Detailed Explanation of Mixed Numbers

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Diego Tue, Jan 25, 2022 02:06 PM UTC

This pdf explains Mixed Numbers. A mixed number is a whole number, and a proper fraction is represented together. It generally represents a number between any two whole numbers.

1. Building Concepts: Mixed Numbers
TEACHER NOTES
Lesson Overview
This TI-Nspire™ lesson uses interactive number lines to Learning Goals
help students investigate mixed numbers as a way to write
Students should understand and be
improper fractions. Division of a whole number by a smaller
able to explain each of the following:
whole number results in an improper fraction that can be
written as a mixed number. 1. A proper fraction is a fraction
whose numerator is less than the
denominator;
2. An improper fraction is any fraction
An improper fraction can be converted to the sum of a whose numerator is equal to or
whole number and a fraction less than one. Converting greater than the denominator;
a mixed number to a fraction can be thought of as 3. An improper fraction can be
fraction addition. expressed as a mixed number that
would be the sum of a whole
number and a fraction;
4. Mixed numbers can be added by
finding the sum of the whole
number parts and adding the sum
of the corresponding proper
fractional parts;
5. To find the number of whole
numbers in an improper fraction,
divide the denominator into the
numerator.
Prerequisite Knowledge Vocabulary
Mixed Numbers is the sixth lesson in a series of lessons  improper fraction: a fraction
that explore fractions. This lesson builds upon the concepts whose numerator is equal to or
explored in the previous lessons. Students should be greater than the denominator
familiar with the terms equivalent fractions, unit fraction,
 mixed number: a whole number
and improper fractions covered in earlier lessons. Prior to
and a fraction combined
working on this lesson students should understand:
 proper fraction: a fraction whose
 the concept of adding and subtracting fractions.
numerator is less than the
 that whole numbers can be expressed as fractions. denominator
 how to add and subtract fractions.
 that an improper fraction can be expressed as a whole
number and a fraction.
©2015 Texas Instruments Incorporated 1 education.ti.com

2. Building Concepts: Mixed Numbers
TEACHER NOTES
Lesson Pacing
This lesson should take 50 minutes to complete with students, though you may choose to extend, as
Lesson Materials
 Compatible TI Technologies:
TI-Nspire CX Handhelds, TI-Nspire Apps for iPad®, TI-Nspire Software
 Mixed Numbers_Student.pdf
 Mixed Numbers_Student.doc
 Mixed Numbers.tns
 Mixed Numbers_Teacher Notes
 To download the TI-Nspire activity (TNS file) and Student Activity sheet, go to
http://education.ti.com/go/buildingconcepts.
Class Instruction Key
The following question types are included throughout the lesson to assist you in guiding students in their
exploration of the concept:
Class Discussion: Use these questions to help students communicate their understanding of the
lesson. Encourage students to refer to the TNS activity as they explain their reasoning. Have students
listen to your instructions. Look for student answers to reflect an understanding of the concept. Listen for
opportunities to address understanding or misconceptions in student answers.
 Student Activity Sheet: The questions that have a check-mark also appear on the Student Activity
Sheet. Have students record their answers on their student activity sheet as you go through the lesson
as a class exercise. The student activity sheet is optional and may also be completed in smaller student
groups, depending on the technology available in the classroom. A (.doc) version of the Teacher Notes
has been provided and can be used to further customize the Student Activity sheet by choosing
additional and/or different questions for students.
Bulls-eye Question: Questions marked with the bulls-eye icon indicate key questions a student
should be able to answer by the conclusion of the activity. These questions are included in the Teacher
Notes and the Student Activity Sheet. The bulls-eye question on the Student Activity sheet is a variation
of the discussion question included in the Teacher Notes.
©2015 Texas Instruments Incorporated 2 education.ti.com

3. Building Concepts: Mixed Numbers
TEACHER NOTES
Mathematical Background
This TI-Nspire™ activity uses interactive number lines to help students investigate mixed numbers as a
way to write improper fractions. Division of a whole number by a smaller whole number results in an
improper fraction that can be written as a mixed number. Mixed numbers can be used to find the
243
magnitude of an improper fraction, and division can help find the integer part of the number (i.e.,
12
3 1
can be thought of as 243 divided by 12, which is 20 and a remainder of 3, 20  or 20 ). One
12 4
approach to adding and subtracting improper fractions is to add the values as fractions and convert the
result to a mixed number. Another approach is to convert each fraction to a mixed number and find the
sum by adding the whole parts of the numbers and the fraction parts. A caution here is that the fraction
parts may produce another mixed number.
1 1
The notation for mixed numbers is confusing to many students: the sum of 2 and is written as 2
2 2
1
which can be misread as the product of 2 and . When a product is intended, the multiplication sign or
2
parentheses should be used to make clear that the operation is not addition.
©2015 Texas Instruments Incorporated 3 education.ti.com

4. Building Concepts: Mixed Numbers
TEACHER NOTES
Part 1, Page 1.3
Focus: Students will explore the
relationship between mixed numbers and
TI-Nspire
improper fractions on the number line.
Technology Tips
Page 1.3 displays two number lines.
Students may find
The arrows at the top and bottom set
it easier to use
the number of equal partitions
the e key to
determined by the denominators with
toggle between
respect to each number line. The top
objects and then
number line displays the number of
copies of the unit fraction determined use the arrow
keys to move or
by the denominator. The bottom
change their
number line displays the whole number
and fractional parts for an improper selections.
fraction. Dragging the dots will To reset the
determine the numerators for each page, select
number. Reset in the
upper right
corner.
Teacher Tip: Lead students in a discussion about the relationship
between the numerator and the denominator in an improper fraction. Have
students name improper fractions. Then have them work together to
predict how the improper fraction will be expressed as a mixed number.
Use the interactive number lines in the file to show both numbers.
As students discuss the concept of mixed numbers, encourage them to explain their reasoning. Have
students work independently, then have them work together to find the solutions using the interactive
number lines.
Class Discussion
Have students… Look for/Listen for…
 Set the top number line to represent the Answer: two whole numbers, one and two, are
7 7
fraction . Use the bottom number line to in .
3 3
identify the number of whole numbers
7
contained in .
3
©2015 Texas Instruments Incorporated 4 education.ti.com

5. Building Concepts: Mixed Numbers
TEACHER NOTES
Class Discussion (continued)
1
 Which of the following could be written as Answer: a could be written as 2 and . In choice c,
5
the sum of 2 and a proper fraction where
9 3
the numerator is less than the is nearly 2 and adding will make the sum
5 4
denominator? Explain your reasoning.
greater than 2 but not yet 3,
(Question #1 on the Student Activity sheet.)
11
2 and .
11 7 3 9 20
a. b. c. 
5 2 4 5
 Order the following from smallest to Answer: Convert all of the fractions to mixed
largest and explain your reasoning: numbers, then compare the whole numbers first
5 16 3 15 17 before comparing the fraction part:
2 , ,2 , ,
6 5 4 6 12 5 1 3 5 1
1  2   2   2   3  ; thus
12 2 4 6 5
17 15 3 5 16
 2 2  .
12 6 4 6 5
Describe two strategies for finding the Answer: Each of the problems can be solved by
solutions written as mixed numbers for each combining the whole number part of the mixed
of the following problems: number and the fraction parts or they can be added
as fractions then converted into mixed numbers.
7 10 1 1 2
  Answer: 2  added to 3  would give 5  . Or
3 3 3 3 3
17 2 2
by adding the fractions, is also 5  ; 5 .
3 3 3
7 5 2 1 3 1 3 1
  Answer:  ; 1  minus 1  results in 
4 4 4 2 4 4 4 4
2 1
or  .
4 2
3 5 8 1 17 12
 2 1 Answer: 3  converts to 4  ;  can be
7 7 7 7 7 7
29 1 1
expressed as 4 ; 4 .
7 7 7
7 2 Answer:
 3 1
8 3 7 2  21 16  37 13 13
4    4    4 5 ; 5
8 3  24 24  24 24 24
©2015 Texas Instruments Incorporated 5 education.ti.com

6. Building Concepts: Mixed Numbers
TEACHER NOTES
Class Discussion (continued)
Have students… Look for/Listen for…
Identify the following as true or false. Explain
your reasoning.
5 3 5 2
 The sum of  is between 2 and 3. Possible answer: True. is 1  and the sum
3 4 3 3
2 3
of and is more than one but less than 2 so
3 4
adding the one will make the sum between
2 and 3.
 A fraction whose denominator is 8 will be Possible answer: False.
19 3
is 2  , which is
between 3 and 4 if the numerator is 8 8
between 18 and 32. less than 3.
(Question #2 on the Student Activity sheet.)
 An improper fraction whose numerator is 5 5 5
Possible answer: True because and are
will be between 1 and 2 if the denominator 4 3
is 4 or 3. both greater than 1 but less than 2.
Answer each and explain your reasoning.
1 1
 A recipe calls for 2 cups of flour. Possible answer: There are three cups in
3 3
1 1
Suppose you only have a -cup each whole, so there would be six cups in 2
3 3
measuring cup. How many times will you 1 1
plus the extra cup. You would fill the cup
1 3 3
have to fill the cup?
3 7 times.
(Question #3 on the Student Activity sheet.)
 How much will each person get if 15 Possible answer: Each person would get three
apples are divided among 4 people? 3
whole apples and then they each would get of
4
another apple. You could cut three of the apples
into half and then half again and give everyone
three of the 12 pieces.
23 23 11
Arnie has traveled of the way towards Answer:  1 so it is between 1 and
12 12 12
his destination. The number of miles he 2 miles.
has traveled will be between what two
whole numbers?
©2015 Texas Instruments Incorporated 6 education.ti.com

7. Building Concepts: Mixed Numbers
TEACHER NOTES
Class Discussion (continued)
10 2 10 4 4
 Explain why is the same as 1 . Possible answer: is 1  and is equivalent
6 3 6 6 6
2
to .
3
Which is greater and why?
12 7 7 1 12
 or Answer: is 3  , which is greater than ,
5 2 2 2 5
2
which is 2  .
5
9 8 9 11 9 11 11
  or  Answer:  has the greater value because
4 7 4 12 4 12 12
8
is less than one and is more than one, so you
7
9
are taking less away from when you subtract
4
11
.
12
15 4 12 19 12 19 12
  or  Answer:  is the greater value because is
12 3 10 9 10 9 10
19
1 plus a proper fraction and is 2 plus a proper
9
fraction so the total is more than 3. In the first sum,
each fraction is 1+ a proper fraction, but the two
1 1
proper fractions are and , and their sum is less
4 3
than 1 making the total sum less than 3.
Solve each. Express the answer as a mixed number.
15 4
 2 Answer: 3 .
11 11
25 3
 7 Answer: 3 .
7 7
31 7
 2 Answer: 1 .
8 8
©2015 Texas Instruments Incorporated 7 education.ti.com

8. Building Concepts: Mixed Numbers
TEACHER NOTES
Class Discussion (continued)
Without solving, identify the statement as true or
false. Explain your thinking in each case.
1 4 1 4
 The sum of 1 and 3 is less than 5. Answer: True because  is less than one
5 9 5 9
and the whole number part only adds to 4.
5 7 Answer: False. The sum of the fraction parts
 The sum of 3 and 2 is less than 6.
9 12 is more than 1 and the whole number parts
add to 5 so the total is more than 6.
7 18 Answer: True. The whole number parts are
 The sum of and is more than 5.
3 5 2 and 3 so the sum will be more than 5
when you add in the proper fraction parts.
Sample Assessment Items
After completing the lesson, students should be able to answer the following types of questions. If
students understand the concepts involved in the lesson, they should be able to answer the following
questions without using the TNS activity.
1
1. Create two fractions, with numerators greater than denominators, equivalent to 3 .
4
26 39
Possible answer: ; .
8 12
2. Order the following from smallest to largest.
10 7 10 10 7 8
; ; ; ; ;
3 4 8 4 3 4
Answer: Thinking in mixed numbers is easier than fractions.
1 3 1 1 5  10  7 8 7 10 10
1   1   2  2   3  so       .
4 4 2 3 4 8  4 4 3 4 3
1
3. Edwin is making chili. The recipe calls for 3 cups of chopped onions. When he measures the
4
3
onions he chopped, he has only 2 cups. How many more cups of onions does he need?
4
Adapted from Ohio, 2006, grade 6
3 1 1 1
Answer: 2 is below 3 and 3 is over 3, so he needs two one fourth cups or one
4 4 4 4
half cup more.
©2015 Texas Instruments Incorporated 8 education.ti.com

9. Building Concepts: Mixed Numbers
TEACHER NOTES
4. Every day Suze collects aluminum for a project. The table shows how much she collected
during one five-day period.
a. Estimate the number of pounds of aluminum she collected during that time. Explain how you
16
made your estimate. Possible answer: Monday and Tuesday add up to or 2; adding
8
in Wednesday and Thursday make it about 5 plus Friday gives about 6. Another
22 6 3
strategy: adding the 8ths gives ,which is 2  or 2  . Add in the whole
8 8 4
numbers 2 and 1 and get about 6.
Day Monday Tuesday Wednesday Thursday Friday
13 3 1 5
Pounds 2 1
8 8 8 8
1
b. Suze recorded the amount she collected on Tuesday incorrectly. It should have been .
2
How does that change your estimate?
Adapted from Ohio, 2006, grade 6
Possible answer: The total is still about 6 because it was close to 6 and the difference
1 4 3 1
between or and is only , so it won’t get much larger.
2 8 8 8
5. Jorge left some numbers off the number line below. Fill in the numbers that should go in A, B,
and C.
3
NAEP grade 4, 2009
©2015 Texas Instruments Incorporated 9 education.ti.com

10. Building Concepts: Mixed Numbers
TEACHER NOTES
Student Activity solutions
Vocabulary In this activity, you will solve problems involving improper
fractions and mixed numbers.
improper fraction:
1. Which of the following could be written as the sum of 2 and
a fraction whose
a proper fraction where the numerator is less than the
numerator is equal to
denominator? Explain your reasoning.
or greater than the
11 7 3 9
a. b. c. 
denominator 5 2 4 5
1 9
Answer: a could be written as 2 and . In choice c, is
5 5
mixed number: 3
nearly 2 and adding will make the sum greater than 2 but
4
a whole number and a 11
not yet 3, 2 and .
20
fraction combined
2. Identify the following statement as true or false: “A fraction
whose denominator is 8 will be between 3 and 4 if the
proper fraction:
numerator is between 18 and 32.” Explain your reasoning.
a fraction whose 3
Possible answer: False. c is 2  , which is less than 3.
8
numerator is less than
1 1
the denominator 3. A recipe calls for 2cups of flour. Suppose you only have a -
3 3
1
cup measuring cup. How many times will you have to fill the
3
cup? Explain your reasoning.
1
Possible answer: There are three cups in each whole, so
3
1 1
there would be six cups in 2 plus the extra cup. You
3 3
1
would fill the cup 7 times.
3
©2015 Texas Instruments Incorporated 10 education.ti.com

11. Building Concepts: Mixed Numbers
TEACHER NOTES
14
4. Arnie hiked miles of a 4-mile hiking trail. Has Arnie hiked more or less than half of
8
the total length of the trail? Complete the number line to show your thinking. Explain your
reasoning.
0 1 2 3 4
14 6 3
Answer: Arnie hiked less than half the distance. is equal to 1 or 1 miles. The half –
8 8 4
3
way point is at 2 miles and 1  2 .
4
©2015 Texas Instruments Incorporated 11 education.ti.com

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