OA3-61 Tape Diagrams (1)
Pages 73–75
Standards: 3.OA.D.8
Goals:
Students will use tape diagrams to find the sum of two numbers or the difference between them.
Given one of two numbers and either the sum or difference, students will use a tape diagram to
find the unknown number.
Prior Knowledge Required:
Can add and subtract within 1,000
Vocabulary: bar, difference, equation, part, sum, tape diagram, total, unknown number
Materials:
BLM Tape Diagrams (1) (p. O-66), 2 copies per student
Review tape diagrams with squares. Draw on the board:
ASK: Does anyone remember what this picture is called? (a tape diagram) Remind students
that a tape diagram is a picture, or diagram, that can help to add, subtract, or compare different
numbers. Each piece in the picture is called a bar. Point to the top bar and ASK: How many
squares did I draw in this top bar? (6) So what number does this top bar show? (6) Write “6”
beside the top bar. Repeat the process for the bottom bar. Finally, write “total”, “difference”, and
the blanks, as shown below:
difference: ______ 6
total: ______ 2
Tell students you want to find the total of 6 and 2. ASK: What is another way to say we want the
total of 6 and 2? (the sum of 6 and 2, 6 + 2 = ?) How do we find the total of 6 and 2? (add them)
Ask students what number belongs in the blank for “total” and have them signal the answer. (8)
O-28
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
ASK: What do we mean by the difference between 6 and 2? Students might say 6 − 2, or how
much larger is 6 than 2. Ensure these two answers come out of the discussion, and ask students
what number belongs in the blank for “difference.” Have students signal the answer. (4)
Tape diagrams without squares. SAY: There is a quicker way to draw tape diagrams—by
drawing bars without squares. Let’s draw a simple tape diagram to compare the numbers 9 and
4. Draw on the board:
9
4
Point to the top bar and ASK: What number does the top bar show? (9) How do you know? (we
wrote the number 9 in the bar) Repeat the questions with the bottom bar, which shows 4.
Explain to students that it does not matter how long they draw the bars for 9 and 4 as long as
the bar for the larger number is longer than the bar for the smaller number. ASK: Which number
is larger, 9 or 4? (9) Is the bar for 9 longer than the bar for 4? (yes)
Tell students that they can show the total and difference in the tape diagram in the same way as
before. The final picture should look like this:
difference: ______ 9
total: ______
4
Ask students how they can find the difference (subtract 9 − 4 = 5) and the total (add 9 + 4 = 13),
and have a volunteer fill in each blank.
The parts, total, and difference in a tape diagram. Point to the two bars in the tape diagram
above and SAY: These two bars show the numbers 9 and 4. The 9 and 4 are called the parts.
So every number in this tape diagram has a name: 5 is the difference, 13 is the total, 9 is the
larger part, and 4 is the smaller part.
Provide each student with a copy of BLM Tape Diagrams (1) to complete the following
exercises.
Exercises: Use a tape diagram to find the total and difference for the given pair of numbers.
Remember to write the larger number in the longer bar.
a) 7, 11
b) 12, 5
c) 17, 13
Bonus: 143, 302
Answers: a) total 18, difference 4; b) total 17, difference 7; c) total 30, difference 4;
Bonus: total 445, difference 159
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-29
Tape diagrams with one part and the total given. Write on the board:
8
___
ASK: Do we know the numbers for both bars? (no) Point to the top bar and SAY: We are given
the number 8. Is that the larger number or the smaller number? (larger) How do you know? (8 is
inside the longer bar) So we have the larger part. Can we figure out the smaller part? Ensure
students understand that they need more information (such as the total or the difference) to find
the unknown, smaller number. Add “13” to the tape diagram, as shown below:
8
__13__ ___
ASK: If we learn that the total is 13, can we figure out the missing number? (yes) SAY: We know
the number for one bar, 8, and we know the total,13. How can we write this as an equation?
(8 + ___ = 13) Write on the board:
8+
= 13
Remind students that the empty box stands for the missing number for one bar. ASK: How do
we find the missing number? Ensure both counting up from 8 to 13 and subtracting 13 − 8 are
mentioned. Students should understand that both methods give the same answer because both
methods calculate 13 − 8. Write “13 − 8 =” on the board and ask students for the answer. (5)
Complete the equation on the board:
8+
= 13
13 − 8 = 5
Add “5” in the blank box, and have students check the answer. To conclude, ASK: What number
belongs in the blank in the tape diagram? (5)
Have students complete Question 1 on AP Book 3.2 p. 73.
Tape diagrams with the difference and larger number given. Write on the board:
__7__ 10
___
O-30
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
ASK: Do we know the numbers in both bars of the tape diagram? (no) Point to the top bar and
SAY: We are given the number 10. ASK: Is that the larger number or the smaller number?
(larger) How do you know? (because it is inside the longer bar) What other number are we
given? (7) Is 7 the total or the difference? (difference) SAY: So we have the larger part and the
difference. Have a volunteer write an equation on the board that uses the larger number (10),
the unknown smaller number, and the difference (7). There are various possible answers. Ask
for more volunteers and guide them to the answers if necessary. (see answers below)
10 −
=7
+ 7 = 10
10 − 7 =
ASK: Which equation makes it easiest to find the unknown number? (the last equation, with the
blank on the right side of the equation) What is 10 − 7? (3) Have a volunteer fill in the blank in
the equation and in the tape diagram.
Have students use the remaining blank tape diagrams in their copies of BLM Tape Diagrams (1)
to complete the exercises below.
Exercises: Use a tape diagram to find the smaller number. Then find the total.
a) Larger number = 10
b) Larger number = 12
Difference = 6
Difference = 9
c) Larger number = 18
d) Larger number = 13
Difference = 2
Difference = 12
(MP.6) Bonus:
e) Larger number = 182
f) Larger number = 397
Difference = 25
Difference = 102
Answers: a) Smaller number = 4, Total = 14; b) Smaller number = 3, Total = 15;
c) Smaller number = 16, Total = 34, d) Smaller number = 1, Total = 14;
Bonus: e) Smaller number = 157, Total = 339; f) Smaller number = 295, Total = 692
Tape diagrams with the difference and smaller number given. Write on the board:
__5__ ____
9
ASK: Do we know the numbers in both bars of the tape diagram? (no) Point to the bottom bar
and SAY: We are given the number 9. Is that the larger number or the smaller number?
(smaller) ASK: How do we know 9 is the smaller number? (because it is in the shorter bar) What
other number are we given? (5) Is 5 the total or the difference? (difference)
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-31
Have volunteers write equations on the board that use the smaller number, 5, the unknown
larger number, and the difference. (see equations below)
−9=5
−5=9
9+5=
ASK: Which equation makes it easiest to find the unknown number? (the last equation, with the
blank on the right side of the equation) What is 9 + 5? (14) Have a volunteer fill in the blank in
the equation and in the tape diagram. (14)
Activity
Give each pair of students another copy of BLM Tape Diagrams (1). Player 1 writes a number in
the blank for one bar of a tape diagram and another number in the blank for the difference. In
choosing numbers, Player 1 must follow these rules:
1. Each number must be less than 300.
2. If a number is chosen for the longer bar, it must be larger than the difference.
Player 2 fills in the two remaining blanks (for the other bar and the total).
Players switch roles. Players can continue until all tape diagrams on the page are filled.
(end of activity)
Extensions
(MP.6) 1. Write an equation that includes the smaller number, the larger number, and the
difference. Use the letter x for the unknown number.
b)
25
a)
328
____
319
103
____
c)
_______
453
219
Sample answers: a) x − 328 = 103 or x = 103 + 328, b) x = 319 − 25, c) x = 453 − 219
(MP.1) 2. Solve the equations in Extension 1.
Answers: a) x = 431, b) x = 294, c) x = 234
3. Find the totals in Extension 1.
Answers: a) 534, b) 613, c) 672
O-32
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
4. a) Explain how would you draw a tape diagram if the difference between the two bars is 0.
How does the length of the top bar compare to the length of the bottom bar? Is there a “larger
number” and a “smaller number”?
b) Complete the tape diagram below.
difference: __0___
9
total: ______ Answers: a) If the difference is 0, the bars should be the same length and the numbers in the
bars are equal. As a result, there is no larger and smaller number; the two numbers are the
same; b) 9 and 9 for a total of 18.
5. Have students complete tape diagrams where the difference and total are given, but neither
of the parts are given. For example:
difference: __4___
y
total: __10__ y
Explain to students that the unknown number that belongs in the bottom bar (labeled y) is the
same as the number that belongs in the matching portion of the longer bar (also labeled y). If
the difference and total are small numbers, students can start by drawing the same number of
dots in each area labeled y, adding one dot at a time to each box until they find the correct
answer (in this example, 3). So the smaller number is 3 and the larger number is 3 + 4 = 7. Ask
students if they can think of other ways of finding the unknown numbers. (For example, find the
total minus the difference: 10 − 4 = 6. Then 2 × y = 6, so y = 3.)
Repeat with the following examples. Draw a tape diagram using the information given, and ask
students to find the numbers for the two parts or bars:
a) Difference = 7, Total = 13
b) Difference = 9, Total = 17
c) Difference = 10, Total = 20
Answers: a) 10, 3; b) 13, 4; c) 15, 5
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-33
OA3-62 Tape Diagrams (2)
Pages 76–79
Standards: 3.OA.D.8
Goals:
Students will use tape diagrams to model and solve two-step word problems involving addition
and subtraction.
Prior Knowledge Required:
Can add and subtract within 1000
Can use a tape diagram to find the sum and difference of two numbers
Can use a tape diagram to find the larger or smaller of two numbers given the other number,
and either the total or the difference
Vocabulary: bar, difference, equation, solve, sum, tape diagram, total, unknown number
Materials:
BLM Tape Diagrams (2) to (3) (pp. O-67–68)
Using tape diagrams for real objects. Write on the board:
7 apples and 4 bananas
SAY: We have 7 apples and 4 bananas. We might want to know how many pieces of fruit we
have in total or how many more apples than bananas we have. We can use a tape diagram to
find out. Draw on the board:
__________
____
_____
__________
____
SAY: One bar will show apples and the other bar will show bananas. ASK: Are there more apples
or more bananas? (more apples) Underline “apples.” ASK: So which bar will show the apples?
(the longer bar) Write “apples” in the blank beside the longer bar. ASK: What do we write beside
the shorter bar? (bananas) Write “bananas” in the blank beside the shorter bar. Have students
signal the number for each bar. (7 for apples, 4 for bananas) The picture should look like this:
7 apples and 4 bananas
_apples____
7
____
_bananas__
O-34
_4_
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
ASK: How many pieces of fruit do we have in total? (11) PROMPT: Add the number of apples
and the number of bananas. (7 + 4 = 11) ASK: Where on the tape diagram do we write this
total? Have a volunteer write the total (11) in the blank to the right of the tape diagram.
Review parts and total in a tape diagram. Point to the tape diagram you just completed, and
ASK: Which number is the total in the tape diagram? (11) Which numbers are the parts? (7 and 4)
Which number is the larger part? (7) What does the longer bar show? (apples) Which number is
the smaller part? (4) What does the shorter bar show? (bananas)
Tape diagrams and word problems. SAY: Remember how an equation and pictures can tell a
story or show a word problem. A tape diagram can be used to tell a story as well. Write and
draw on the board:
10 fish in total. 2 are green and the rest are blue.
___________
____
____ ___________
____
Read the story and have students identify what the number 10 shows. (the total number of fish)
ASK: Where do we write the total in a tape diagram? Have students signal with thumbs up for
“yes” and thumbs down for “no” as you point to the various blanks. When students have
signaled “yes” to the blank to the right of the diagram, write “10” in the blank.
ASK: In this story, what do the bars show? (green fish and blue fish) How many green fish are
there? (2) Does the example say how many blue fish there are? (no) Explain that since we don’t
know how many blue fish there are, we don’t know which part is larger. ASK: How do we find
out the number of blue fish? (total fish minus the number of green fish) Write “10 − 2 =” on the
board and have students signal the answer. (8) SAY: So there are 8 blue fish and 2 green fish.
ASK: Which is the larger part of the total? (blue fish) Underline the word “blue” in the second
sentence on the board, and ASK: Where should we write “blue fish”? (beside the longer bar)
Write “blue fish” beside the longer bar. ASK: Where should we write “green fish”? (beside the
shorter bar) Write “green fish” beside the shorter bar, and then have students signal the
numbers that belong in each bar. The final picture should look like this:
10 fish in total. 2 are green and the rest are blue.
__blue fish_____
_8_
_10__ __green fish____
_2_
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-35
Explain to students that, if they accidentally write the smaller part in the longer bar or vice
versa, they can simply erase and switch the numbers. Have students complete Question 1 on
AP Book 3.2 p. 76.
Finding which is the larger part given one part and the difference. Write on the board:
3 more girls than boys. 13 boys.
Tell students you are going to create a tape diagram for this example. ASK: What will the bars in
the tape diagram show? (girls and boys) Which bar should be longer, the one for girls or the one
for boys? (girls) How do you know? (it says “more girls than boys,” not “more boys than girls”)
Underline “girls.”
Before you continue with the girls and boys example, find the larger part again in a new
example. Write on the board:
12 cats. 6 fewer cats than dogs.
ASK: What are the two parts in this example? (cats and dogs) Which part is larger, cats or
dogs? (dogs) PROMPT: Does it say there are fewer cats than dogs or fewer dogs than cats?
(fewer cats than dogs) SAY: So, if there are fewer cats, then there must be more dogs. Have a
volunteer underline “dogs.”
Exercises: Write the two parts. Underline the larger part.
a) 10 knives. 5 more forks than knives.
b) 6 fewer comic books than art books. 10 comic books.
c) 9 red apples. 6 more red apples than green.
Answers: a) knives, forks; b) comic books, art books; c) red apples, green apples
Go back to the example with girls and boys in order to draw a tape diagram. Draw on the board:
3 more girls than boys. 13 boys.
______
__________
____
_____
__________
____
SAY: Remember, we figured out that the parts in this question are “girls” and “boys,” and the
larger part is “girls.” Have a volunteer label the bars. (girls for the longer bar, boys for the shorter
bar) ASK: Does the example say the number of girls? (no, it just says “3 more girls than boys”)
SAY: So we will leave the number of girls blank for now. That is an unknown number.
ASK: Does the example say how many boys there are? (yes, 13) Write “13” in the shorter bar.
Pointing to the first sentence on the board, SAY: It says, “3 more girls than boys.” Is the number
O-36
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
3 the difference or the total? (difference) Write “3” in the blank for difference. Point out that you
now have the smaller part (13) and the difference (3). ASK: How would you find the larger part?
Through the discussion, ensure that students see that they need to add the smaller number (13)
and the difference (3) to find the larger number. Write on the board:
13 + 3 =
Ask for a volunteer to give the answer. (16) Complete this equation and have a volunteer write
“16” in the appropriate blank in the tape diagram. Then ask students how they would find the
total (add the two parts, 13 + 16) and have a volunteer fill in the remaining blank on the tape
diagram. The final diagram should look like this:
3 more girls than boys. 13 boys.
__3__
__girls____
__16__
__29__
___boys___
__13__
Have students complete Questions 2–3 on AP Book 3.2 pp. 77–78.
Tape diagrams and word problems. Write on the board:
Ed walked 34 miles in June. Beth walked 52 miles in June.
a) How much farther did Beth walk than Ed?
b) How much did they both walk in total?
SAY: We can use a tape diagram to answer these questions. Draw on the board:
______
__________
____
_____
__________
____
Have students identify the parts in this problem. (the distance Ed walked, the distance Beth
walked) ASK: Who walked more? (Beth) So what do we write beside the longer bar? (Beth)
Write “Beth” beside the longer bar and “Ed” beside the shorter bar. Then have students identify
the numbers of miles Beth and Ed each walked and write them in the correct blanks (52 in the
longer bar for Beth, 34 in the shorter bar for Ed)
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-37
Continue filling in the rest of the tape diagram. Pointing to the blank to the right of the tape
diagram, ASK: What number belongs in this blank? (the total) How do we find the total? (add
the two parts, 52 + 34) Ask a volunteer to add the numbers and insert the total. (86) Pointing to
the blank above the tape diagram, ASK: What belongs in this blank? (the difference) How do we
find the difference? (the larger part minus the smaller part, 52 − 34) Ask a volunteer to subtract
and write the difference on the tape diagram. (18) Note that the student will need to regroup.
The final picture should look like this:
__18__
__Beth___
_52__
__86__
__Ed_____
_34__
+
5
3
8
2
4
6
−
4
5 3
1
12
2
4
8 Now tell students that they can find the answers to a) and b). Have a volunteer read part a),
then ASK: Where can we find the answer on the tape diagram? (the difference, at the top) What
is the answer? (18) Write “18 miles” beside part a). Have a volunteer read part b), then
ASK: Where can we find the answer on the tape diagram? (the total, at the right) What is the
answer? (86) Write “86 miles” beside part b). SAY: So, we can use tape diagrams to help us
find the answers to word problems.
For the exercises below, ask students to draw their own tape diagrams or give each student a
copy of BLM Tape Diagrams (2).
Exercises: Use a tape diagram to solve the problem.
(MP.4) 1. Alexa has 7 pets. 3 are cats and the rest are dogs.
a) How many dogs does Alexa have?
b) How many more dogs than cats does Alexa have?
Answers: a) 4, b) 1
(MP.4) 2. Ben has 417 red marbles. He has 108 fewer green marbles than red marbles.
a) How many green marbles does Ben have?
b) How many green and red marbles does Ben have altogether?
Answers: a) 309, b) 726
Tape diagrams with 3 bars. Write on the board:
Ross has 14 red apples.
He has 5 more green apples than red apples.
He has 2 fewer yellow apples than green apples.
How many apples does he have altogether?
O-38
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
SAY: This story has different parts. ASK: What are the 3 different objects or parts in this problem?
(red, green, and yellow apples) SAY: We can use a tape diagram to solve this problem, but 2 bars
are not enough. ASK: How many bars do you think we’ll need? (3) Draw on the board:
______________
____
______________
____
______________
____
_____
(MP.1) SAY: We know that Ross has 14 red apples but he has more green apples than red
apples. ASK: What is the larger part, green or red apples? (green apples) SAY: So, for now we
will underline “green apples.” Now, we know that there are yellow apples as well as green and
red apples. The story says that there are 2 fewer yellow apples than green apples. So there are
more green apples than red apples, and there are more green apples than yellow apples. This
makes the green apple bar the longest of all 3 bars. Label the top bar “green apples.”
SAY: Ross has 14 red apples, and it says he has 5 more green apples than red apples.
ASK: How can we find the number of green apples? (14 + 5) What is 14 + 5? (19) Write “19” in
the bar for green apples. SAY: Now let’s think of green and yellow apples. There are 19 green
apples, and it says that Ross has 2 fewer yellow apples than green apples. ASK: How can we
find the number of yellow apples? (19 − 2) What is 19 − 2? (17) SAY: So there are 14 red
apples (point to the original story on the board) and 17 yellow apples. ASK: Are there more
yellow apples or more red apples? (more yellow) So where do we write “red apples”? (beside
the shortest bar) Write “red apples” beside the shortest bar, then “yellow apples” beside the
middle bar. Fill in the numbers for yellow (17) and for red (14).
Draw brackets for the differences, as shown below. Point to the blank at the top and SAY: This
blank shows the difference between the largest part (green apples, 19) and the smallest part
(red apples, 14). ASK: What number belongs in this blank? (5) Repeat for the difference
between the red and yellow apples (17 − 14 = 3) and between the yellow and green apples (2).
ASK: How do we find the total number or red, green and yellow apples? (add the 3 parts:
19 + 17 + 14) Have a volunteer line up the digits and add the 3 numbers on the board by
regrouping. (50) Write “50” in the blank for the total. The final picture should look like this:
__5__
green apples___
_19__
_yellow apples___
_17__
_red apples_____
_14__
__50__
__3__
__2__
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking
O-39
Give each student a copy of BLM Tape Diagrams (3) to complete the following exercise.
Exercise: Rani has 21 red marbles. She has 4 more blue marbles than red marbles. She has
7 fewer yellow marbles than red marbles. How many red, blue, and yellow marbles does she
have altogether?
Answer: 21 red, 25 blue, and 14 yellow; 60 marbles in total
NOTE: When students complete Question 6 and the Bonus on AP Book 3.2 p. 79, they will
need to draw tape diagrams in their notebooks with two or three bars. Alternatively, students
can use BLM Tape Diagrams (2) for Question 6 and BLM Tape Diagrams (3) for the Bonus.
Extensions
(MP.4) 1. Phil bought a book for $18 and a magazine for $7. Ella bought a book for $15 and a
magazine for $8.
a) How much more did Phil spend than Ella?
b) How much did Phil and Ella spend together?
Hint: Use two tape diagrams, one for Phil and one for Ella. Then compare the totals.
Answers: a) 25 − 23 = 2 dollars, b) 25 + 23 = 48 dollars
(MP.1) 2. Ross has red, green, yellow, blue, and purple marbles. He has 10 red marbles. He
has 5 more green marbles than red marbles. He has 5 more yellow marbles than green. He has
5 more blue marbles than yellow. He has 5 more purple marbles than blue.
a) How many of each type of marble does he have?
b) How many marbles does he have altogether?
Hint: You can solve this problem without using a tape diagram.
Solution: red = 10, green = 10 + 5 = 15, yellow = 15 + 5 = 20, blue = 20 + 5 = 25,
purple = 25 + 5 = 30, total = 10 + 15 + 20 + 25 + 30 = 100
3. Make up a problem that you would solve using a tape diagram with 3 bars.
Sample answer: A farmer has cows, chickens, and horses. She has 5 cows. She has 7 more
chickens than cows, and she has 4 fewer horses than chickens. How many cows, chickens, and
horses does the farmer have in total?
O-40
Teacher’s Guide for AP Book 3.2 — Unit 3 Operations and Algebraic Thinking