Adding and Subtracting Mixed Numbers
Confirming Calculations
ACTIVITY
ACTIVITY 2.2 Guided
2.2
Adding and Subtracting
Mixed Numbers
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Create Representations
My Notes
Activity Focus
After working with the fabric lengths in Activity 2.1, Selene noticed
that she now has a method for adding mixed numbers. She can
add the whole number parts and the fractional parts of the mixed
numbers separately and then add those sums together.
• Estimation with fractions
• Adding and subtracting mixed
numbers
3 + 3 __
2 + 5 __
1
1. Use Selene’s method to evaluate 4 __
4
3
2
Materials
4 + 5 + 3 = 12
3 + __
8 + ___
9 + ___
6 = ___
23 = 1 ___
2 + __
1 = ___
11
__
3
4
2
12
12
12
12
• Fraction bars or fraction circles
(optional)
12
11 = 13 ___
11
12 + 1 ___
12
12
Chunking the Activity
2. Use models to confirm your work.
#1–2
#3–4
a. Represent each mixed number with a model.
Answers may vary. Sample models shown below.
2
4 __
3
3
5 __
4
1
3 __
2
#5–6
#7–8
#9–12
#13
Paragraph Summarize/
Paraphrase/Retell
1 Students realize they know
how to add mixed numbers.
They make connections to prior
knowledge and apply skills to
a new situation. They will use
pictorial and algorithmic methods
for making the connection
between adding fractions and
adding mixed numbers.
© 2010 College Board. All rights reserved.
b. Shade the models below to represent the sum of the mixed
numbers in Question 1. Use a different color for each mixed
number. Fill each line before starting the next one.
2 Create Representations
Students draw their own models
to make sense of the algorithmic
method they have just developed.
Then they combine fraction bar
models to continue verifying the
algorithm from Question 1.
Unit 2 • Operations with Numbers
© 2010 College Board. All rights reserved.
073-080_SB_MS1_2-2_SE.indd 73
73
12/16/09 5:51:04 PM
S
Students
are asked to use different colors for the lengths
b
because it can help them see the individual addends and
sum Remind
R
the sum.
students to shade an entire figure before moving
on to the next.
TEACHER TO
TEACHER
Unit 2 • Operations with Numbers
73
ACTIVITY 2.2 Continued
ACTIVITY 2.2
continued
Adding and Subtracting Mixed Numbers
Confirming Calculations
3 Quickwrite Stress that
students can write the problems
horizontally or vertically or as
improper fractions and that they
can add the whole numbers first
or add the fractions first. It is
simply personal preference.
SUGGESTED LEARNING STRATEGIES: Quickwrite
My Notes
7 .
3 + 7 ___
Selene tested her method by using it to add 6 __
5
10
She wrote the numbers horizontally.
7 = (6 + 7) + __
7
3 + ___
3 + 7 ___
6 __
5
5 10
10
7
6 + ___
= (6 + 7) + ___
10 10
3 = 14 ___
3
13 = 13 + 1___
= 13 + ___
10
10
10
(
(
)
)
She wrote them vertically.
3 = 6 + ___
6
6 __
5
10
7 = 7 + ___
7
+ 7 ___
10
10
3 = 14 ___
3
13 = 13 + 1 ___
13 + ___
10
10
10
She wrote them as improper fractions.
© 2010 College Board. All rights reserved.
7 = ___
77
3 + 7 ___
33 + ___
6 __
5
5
10
10
66 + ___
77
= ___
10 10
3
143 = 14 ___
= ____
10
10
She found that the answer was the same whatever way she used.
3. Look back at Selene’s work. Does it matter which of the
methods above she uses to add the mixed numbers? Explain.
Explanations may vary. Sample answer: No; the method does
not matter because the answer is the same either way.
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073-080_SB_MS1_2-2_SE.indd 74
74 SpringBoard® Mathematics with Meaning™ Level 1
Adding and Subtracting Mixed Numbers
ACTIVITY 2.2
Confirming Calculations
ACTIVITY 2.2 Continued
continued
EXAMPLE 1 Estimating to the
nearest _12_ is difficult for many
students, especially with fractions that are an equal distance
SUGGESTED LEARNING STRATEGIES: Create Representations,
Close Reading
My Notes
Rounding mixed numbers can help you use mental math when you
want to estimate.
between 0 and _12_, such as _14_, or
between _12_ and 1, such as _34_. Use
EXAMPLE 1
7 , and 4 __
4.
2 , 2 ___
Round these mixed numbers: 3 __
Step 1:
number lines to help students
determine how to round, and
relate back to rounding whole
numbers like 35 to the next 10.
5
9 15
Look at the fraction portion of the number first, then round.
2 is close to 0, 3 __
2 rounds to 3.
Since __
9
9
7 rounds to 2 __
7 is close to __
1 , 2 ___
1.
Since ___
15
2 15
2
Paragraph Close Reading
4 is close to 1, 4 __
4 rounds to 5.
Since __
5
5
4 Debriefing This is the first
time students are asked to
add mixed numbers and use
estimation to check. If necessary,
students may draw models.
TRY THESE A
Round each mixed number. Show your work in the My Notes space.
3
7
1
2
2
a. 2 __
b. 3 __
c. 10 __
d. 9 __
e. 4 __
7
3
8
8
3
© 2010 College Board. All rights reserved.
You can use estimation before you do a calculation to see
about what your answer should be or afterwards to check the
7
2 + 2 ___
reasonableness of your answer. To estimate the sum of 3 __
9
15
4
1
1
__
__
__
+ 4 , think 3 + 2 + 5 = 10 . This estimate is close to the actual
5
2
2
22 .
sum of 10 ___
45
Try These A Answers
a. 2
1 or 3
b. 3 __
2
c. 11
1 or 10
d. 9 __
2
1
e. 4 or 4 __
2
Point out that finding an exact
answer and then rounding it is
not an acceptable method of
estimation.
Math Tip A common error is
for students to forget the whole
number when writing equivalent
fractions. Do not allow students
to do this as a shortcut. It is
important that their work is
mathematically correct. This
will also prevent students from
forgetting the whole numbers
when computing later.
4. Find each sum. Then use estimation to determine the reasonableness of each of your answers in Question 4.
1
2 + 2 __
a. 4 __
5
5
3 ; which is close to 6 __
1 , my estimate.
6 __
5
2
2 + 1 __
1
b. 3 __
3
6
5 ; a little less than my estimate of 5.
4 __
6
7
3 + 2 __
c. 2 __
4
8
5 ; which is near my estimate of 6.
5 __
8
Be sure you do not forget the
whole number when writing
equivalent fractions.
4
1 = 3 ___
3 __
3
12
but
4
1 ≠ ___
3 __
3 12
Unit 2 • Operations with Numbers
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Differentiating Instruction
For students who need extra support, use manipulatives such
as fraction bars or fraction circles in addition to the pictorial
representations.
For students who need a challenge and/or extra practice:
• Provide recipes and have students use addition to double them.
• Write the following equations on sticky notes and give them to
students to solve.
5
2 = 3__
x - 2__
7
3
3
11
= 7___
x - 1__
5
13
5
7
= 2__
x - 5__
6
8
Unit 2 • Operations with Numbers
75
5 Quickwrite, Debriefing,
Activating Prior Knowledge
The purpose of this question is
to introduce subtracting mixed
numbers, and provide alternative
problem-solving strategies.
ACTIVITY 2.2
continued
Adding and Subtracting Mixed Numbers
Confirming Calculations
SUGGESTED LEARNING STRATEGIES: Activate Prior
Knowledge, Identify a Subtask, Create Representations,
Quickwrite
My Notes
When comparing the amount of fabric they had to the amount
they needed, Selene and Gregg subtracted the fractional part of
their mixed numbers. They wonder whether the methods they used
when adding mixed numbers will work when subtracting.
6 Create Representations Ask
students to share their models
either in a small group or as a
class. Some students may cross
out _13_ of a whole figure, whereas
others may cross out _13_ of the
partially shaded figure. Encourage
a discussion on which method
is more efficient for finding the
answer.
5. Subtract. Use methods like those you used to add fractions.
Show your work.
( )
9 - ___
4
= (5 - 3) + (___
12 12 )
3 - 3 __
1 = (5 - 3) + __
3 - __
1
a. 5 __
4 3
4
3
5
5 = 2___
= 2 + ___
12
12
b.
3 = 5 + ___
9
5 __
12
4
1 = 3 + ___
4
- 3 __
12
3
5 = 2 ___
5
= 2 + ___
12
12
O
Observe the students’
figures as they work and
select 2 st
students who crossed out
differently. Provide these students
with an overhead of the question,
and ask them to use it to explain
their work.
TEACHER TO
TEACHER
23 - ___
10 = ___
40 = ___
29 = 2 ___
5
69 - ___
c. ___
12 12 12
12
4
3
6. Use bar models to confirm your work.
5
2 ___
12
© 2010 College Board. All rights reserved.
ACTIVITY 2.2 Continued
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 80, #1–3
UNIT 2 PRACTICE
p. 133, #6–8
TRY THESE B
Subtract these numbers. Show your work in the My Notes space.
5 - 2 __
3 - 2 __
4 - 6 __
1 __
2 __
2 5 ___
1
2
1
c. 11 __
b. 9 __
a. 5 __
5
4
4 32
3 76
3 15
6
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MINI-LESSON: Rounding and Estimation
Ask students to place each of the following mixed numbers on a
number line one at a time and discuss how each rounds and why.
Round each mixed number to both the nearest whole number as
well as the nearest half.
3 __
1 , 3__
2 , 1__
7 , 6__
2
, 5 8 , 4___
2__
3 5 4 9 12 7
Ask students the following questions:
• What is the difference between estimating and rounding?
• How do you know what a “reasonable” estimate is?
6
1 + 23___
? Explain.
• Is 37 a reasonable estimate for 12__
2
10
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073-080_SB_MS1_2-2_SE.indd 76
Adding and Subtracting Mixed Numbers
ACTIVITY 2.2
Confirming Calculations
SUGGESTED LEARNING STRATEGIES: Activate Prior
Knowledge, Quickwrite, Discussion Group, Self Revision/
Peer Revision
ACTIVITY 2.2 Continued
continued
78 Activating Prior Knowledge,
Debriefing Guide students
through the worked-out example
to review regrouping. Use guided
questioning to help students recall
whole number subtraction with
regrouping:
My Notes
Selene remembered that when she learned to subtract sometimes
she had to regroup to be able to subtract.
7. Give an example of a subtraction problem that requires
regrouping. Use 2-digit whole numbers in your example.
• Have you subtracted numbers
Answers may vary. Sample answer: 63 - 29
where the subtrahend was
bigger than the minuend?
Give an example. (34-18)
When subtracting mixed numbers, you may have to regroup. Look
1 - 3 __
4.
at this example for 6 __
5
5
6 = 5 __
6
1 = 5 + 1 __
1 = 5 + __
6 __
5
5
5
5
4
4
- 3 __
- 3 __
5
5
__
= 22
5
• What did you do?
• Would that work with fractions
too?
This discussion allows students to
connect what they know about
whole number subtraction to
subtraction with mixed numbers.
8. Describe the process used to subtract fractions in the example
above.
4
Answers may vary. Sample answer: You cannot subtract __
5
5
1
1
1
__
__
__
from , so you change 6 to 5 + 1 ; next you change the 1 to __
5
5
5
5
6
1
__
__
and add it to to get ; then you can subtract.
5
5
9 Discussion Group, Self
Revision/Peer Revision Help
students see that there may be
problems in which they must both
rename and regroup the fractions
before they can subtract.
© 2010 College Board. All rights reserved.
Sometimes you may need to do more than regroup. Look at this
3.
1 - 2 __
example for 8 __
4
3
16
16 = 7 ___
1 = 8 ___
4 = 7 + 1 ___
4 = 7 + ___
8 __
3
12
12
12
12
9
9
3 = 2 ___
- 2 __
- 2 ___
4
12
12
7
= 5 ___
12
9. How are these two examples the same and how are they
different?
Answers may vary. Sample answer: They are the same
because you cannot subtract right away since the top
fraction is smaller than the one you are subtracting; they
are different because in Question 8 you have to start by
making the denominators the same.
Unit 2 • Operations with Numbers
© 2010 College Board. All rights reserved.
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MINI-LESSON: Rounding and Estimation (continued )
Students should see that “reasonable” estimates with fractions and mixed
1 or 1 of the actual answer. However, there
numbers usually come within __
2
is no exact rule for rounding and estimation. Depending on the situation,
it is sometimes more reasonable to round both numbers in the same
direction, such as when subtracting, even though one number may round
in the other direction given the traditional rules of rounding. Students
must realize that rounding is only a tool for making estimates.
Unit 2 • Operations with Numbers
77
ACTIVITY 2.2 Continued
ACTIVITY 2.2
continued
Adding and Subtracting Mixed Numbers
Confirming Calculations
0 Create Representations
SUGGESTED LEARNING STRATEGIES: Quickwrite, Create
Representations
a Quickwrite This is a good
My Notes
place for whole group debriefing.
Be sure students understand
there is more than one way to
subtract, and that any method is
acceptable.
TRY THESE C
Subtract these numbers. Show your work in the My Notes space.
7 - 7 ___
1 - 3 __
4 __
14 3 ___
8
4
a. 9 __
b. 11 ___
7
7 57
15
15
15
10. You can use models of improper fractions to evaluate
subtraction expressions with mixed numbers that require
regrouping.
7 - 1 ___
11 . Use bar models.
a. Evaluate 2 ___
12
12
8 or __
2
___
12
3
1 - 2 __
2 . Use improper fractions.
b. Evaluate 5 __
4
3
63 - ___
32 = ___
31 = 2 ___
8 = ___
7
21 - __
___
4
3
12
12
12
12
2 . Use the method you prefer and explain
1 - 1 __
11. Evaluate 4 __
3
6
why you like that method.
© 2010 College Board. All rights reserved.
If you are having trouble using a
numeric method, draw models to
help you.
Explanations and methods may vary. Sample
1 ; I changed the mixed numbers to
answer: 2 __
2
improper fractions because you do not need to
regroup when you use mixed numbers.
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0
MINI-LESSON: Simplifying Fractions to Lowest Terms
Students often struggle with knowing when a fraction is in lowest
terms. When there are no common factors between the numerator and
denominator, a fraction is in lowest terms. Students may find the GCF
and divide once, or they may divide by smaller factors multiple times until
they obtain the lowest terms. Have students try and discuss both ways:
8
3
24 ÷ __
= __
GCF: ___
4
32
8
6
3
24 ÷ __
2 = ___
12 ÷ __
2 = __
2 = __
Several factors: ___
÷ __
4
32
2
16
2
8
2
Then ask students to decide whether each of the following fractions is in
lowest terms. If any are not, have them find the lowest terms.
21
1. ___
36
78 SpringBoard® Mathematics with Meaning™ Level 1
8
2. ___
15
10
3. ___
27
48
4. ___
60
© 2010 College Board. All rights reserved.
073-080_SB_MS1_2-2_SE.indd 78
Adding and Subtracting Mixed Numbers
ACTIVITY 2.2
Confirming Calculations
ACTIVITY 2.2 Continued
continued
b Think/Pair/Share Students
are asked to write solutions in
simplest form, or lowest terms.
This is helpful while they are
just learning to compute with
fractions.
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
12. Evaluate each expression. Write answers in lowest terms and
show your work.
MATH TERMS
A fraction is in lowest terms
or simplest form when the
numerator and denominator do
not have any common factors
other than 1.
3 __
1 - 1 __
1
a. 2 __
4
4 2
Most multiple-choice problems
list fractions in simplest form as
well, so students must be able to
simplify or recognize a simplified
fraction in order to find the
correct answer.
2 - 1 __
1 2 __
1
b. 3 __
3
6 2
c Think/Pair/Share Remind
students that finding an exact
answer and then rounding it is not
an acceptable way to estimate.
4 __
1 - 2 __
2
c. 3 __
5
5 5
2 1 __
1
d. 4 - 2 __
3 3
Differentiating Instruction
In small group settings, work
with students who are struggling.
Use manipulatives and pictorial
representations to help students
understand subtraction of mixed
numbers.
© 2010 College Board. All rights reserved.
13. Use estimation to check whether each of your answers in
Question 12 is reasonable. Show your estimates.
3 __
1 - 1 __
1 is close to the estimate of 1.
a. 2 __
4
4 2
G the following
Give
p
problems to students
who need more practice and/or
challenge:
TEACHER TO
TEACHER
2 - 1 __
1 2 __
1.
1 is the same as the estimate of 2 __
b. 3 __
2
2
3
6
1 - 2 __
4 __
2 is near the estimate of 0.
c. 3 __
5
5 5
5_34_ - x = 3_27_
2 1 __
1 is a little more than the estimate of 1.
d. 4 - 2 __
3
3
2_23_ + x = 10_15_
6
x + 9___
= 15_31_
13
Suggested Assignment
Unit 2 • Operations with Numbers
© 2010 College Board. All rights reserved.
PM
073-080_SB_MS1_2-2_SE.indd 79
CHECK YOUR UNDERSTANDING
p. 80, #4–7
79
12/16/09 5:51:26 PM
UNIT 2 PRACTICE
p. 133–134, #9–11
MINI-LESSON: Using Fractions and Mixed Numbers
This short lesson can be used as a culminating practice for adding and
subtracting fractions and mixed numbers or to extend the Embedded
Assessment.
1 "× 11" scrapbook page You may choose from
Design your own 8__
2
3
3
3
11 "× 6__
1 ", 6__
1 "× 2__
", 3__
"× 3__
",
photographs in the following sizes: 2___
4
4
16
2
8
8
7 "× 3", and 5"× 6__
1 ".
1__
4
8
a. Sketch the layout of your page. You must use at least three photos
and they may not overlap.
b. Label your sketch. Show the size of each photo. Calculate and
label the distances between the photos and from the photos to the
edges of the page.
Unit 2 • Operations with Numbers
79
ACTIVITY 2.2 Continued
ACTIVITY 2.2
continued
Adding and Subtracting Mixed Numbers
Confirming Calculations
1a. 4_25_ + 5_15_ = 9_35_
3
7 + 3_1_ = 7___
7 + 3___
b. 7___
=
15
5
15
15
10
2
___
_
_
10 15 = 10 3
CHECK YOUR UNDERSTANDING
13
12 +
c. 34_67_ + 22___
= 34___
14
14
13
25
11 =
___
=
56
=
56
+ 1___
22___
14
14
14
Write answers in simplest form.
11
57___
14
16
15
d. 9_45_ + 8_34_ = 9___
+ 8___
=
20
20
31
11
11
___
___
17 20 = 17 + 1 20 = 18___
20
2. Explanations may vary. Sample
answer: Yes, 1_13_ ≈ 1, 2_38_ ≈ 2_12_,
and 1_12_ = 1_12_. So adding them
you get: 1 + 2_12_ + 1_12_ = 5,
which is less than the 5_34_ cups
she has.
Write your answers on notebook paper.
Show your work.
1. Compute.
7 + 3 __
1
2 + 5 __
1
b. 7 ___
a. 4 __
5
5
5
15
6 + 22 ___
13 d. 9 __
3
4 + 8 __
c. 34 __
7
5
4
14
3
__
2. Hannah has 5 cups of salt. She is
4
coloring the salt to make an art project.
3 cups of
1 cups of red salt, 2 __
She needs 1 __
3 1
8
cups of green salt. Use
blue salt, and 1 __
2
estimation to determine if she has enough
salt. Explain your reasoning.
3. Shane and his friends had a movie-a-thon.
3 hr, the second
The first movie lasted 2 ___
10
17
___
movie lasted 1 hr, and the final movie
20
2 hr. Exactly how long did Shane
lasted 2 __
5
and his friends spend watching movies?
4. Evaluate each expression.
9 - 1 ___
8 - 4 __
2
2
b. 7 ___
a. 3 ___
7
11
11
21
3
3
13
__
___
__
d. 9 - 2
c. 16 - 9
4
8
16
7
3 ft tall.
__
5. Jacob is 4 ft tall, and Emily is 5 ___
8
16
1
__
Emily claims she is at least ft taller than
2
Jacob. Is she? Explain.
17 - 6 ___
4,
6. Which is a better estimate for 13 ___
16
1 ? Explain your reasoning.32
7 or 7 __
2
7. MATHEMATICAL How is adding and
R E F L E C T I O N subtracting mixed
numbers similar to and different from
adding and subtracting fractions?
© 2010 College Board. All rights reserved.
3
6
17 + 2_2_ = 2___
+ 1___
+
3. 2___
5
10
20
20
8
31
17 + 2___
11 hr =
= 5___
= 6___
1___
20
20
20
20
6 hr 33 min
9
2 = 2___
7
- 1___
4a. 3___
11
11
11
8
8
6
2
b. 7___
- 4_27_ = 7___
- 4___
= 3___
21
21
21
21
13
6
13
c. 16_38_ - 9___
= 16___
- 9___
=
16
16
16
13
9
22
___
___
___
15 16 - 9 16 = 6 16
d. 9 - 2_43_ = 8_44_ - 2_34_ = 6_14_
5. Explanations may vary. Sample
3
- 4_78_ =
answer: No, 5___
16
6. Answers may vary. Sample
response: To get 7_21_, you
17 to 13_1_ and 6___
4 to
round 13___
32
2
16
6, then subtract. To get 7, you
17 to 13_1_ and 6___
4
round 13___
32
2
16
1
to 6_2_ and then subtract. The
9
actual answer is 7___
which is
32
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7. Answers may vary. Sample answer: You still need common
denominators to add and subtract, but you also need to
compute with the whole numbers too. Sometimes you need to
regroup before you can subtract.
closer to 7_12_ than to 7.
80 SpringBoard® Mathematics with Meaning™ Level 1
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© 2010 College Board. All rights reserved.
3
19
5
14 = 4___
14 = ___
- 4___
- 4___
,
5___
16
16
16
16
16
5
___
so Emily is 16 ft taller, which is
less than _12_ ft.