-six” stands for? What addition sentence can you write from
this? (30 + 6 = 36) Have a volunteer write the numeral for thirty-five with the addition
sentence. (35 = 30 + 5)
Exercises: Write the numeral and the addition sentence.
a) thirty-three
b) thirty-two
c) thirty-eight
d) thirty-four
e) forty-two
f) eighty-one
g) fifty-four
h) ninety-seven
Answers: a) 30 + 3 = 33, b) 30 + 2 = 32, c) 30 + 8 = 38, d) 30 + 4 = 34, e) 40 + 2 = 2,
f) 80 + 1 = 81, g) 50 + 4 = 54, h) 90 + 7 = 97
Write on the board:
73 = 70 + 3
seventy-three
32 =
30 + 2
thirty-two
61 = 60 + 1
sixty-one
54 =
50 + 4
fifty-four
15 = 5 + 10
fifteen
18 =
8 + 10
eighteen
13 = 3 + 10
thirteen
16 =
6 + 10
sixteen
If available, use colored chalk or an overhead projector and write the parts in bold in a different
color. Point to each question and ASK: Where do you see the first digit of the number in the
number word—at the beginning or at the end? Which number words have the first digit at the
beginning? (twenty and higher) Which number words have the first digit at the end? (thirteen to
nineteen)
Exercises: Write the numeral for the number word.
a) thirty-eight
b) forty-five
c) twenty-six
e) sixty-seven
f) eighty-nine
g) seventy-four
Answers: a) 38, b) 45, c) 26, d) 51, e) 67, f) 89, g) 74, h) 91
d) fifty-one
h) ninety-one
Exercises: Write the numeral for the number word.
a) twenty-eight
b) eighteen
c) sixteen
d) four
e) forty
f) forty-three
g) zero
h) fifty
i) fifty-eight
j) thirteen
k) twelve
l) nineteen
m) twenty-nine
n) fifty-nine
o) forty-eight
p) twenty
q) thirty-one
r) eleven
Answers: a) 28, b) 18, c) 16, d) 4, e) 40, f) 43, g) 0, h) 50, i) 58, j) 13, k) 12, l) 19, m) 29, n) 59,
o) 48, p) 20, q) 31, r) 11
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-17
Writing number words from 0 to 99. Tell students you want to write the number word for 45.
PROMPT: What does the 4 stand for? (40) Write on the board:
45
forty-_____
ASK: what does the 5 stand for? Have a volunteer fill in the blank. (five)
Exercises: Write the number word for the numeral.
a) 41
b) 32
c) 90
d) 9
e) 89
f) 74
g) 99
h) 0
i) 50
j) 25
k) 17
l) 11
Bonus: Is the number word written correctly? If not, find the mistake and write the number word
correctly.
m) forty-zero
n) forty-three
o) twenty-eight
p) thirty nine
q) eight-five
r) seventy-six
Answers: a) forty-one; b) thirty-two; c) ninety; d) nine; e) eighty-nine; f) seventy-four; g) ninetynine; h) zero; i) fifty; j) twenty-five; k) seventeen; l) eleven; Bonus: m) should be “forty”, no “zero”; n) correct; o) correct; p) the “-” is missing, should be “thirty-nine”; q) should be “eightyfive”, the “y” is missing; r) correct
Number words for multiples of 100. Once students have mastered writing numbers up to 99,
tell them that writing hundreds is even easier. There is no special word for three hundreds like
there is for three tens: 30 = 10 + 10 + 10 = thirty, but 300 = 100 + 100 + 100 = three hundred
(not three hundreds) SAY: You just write what you see: three hundred. There’s no special word
to remember.
Have volunteers write the number words for 3-digit multiples of 100: 200, 300, 500. (two
hundred, three hundred, five hundred) Remind them not to include a final “s” even when there is
more than one hundred. Have volunteers write the numerals given number words for multiples
of 100: two hundred, six hundred, nine hundred, one hundred, five hundred, and eight hundred.
(200, 600, 900, 100, 500, 800)
Exercises:
1. Write the number word for the numeral.
a) 400
b) 600
c) 900
d) 800
e) 700
f) 100
Answers: a) four hundred, b) six hundred, c) nine hundred, d) eight hundred, e) seven hundred,
f) one hundred
2. Write the numeral for the number word.
a) three hundred
b) one hundred
Answers: a) 300, b) 100, c) 400, d) 700
c) four hundred
d) seven hundred
Number words for 3-digit numbers. Tell students that they can write out 3-digit numbers like
532 by breaking them down. Say the number out loud and invite students to help you write what
they hear: five hundred thirty-two. Point out that there is no dash between “five” and “hundred.”
C-18
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Also, emphasize that the word and should not appear in these number words: for example, 301
is written as “three hundred one,” not as “three hundred and one.”
Exercises: Fill in the blanks.
1. a) 134 = one hundred ______________
b) 904 = nine hundred ___________
c) 650 = six hundred _________________
d) 718 = seven hundred __________
Answers: a) thirty-four, b) four, c) fifty, d) eighteen
2. a) 761 = ______________ sixty-one
b) 840 = ___________ forty
c) 327 = ________________ twenty-seven
d) 205 = _______________ five
Answers: a) seven hundred, b) eight hundred, c) three hundred, d) two hundred
3. a) 898 = _________________________
b) 740 = ___________________
c) 756 = __________________________
d) 601 = ___________________
Answers: a) eight hundred ninety-eight, b) seven hundred forty, c) seven hundred fifty-six,
d) six hundred one
Number words from 0 to 999. Write some typical text from signs and banners and have
students replace any number words with numerals and vice versa.
Exercises: Write the number word for the numeral, or the numeral for the number word.
a) New York 181 miles
b) Speed Limit: 65 miles/hour
c) Maximum Height 6 feet
d) Seventy-Four Main Street
e) Bulk Sale! Buy Ten for the Price of Five!
f) Bus Stop: Route 108
Answers: a) one hundred eighty-one; b) sixty-five; c) six; d) 74; e) 10, 5; f) one hundred eight
Exercises: Write the correct number word to complete the sentence.
a) There are ___________ months in a year.
b) There are ___________ days in a week.
c) There are ___________ weeks in a year.
d) February normally has ____________ days.
e) A year normally has ___________ days.
f) A leap year has ____________ days.
Answers: a) twelve, b) seven, c) fifty-two, d) twenty-eight, e) three hundred sixty-five, f) three
hundred sixty-six
Activity
At www.funbrain.com/numwords, students can use “Method 1” to write the number word
corresponding to a given numeral in the correct place on a check, or use “Method 2” to read a
given number word and write the corresponding numeral. You may select the number range
(e.g., 0 to 100, 0 to 1000). Alternatively, you can draw checks on the board, where you fill in the
numeral amount and have students write the number in words, or vice versa, and then do the
same with BLM Blank Checks.
(end of activity)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-19
Extensions
1. Provide copies of BLM Number Word Search. Encourage students to use the message they
find after finishing the puzzle to check that they did the puzzle correctly.
Selected Answer: 2. winter, spring, and, summer
2. Give students BLM Number Word Crossword Puzzle.
Answers: Across: 2. six, 4. nine, 7. seventeen, 8. eighty, 10. forty, 11. zero; Down: 1. one, 2.
seventy. 3. fifteen, 5. twenty, 6. ten, 9. three
(MP.1) 3. Give students BLM Crossword without Clues.
Answers:
1. 4 letters: nine, zero; 5 letters: forty, three; 6 letters: eighty, twenty; 7 letters: fifteen, seventy; 9
letters: seventeen;
2. seventeen is the only word in a group by itself (9 letters)—see solved puzzle below for where
it fits;
3.
s
t
w
t
t
w
e
e
o
n
n
t
e
i
g
h
t
v
e
n
z
t
e
e
n
r
f
o
r
t
y
y
h
r
s
f
i
e
f
t
e
e
x
n
i
s
e
v
e
n
t
y
e
C-20
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-5 Number Representations Summary
Pages 25–27
Standards: preparation for 3.NBT.A.2
Goals:
Students will practice interchanging between the various number representations covered thus
far (number words, base ten blocks, and expanded form) for 1-, 2-, and 3-digit numbers.
Prior Knowledge Required:
Understands expanded form for 2- and 3-digit numbers
Can read and write number words for 2- and 3-digit numbers
Can draw base ten models for 2- and 3-digit numbers
Materials:
BLM Blank Checks (p. C-83; see Extension 4)
Review of expanded form. Have a volunteer do Exercises 1.a) and 2.a) below and then have
all students do the remaining exercises.
Exercises:
1. Fill in the blank.
a) 400 + 30 + 7 = _____
b) 300 + 50 + 9 = _____
d) 40 + 3 = _____
e) 600 + 80 + 8 = _____
Answers: a) 437, b) 359, c) 807, d) 43, e) 688, f) 940
c) 800 + 7 = _____
f) 900 + 40= _____
2. For each part of Exercise 1, write the number in expanded form using numerals and words.
Selected answer: a) 4 hundreds + 3 tens + 7 ones
Demonstrate the first exercise below and then have students do the remaining exercises.
(MP.7) Exercises: Fill in the blank.
a) 600 + ____ + 9 = 679
b) ____ + 40 + 8 = 548
d) 200 + ____ = 250
e) 200 + ____ = 205
Answers: a) 70, b) 500, c) 5, d) 50, e) 5, f) 10
c) 500 + 50 + ____ = 555
f) 900 + ____ + 6 = 916
Review of number words. Have a volunteer do the first exercise below and then have all
students do the remaining exercises.
Exercises: Write the number word for the given numeral.
a) 578
b) 611
c) 818
d) 923
e) 650
f) 605
Answers: a) five hundred seventy-eight, b) six hundred eleven, c) eight hundred eighteen,
d) nine hundred twenty-three, e) six hundred fifty, f) six hundred five
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-21
Exercises: Write the numeral for the given number word.
a) four hundred eighty-nine
b) three hundred twenty
c) one hundred four
d) sixty-seven
e) eleven
f) nine hundred twenty-five
Answers: a) 489, b) 320, c) 104, d) 67, e) 11, f) 925
Review of base ten blocks. Write 1-, 2-, and 3-digit numbers on the board and have student
volunteers draw base ten models (sketches). Remind students of a fast way to make sketches:
Hundreds Block
Tens Block
Ones Block
Exercises: Draw base ten sketches for the following numbers:
a) 314
b) 278
c) 83
d) 401
e) 189
f) 15
Selected answer:
a)
Extensions
(MP.6) 1. Ask students to explain and show with base ten blocks the meaning of each digit in a
number with all digits the same (e.g., 333).
2. Have students solve these puzzles using base ten blocks:
a) I am greater than 20 and less than 30. My ones digit is one more than my tens digit.
b) I am a 3-digit number. My digits are all the same. Use 12 blocks to make me.
c) I am a 2-digit number. My tens digit is 5 more than my ones digit. Use 7 blocks to make me.
d) I am a 3-digit number. My tens digit is one more than my hundreds digit and my ones digit is
one more than my tens digit. Use 6 blocks to make me.
Answers: a) 23, b) 444, c) 61, d) 123
3. Have students solve these puzzles by only imagining the base ten blocks. NOTE: Parts
a) through d) have more than one answer—emphasize this by asking students to share their
answers.
a) I have more tens than ones. What number could I be?
b) I have the same number of ones and tens blocks. What number could I be?
c) I have twice as many tens blocks as ones blocks. What 2-digit number could I be?
d) I have six more ones than tens. What number could I be?
(MP.3) e) You have one set of blocks that make the number 13 and one set of blocks that make
the number 22. Can you have the same number of blocks in both sets?
(MP.3) f) You have one set of blocks that make the number 23 and one set of blocks that make
the number 16. Can you have the same number of blocks in both sets?
C-22
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Selected answers: e) Yes, standard models—i.e., 4 blocks each; f) No. There are only two
possibilities for 16: 7 blocks and 16 blocks. There are only three possibilities for 23: 5 blocks, 14
blocks, 23 blocks. So there is no match.
4. See Activity from Lesson NBT3-4. Show students a copy of a check and explain why it’s
important to write the amount using both words and numerals. Show them how easy it is to
change a number such as “48.00” to “148.00” by adding the digit 1. On the other hand, it would
be very difficult to add “one hundred” before the word “forty-eight,” especially if you write “fortyeight” starting at the far left. Write sample checks on the board with the amount written in
numeral form and have students fill in the number word in the appropriate spot, and vice versa.
Or do this exercise using BLM Blank Checks, as in the Activity from Lesson NBT3-4.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-23
NBT3-6 Comparing Numbers with Base Ten Models
Pages 28–29
Standards: preparation for 3.NBT.A.1
Goals:
Students will use base ten blocks to compare pairs of 1-, 2-, and 3-digit numbers and determine
which number is greater.
Prior Knowledge Required:
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Vocabulary: base ten blocks, greater than, hundreds, less than, ones, ones place, tens,
regrouping, tens place
Materials:
base ten blocks
Less than and greater than. Emphasize that to say that one number is greater than another
means the first number represents more objects than the second—e.g., 4 dollars is more money
than 3 dollars, 4 meters is longer than 3 meters, 4 goals is more than 3 goals, 4 minutes is more
time than 3 minutes. It is crucial that students understand that 4 of anything is more than 3 of
the same thing, and so it makes sense to compare the numbers 3 and 4 by saying that 4 is
“more than” 3. The correct mathematical expression is “4 is greater than 3,” and students should
get used to using the phrases “greater than” and “less than.”
Write on the board:
4 is ___________ 5
Have volunteers say either “greater than” or “less than” as appropriate (in this case, “less than”
is correct). Repeat with several pairs of 1-digit numbers to ensure that students are comfortable
with the phrases “greater than” and “less than.”
Comparing 2-digit numbers by matching tens. Model the numbers 43 and 26 using base ten
materials:
43
C-24
26
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten ASK: Which number has more tens blocks, 43 or 26? (43) Which has more ones blocks? (26)
Hmmm, 43 has more tens blocks, but 26 has more ones blocks—how can we know which
number is bigger? Show students how to match the 2 tens blocks of 26 with 2 tens blocks of 43.
ASK: After matching the 2 tens blocks of 26, how many ones are left over in the number 26? (6)
Can we use these 6 ones blocks to match with one of the remaining tens blocks of 43? (no)
Point out that the remaining 6 ones blocks of 26 are not enough to make another tens block.
ASK: Which number is greater, 43 or 26? (43) Why? (because 43 has more tens)
ASK: If two 2-digit numbers have different tens digits, which number is greater? Ensure students
understand that the number with more tens is the larger number.
Exercises: Use base ten blocks to model the numbers. Circle the greater number.
a) 56 38
b) 39 45
c) 17 46
d) 38 55
Answers: a) 56, b) 45, c) 46, d) 55
Model the numbers 54 and 57. ASK: Which number has more tens, 54 or 57? Students should
see that the number of tens match. ASK: How do we know which number is greater? Explain to
students that if the number of tens match, you need to compare the number of ones blocks.
ASK: Which number has more ones blocks, 54 or 57? (57) So which number is greater, 54 or
57? (57)
Exercises: Use base ten blocks to model the numbers. Circle the greater number.
a) 56 58
b) 59 62
c) 37 33
d) 28 51
Answers: a) 58, b) 62, c) 37, d) 51
Summarize the above exercises. ASK: If two 2-digit numbers have the same tens digit, how
can you tell which number is greater? (the number with the greater ones digit will be greater) If
the numbers have different tens digits, how can you tell which number is greater? (the number
with a greater tens digit will be greater).
2-digit numbers are larger than 1-digit numbers. Guide students to see that 2-digit numbers
are larger than 1-digit numbers. Model the numbers 7 and 15. SAY: 7 is a 1-digit number. Does
7 have any tens blocks? (no) ASK: Does any 1-digit number have enough ones to make a tens
block? (no) SAY: 15 is a 2-digit number. How many tens blocks are in 15? (1) Does a 2-digit
number always have at least one tens block? (yes) Which number is bigger, 7 or 15? (15) Why?
(15 has more tens) Why is a 2-digit number always greater than a 1-digit number? Students
should see that 2-digit numbers have 1 or more tens, while 1-digit numbers have no tens.
Exercises: Use base ten blocks to model the numbers. Circle the greater number.
a) 6
13
b) 49 62
c) 12 9
d) 48 31
Answers: a) 13, b) 62, c) 12, d) 48
Comparing 3-digit numbers. Ask volunteers to make (or draw rough sketches of) base ten
models for the numbers 238 and 153. ASK: Which number has more hundreds blocks? Tens
blocks? Ones blocks? Which number do you think is greater? Match the hundreds block of 153
with one of the hundreds blocks of 238. ASK: How many tens are left over in 153? (5) How
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-25
many ones blocks are there? (3) Can this be enough to make another hundreds block? (no) So,
which number is greater, 238 or 153? (238)
Teach the students how to compare two 3-digit numbers by comparing first the hundreds, then
the tens (if the hundreds are the same), and then the ones (if the hundreds and tens are the
same). Make or draw base ten models to compare 153 and 147:
153
same more tens
147
SAY: We first compare the hundreds. Which number has more hundreds, 153 or 147? (they
both have 1 hundred) Since they have the same number of hundreds, how do we check which
number is greater? (compare the tens) Which number has more tens? (153 has more tens than
147) Do we need to compare the ones now? (no, we already know 153 is the greater number)
Repeat with several examples where the numbers differ in the hundreds, tens or ones digits.
Examples: 235 and 239, 125 and 213, 136 and 152, 108 and 105
Exercises: Use base ten blocks to model the numbers. Circle the greater number.
a) 236 149
b) 249 261
c) 129 127
d) 148 184
Answers: a) 236, b) 261, c) 129, d) 184
(MP.8) Summarize the above exercises. ASK: If two 3-digit numbers have different hundreds
digits, which number is greater? (the one with more hundreds) If two 3-digit numbers have the
same hundreds digit, how do we check which one is greater? (look at the tens) What if the tens
are different, then which number is greater? (the one with more tens) What if the tens are also
the same? How do we check which number is greater? (look at the ones) If the ones are
different, which number is greater? (the number with more ones) If all 3 digits are the same,
what can we say about the numbers? (the numbers are equal)
3-digit numbers are larger than 2-digit numbers. Guide students to see that 3-digit numbers
are larger than 2-digit numbers. Model the numbers 47 and 125. SAY: 47 is a 2-digit number.
Does 47 have any hundreds blocks? (no) ASK: Does any 2-digit number have enough tens and
ones to make a hundreds blocks? (no) SAY: 125 is a 3-digit number. How many hundreds
blocks are in 125? (1) Does a 3-digit number always have at least one hundreds block? (yes)
Which number is bigger, 47 or 125? (125) Why? (125 has more hundreds) Why is a 3-digit
number always greater than a 2-digit number? Students should see that 3-digit numbers have 1
or more hundreds, while 2-digit numbers have no hundreds.
C-26
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises: Use base ten blocks to model the numbers. Circle the greater number.
a) 67 143
b) 439 612
c) 212 98
d) 148 131
Answers: a) 143, b) 612, c) 212, d) 148
Extensions
1. Have students compare numbers with base ten models where the blocks are not in order.
Students should write the number represented and then circle the larger number. Include
examples where the two models represent the same number. In such cases, students should
write the word “equal” or the symbol = rather than circle either number. Example:
123
=
123
2. Create base ten models of a pair of two-digit numbers. Ask students to say how they know
which number is greater. You might make one of the numbers in non-standard form, as shown
in the first number in the example below:
To compare the numbers, students could remodel the first number in standard form by
regrouping 10 ones blocks as 1 tens block.
3. Ask students to create base ten models of two numbers where one of the numbers …
a) is 30 more than the other.
b) is 50 less than the other.
c) has a hundreds digit equal to 6 and is 310 more than the other.
Sample answers: a) 480, 450; b) 214, 264; c) 628, 318
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-27
NBT3-7 Comparing Numbers by Place Value
Pages 30–32
Standards: preparation for 3.NBT.A.1
Goals:
Students will compare pairs of 1-, 2-, and 3-digit numbers using place value to determine which
number is greater.
Students will use words and symbols (<, >, =).
Prior Knowledge Required:
Understands expanded form for 2- and 3-digit numbers
Vocabulary: expanded form, symbols for “less than” (<), “greater than” (>), and “equals” (=)
Comparing numbers using the value of digits. Write 52 and 42 on the board as follows:
5
2
4
2
Have volunteers write the value of each digit in the appropriate boxes and tell you which number
is greater. Explain that since 50 is larger than 40, and 2 is the same as 2, we know 52 is larger
than 42. Write several similar questions on the board comparing either 2- or 3-digit numbers that
differ by only one digit and have students determine the greater number in each pair.
Exercises: Circle the greater number.
a) 63 83
b) 28 25
d) 529
729
e) 306 206
Answers: a) 83, b) 28, c) 71, d) 729, e) 306, f) 543
c) 71
f) 543
61
243
Comparing numbers using expanded form. Compare numbers by writing them in expanded
form instead:
52 = 50 + 2
42 = 40 + 2
SAY: Since 50 is larger than 40, and 2 is the same as 2, 52 is larger than 42. Repeat with the
following pairs of numbers: 36 and 26, 68 and 98, 46 and 43. C-28
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Ask students which of the 3-digit numbers is greater, 475 or 465:
475 = 400 + 70 + 5
465 = 400 + 60 + 5
Since 70 is greater than 60, we know that 400 + 70 + 5 is greater than 400 + 60 + 5.
SAY: Without writing the expanded form, we can see this from the digits themselves.
475
465
Since the 7 means 70 and the 6 means 60, we can see that 475 is greater than 465.
(MP.6) Have students compare two 3-digit numbers (that differ by one digit) by circling the digit
that is different and writing the greater number in the space below:
a)
475
465
b)
356
358
c)
297
497
d)
493
490
e)
527
507
475
(MP.3) Comparing two numbers that differ by more than one digit. Tell students you want
to compare 342 and 257. ASK: Which number has more hundreds? More tens? More ones?
Which number do you think is greater, the one with the most hundreds, the most tens or the
most ones? (the one with the most hundreds) Why? (because hundreds are worth more than
tens and ones)
Remind the students how to compare two 3-digit numbers by comparing first the hundreds, then
the tens (if the hundreds are the same), and then the ones (if the hundreds and tens are the
same). SAY: How do we compare 246 and 239? We first compare the hundreds. Which number
has more hundreds, 246 or 239? (they both have 2 hundreds) Since they have the same
number of hundreds, how do we check which number is greater? (compare the tens) Which
number has more tens? (246 has more tens than 239) Do we need to compare the ones now?
(no, we already know 246 is the greater number)
(MP.8) Summarize the method for comparing 3-digit numbers. ASK: If two 3-digit numbers
have different hundreds digits, which number is greater? (the one with more hundreds) If two 3digit numbers have the same hundreds digit, how do we check which one is greater? (look at
the tens) What if the tens are different, then which number is greater? (the one with more tens)
What if the tens are also the same? How do we check which number is greater? (look at the
ones) If the ones are different, which number is greater? (the number with more ones) If all 3
digits are the same, what can we say about the numbers? (the numbers are equal)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-29
Demonstrate the first exercise below before having all students complete the rest. Remind
students to read the numbers from left to right when they are looking for the highest place value
where the digits are different.
Exercises: Circle the highest place value where the digits are different. Write the larger number
in the blank.
a) 372
b) 156
c) 297
d) 193
e) 527
368
351
296
222
509
372
Answers: b) 351, c) 297, d) 222, e) 527
Comparing numbers that don’t have the same number of digits. Write on the board:
7
13
ASK: Which of these numbers is less than 10? (7) Which is greater than 10? (13) So which
number is greater, 7 or 13? (13) Is a 2-digit number always greater than a 1-digit number? (yes)
Why? PROMPT: Are all 2-digit numbers greater than or equal to 10? (yes) Are all one-digit
numbers less than 10? (yes)
Write on the board:
87
123
ASK: Which of these numbers is less than 100? (87) Which is greater than 100? (123) So which
number is greater, 87 or 123? (123) Is a 3-digit number always greater than a 2-digit number?
(yes) Why? PROMPT: Are all 3-digit numbers greater than or equal to 100? (yes) Are all twodigit numbers less than 100? (yes)
Exercises: Circle the greater number.
a) 108
97
b) 88 800
Answers: a) 108, b) 800, c) 14
c) 14
7
Comparing two numbers between 0 and 999. Write on the board:
456
309
34
43
280
219
Have volunteers circle the greater number in each pair. (456, 43, 280) Write on the board:
456 is __________ 309.
34 is ___________ 43.
280 is ____________ 219.
Have volunteers complete the sentences with either the phrase “less than” or “greater than.”
(greater than, less than, greater than)
C-30
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises: Write “less than” or “greater than” in the blank.
a) 359 is ______________ 601.
b) 163 is ______________ 139.
d) 134 is _______________ 75.
d) 859 is ______________ 895.
e) 480 is ______________ 419.
f) 437 is ______________ 441.
Answers: a) less than, b) greater than, c) greater than, d) less than, e) greater than, f) less than
The meaning of the symbols < and >. Tell students that there is a short form for writing the
phrases “less than” and “greater than.” You can write the following examples on the board to
illustrate:
< means “less than”
3<8
3 is less than 8.
> means “greater than”
6>1
6 is greater than 1.
Since some students will have difficulty reading and writing sentences using < and >, take them
through the following progression using only 1-digit numbers at first.
Reading < and > sentences. Write true sentences on the board with the symbols < or >
involving 1-digit numbers:
6<9
3>1
5<7
4>3
Have volunteers read the sentences out loud and then write the corresponding sentence in
words underneath. (6 is less than 9, 3 is greater than 1, 5 is less than 7, 4 is greater than 3)
Exercises: Write the sentence in words.
a) 4 < 5
b) 7 > 2
c) 2 > 0
d) 8 < 9
Answers: a) 4 is less than 5, b) 7 is greater than 2, c) 2 is greater than 0, d) 8 is less than 9
Verifying < and > sentences. Write true and false sentences on the board involving <, >, and
1-digit numbers. Have the students read the sentences out loud, write the sentence in words,
and then decide whether the sentence is true or false:
4>8
4 is greater than 8
TRUE FALSE
5>3
____________
TRUE FALSE
9<2
___________
TRUE FALSE
1<7
______________
TRUE FALSE
Exercises: Write the sentence in words. Is it true or false?
a) 6 > 8
b) 5 < 9
c) 3 < 2
d) 1 > 0
Selected answers: a) 6 is greater than 8, FALSE; b) 5 is less than 9, TRUE
Writing < and > sentences. Write true sentences on the board with the phrases “less than” or
“greater than” involving 1-digit numbers:
3 is less than 7
4 is greater than 1
5 is less than 8
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-31
Have volunteers read the sentences out loud and then write the corresponding sentences using
the symbols < or >. (3 < 7, 4 > 1, 5 < 8)
Exercises: Write the sentence using the symbols < and >.
a) 5 is greater than 2
b) 1 is less than 2
c) 9 is greater than 7
Answers: a) 5 > 2, b) 1 < 2, c) 9 > 7
Choosing and writing the correct symbol. Write on the board:
5
<
8
>
7
<
3
>
9
<
>
4
2
<
>
6
Have volunteers circle the correct symbol in each question. (<, >, >, <) Write on the board:
4
8
6
0
9
7
5
6
2
1
Have volunteers write in the correct symbol in each case. (<, >, >, <, >)
Exercises: Write < or > between the numbers.
a) 3
7
b) 1
5
c) 7
3
d) 5
Answers: a) <, b) <, c) >, d) >, e) <
2
e) 8
9
When students can confidently write the correct symbols, progress to 2- and 3-digit numbers.
A trick for using the symbols < and > correctly. Students might still have a hard time
remembering which of the two symbols means less than and which means greater than. Draw
the following face on the board:
SAY: This hungry person can have only one pile of bananas. Which pile should the person take?
45 bananas
31 bananas
45
31
Have a volunteer draw the face with the mouth opening toward the pile with more bananas, as
above (left). Erase the face and leave only the mouth (as above, right).
SAY: 45 is greater than 31 (write: 45 > 31), and 31 is less than 45 (write: 31 < 45). Think of the
< and > symbols as a hungry mouth that always opens toward the larger number.
C-32
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Forming the least and greatest numbers from different digits. Write on the board:
5, 8
Tell students that you want to make a 2-digit number using each of these digits once. ASK:
What numbers can we make? (58 and 85) Write these numbers on the board. ASK: Which is
greater? (85) Why? How do you know 85 is greater than 58? Emphasize that the greater
number, 85, has the most tens possible.
(MP.3) Write on the board:
1, 4, 7
Tell students that you want to make a 3-digit number using these digits. Have two students
volunteer different answers. Write their answers on the board. ASK: Which is greater? Tell
students you want the greatest 3-digit number they can think of that uses these 3 digits. (741)
ASK: How do you know it’s the greatest? Wait for an answer before giving any hints. Emphasize
that the greatest number has the most hundreds possible, and then the most tens possible. Ask
students to make the smallest 3-digit number possible with the same digits (147) and to explain
how they know it’s the smallest. ASK: Can you make a smaller number with those digits?
Repeat with several examples of 3-digit numbers: 2, 5, 8; 3, 7, 9; 1, 2, 4.
Comparing and ordering 2- and 3-digit numbers. SAY: I want to compare two 2-digit
numbers. ASK: If the number of tens is different, how can you tell which number is greater? (the
number with the greater tens digit) If the number of tens is the same, how can you tell which
number is greater? (the number with the greater ones digit)
Exercises: Arrange the numbers in order from smallest to greatest.
a) 27, 24, 29
b) 69, 99, 89
c) 48, 49, 44
d) 83, 53, 63
e) 49, 53, 29
f) 57, 43, 60
g) 43, 50, 29
h) 30, 25, 23
Selected answers: a) 24, 27, 29; d) 53, 63, 83; e) 29, 49, 53; h) 23, 25, 30
When students are comfortable comparing 2-digit numbers, ask them to recall how they would
compare two 3-digit numbers. ASK: Which digit should you look at first, the ones digit, the tens
digit, or the hundreds digit? Why?
Exercises: Arrange the numbers in order from smallest to greatest.
a) 134, 127, 198
b) 354, 350, 357
c) 376, 762, 480
Answers: a) 127, 134, 198; b) 350, 354, 357; c) 376, 480, 762
(MP.8) Two ways of forming the least and greatest numbers with different digits. Write the
digits 3, 5, and 8. Ask students to make all the possible 3-digit numbers from these digits. Tell
them to try to do it in an organized way. Take suggestions for how to do that. For example, they
could start with the hundreds digit: write all the numbers that have hundreds digit 3 first, then
work on the numbers with hundreds digit 5, and then 8. How many numbers are there
altogether? (6 numbers; 2 starting with each digit) Which of these numbers is greatest? Which is
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-33
least? Have students reflect: Could they have found the greatest number using these digits
without listing all the possible numbers? (yes, write the digits in order from greatest to least)
Why does this work? (because to make the number as large as possible, we want to use the
larger digits in the place values that are worth more)
Extensions
(MP.4) 1. Do some comparison questions involving real-life scenarios. For example:
a) Rita’s mother is 43 years old. Anna’s mother is 51 years old. Whose mother is older?
b) Raj has $540. Karen has $259. Who has more money?
Answers: a) Anna’s mother is older, b) Raj has more money
2. a) Use the digits 5, 6, and 7 to create as many 3-digit numbers as you can. Use each digit
only once when you create a number. Write your answers from least to greatest.
(MP.7) b) Use the digits 5, 6, 7 and 8 to create as many 3-digit numbers as you can. Use each
digit only once when you create a number. Note that each 3-digit number will leave out one of
the digits. Write your answers from least to greatest.
Answers:
a) 567, 576, 657, 675, 756, 765;
b) 567, 568, 576, 578, 586, 587, 657, 658, 675, 678, 685, 687, 756, 758, 765, 768, 785, 786,
856, 857, 865, 867, 875, 876
3. List 4 numbers that come between 263 and 527.
Sample answer: 269, 300, 400, 501
4. Name 2 places where you might see more than 1,000 people.
5. Have students determine whether the following sentences are true or false.
a) 600 + 40 + 7 = 674
b) 300 + 4 + 30 = 334
c) 586 < 50 + 800 + 6
d) seven hundred one > 710
Answers: a) false, b) true, c) true, d) false
(MP.1) 6. Have students compare pairs of numbers represented in various forms, including
numbers written in words, numbers written in expanded form, and numbers drawn with base ten
models (with blocks possibly out of order). Include examples where the two numbers are equal.
Students should use the symbols <, > and = to fill in the blanks. Examples:
a) 800 + 10 + 7 _____ eight hundred seventy
b) 613 _____ 6 hundreds + 1 ten + 3 ones
c)
_____
You might have students write each number as a numeral before comparing.
Answers: a) <, b) =, c) <
C-34
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten 7. Ask students where they tend to see many numbers in increasing order.
Sample answers: page numbers in a book, house numbers, mailboxes, apartment numbers,
store lineups when people need to take a number to be served
8. Have students fill in the blanks with the given numbers.
a) 3, 9
b) 10, 4
c) 6, 5
d) 1, 8
e) 12, 19
__ > __
__ > __
__ < __
__ < __
___ > ___
f) 436, 379
____ < ____
Answers: a) 9, 3; b) 10, 4; c) 5, 6; d) 1, 8; e) 19, 12; f) 379, 436
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-35
NBT3-8 Differences of 10 and 100
Pages 33–35
Standards: 3.OA.D.8, 3.OA.D.9
Goals:
Students will recognize when two numbers differ by 10 or 100.
Students will be able to say that one number is 10 (or 100) more (or less) than another.
Prior Knowledge Required:
Can compare and order 1-, 2-, and 3-digit numbers
Understands place value
Understands expanded form
Vocabulary: expanded form, less than, more than
Materials:
BLM Hundreds Chart Pieces (pp. C-87–88, see Extension 2)
Comparing numbers that differ by 1. Write on the board:
a) 8 is _________ than 7
b) 4 is _________ than 5
c) 2 is _________ than 3
ASK: Is 8 more than or less than 7? (more than) How much more? (8 is 1 more than 7). Write “1
more” in the blank for a). Repeat with b) and c), writing “1 less” in the blanks for b) and c).
Exercises: Write “1 more” or “1 less” in the blank.
a) 9 is _________ than 8
b) 10 is _________ than 11
c) 6 is _________ than 7
d) 5 is _________ than 4
e) 12 is _________ than 13
f) 40 is _________ than 39
Answers: a) 1 more, b) 1 less, c) 1 less, d) 1 more, e) 1 less, f) 1 more
Comparing numbers that differ by 10. Write on the board:
a) 50 is ________ than 60
b) 40 is ________ than 30
c) 20 is ________ than 30
ASK: Is 50 more than 60 or less than 60? (50 is less than 60) How much less? (50 is 10 less
than 60) PROMPT: How many tens are there in 50? (5) In 60? (6) Write “10 less” in the blank
for a). Repeat with b) and c), writing “10 more” for b) and “10 less” for c).
Exercises: Write “10 more” or “10 less” in the blank.
a) 90 is ________ than 80
b) 10 is ________ than 20
c) 60 is ________ than 70
d) 50 is ________ than 40
e) 70 is ________ than 80
f) 30 is ________ than 40
Answers: a) 10 more, b) 10 less, c) 10 less, d) 10 more, e) 10 less, f) 10 less
C-36
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Comparing numbers that differ by 100. Write on the board:
a) 500 is ______ than 600
b) 400 is ______ than 300
c) 200 is _______ than 300
ASK: Is 500 more than 600 or less than 600? (500 is less than 600) How much less? (500 is
100 less than 600) PROMPT: How many hundreds are there in 500? (5) In 600? (6) Write “100
less” in the blank for a). Repeat with b) and c), writing “100 more” for b) and “100 less” for c).
Exercises: Write “100 more” or “100 less” in the blank.
a) 700 is ______ than 800 b) 400 is ______ than 500 c) 600 is ______ than 500
d) 100 is ______ than 200 e) 700 is ______ than 600 f) 300 is ______ than 200
Answers: a) 100 less, b) 100 less, c) 100 more, d) 100 less, e) 100 more, f) 100 more
Comparing numbers using the expanded form. Write “345” and “335” on the board, and
have volunteers write the numbers in expanded form using numerals only:
345 = 300 + 40 + 5
335 = 300 + 30 + 5
345 is _________ than 335
Ensure the numbers are aligned, as above. ASK: Do these numbers have the same expanded
form? (no) Which part is different? (40 is different from 30). Which number is larger, 40 or 30?
(40) How much more is 40 than 30? (40 is 10 more than 30) So, how much more is 345 than
335? (345 is 10 more than 335) Fill in the blank above with “10 more.”
Comparing numbers using place value. Write on the board.
259
269
259 is __________ than 269
SAY: Let’s compare these two numbers just by looking at the digits. In which place value are the
numbers different? PROMPT: Do the hundreds digits match? (yes) Do the ones digits match?
(yes) Do the tens digits match? (no) Circle the tens digits, as shown below:
259
269
259 is __________ than 269
Which number is more? (269) How much more? Ensure students understand that 269 has one
more ten than 259, so 269 is 10 more than 259, and 259 is 10 less than 269. Write “10 less” in
the blank above. Repeat this process with the first two exercises below.
Exercises: Circle the digits that are different. Then fill in the blank.
a) 273
b) 401
263
273 is _______ than 263
411
401 is _______ than 411
c) 560
460
560 is _______ than 460
d) 812
813
812 is _______ than 813
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-37
(MP.8) Bonus:
e) 563
f) 412
263
563 is _______ than 263
432
412 is _______ than 432
Answers: a) 10 more, b) 10 less, c) 100 more, d) 1 less, Bonus: e) 300 more, f) 20 less
Have volunteers complete the first three questions of the exercise below on the board.
(MP.7) Exercises: Circle the digits that are different. Then fill in the blanks.
a) 345
b) 429
c) 327
335
529
328
_335_ is _10_ less than _345_
___ is __ greater than ___
___ is __ less than ___
d) 245
246
___ is __ less than ___
e) 639
649
___ is __ greater than ___
f) 718
618
___ is __ less than ___
g) 127
h) 325
i) 108
137
326
208
___ is __ greater than ___
___ is __ greater than ___
___ is __ less than ___
Answers: b) 529 is 100 greater than 429, c) 327 is 1 less than 328, d) 245 is 1 less than 246,
e) 649 is 10 greater than 639, f) 618 is 100 less than 718, g) 137 is 10 greater than 127, h) 326
is 1 greater than 325, i) 108 is 100 less than 208
Addition and subtraction with 10 and 100. Write on the board:
______ is 100 more than 352
ASK: How can we find which number is 100 more than 352? Is there an easy way? PROMPT:
Which digit in 352 needs to be increased? (the hundreds digit) So what is the hundreds digit of
the number we’re looking for? (4) So, what is the number? (452) Write “452” in the blank.
Write on the board:
______ is 10 less than 352
ASK: How can we find a number that is 10 less than 352? PROMPT: Which digit in 352 needs
to be decreased? (the tens digit) So, what is the tens digit of the number we’re looking for? (4)
Okay, so what is the number? (342) Write “342” in the blank.
Exercises: Fill in the blank.
a) ____ is 10 more than 823
b) ____ is 10 less than 526
c) ____ is 10 less than 427
d) ____ is 10 more than 185
e) ____ is 100 more than 431
f) ____ is 100 less than 701
g) ____ is 100 more than 516
h) ____ is 100 less than 208
Answers: a) 833, b) 516, c) 417, d) 195, e) 531, f) 601, g) 616, h) 108
C-38
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Write on the board:
385 + 10 = ____
ASK: What is an easy way we can find the answer to 385 + 10? PROMPT: When we add a
number to 385, is the answer more or less than 385? (more) Which number is 10 more than
385? (395) PROMPT: Find the tens digit of 385, and increase it by 1. Write “395” in the blank
above. Write on the board:
723 − 100 = ____
ASK: What is an easy way we can find the answer to 723 − 100? PROMPT: When we subtract
from 723, will the answer be more or less than 723? (less) What number is 100 less than 723?
(623) PROMPT: Find the hundreds digit of 723 and decrease it by 1. Write “623” in the blank
above.
(MP.8) Exercises: Fill in the blank.
a) 436 + 10 = _____
b) 645 + 10 = _____
d) 327 − 10 = _____
e) 298 + 100 = ____
Answers: a) 446, b) 655, c) 508, d) 317, e) 398, f) 681
c) 518 − 10 = _____
f) 781 − 100 = _____
Demonstrate the first two exercises below and have all students complete the remaining ones.
Exercises: Fill in the blank. The answer will be 10 or 100.
a) 314 + ___ = 324
b) 685 + ___ = 695
c) 229 − ___ = 129
d) 327 − ___ = 227
e) 598 + ___ = 698
f) 781 − ____= 771
g) 431 − ___ = 421
h) 138 + ___ = 238
i) 487 + ____= 497
Answers: a) 10, b) 10, c) 100, d) 100, e) 100, f) 10, g) 10, h) 100, i) 10
Adding and subtracting 10 over multiples of 100. Write on the board:
304 − 10 = ____
ASK: How many tens are in 304? (30) If students do not see this immediately, you may need to
ask a series of questions to guide them: How many tens are in 20? (2) 30?(3) 80? (80) 90? (9)
100? (10) 110? (11) 120? (12) 150? (15) 190? (19) 200? (20) 210? (21) Point out the difference
between the number of tens in a number versus the tens digit. For example, the tens digit of 210
is 1, but 210 has 21 tens. ASK: How many tens are in 27? (2) 37? (3) 84? (8) 96? (9) 105? (10)
115? (11) 122? (12) 151? (15) 196? (19) 201? (20) 213? (21)
Return to the example on the board. ASK: Which digit should I cover up to see how many tens
are in 304? (the ones digit) Demonstrate covering up the ones digit, using your hand or a piece
of paper. Ensure students understand there are 30 tens in 304 and ASK: If I take away 1 ten,
how many tens are left? (29 tens) Do I change the ones digit by subtracting 10? (no) What
number has 29 tens and 4 ones? (294) Fill in the blank on the board: 304 − 10 = 294. Similarly,
students can add 496 + 10 by realizing that 49 tens plus 1 more ten is 50 tens, so 496 + 10 is
506. Demonstrate the first two exercises below before having all students complete the rest.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-39
(MP.8) Exercises: Fill in the blank. Cover up the ones digits of both numbers.
a) 590 + 10 = ___
b) 900 − 10 = ___
c) 390 + 10 = ___
d) 102 − 10 = ___
e) 195 + 10 = ___
f) 609 − 10 = ___
g) 807 − 10 = ___
h) 899 + 10 = ___
Answers: a) 600, b) 890, c) 400, d) 92, e) 205, f) 599, g) 797, h) 909
Demonstrate the first two exercises below, and then have all students do the rest.
Exercises: Fill in the blanks for the number pattern. You will need to add 10 or 100.
a) 105, 205, 305, _____, _____
b) 473, 483, 493, ____, _____
c) 276, 286, 296, _____, _____
d) 581, 591, ____, 611, _____
e) 273, 373, ____, 573, ______
f) 297, ____, 317, 327, _____
Answers: a) 405, 505; b) 503, 513; c) 306, 316; d) 601, 621; e) 473, 673; f) 307, 337
Extensions
1. Fill in the blanks for the number pattern. You will need to subtract 10 or 100.
a) 805, 705, 605, _____, _____
b) 473, 463, 453, ____, _____
c) 326, 316, 306, _____, _____
d) 581, 481, ____, 281, _____
e) 233, 223, ____, 203, ______
f) 207, ____, 187, 177, _____
Answers: a) 595, 585; b) 443, 433; c) 296, 286; d) 381, 181; e) 213, 193; f) 197, 167
(MP.7) 2. Have students complete BLM Hundreds Chart Pieces.
Answers:
1. a) 14, b) 48, c) 71, d) 91
2. a) 17, b) 28, c) 13, d) 40, e) 37, f) 46, g) 79, h) 62, i) 71, j) 89
3.
a)
3
4
b)
24
c)
34
4.
a)
18 19
b)
39
d)
19 20
e)
48 49
49
c)
26
a)
33 34
d)
46 47
e)
32 32
g)
26
35
C-40
c)
21 22
18
i)
29
d) 32
32
42
h)
f)
60
69 70
70
b) 31 32
22
60
70
36
5.
f)
57
63
72
f)
18
28 29
42 43
j)
46
e)
65
74
23 24
34
67
76
78
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten k)
2
3
4
5
6
7
8
9
l)
2
12
19
13
22
29
32
39
42
49
44
47
52
59
54
57
62
69
72
79
82
89
24
29
35 36
63
72
38
65 66
77
81
88
92 93 94 95 96 97 98 99
99
3. Write the tens digit and the number of tens. Cover up the ones digit.
a) 905
tens digit: _____
number of tens: _______
b) 416
tens digit: _____
number of tens: _______
c) 523
tens digit: _____
number of tens: _______
d) 62
tens digit: _____
number of tens: _______
e) 5
tens digit: _____
number of tens: _______
f) 897
tens digit: _____
number of tens: _______
Answers: a) 0, 90; b) 1, 41; c) 2, 52; d) 6, 6; e) 0, 0; f) 9, 89
4. Write the hundreds digit and the number of hundreds. Cover up the ones and tens digits.
a) 915
hundreds digit: _____
number of hundreds: _______
b) 802
hundreds digit: _____
number of hundreds: _______
c) 123
hundreds digit: _____
number of hundreds: _______
d) 47
hundreds digit: _____
number of hundreds: _______
Answers: a) 9, 9; b) 8, 8; c) 1, 1; d) 0, 0
5. Write how much more or less.
a) 963 is _______ than 463
b) 225 is _______ than 255
c) 489 is _______ than 509
c) 422 is _______ than 372
Answers: a) 500 more, b) 30 less, c) 20 less, d) 50 more
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-41
NBT3-9 Regrouping of Ones, Tens, and Hundreds
Pages 36–38
Standards: 3.NBT.A.2
Goals:
Students will regroup to write numbers as sums of ones, tens, and hundreds in different ways.
Prior Knowledge Required:
Understands place value
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Vocabulary: regrouping, trading
Materials: base ten blocks (optional, since sketches can be drawn instead)
Regrouping ones to form tens. Draw on the board:
ASK: How many ones blocks are there altogether? (34) Do we have enough to trade for a tens
block? (yes) How do you know? (there are more than 9 ones blocks) How many tens blocks can
we trade for? (3) How do you know? (there are 3 groups of ten ones blocks) Have a volunteer
group sets of 10 by circling in the above picture. Where do you see the number of tens in “34”?
(the “3” represents the number of tens) What does the “4” tell you? (the number of ones left over)
Draw the following sketches on the board, where dots represent ones blocks. Have volunteers
circle the groups of ten ones blocks, and write how many ones blocks are left over. Show them
how to do the first example.
a)
b)
c)
d)
__3_ ones left
____ one left
____ ones left
____ ones left
(b), circle first and second rows, 1 one left; c), circle first and second rows, circle third and fourth
rows, 4 ones left; d), circle first and second rows, circle third and fourth rows, 5 ones left)
C-42
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Draw base ten representations with more than ten ones and have student volunteers practice
trading 10 ones blocks for a tens block by circling the groups of 10 ones blocks, and drawing the
new number after regrouping. Remind your students that this process of grouping 10 ones
blocks into one stack of ten is called regrouping because you are rearranging the blocks into a
group. They can draw an arrow from a group of 10 ones blocks to the new tens block.
a)
b)
c)
d)
(b) circle first and second rows, regroup to 5 tens and 7 ones; c) circle first and second rows,
regroup to 4 tens and 2 ones; d) circle first and second rows, regroup to 5 tens and 8 ones)
Draw the following example on the board. Have volunteers write the number of tens and ones
represented before and after regrouping. (3 tens + 12 ones  4 tens + 2 ones)
____ tens + ____ ones
____ tens + ____ ones
Repeat with the following examples on the board:
5 tens + 17 ones
2 tens + 15 ones
(regroup to 6 tens + 7 ones, regroup to 3 tens + 5 ones)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-43
Have students regroup numbers (to get less than 10 of each place value) without using base ten
sketches. Explain that you can figure out how to regroup by using the digits of the number that
has more than 10. Example:
3 tens + 16 ones
= 4 tens + 6 ones
SAY: The “1” from “16” stands for “10”, so we have 3 tens + 1 ten = 4 tens. The “6” from “16”
tells us that there are “6” ones left over. (Point to the example above while explaining.)
Exercises: Write single digits in the blanks.
a) 2 tens + 15 ones
b) 6 tens + 19 ones
c) 7 tens + 17 ones
= __ tens + __ ones
= __ tens + __ ones
= __ tens + __ ones
Answers: a) 3 tens + 5 ones, b) 7 tens + 9 ones, c) 8 tens + 7 ones
When students have mastered this regrouping vertically, have them practice horizontally
(without arrows). Example: 4 tens + 16 ones = _5_ tens + _6_ ones
Exercises: Fill in the blanks.
a) 8 tens + 13 ones = ___ tens + ___ ones
b) 5 tens + 14 ones = ___ tens + ___ ones
Bonus:
c) 2 tens + 27 ones = ___ tens + ___ ones
d) 6 tens + 31 ones = ___ tens + ___ one
Answers: a) 9 tens + 3 ones, b) 6 tens + 4 ones, Bonus: c) 4 tens + 7 ones, d) 9 tens + 1 one
Regrouping so that there are less than 10 of each place value allows you to write the
number. Write on the board:
3 tens + 5 ones = ______
Point out that the number can be written by writing the digits from left to right. Write 35 in the
blank. Write on the board:
3 tens + 15 ones = _____
Point out that we cannot simply read this number from left to right—the number is not 315! We
have to regroup the ones before we can find the answer. Demonstrate the solution:
3 tens + 15 ones
C-44
=
=
=
3 tens + 1 ten + 5 ones
4 tens + 5 ones
45
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten SAY: Now that there are less than 10 in each place value, we can write the number.
Regrouping tens to form ones. SAY: Sometimes we want to trade a tens block for 10 ones
blocks. For example, if we have 2 tens and 3 ones, how many ones do we have altogether?
Write on the board:
2 tens + 3 ones
=
_____ ones
PROMPT: Remember that each tens block is worth 10 ones. Have a volunteer fill in the blank.
(23) Repeat with more examples: 3 tens + 5 ones (35 ones), 1 ten + 8 ones (18 ones), 4 tens +
7 ones (47 ones). ASK: Do you see an easy way to find the answer? (just write the digits in
order)
Regrouping tens to form hundreds. Remind students that 10 tens blocks can be traded for, or
“regrouped” as, 1 hundreds block. ASK: How can we trade tens for hundreds if we have 13
tens? Draw:
13 tens
=
1 hundred + 3 tens
Draw base ten representations with more than 10 tens and have volunteers practice trading 10
tens blocks for a hundreds block by circling the groups of 10 tens blocks, and drawing the new
number after regrouping. They can draw an arrow from a group of 10 tens blocks to the new
hundreds block. Example:
Have volunteers write the number of hundreds and tens represented before and after
regrouping:
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-45
Exercises: Draw a base ten sketch before and after regrouping. Then fill in the blanks.
a) 2 hundreds + 15 tens = ___ hundreds + ___ tens
b) 1 hundred + 14 tens = ___ hundreds + ____ tens
Answers: a) 3 hundreds + 5 tens, b) 2 hundreds + 4 tens
Have students regroup numbers (to get less than 10 of each place value) without using base ten
sketches. Explain that, as before, you can figure out how to regroup by using the digits of the
number that has more than 10. Example:
3 hundreds + 16 tens
= 4 hundreds + 6 tens
SAY: The “1” from “16 tens” stands for 10 tens, or 1 hundred, so we have 3 hundreds +
1 hundred = 4 hundreds. (Point to the example above while explaining.) The “6” from “16 tens”
tells us that there are “6” tens left over.
Exercises: Draw arrows and circles to show the regrouping. Then fill in the blanks.
a) 2 hundreds + 15 tens
b) 6 hundreds + 19 tens
= __ hundreds + __ tens
= __ hundreds + __ tens
Answers: a) 3 hundreds + 5 tens, b) 7 hundreds + 9 tens
When students have mastered this regrouping vertically, have them practice horizontally
(without arrows). Example: 4 hundreds + 16 tens = _5_ hundreds + _6_ tens
Exercises: Fill in the blanks.
a) 8 hundreds + 13 tens = ___ hundreds + ___ tens
b) 5 hundreds + 14 tens = ___ hundreds + ____ tens
Bonus:
c) 2 hundreds + 27 tens = ____ hundreds + ____ tens
d) 3 hundreds + 42 tens = ____ hundreds + ____ tens
Answers: a) 9 hundreds + 3 tens, b) 6 hundreds + 4 tens, Bonus: c) 4 hundreds + 7 tens
d) 7 hundreds + 2 tens
Regrouping hundreds to form tens. SAY: Sometimes, we want to trade a hundreds block for
10 tens blocks. For example, if we have 2 hundreds and 3 tens, how many tens do we have
altogether? In other words, if you are not allowed to use hundreds blocks, how many tens
blocks do you need to make the same number? Write on the board:
2 hundreds + 3 tens = _____ tens
C-46
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten PROMPT: Remember that each hundred is worth 10 tens. Have a volunteer fill in the blank. (23)
Repeat with more examples: 4 hundreds + 6 tens, 1 hundred + 5 tens, 3 hundreds + 9 tens.
ASK: Do you see an easy way to find the answer? (just write the digits in order)
Regrouping ones, tens and hundreds.
Exercises: Fill in the blanks.
a) 5 hundreds + 4 tens + 13 ones = 5 hundreds + ____ tens + ____ ones
b) 6 hundreds + 14 tens + 9 ones = ___ hundreds + ____ tens + 9 ones
Answers: a) 5, 3; b) 7, 4
(MP.7) Have students mentally trade tens for ones in order to fill in the following blank, where
the blank is a 2-digit number between 10 and 19. Write on the board:
5 tens + ____ ones = 6 tens + 7 ones
PROMPT: How many tens are there on the right side of the = sign? (6) How many are on the
left side? (5) So, what if we wrote 10 in the blank? (we’d get 6 tens, but we still need 7 ones) So,
how many more ones do we need? (7) So, how do we find the total? (10 + 7 = 17)
(MP.7) Exercises: Fill in the blank.
a) 2 hundreds + ____ tens = 3 hundreds + 4 tens
b) ___ hundreds + 11 tens = 5 hundreds + 1 ten
c) ___ tens + 13 ones = 3 tens + 3 ones
Answers: a) 14, b) 4, c) 2
Extensions
(MP.1, MP.5) 1. Have students regroup to get less than 10 in each place value. Students will
need to regroup more than once. Emphasize the importance of regrouping the ones first, so that
you can tell how many tens you need to regroup into hundreds. You could ask students to try
regrouping the tens first and then the ones, so that they see this method is less efficient.
a) 2 hundreds, 9 tens, 14 ones
b) 4 hundreds, 19 tens, 13 ones
c) 3 hundreds, 8 tens, 31 ones
d) 6 hundreds, 7 tens, 52 ones
Answers: a) 3 hundreds, 0 tens, 4 ones; b) 6 hundreds, 0 tens, 3 ones; c) 4 hundreds, 1 ten, 1
one; d) 7 hundreds, 2 tens, 2 ones
2. Write on the board:
73 tens + 5 ones
= ___ hundreds + ___ tens + ___ ones
Have a volunteer fill in the blanks. Point out that now we can write the number by writing the
digits from left to right. ASK: What would happen if we did that with the original representation,
73 tens + 5 ones? Would we still get the same answer? (yes!) Discuss why that is the case.
Emphasize that because hundreds are the largest place value, regrouping the hundreds won’t
affect how we write the number.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-47
3. If you taught your students Egyptian writing (see Extension 3 in Lesson NBT3-1), you could
ask them to show regrouping using Egyptian writing. Example:
(MP.7) 4. Have students do these more challenging fill in the blanks with regrouping, where
there are two blanks to fill in. The blanks on the right hand side of the equations are all 1-digit
numbers.
a) ___ tens + 15 ones = 6 tens + ____ ones
b) ___ hundreds + 13 tens = 7 hundreds + ____ tens
c) ___ hundreds + 3 tens + ___ ones = 2 hundreds + 4 tens + 5 ones
d) ___ hundreds + 11 tens + 4 ones = 3 hundreds + ____ tens + 4 ones
e) 8 hundreds + ____ tens + 5 ones = 9 hundreds + 4 tens +____ ones
f) 7 hundreds + ____ tens + 14 ones = 7 hundreds + 5 tens + ____ ones
Answers: a) 5, 5; b) 6, 3; c) 2, 15; d) 2, 1; e) 14, 5; f) 4, 4
C-48
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-10 Addition with Regrouping—Tens
Pages 39–41
Standards: 3.NBT.A.2
Goals:
Students will add 2-digit numbers (using base ten drawings and the standard algorithm)
involving the regrouping of 10 ones blocks as 1 tens block.
Prior Knowledge Required:
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Understands place value
Can add within 20
Vocabulary: ones place, regrouping, sum, tens place
Use base ten drawings to teach addition of 2-digit numbers. Draw on the board:
Tens Blocks
Ones Blocks
15
43
Have volunteers draw base ten representations of the numbers 15 and 43 using the chart.
Students can draw sticks for tens and dots for ones. If students have trouble representing 15 (or
43), ASK: Which digit is the ones digit, the 1 or the 5? (5) Which is the tens digit? (1)
Tell students you want to add these two numbers. Use the above chart with the base ten
drawings to perform the addition, by adding a row to the table. ASK: How many ones blocks do
we have in total? (8) How many tens in total? (5) Draw the total ones, and then the total tens.
Tens Blocks
Ones Blocks
15
43
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-49
ASK: What number has 5 tens and 8 ones? (58) So what is 15 + 43? (58). Write “15 + 43 = 58”
underneath the chart.
The meaning of the word sum. Teach students that if you are adding numbers together, the
total is called the sum. Write the word “sum” in the chart underneath “43.” SAY: What is the sum
of 15 and 43? (the sum is 58) Point out that the word “sum” sounds just like the word “some,”
except they are spelled differently. Ask students to come up with a sentence using the word
“some” and with another sentence using the word “sum.” (Examples: “I would like some apples.”
“The sum of 2 and 3 is 5.”)
Exercises: Draw base ten sketches to add the numbers.
a) 41 + 50
b) 38 + 21
c) 54 + 34
Answers: a) 91, b) 59, c) 88, d) 75
d) 73 + 2
The standard algorithm for addition (no regrouping). Tell students there is a faster way to
find a sum of 2-digit numbers by just using the digits. Add the chart on the right to the board:
Tens
Ones
Tens
Ones
15
1
5
43
4
3
Sum
15 + 43 = 58
Direct the students’ attention to the new ones column. ASK: What is 5 + 3? (8) Write “8” under
the 3. ASK: What is 1 + 4? (5) Write “5” under the 4. Point out that the answer from the addition
of the digits, 58, matches the sum obtained using the base ten drawings.
Have students try a question where no base ten models are drawn. Write on the board:
57
+ 21
ASK: How many ones blocks are in 57? (7) How many ones blocks are in 21? (1) Do we need
base ten models to find out how many there are in total? (No, there are 8 in total. We can just
add the 7 and the 1.) How many tens are in 57? (5) How many tens are in 21? (2) How many
are there altogether? There are 7 tens and 8 ones altogether—what number is that? As
students answer your questions, write the digits in the correct position beneath the line to
demonstrate the standard algorithm for addition:
57
+ 21
78
C-50
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten ASK: Why did I write the 8 under the 7 and the 1? Why did I write the 7 under the 5 and the 2?
Ensure that students understand that the ones digit of the answer is written under the ones
digits of the addends (i.e., the numbers being added), and similarly for the tens digit. Use the
phrases “ones place” and “tens place.” (You can also use the phrases “ones column” and “tens
column” if students have been taught the word “column.”)
Exercises: Add.
a) 45
b) 37
c) 56
+21
+32
+43
Answers: a) 66, b) 69, c) 99, d) 89, e) 27
d) 18
+71
e) 14
+13
(MP.6) Lining up digits correctly for addition. Emphasize the importance of lining up the
ones digits in the ones place and the tens digits in the tens place. To demonstrate the
importance, ask students to find the sum of 21 and 3 by counting on. When they all agree that
21 + 3 = 24, write on the board:
21
+3_
51
(MP.3) ASK: What is wrong with this addition? (the 3 is written in the tens place instead of the
ones place) Do we get the right answer? (no) Have a volunteer rewrite the question with the
digits properly aligned and find the sum.
Teach students how to line up digits for addition with the standard algorithm using grid paper.
Draw the following blank grid on the board:
Tell the students that you want to add 46 + 32. Ask a volunteer to line up the digits correctly,
and have them write the + symbol and the answer line in the correct place. Explain that
the + symbol is always written to the left of the bottom number. Have a different volunteer find
the sum. Repeat with more examples. Have students line up and add 2-digit numbers
individually (where no regrouping is required) using grid paper.
Exercises: Line up the digits using grid paper. Add.
a) 24 + 32
b) 14 + 73
c) 25 + 41
d) 34 + 4
(MP.8) Bonus:
f) 41 + 23 + 15
g) 30 + 44 + 23
h) 21 + 21 + 21
Answers: a) 56, b) 87, c) 66, d) 38, e) 39, Bonus: f) 79, g) 97, h) 63
e) 26 + 13
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-51
Use base ten blocks to represent regrouping. Tell students you want to add 27 and 15.
Begin by drawing base ten pictures of 27 and 15 on the board:
Write the addition and combine the two pictures to represent the sum:
ASK: How many ones blocks do we have in the total? (12) How many tens? (3) Replace 10
ones blocks with 1 tens block. ASK: Now how many ones blocks do we have? (2) How many
tens? (4) How many do we have altogether? (4 tens and 2 ones, or 42)
Use a tens and ones table to summarize how you regrouped the ones:
Exercises: Add by using a base ten sketch and a tens and ones table.
d)
b)
a)
c)
Bonus:
e)
f)
Answers: a) 81, b) 65, c) 82, d) 47, Bonus: e) 91, f) 74
C-52
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Ask students if they really need to draw the base ten materials or if they can add without them.
Use a table to add 37 and 46:
SAY: When you use a table, you can add the tens and ones first and then regroup. When you
do the sum directly, you regroup right away: 7 + 6 = 13, which is 1 ten + 3 ones, so you put the
3 in the ones place and add the 1 to the tens place. Write on the board:
Ask students how many ones there are when you add the ones digits and how many there are
after you regroup the ones. Tell them that when we regroup 10 ones as a ten, we put the 1 on
top of the tens place.
Have students do the first step for several problems (as in the previous example). Write the
second step:
Give students exercises where the first step has been done and they just need to do the second
step. Have students do problems where they need to do both steps. Give students questions
written horizontally, and have them write the questions vertically in their notebooks (using grid
paper). Remind them to leave space above each question for regrouping ones as tens.
Examples: 25 + 48, 37 + 59, 69 + 13, 47 + 19, 58 + 22. (73, 96, 82, 66, 80)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-53
(MP.3) The importance of starting in the ones place. Tell your students that you had a
student once who always added the tens before the ones. Show them the student’s work and
challenge them to find the answers that the student got wrong:
a)
b)
c)
d)
(part a) is correct; part b) is incorrect—should be 44; part c) is incorrect—should be 52; part d) is
correct)
Discuss why it is important to add the ones first. Explain that if you add the tens first, you might
forget to add the extra 1 that was traded for 10 ones. Tell them that it is a bit tricky because they
have to add from right to left instead of from left to right. Tell them that even many Grade 4
students will sometimes have trouble remembering to add from right to left because it is so
different from what they are used to, so that’s why it’s important to practice a lot.
Extensions
(MP.2) 1. Fill in the missing numbers to make each sum correct. Part c) has more than one
answer. How many can you find?
Answers: a) 52 + 29, b) 25 + 18, c) 17 + 29 or 18 + 28 or 19 + 27
2. Have students add three 2-digit numbers at a time. They will need to regroup 10 ones as 1
ten.
a) 23 + 45 + 15
b) 24 + 24 + 24
c) 18 + 13 + 10
d) 34 + 17 + 21
Answers: a) 83, b) 72, c) 41, d) 72
(MP.7) 3. Have students add three numbers at a time. They will need to regroup 20 ones as 2
tens.
a) 43 + 19 + 29
b) 28 + 28 + 28
c) 38 + 17 + 15
d) 19 + 18 + 17
Answers: a) 91, b) 84, c) 70, d) 54
C-54
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-11 Addition with Regrouping—Hundreds
Pages 42–44
Standards: 3.NBT.A.2
Goals:
Students will add 3-digit numbers (using base ten models and the standard algorithm) involving
the regrouping of ones as tens and tens as hundreds.
Prior Knowledge Required:
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Understands place value
Can add within 20
Understands expanded form
Vocabulary: hundreds place, ones place, sum, tens place
Materials:
dice
Adding 3-digit numbers with base ten models where no regrouping is needed. Have
volunteers draw base ten models for 143 and 254 on the board and tell students that you want
to add these numbers. ASK: How many hundreds, tens, and ones are there altogether? What
number is that?
Draw the following and then fill in the blanks to add 143 and 254:
143
+ 254
___ hundred
+ ___ tens + ___ ones
___ hundreds + ___ tens + ___ ones
___ hundreds + ___ tens + ___ ones
(1 hundred + 4 tens + 3 ones, 2 hundreds + 5 tens + 4 ones, 3 hundreds + 9 tens + 7 ones)
Adding 3-digit numbers by regrouping. Have volunteers draw base ten models for 152 and
273 on the board. Tell students that you want to add these numbers. ASK: How many hundreds,
tens, and ones are there altogether? Do we need to regroup? How do you know? How can we
regroup? (Since there are 12 tens, we can trade 10 of them for 1 hundred.) After regrouping,
how many hundreds, tens, and ones are there? What number is that?
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-55
Draw the following and then fill in the blanks to add 152 and 273:
(1 hundred + 5 tens + 2 ones, 2 hundreds + 7 tens + 3 ones, 3 hundreds + 12 tens + 5 ones,
4 hundreds + 2 tens + 5 ones)
Have students add more pairs of 3-digit numbers. Provide examples in the following sequence:
• No regrouping is needed (e.g., 416 + 322, 536 + 123)
• The ones need to be regrouped (e.g., 238 + 147, 426 + 165)
• The tens need to be regrouped (e.g., 456 + 381, 277 + 392)
• Either the ones or the tens need to be regrouped (e.g., 349 + 229, 191 + 440)
• Both the ones and the tens need to be regrouped (e.g., 195 + 246, 186 + 593)
(MP.6) • You have to regroup the tens, but you don’t realize it until you regroup the ones
(e.g.,159 + 242, 869 + 237) Use this to emphasize the importance of regrouping the ones first.
Adding 3-digit numbers with the standard algorithm. Show students the standard
algorithm alongside a hundreds, tens, and ones table for 152 + 273:
In the standard algorithm, point out that after regrouping the tens to get 1 hundred, you add that
1 hundred to the other hundreds, so you get 1 + 1 + 2 = 4 hundreds.
Demonstrate both the chart and the algorithm for adding a 3-digit number to a 2-digit number and
emphasize the importance of lining up the digits—ones with ones, tens with tens and hundreds
with hundreds. If the digits are not aligned, the sum will be incorrect. Example: 459 + 35. (494)
Have students add more 3-digit numbers using the standard algorithm. Use the same
progression of examples as above:
• No regrouping is needed (e.g., 613 + 212, 436 + 323)
• The ones need to be regrouped (e.g., 347 + 526, 29 + 165)
• The tens need to be regrouped (e.g., 163 + 481, 387 + 392)
• Either the ones or the tens need to be regrouped (e.g., 219 + 429, 378 + 540)
• Both the ones and the tens need to be regrouped (e.g., 287 + 137, 387 + 493)
(MP.6) • The tens need to be regrouped after the ones have been regrouped (e.g., 278 + 322,
567 + 137)
C-56
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Activity
Students play in pairs. Each student makes a copy of these grids:
+
Players take turns rolling a die and writing the number rolled in the 6 boxes at left. Without
looking at each other’s grids, each player arranges those numbers in the boxes at right to create
different sums of 3-digit numbers: Player 1 tries to create the greatest sum possible and Player
2 tries to create the least sum possible. The pair wins if Player 1’s sum is larger than Player 2’s
sum. Players switch roles and repeat.
(end of activity)
(MP.4) Simple word problems involving addition. Write on the board:
Kim has 176 paper clips. Jin has 258 paper clips.
How many paper clips do they have in total?
Have a volunteer read the problem. ASK: How can we find out how many paper clips they have
in total? (add) Have a volunteer line up the digits and add by regrouping. (176 + 258 = 434)
Do a few more word problems involving the addition of two 3-digit numbers.
Exercises: Add.
a) Abdul has 273 rocks in his collection. Alexa has 356 rocks in her collection.
How many rocks do they have in total?
b) Sandy has 327 baseball cards in her collection. She gets 185 more from her friend.
How many cards does she have now?
Answers: a) 629, b) 512
Extensions
(MP.1) 1. Fill in the missing numbers to make each sum correct.
a)
b)
Answers: a) 392 + 361, b) 295 + 531
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-57
2. Have students make puzzles like the ones in Extension 1.
3. Have students add three 3-digit numbers where they will need to regroup 10 ones as 1 ten, or
10 tens as 1 hundred, or both.
a) 345
b) 416
c) 348
d) 509
e) 511
417
381
239
26
222
+123
+150
+172
+194
+168
Answers: a) 885, b) 947, c) 759, d) 729, e) 901
C-58
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-12 Subtraction without Regrouping
Pages 45–46
Standards: 3.NBT.A.2
Goals:
Students will subtract 2- and 3-digit numbers without regrouping.
Prior Knowledge Required:
Understands subtraction as taking away
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Understands place value
Can write 2- and 3-digit numbers in expanded form
Understands expanded form
Vocabulary: difference, expanded form, hundreds place, ones place, tens place
Subtracting with base ten sketches. Tell students that you want to subtract 48 − 32 using
base ten materials. Write on the board:
48 − 32
Have a volunteer draw a base ten representation of 48 on the board. SAY: I want to take away
32. How many tens blocks should I remove? (3) Point to the tens digit “3” in “32.” Demonstrate
crossing out the 3 tens blocks. ASK: How many ones blocks should I remove? (2) Point to the
ones digit “2” in “32.” Cross out 2 ones blocks. ASK: What do I have left? How many tens? (1)
How many ones? (6) What is 48 − 32? (16) Add this to the board:
48 − 32 = 16
Review the word difference. Tell students that if you are subtracting one number from another
number, the answer is called the difference. For example, if you subtract 48 − 32, the answer
(16) is called the difference.
Have volunteers do the exercises below with no regrouping on the board. Have classmates
explain the steps the volunteers are taking.
Exercises: Subtract.
a) 97 − 46
b) 83 − 21
d) 26 − 13
e) 59 − 38
Answers: a) 51, b) 62, c) 41, d) 13, e) 21
c) 75 − 34
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-59
Subtraction using tens and ones tables. Give students examples of base ten drawings that
show subtraction and have them complete tens and ones tables. Example:
Have students subtract more 2-digit numbers (no regrouping) using both the table and base ten
sketches.
Exercises: Use base ten sketches and a tens and ones table to subtract.
a) 74 − 32
b) 85 − 52
c) 39 − 27
d) 55 − 23
e) 48 − 36
f) 96 − 54
Answers: a) 42, b) 33, c) 12, d) 32, e) 12, f) 42
Subtraction using expanded form. Demonstrate subtracting 46 − 13 by using the expanded
form. Write on the board:
Exercises: Write the number of tens and ones in each number. Then subtract.
a) 57 = _____ tens + ____ ones
b) 89 = _____ tens + ____ ones
− 31 = _____ tens + ____ one
− 64 = _____ tens + ____ ones
--------------------------------------------------------------------------------------= _____ tens + ____ ones
= _____ tens + ____ ones
= ______
= ______
Answers: a) 5 tens + 7 ones, 3 tens + 1 one, 2 tens + 6 ones, 26; b) 8 tens + 9 ones, 6 tens + 4
ones, 2 tens + 5 ones, 25
Remind students that this is the expanded form using numerals and words. Have students
separate the tens and ones using only numerals—in other words, have students use the
expanded form involving numerals only:
C-60
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises: Write in expanded form using numerals only. Then subtract.
a) 48 = 40 + 8
b) 97 = ____ + _____
- 23 = 20 + 3
- 61 = ____ + _____
------------------------------------------------------= ____ + ____
= ____ + _____
= ______
= ______
Answers: a) 20 + 5, 25; b) 90 + 7, 60 + 1, 30 + 6, 36
(MP.5) Exercise: Do the following subtraction question using any method you like:
54
− 23
Answer: 31
Compare methods of subtraction. ASK: Which strategy did you use? Is there a quick way to
subtract without using base ten materials, or a tens and ones table, or separating the tens and
ones? (yes—subtract each digit from the one above) Point out that by subtracting each digit
from the one above, they are really subtracting the ones from the ones and the tens from the
tens.
Subtraction with the standard algorithm. Write the following questions on the board:
43 − 21
87 − 52
97 − 23
84 − 12
Have volunteers line up the digits correctly (vertically) to subtract using the standard algorithm.
Emphasize the importance of lining up the ones with the ones, and tens with the tens. Teach
students that the minus sign (−) goes to the left of the bottom number. Now have volunteers
subtract using the standard algorithm, starting in the ones place. (22, 35, 74, 72)
Have students do similar questions individually in their notebooks. They might use grid paper to
help them line up the digits.
Subtracting 3-digit numbers. Have students draw a base ten picture of 624 and show how to
subtract 310, just as before, by crossing out hundreds, tens, and ones. Have them subtract
using the standard algorithm (by lining up the digits) and check to see if they got the same
answer both ways. Give them more practice subtracting 3-digit numbers using the standard
algorithm. Remind them to subtract from right to left (i.e., subtract ones, then tens, then
hundreds). This will be necessary for the subsequent lessons on subtraction with regrouping.
Exercises: Subtract. Start in the ones place.
a) 7 2 9
b) 8 9 5
c) 5 2 4
d) 3 9 8
−316
−254
−401
−163
e) 5 9 2
−170
Answers: a) 413, b) 641, c) 123, d) 235, e) 422
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-61
Extensions
1. Have students subtract large numbers using the standard algorithm, where no regrouping is
needed. Although they might not know how to read these large numbers yet, they will be excited
to extend the standard algorithm to these large numbers. Remind them to work from right to left
(since they will need to do this later when they do subtraction involving regrouping).
a) 546,983,201
− 431,620,100
b) 9,964,387
− 5,541,263
c) 678,439,841
− 138,210,731
Answers: a) 115,363,101; b) 4,423,124; c) 540,229,110
2. Fill in the missing numbers to make each subtraction question correct
a)
9
b)
6
8
c)
−
5
4
−
2
−
4
2
3
0
3
3
3
4
Answers: a) 984 − 754, b) 658 − 325, c) 898 − 426
8
2
7
2
(MP.6) 3. There are errors in the following answers. Circle the errors and correct them.
a)
6
7
3
b)
3
9
8
c)
5
9
8
−
2
5
1
−
1
2
5
−
4
2
5
4
4
0
2
5
3
4
7
2
4. Teach students to subtract numbers like 100 − 30 by counting the number of tens in each
number:
10 tens − 3 tens = 7 tens = 70
Give several practice problems of this type. ASK: What is 100 − 40? What is 40 − 36? (students
can count up to find this answer) How does this help to find 100 − 36? Show a number line to
help students see the addition they need to do.
(MP.1) 5. Have students subtract by first changing each problem to a problem they already
know how to do and then adjusting their answer:
a) 61 − 28
b) 34 − 15
c) 68 − 39
Selected sample answer: a) 58 − 28 = 30, so 61 − 28 = 30 + 3 = 33
C-62
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-13 Subtraction with Regrouping—Tens
Pages 47–49
Standards: 3.NBT.A.2
Goals:
Students will subtract 2-digit numbers by regrouping 1 tens block as 10 ones blocks, using base
ten sketches and the standard algorithm.
Prior Knowledge Required:
Understands subtraction as taking away
Can identify the numeral for a base ten block representation
Can make or sketch a base ten model for a number
Understands place value
Can subtract 2-digit numbers without regrouping
Vocabulary: difference, hundreds place, notation, ones place, tens place
Materials:
base ten blocks (optional, since sketches can be drawn instead)
(MP.3) Introduce subtractions that require regrouping. Ask students how they learned to
subtract 45 − 21. Have a volunteer demonstrate on the board. Ask the class if you can use the
same method to subtract 45 − 28. What goes wrong? Should you be able to subtract 28 from
45? If you have 45 things, does it make sense to take away 28 of them? Sure it does!
Subtract with regrouping. ASK: What if we regroup the larger number, 45, by trading 1 tens
block to get 10 more ones blocks? Use base ten materials or a sketch to make a standard
representation of 45 and then regroup 1 ten as 10 ones:
45
− 28
SAY: Now there are 15 ones, so we can take away 8 of them. Since we didn’t change the value
of 45—we just traded blocks—there is no need to adjust the answer. Removing the crossed-out
blocks (2 tens and 8 ones), we are left with 1 ten and 7 ones, so 45 − 28 = 17.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-63
Notation for regrouping in subtraction. Tell your students that mathematicians have come up
with a standard notation to say that you are borrowing a ten from the tens digit and trading it for
ten ones:
54 = 5 tens + 4 ones
= 4 tens + 14 ones
4 14
5 4
Emphasize that you are taking 1 ten away and replacing it with 10 ones. Have volunteers come
to the board to show this standard notation for various examples alongside the trading of base
ten materials. First, have volunteers show each number using base ten materials (or sketches
on the board) and then have another volunteer show the trading and the notation for trading at
the same time.
Exercises: Regroup 1 ten as 10 ones. Show the notation.
a)
b)
Selected answer: a)
c)
6
7
d)
e)
15
5 Subtracting with the standard algorithm. When students have mastered this, move on to
subtracting using the standard algorithm. Draw on the board just the grid on the left:
−
7
4
5
8
−
6
7
4
15
5
8
ASK: Can you take away 48 from 75? (yes, 75 is bigger than 48, so if I have 75 objects, I can
just remove 48 of them and count how many are left) Can you take away 8 ones from the 5
ones, or do you need to regroup? Have a volunteer show using the standard notation how they
would take a ten from the tens place and replace it with ten ones—emphasize that the 48 stays
the same; only the 75 is changing. When they are done with the regrouping, the result should
look like the grid above on the right.
Students can check their borrowing by adding 60 + 15 to make sure they get 75. ASK: Can we
take 8 ones from 15 ones? Can we take 4 tens away from 6 tens? Have a volunteer show this
and say the answer to 75 − 48. (27)
Exercises: Regroup 1 ten as 10 ones. Then subtract. Start in the ones place.
a) 84
b) 36
c) 47
d) 51
− 67
− 19
− 29
− 24
Answers: a) 17, b) 17, c) 18, d) 27
C-64
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten (MP.2) Deciding whether regrouping is necessary in subtraction. Tell the students that
sometimes regrouping is not required, and they have to decide when to use regrouping. Show
them what would happen if you borrowed when you didn’t need to, as in the left grid below:
_
after regrouping:
6
15
7
5
5
2
1
13
2
3
They could do the question this way if they remember to trade the ten ones back for a ten to end
up with 2 tens and 3 ones, but it is a lot faster not to regroup in the first place, as in the right grid
above. ASK: Why can we do this question without regrouping a ten as 10 ones? Have the
students look in the ones place. Notice that 5 is larger than 2, so we can subtract directly in the
ones place without regrouping. Write the following questions on the board:
a)
−
4
1
5
8
b)
−
6
5
7
4
SAY: Look at the ones place in part a). Which digit is larger, the 5 or the 8? Circle the 8. ASK:
Do we need to regroup? (yes, because 5 is less than 8) Write on the board, beside part a),
“regroup (5 is less than 8).” Have a volunteer show the regrouping, but not the subtraction.
Move on to part b). ASK: Which digit is greater in the ones column, the 7 or the 4? Circle the 7.
ASK: Do we need to regroup? (no) Write “OK” beside part b) to indicate that no regrouping is
needed.
Exercises: Regroup if needed. If you don’t need to regroup, write “OK.”
a)
b)
c)
d)
3
6
5
8
4
4
− 2
9
−
2
3
−
2
8
−
Answers:
a)
2
3
16
6
, b) OK,
c)
5
2
9
1
7
4
8 , d) OK
3
Once students have mastered deciding whether regrouping is needed, give them exercises
where they need to decide whether to regroup, and then complete the subtraction (by
regrouping, if needed). Make sure some of the examples require regrouping and some do not.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-65
Exercises: Subtract. Regroup when needed.
a) 53
b) 38
c) 25
d) 63
− 27
− 12
− 19
− 21
e) 48
− 29
f) 95
− 34
Answers: a) 26, b) 26, c) 6, d) 42, e) 19, f) 61
Extensions
(MP.1, MP.5) 1. Have students subtract first by using the method below and then by using the
standard algorithm for several problems. Example:
Sample answer: 4 + 20 + 4 = 28, so 54 − 26 = 28
When is the standard algorithm easiest? (when there is no regrouping needed in the
subtraction) When is subtraction by adding easiest? (when the adding does not require
regrouping) Notice that the addition will require regrouping precisely when the subtraction does
not require regrouping. For example, 54 − 26 = 4 + 20 + 4 = 20 + 8 = 28 and no regrouping is
required, whereas 54 − 23 = 4 + 20 + 7 = 20 + 11 = 31 and regrouping is required.
(MP.8) 2. Make a 2-digit number using consecutive digits (e.g., 23). Reverse the digits of your
number to create a different number, and subtract the smaller number from the larger one (e.g.,
32 − 23). Repeat this several times. What do you notice? (the answer is always 9) Some
students may wish to investigate what happens when we don’t use consecutive digits (e.g.,
42 − 24 = 18, 63 − 36 = 27, 82 − 28 = 54; the result in this case is always from the 9 times
tables).
(MP.7) 3. What is the smallest 2-digit number you can subtract so that you need to regroup 1
ten as 10 ones?
a)
b)
c)
d)
4
7
3
5
2
4
8
8
−
−
−
−
Answers: a) 18, b) 16, c) 15, d) 19
(MP.7) 4. What is the largest 2-digit number you can subtract so that you need to regroup 1 ten
as 10 ones?
a)
b)
c)
d)
4
7
3
5
2
4
8
8
−
−
−
−
Answers: a) 39, b) 29, c) 19, d) 79
C-66
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-14 Subtraction with Regrouping—Hundreds
Pages 50–53
Standards: 3.NBT.A.2
Goals:
Students will subtract 3-digit numbers by regrouping 1 hundreds block as 10 tens blocks, or 1
tens block as 10 ones blocks, or both.
Students will use base ten sketches and the standard algorithm.
Prior Knowledge Required:
Can subtract 2-digit numbers by regrouping using base ten materials
Can subtract 2-digit numbers by regrouping using the standard algorithm
Can write the standard notation for regrouping
Vocabulary: difference, hundreds place, ones place, tens place
Subtracting 3-digit numbers by regrouping once, using a base ten sketch. Tell the
students you want to subtract 325 − 172. Draw a base ten sketch of 325 on the board, as in the
left table below.
Hundreds
Tens
Ones
Hundreds
Tens
Ones
Hundreds
Tens
Ones
3
2
5
2
12
5
1
5
3
ASK: Since we are subtracting 172, how many ones will we need to cross out? (2) Do we have
enough ones, or do we need to regroup 1 tens block as 10 ones? (no regrouping needed, since
5 is greater than 2) What about in the tens place? What do we need to subtract? (7) Do we have
enough tens blocks to subtract, or do we need to regroup? (regrouping is needed, since 2 is
less than 7) How should we regroup to get more tens blocks? (regroup 1 hundred as 10 tens)
Have a volunteer show the regrouping to obtain the middle table above. ASK: How can we do
the subtraction now? Have a volunteer erase (or cross out) ones, tens, and then hundreds to
show the subtraction, and count the remaining blocks to obtain the answer 153, as in the right
table above. Write on the board:
325 − 172 = 153
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-67
Subtracting 3-digit numbers by regrouping once, using the standard algorithm. Tell the
students that there is a faster way to do this subtraction without drawing base ten materials, by
just regrouping and subtracting using the digits (i.e., the standard algorithm). Copy the left grid
below on the board:
2
12
3
2
5
3
2
5
− 1
7
2
−
1
7
2
Repeat the process for subtraction, but now with the standard algorithm. ASK: Can we do this
subtraction? Is 325 greater than 172? (yes) Remind the students of the order for subtracting
(using the standard algorithm): subtract ones, then tens, then hundreds. Tell students you want
to check whether regrouping is needed, just like before. ASK: Can we subtract in the ones
place, or do we need to regroup a tens block as 10 ones first? (no regrouping needed, because
5 is greater than 2) Move to the tens place and ASK: Can we subtract in the tens place? (no,
because 2 is less than 7). How can we get more tens? (regroup 1 hundred as 10 tens). Show
this regrouping, as in the right grid above. Have a volunteer do the subtraction, starting in the
ones place.
Do several similar examples of 3-digit subtraction where 1 hundred must be regrouped as 10
tens, but have the students do the regrouping only, not the subtraction, so that they can practice
regrouping.
Exercises: Regroup 1 hundred as 10 tens. Do not subtract.
a) 547
b) 639
c) 465
− 281
− 376
− 194
d) 327
− 164
Once students have mastered regrouping, provide similar examples and have students regroup
and subtract.
Exercises: Subtract. You will need to regroup 1 hundred as 10 tens.
a) 226
b) 406
c) 947
d) 801
− 153
− 294
− 482
− 180
Answers: a) 73, b) 112, c) 465, d) 621
Move to examples where students need to regroup once (either 1 hundred as 10 tens or 1 ten
as 10 ones), and they have to figure out where the regrouping is needed.
Exercises: Subtract. You will need to regroup 1 hundred as 10 tens, or 1 ten as 10 ones.
a) 647
b) 339
c) 541
d) 236
e) 386
− 238
− 182
− 125
− 128
− 193
Answers: a) 409, b) 157, c) 416, d) 108, e) 193
C-68
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Subtracting 3-digit numbers by regrouping twice. Write on the board:
842
− 567
ASK: How is this problem different from the problems we have done before? (both the ones digit
and the tens digit need regrouping). Demonstrate how to regroup in this case:
Point out that the subtraction looks different because we had to regroup the same place value
twice, but we are really doing the same thing. Once all the regrouping is done, subtracting is
easy—just subtract one place value at a time:
Do several more examples with the whole class of this subtraction with double regrouping.
(Students might find it easier to subtract each digit as it becomes available, from right to left,
starting in the ones place, rather than doing all the regrouping first.). Include an example (such
as 453 − 258) where you have to regroup 1 hundred as 10 tens, but you realize this only after
regrouping 1 ten as 10 ones in the tens place. Use this to emphasize the importance of
regrouping from right to left. Have the students do the following exercises that require
regrouping twice.
Exercises: Subtract. You will need to regroup twice.
a) 547
b) 238
c) 632
d) 236
− 268
− 189
− 184
− 138
e) 516
− 199
Answers: a) 279, b) 49, c) 448, d) 98, e) 317
Finally, have the students do subtraction questions (involving 1-, 2-, and 3-digit numbers) where
they might have to regroup once, twice, or not at all. Emphasize that if, in any place value, they
don’t have as many in the larger number as in the smaller number, they will need to regroup.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-69
Also, emphasize the need to line up the place values: ones with ones, and so on.
Exercises: Subtract. Regroup when needed.
a) 359
b) 458
c) 356
d) 236
− 68
− 125
− 47
− 9
e) 478
− 299
Answers: a) 291, b) 333, c) 309, d) 227, e) 179
(MP.4) Simple word problems involving subtraction. Write on the board:
Amy’s book has 423 pages. Ravi’s book has 275 pages.
How many more pages does Amy’s book have?
Have a volunteer read the problem. ASK: How do we know Amy’s book has more pages?
(because 423 is greater than 275) How can we find out how many more pages? (subtract) Have
a volunteer line up the digits, and subtract by regrouping. Do a few more word problems
involving subtraction of 3-digit numbers.
Exercises: Subtract.
a) Sandy had 327 baseball cards in her collection. She gave 145 to her friend.
How many cards does she have left?
b) Abdul walks 873 feet to get to school. Alexa walks 952 feet.
How much more does Alexa walk than Abdul?
Answers: a) 182 cards, b) 79 feet
NOTE: Since this lesson is long, you might choose to break the lesson into two parts here. The
remaining part of the lesson teaches a mental math skill that is needed only on AP Book 3.1, p.
53
Subtracting from 100 by regrouping. Write on the board:
100
− 57
ASK: Do we need to regroup in the ones place? (yes, since 0 is less than 7) Do we have any
tens to regroup as 10 ones? (no, we have 0 tens) How can we get tens in order to regroup them
as ones? (regroup from the hundreds first)
Show them the steps for regrouping:
9
Step 1:
0 10
Step 2:
0
10 10
1
0
0
1
0
0
−
5
7
−
5
7
SAY: Now the subtraction can be done by simply subtracting the digits. Have a volunteer do the
subtraction, starting in the ones place. (43)
C-70
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises: Subtract by regrouping.
a) 100
b) 100
− 32
− 91
c) 100
− 29
Answers: a) 68, b) 9, c) 71
Subtracting from 100 by subtracting from 99 first and then adding 1 to the answer. When
students have mastered subtracting from 100 by regrouping from the hundreds first, show them
another way to do 100 − 57. ASK: What if we first did 99 − 57? Write on the board:
99
− 57
Have a volunteer find the answer by subtracting ones, then tens. (42) Note that no regrouping is
needed. Remind the students that the answer to 100 − 57 (done previously on the board) is 43.
ASK: How do these answers compare? (43 is one more than 42) Why? (because 100 is one
more than 99) So how could we find the answer to 100 − 57 using 99 − 57? (just add 1)
Exercises: Subtract from 99. Use your answer to subtract from 100.
a) 99
b) 99
c) 99
d) 99
− 68
− 25
− 37
− 82
So 100 − 68 =
So 100 − 25 =
Answers: a) 32, b) 75, c) 63, d) 18
So 100 − 37 =
So 100 − 82 =
(MP.3) Ask students which of the two methods for subtracting from 100 mentioned above they
find easier. Why? Some students might find the second method easier (subtracting from 99 and
then adding 1 to the answer) since no regrouping is needed.
Extensions
(MP.2) 1. Create a 3-digit number using consecutive digits (example: 456). Reverse the digits of
your number to create a different number, and subtract the smaller number from the larger one.
Repeat this several times. What do you notice? (The result is always 198.) Some students may
wish to investigate what happens when you don’t use consecutive digits (example: 368).
(MP.8) 2. Show students a shortcut for regrouping for subtraction from a 3-digit number when
the tens and ones digits are zero (i.e., the number is a multiple of 100). Tell students that when
they regroup, they will always reduce the hundreds digit by 1, write a 9 over the tens digit, and a
10 over the ones digit. For example, if the number is 300 …
2
3
9
0
10
0
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
C-71
Demonstrate that this is correct by doing the regrouping the longer way, just as they learned to
do for subtracting from 100:
9
2 10
2 10 10
3 0 0
3 0 0
3 0 0
Demonstrate that this regrouping is correct by adding:
200 + 90 + 10
= 200 + 100
= 300
Use this regrouping to subtract 300 − 176. (answer: 124)
Have students verify the regrouping for the following numbers:
a)
1 9 10
b)
4 9 10
2 0 0
5 0 0
Have students use the regroupings to do the following subtractions:
d) 200 − 183
e) 500 − 237
c)
6
7
9
0
10
0
f) 700 − 352
Answers: a) 100 + 90 + 10 = 100 + 100 = 200, b) 400 + 90 + 10 = 400 + 100 = 500,
c) 600 + 90 + 10 = 600 + 100 = 700, d) 17, e) 263, f) 348
(MP.7) 3. What is the smallest 3-digit number you can subtract so that you need to regroup 1
hundred as 10 tens, and 10 tens as 10 ones?
a)
b)
c)
d)
4
3
7
3
2
5
2
6
4
8
8
−
−
−
−
8
Answers: a) 138, b) 126, c) 165, d) 189
(MP.7) 4. What is the largest 3-digit number you can subtract so that you need to regroup 1 ten
as 10 ones?
a)
b)
c)
d)
4
3
7
3
2
5
2
6
4
8
8
8
−
−
−
−
Answers: a) 429, b) 319, c) 259, d) 879
C-72
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten NBT3-15 Puzzles and Problems
Pages 54–55
Standards: 3.NBT.A.2, 3.OA.D.8
Goals:
Students will solve word problems of various kinds involving concepts taught in this unit.
Prior Knowledge Required:
Comparing numbers up to 999
Ordering numbers up to 999
Adding and subtracting numbers up to 999
Vocabulary: difference, sum
NOTE: The AP Book questions for this lesson will give students a chance to apply some of the
concepts they have learned in this unit. Some students might find these word problems very
challenging. This lesson will help students to learn strategies needed to solve various types of
word problems.
Comparing and ordering numbers within 999 in word problems. Write on the board:
Raj has 437 stamps. Yu has 452 stamps. Who has more stamps?
Have a volunteer read the problem. ASK: How can we solve this problem? Ensure students
understand that the two 3-digit numbers (437 and 452) must be compared to see which one is
greater. ASK: Which number is greater, 437 or 452? (If students are struggling with comparing
3-digit numbers, you can begin with 2-digit numbers, or review the steps on p. C-28 of Lesson
NBT3-7.) Write on the board:
Yu has more stamps than Raj because 452 is greater than 437.
Repeat with similar examples. When students have mastered comparing two 3-digit numbers in
a word problem, challenge them to compare and order three numbers. Begin with 2-digit
numbers.
Heights of Students
Jon
47 inches
Rita
51 inches
Sam
49 inches
ASK: Which student is the tallest? (Rita) Who is the shortest? (Jon) Ask the students to order
the heights from least to greatest. (47, 49, 51) Repeat with similar examples.
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
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When students have mastered comparing and ordering three 2-digit numbers, proceed to 3-digit
numbers:
Distances Walked
Abdul
435 feet
Sara
517 feet
Wendy
482 feet
ASK: Who walked the most? (Sara) Who walked the least? (Abdul) Ask the students to order
the distances from least to greatest. (435, 482, 517) Repeat with similar examples.
Adding or subtracting in word problems. Write on the board:
There are 15 blue pens and 7 red pens.
How many pens are there in total?
ASK: How can we solve this problem? Should we add or subtract? (add) How do you know?
(because of the word “total”) Emphasize that the word “total” tells us that we want the number of
all pens, both red and blue, so we need to add. In other words, we want the sum. Have a
volunteer do the addition: 15 + 7 = 22. ASK: Does the answer make sense? PROMPT: Is 22
larger than 15 and 7? Ask the students what would happen if we accidentally did subtraction
instead of addition: 15 − 7 = 8. ASK: How could we know that we made a mistake? (8 is smaller
than 15) PROMPT: Is 8 larger than 15? (no) Should the total be larger than each part? (yes) If
we had only 8 blue and red pens in total, could we have 15 blue pens? (no)
Change the problem. Write:
There are 15 blue pens and 7 red pens.
How many more blue pens than red pens are there?
(MP.5) ASK: How can we solve this problem? Should we add or subtract? (subtract) How do
you know? (because of the words “how many more”) Emphasize that the words “how many
more” tell us that we want the difference. SAY: We want to know the difference between the
number of red pens and the number of blue pens. So we need to subtract to find the difference.
ASK: Which number is larger, 15 or 7? (15) So which number do we write first for the
subtraction? (15) Have a volunteer do the subtraction: 15 − 7 = 8. ASK: Does the answer make
sense? (yes) PROMPT: Which number is larger, 8 or 15? (15) Emphasize that the difference
between 15 and 7 cannot be larger than 15. In other words, the difference between two
numbers cannot be more than the bigger number. Ask students what would happen if we
accidentally did addition instead of subtraction: 15 + 7 = 22. ASK: How could we know that we
made a mistake? (22 is more than 15) PROMPT: Is 22 larger than 15? (yes) Can the difference
be more than the bigger number, 15? (no)
Have students practice with many similar problems where they have to either add or subtract.
Keep the numbers small (within 20) at first so that they focus on the concepts.
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Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten Exercises: Use addition or subtraction to solve the problems.
1. There are 13 green apples and 4 red apples.
a) How many more green apples than red apples are there?
b) How many apples in total?
Answers: a) 9, b) 17
2. Ava has 9 books, and Jay has 6 books.
a) How many books do they have in total?
b) How many more books does Ava have than Jay?
Answers: a) 15, b) 3
When students have mastered knowing whether to add or subtract in these types of simple
word problems, give them similar problems with 3-digit numbers. Remind them to line up their
digits to add and subtract using the standard algorithms.
Exercise: Add or subtract to answer the questions.
1. Anna likes to read many books. In May, she read 325 pages; in June, she read 384 pages;
and in July, she read 295 pages.
a) How many more pages did Anna read in June than in May?
b) How many pages did Anna read in total in June and July?
Answers: a) 59 pages, b) 679
(MP.3, MP.6) Multi-digit addition and subtraction algorithms. Have students reinforce their
understanding of the standard algorithms for addition and subtraction of 2- and 3-digit numbers
by spotting errors in problems done incorrectly. Students who struggle with this task can do the
problems themselves (correctly) and then find the difference between what they’ve done and
what they see.
Exercises: Find the mistakes.
a)
3
b)
5
16
c)
d)
3
17
4
6
6
5
2
8
3
7
+
1
7
−
3
8
+
7
−
1
9
8
1
2
8
9
8
2
8
Answers: a) 6 + 7 = 13, but 3 was written in the tens place instead of 1; b) the regrouping is
incorrect, and there should be 15 ones; c), the 7 should be in the ones place; d) after regrouping
3 tens, only 2 tens should be left
(MP.7) Maximizing sums and differences. Draw on the board:
4
1
+
Challenge the students to place the digits 5 and 3 in the boxes (below 4 and 1) to make the
largest possible sum. (53) PROMPT: Which are worth more, ones or tens? (tens) How can we
make the most tens? (put 5 in the tens, because 5 is more than 3)
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
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Draw on the board:
3
+
Challenge the students to place the digits 2, 4, and 5 in the three blanks to make the largest
possible sum. (32 + 54 or 34 + 52).
Exercise: Place the digits 1, 2 ,4, and 5 in the blanks to make the largest sum possible.
Hint: Put the higher digits in the tens place.
+
Answer: 52 + 41 or 51 + 42 or 41 + 52 or 42 + 51
Draw on the board:
−
7
2
Challenge the students to place the digits 6 and 3 in the blanks to make the difference as large
as possible. PROMPT: When do you get a larger difference, when the numbers are close
together or far away? Repeat with the grid below:
7
2
−
Exercise: Place the digits 2, 3, 6, and 7 in the blanks to make the largest difference possible.
Hint: Make the top number as large as possible and the bottom number as small as possible.
+
Answer: 76 − 23
Extensions
1. Arrange the digits 1, 3, 5, and 6 in the boxes to get the smallest possible sum or difference.
Hint: Put 6 in the tens place of the top number in b).
a)
b)
+
−
Answers: a) 15 + 36 or 16 + 35, b) 61 − 53
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Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten 2. a) What is the greatest 3-digit number you can add to 468 without having to regroup?
b) What is the largest number you can subtract from 468 without needing to regroup?
Answers: a) 531, b) 468
(MP.7) 3. What is the smallest digit you can write in each blank so that you have to regroup in
the ones place and tens place?
a)
b)
c)
d)
4
7
3
8
9
+
+
6
+
2
2
+
Answers: a) 47 + 53, b) 34 + 66, c) 78 + 22, d) 89 + 11
(MP.4) 4. The table shows the distances that 5 students walk during recess.
Distances Walked
Abdul
435 feet
Sara
517 feet
Wendy
482 feet
Amit
347 feet
Sun
567 feet
a) Arrange the distances from least to greatest.
b) Who walks the least? Who walks the most?
c) How much more than Wendy does Sun walk?
d) How much do Amit and Sara walk in total?
Answers:
a) 347, 435, 482, 517, 567; b) Amit walks the least, Sun walks the most; c) 567 − 482 = 85 feet;
d) 347 + 517 = 864 feet
Teacher’s Guide for AP Book 3.1 — Unit 2 Number and Operations in Base Ten
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