Grade 4 Fractions on the Number Line, Fraction Operations & Decimal Fractions- Conceptual Lessons
Type of
Lesson Title and Objective/Description
Suggested
Math Practice
Knowledge
Time Frame
embedded
& SBAC
Claim
C, RK- 1 ,2, 3 Defining Fractions (Blue/White Strips and letters A-D are “pre-unit”)
2 class
3, 6, 7
Students will define a fraction by trying to determine what fraction of a periods
whole a given strip of paper represents. Students will use the definition
to plot fractions and mixed numbers on the number line.
C- 1, 3
On The Line Fractions & Mixed Numbers
1-2 class
3, 7, 8
Students will physically move on a number line to iterate a unit fraction
periods
and describe the fraction as repeated addition of the unit fraction or
multiplication of the unit fraction. Students will give two names for mixed
numbers by first seeing the fraction as an iteration of a unit fraction and
then reading the name for the mixed number representing the same point.
C, P- 1, 3
What’s the Name of That Point?
1 period
3, 7, 8
Students will use a number line outside with their bodies and inside with
pencils to represent and then rename points as fractions greater than 1
and mixed numbers.
C, P- 1, 3
Which Measuring Cup Should I Use?
1 period
3, 7, 8
Students will measure out water using whole and fractional measuring
cups and then measure the same amount using only fractional cups to
understand how to rename mixed numbers as fractions greater than one
and vice versa. Students will apply this understand to find sums of
fractions greater than 1 and mixed numbers and represent the sum with
two names.
P, RK- 1, 2, 3 What’s the Point? I and II **
20-30
3, 5, 6
Students will study a number line to determine how it was partitioned and minutes
use this information to determine the value of a certain point.
C, P- 1, 3
Multiplying Fractions with Whole Numbers
1 period
3, 7, 8
1
P, RK- 1, 2
RK- 2
P, RK- 1, 2
M, C- 1, 3
C, P- 1
C, RK- 1, 2, 3
Students will represent and simplify problems involving multiplication of a
fraction and a whole number using repeated addition and groups of.
Students will study the results to conclude that one method that works is
to multiply the whole number by the numerator and keep the same
b a b
denominator; i.e., a  
.
c
c
Hopping Along the Number Line
Students will use a number line (on paper and with their bodies) to
represent multiplication of a whole number with a fraction.

The Factor Game- Fraction Multiplication
Students will play a game to practice multiplication of whole numbers and
fractions by choosing two factors and placing a counter on a square in an
attempt to cover 4 squares in a row.
Practice/ Problem Solving: Comparing fractions with same and different
numerators and denominators.
Fractions and New Place Value Cards
Students will look at fractions shaded to represent tenths and hundredths
and record their fraction name and the name in words. Students will then
be introduced to decimal fractions and extend the place value chart to
learn how to record tenths and hundredths in decimal notation. Students
will use place value cards to build decimal fractions that involve both
tenths and hundredths.
Fractions and Decimal Fractions
Students will shade given fractions or decimal fractions on ten and
hundred grids and record the name the number represents both as a
fraction and a decimal fraction. Students will explain how fractions and
decimal fractions are both valid ways of expressing tenths and hundredths.
Meter Stick Decimals
1 period
1, 3, 5, 6
1 period
1, 2
2-3 class
periods
1 class period
5, 6
1 class period
3, 5, 6, 7
1-2 class
periods
1, 3, 7
2
By building and using a meter stick to measure various objects, students
will develop an understanding of tenths and hundredths. Students will
explain the meaning of tenths and hundredths by writing out numbers
using expanded notation.
What’s the Point- Decimals
Students will study a number line to determine how it was partitioned and
use this information to determine the value of a certain point.
P- 1
RK- 2, 4
Problem Solving: Writing and comparing decimal fractions
15- 20
minutes
3, 6
2-3 class
periods
Summative Assessment
NOTES:
5 Week Unit Addresses Standards: NF 1, 2, 3b, 4a, 5, 6, 7, MD 4 (part)
** optional and available upon request.
Key:
Types of Knowledge:
Facts (F)
Procedures (P)
Concepts (C )
Relational Knowledge (RK)
SBAC Claims:
1) Concepts & Procedures
2) Problem Solving
3) Communicating & Reasoning
4) Modeling & Data Analysis
3
IUSD Grade 4 Irvine Math Project Lessons
Fractions on the Number Line, Fraction Operations, Decimal & Place Value
Lesson Title
Defining Fractions
Topic, MP, Claims
Fractions on the # Line; Meaning of
Numerator & Denominator
MP 3,6,7; Claims 1,2,3
On the Line Fractions
& Mixed Numbers
Physically moving on a number line to
iterate a unit fraction & fractions
greater than 1
MP 3,7,8; Claims 1,3
Fractions greater than 1 on the # Line
MP 3,7,8; Claims 1,3
What’s the Name of
that Point?
Which Measuring
Cup Should I Use?
Renaming Fractions Greater than 1 as
Mixed Numbers and Vice Versa
MP 3,7,8; Claims 1,3
Multiplying Fractions
with Whole Numbers
Representing and simplify problems
involving multiplication of a fraction
and a whole number using repeated
addition and groups of
MP 3,7,8; Claims 1,3
Using a number line (on paper and
with their bodies) to represent
multiplication of a whole number
with a fraction
MP 1,3,5,6; Claims 1,2
Hopping Along the
Number Line
When will I use this?
Modifications to be made
FD_T1
The Factor Game:
Fraction
Multiplication
Fractions & the New
Place Value Cards
Fractions & Decimal
Fractions
Meter Stick Decimals
What’s the Point? Decimals
Practicing multiplication of whole
numbers and fractions by choosing
two factors and placing a counter on a
square in an attempt to cover 4
squares in a row
MP 1,2; Claim 2
Looking at grids to determine the
name of fractions with denominators
of 10 and 100; connecting these
fractions to decimal fractions and
introduction of new PV cards and
extended PV chart
MP 5,6; Claims 1,3
Shading Fractions and Decimal
Fractions
MP 3, 5, 6, 7; Claim 1
Building and using a meter stick to
measure various objects, students will
develop an understanding of tenths
and hundredths
MP 1,3,7; Claims 1,2,3
Studying a number line to determine
how it was partitioned and use this
information to determine the value of
a certain point
MP 3,6; Claim 1
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Teacher Directions
Materials:
◊ Blue strips copied onto blue paper and cut out- 1 per student
◊ White Strips copied onto white paper and cut out- 1 per student
◊ Optional- play-doh, plastic knives and plates (1 per student)
Objective: Students will define a fraction by trying to determine what fraction of a
whole a given strip of paper represents. Students will use the definition to plot
fractions and mixed numbers on the number line.
Opening
Write up the question, “What is a Fraction?” Pass out play-doh, a plate and a plastic
3
knife and ask each student to show with their play-doh. Give them 2 minutes to do
5
this. Have the class share using inside-outside line, allowing for 3 rotations. Come
back together and ask a few students to define what a fraction is.
Blue Strip/White Strip
Pass out one blue and one white strip (from pages that follow) to each student. Set the
timer for 1 minute and have them work silently, alone to try to figure out “What fraction
of the white strip is the blue strip?” (Write this question on the board.) After 1 minute,
allow the students to work with their group for the next 5-8 minutes to try to determine
the answer (note: they can combine strips to help, mark on the strips, etc). When most
groups have an answer, bring the class together and have volunteers come up to share
what fraction they got and why (have them show on the elmo or with their papers).
After each person presents, ask the rest of the class who thought about it the same way
and who thought about it differently. Call on those who thought about it differently to
explain their thinking. Note: The goal of this activity is NOT about getting the right
answer, but about the students seeing that they need to figure out how many of the blue
fit along the white (i.e., they need to see that they must divide the whole into blue
sections). This will be different than the definition/picture most students shared during
the roundtable.
Introducing the Definition of a Fraction
Point out to the class that the groups who shared were all trying to find a way to divide
up the white strip into “blue strip” units or they were trying to find who many blues
covered a white, etc. Tell them that this idea of dividing the whole into smaller pieces
will be the way they will think about fractions for this unit with the number line. Bring
out the Word Wall definition for fraction. Pick one of the groups’ fractions for the
blue/white strip activity to model the definition. For example, if a group came up with
3/10, read the definition (while pointing to the strips) “Start with one whole and divide
it into 10 equal pieces. We’re talking about 3 of those 10 equal pieces when we name
the fraction 3/10.” Repeat the same process, having the class help you fill in the blanks,
for another fraction a group shared (perhaps 2/7 or 1/3, etc).
IMP Activity Defining Fractions
5
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Defining Fractions I
The students will now work with the concept of dividing a whole into equal pieces (the
denominator) and counting the number of pieces as the numerator of the fraction. Pass
out the activity sheet. Allow students to work with a partner, but make sure each
student completes their own activity sheet. Work through problem A as a class.
START with completing the sentence frame for the definition. Then ask the class “How
many parts do we need to break the whole into?” Use your fingers pinched as if
showing distance, to estimate where to draw the lines, so that you can fit “5” of your
finger widths across. Once you mark the sections to show 5 equal pieces (note: it is
okay if the students don’t know they need 4 lines- they will figure this out after a
while), ask the students how many pieces they need. Show them where 1 piece is and
mark a point there and label it 1/5. (See problem A below). Begin problem B together,
asking the students what to write in the blanks for the definition. Then let them try to
mark the 1/7. Note: it is okay if the sevenths are not perfect, so long as they are getting
the concept. Set the timer for 10 minutes to have the students complete problems that
are fractions less than one. Bring the class back together and use random selection to
have students come show and explain their work on the fractions less than one on the
elmo. Then go over problem I (first fraction that is greater than 1) together. Begin, as
before, by completing the sentence frame. Once the class decides they need to divide
the wholes (this time plural) into five equal pieces and as you do this on your number
line, ask them how many pieces they need. Show them how seven “fifths” is the same
as five “1/5’s” by and two more “1/5’s”. Give the students 10 minutes to complete the
remaining problems and then have students come share their work and thinking.
F.
Definition: Start with one whole and divide it into 5 equal pieces. We’re talking
about 3 of those 5 pieces when we name the fraction 3
3
5
0
1
5
1
5
1
5
5
1
3
1
= 3•
5
5
“Improper fractions” and mixed numbers.
Note that “improper fractions” should not be labeled as such, as there is nothing
improper about them! They can be called “fractions greater than 1”.
IMP Activity Defining Fractions
6
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Blue Strips
Copy onto blue paper and then cut 1 strip per student
IMP Activity Defining Fractions
7
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White Strips
Copy onto white paper and then cut 1 strip per student
IMP Activity Defining Fractions
8
FD_T6
Teacher Directions Materials: • Number Line drawn in chalk (or tape) marked from 0-­‐3 in thirds and sixths (1 per pair of students) Note: Number lines should be drawn parallel to each other so that the teacher can stand in the middle and all pairs can hear the directions. (See picture below) It will be easiest to label 0, 1, 2 and 3 and then divide each whole into thirds and then divide each third into halves. Objective: Students will physically move on a number line to iterate a unit fraction and describe the fraction as repeated addition of the unit fraction or multiplication of the unit fraction. Students will give two names for mixed numbers by first seeing the fraction as an iteration of a unit fraction and then reading the name for the mixed number representing the same point. Directions: Simon Says-­ Outside Take the class outside to where you have drawn a number line (0-­‐3, marked to show thirds and sixths, for each pair of students) and let them know that today you will playing Simon Says on the number lines. One team member will begin on 0 and be playing Simon Says and the other team member will stand just past the 3 and be the “referee”. If the team member playing Simon Says makes a mistake, the referee points it out and the students trade roles. Make sure the class understands the rules of Simon Says. Once students understand, have each pair go to their number line. Play Simon Says using the following statements. 1. Simon Says the number line is marked to show halves, thirds and sixths. Stand on the point representing one-­‐half. This is called one-­‐half of a unit. 2. Stand on the point representing one-­‐third. 3. Simon Says, Stand on the point representing one-­‐third. This is called one-­‐third of a unit. 4. Simon says, Stand on the point representing one-­‐sixth. This is called one-­‐sixth of a unit. 5. Go back to 0. 6. Simon Says, Go back to 0. 7. Walk one-­‐third of a unit, two times. 8. Simon Says, Walk one-­‐third of a unit, two times. 9. What point are you standing on now? 2
10. Simon Says, What point are you standing on now? (Answer: ). 3
11. Go back to zero. 12. Simon Says, Go back to 0. IMP Activity: On The Line- Fractions & Mixed Numbers
4
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13. Walk one-­‐third of a unit, three times. 14. Simon Says, Walk one-­‐third of a unit, three times. 15. What point are you standing on now? 3
16. Simon Says, What point are you standing on now? (Answer: ). 3
17. Simon Says, Tell me another name for this point. (Answer: 1) 18. Go back to zero. 19. Simon Says, Go back to 0. 20. Walk one-­‐third of a unit, five times. 21. Simon Says, Walk one-­‐third of a unit, five times. 22. What point are you standing on now? 5
23. Simon Says, What point are you standing on now? (Answer: ). 3
2
24. Simon Says, Tell me another name for this point. (Answer: 1 ). 3
25. Simon Says, Go back to 0. 26. Simon Says, Walk one-­‐third of a unit, six times. 27. What point are you standing on now? 6
28. Simon Says, What point are you standing on now? (Answer: ). 3
29. Simon Says, Tell me another name for this point. (Answer: 2). 30. Simon Says, Go back to 0. 31. Simon Says, Walk one-­‐third of a unit, 8 times. 32. What point are you standing on now? 8
33. Simon Says, What point are you standing on now? (Answer: ). 3
2
34. Simon Says, Tell me another name for this point. (Answer: 2 ). 3
35. Tell me how many one-­‐third of a unit steps you would need to take to land on 3. 36. Simon Says, Tell me how many one-­‐third of a unit steps you would need to take to land on 3 37. Go back to zero. 38. Simon Says, Go back to 0. 39. Simon Says, Walk one-­‐sixth of a unit, 4 times. 40. Tell me two names for this point. 4 2
41. Simon says, Tell me two names for this point (Answer: , ). 6 3
42. Simon Says, Go back to 0. 39. Simon Says, Walk one-­‐sixth of a unit, 10 times. 40. Tell me three names for this point. 10 2 4
41. Simon says, Tell me three names for this point (Answer: ,1 ,1 ). 6 3 6
42. How many one-­‐sixths of a unit steps you would need to take to land on 3. 43. Simon Says, Tell me how many one-­‐sixths of a unit steps you would need to take to land on 3 (Answer 8 one-­‐sixth steps) 44. Simon Says, Go back to 0. IMP Activity: On The Line- Fractions & Mixed Numbers
5
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45. Simon says, Tell me how many one-­‐half unit steps you would need to take to get to 4. 1
46. How many one-­‐half unit steps would you need to take to get to 2 ? 2
1
47. Simon Says, Tell me how many one-­‐half unit steps would you need to take to get to 2 . 2
(Answer: 5) On the Line-­ Inside Take the class back inside and pass out the activity sheet. Explain to the class that they will be doing what they did outside, but without Simon Says and with their pencil and paper number lines. Walk through problem 1 together. Have a student come up to model three “steps” of one-­‐eighth unit on the number line (see below). Ask the class the name of the 3
point ( ). Then give students a minute to try to complete the addition and multiplication 8
sentence to represent this scenario. Allow students to share with a partner and then select students to share with the class. 1 1 1 3
1
3
Addition sentence: + + = Multiplication sentence: 3( ) = 8 8 8 8
8 8
1
3
1
0 1
2
2
8
2
8
Once the class understands, let them work alone or in pairs on the remaining problems. Make sure to point out that some problems will ask for a second name for the same point (and in some cases, there are more than two correct names, so students can pick!). After about 15-­‐25 minutes, choose volunteers to write out their work on the whiteboard. Once they are done, have each student explain their work to the class and ask if any student had a different name for that fraction. For students who finish early, have them work on the challenge problems (note there are multiple correct answers). Bring the class back together and give the students 5 minutes to complete the summary. Choose students to read what they wrote aloud to the class. IMP Activity: On The Line- Fractions & Mixed Numbers
6
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Teacher Directions
Materials:
• Number lines marked from 0 to 4 in thirds and sixths (1 per pair)
• Copies of the number lines marked from 0 to 4 in fourths and eighths for each student.
Objective: Students will use a number line outside with their bodies and inside with pencils to
represent and then rename points as a fraction greater than 1 and mixed numbers.
Directions:
Pass out the activity sheet and have a volunteer read the opening scenario. Give the students a
few minutes to think and record their ideas. Note that you will not share ideas as a class, but
students will verify their thinking when they go outside. Put the students into pairs and have
each pair walk outside to a number line. Students will need to take this activity sheet and a
pencil outside with them. Model the first problem for the class, by using two volunteers. Ask
the first student to walk eleven one-third steps forward from 0. Ask them what point they are
standing on. Then proceed through the questions for #1 (see below for an answer). Once the
class understands, give them about 10-15 minutes to complete the 8 problems using their number
lines. Note that students that can do this in their head will be greatly helped by using their bodies
to model as this connects to different aspects of their brain and memory!
11
1. Walk to the point . Have your partner now walk to meet you, but by taking as many whole
3
1
unit steps as possible first and then taking steps for the remaining distance. Record the work
3
using the sentence frame below.
11
1
, I can take 3 whole steps and then I need to walk 2 steps.
3
3
11
2
Another name for
is 3 .
3
3
To get to
Bring the class back inside and explain that they will follow the same process, but this time with
paper number lines. Pass out the number lines marked in fourths and eighths to each student. As
before, encourage students to physically model the process. Allow students about 10-15 minutes
to work on these alone and then give them a few minutes to compare answers with a partner
before selecting students to come share with the class.
Close out the lesson by having students complete the summary box. Give students a few minutes
to write, and then select a few students to share their ideas. Note that it is more important that
students understand the idea than having an algorithm!
IMP Activity: What’s the Name of That Point?
5
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Number Lines for Students to Use Inside
0
1
0
2
1
IMP Activity: What’s the Name of That Point?
2
3
4
3
4
6
FD_T11
Teacher Directions
Materials:
◊
◊
◊
◊
1
1
Measuring cups (1 cup, cup, cup)-­‐ 1 per pair (ask students to bring in!) 4
3
1
Measuring spoons (1 tsp, tsp)-­‐ 1 per pair (ask students to bring in!) 8
Water for students to use and pour (any sized container will work) Paper towels Objective: Students will measure out water using whole and fractional measuring cups and then
measure the same amount using only fractional cups to understand how to rename mixed
numbers as fractions greater than one and vice versa. Students will apply this understand to find
sums of fractions greater than 1 and mixed numbers and represent the sum with two names.
Directions
Pass out the activity sheet and have a student read scenario 1. Explain to the class that they will
use water and measuring cups to measure and pour using both a whole cup and the quarter cup
and then using only quarter cups. Have pairs of students get a container with some water in it
and paper towels. Ask them to bring out the measuring cups they brought from home. Allow the
pairs to figure out the problems on their own. Ask guiding questions if students are stuck, but let
them experiment to come to their own understanding of how to rename fractions greater than 1
and mixed numbers.
Give pairs about 10 minutes to complete the three scenarios. Collect the water and select
students to share how they measured and what names they used for each fraction. Lead a
discussion about which method would be better and why (there is not a correct answer!).
Practice Time
Students will now practice renaming fractions greater than 1 and mixed numbers. If some
students are struggling, model how to use the guiding questions for the first problem. Have
students work alone on the remaining problems. Come back as a class to have students share
their methods for problems 2-4. If needed, model how to use the new set of guiding questions
for problem 5 and then give students a few minutes to complete problems 6-8 on their own.
Come back again and have students share their methods. Finally, have students work on part 3,
which combines addition with renaming. Give students a few minutes to work alone and then
come back as a class to have students present solutions and explain their reasoning.
Note: Allow students to use number lines, pictures or any other method to help them add, if
needed.
IMP Activity: Which Measuring Cup Should I Use?
3
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Teacher Directions
Materials:
◊
◊
◊
Colored Pencils Measuring Cups (representing thirds)-­‐ for class demo Water -­‐ about 6 cups-­‐ for class demo Objective: Students will represent and simplify problems involving multiplication of a fraction
and a whole number using repeated addition and groups of. Students will study the results to
conclude that one method that works is to multiply the whole number by the numerator and keep
b a• b
the same denominator; i.e., a • =
.
c
c
Directions
Pass out the activity sheet and have a student read the opening scenario. Give students about 3-5
minutes to think and record their ideas independently. Pass out colored pencils to encourage the
use of drawing representations of the fractions. After most students have an idea written down,
select a student to come up front and model the scenario by using a measuring cup and water.
Have the student explain his/her thinking and ask the class if they can understand and then who
thought about it the same way or differently. Select all students who have a different idea to
come share. Some possible methods include: a) pouring one-third of a cup, 16 times; b) pouring
two-thirds of a cup, eight times; c) seeing that every three two-thirds makes 2 whole cups and
pouring whole cups and then the remaining with third cups.
Note: while this lesson will guide students to discover an “algorithm” to multiply fractions by
whole numbers, student intuitive understanding of this topic should be the focus. So, spend time
on the opening scenario to make meaning and reason through what it means to multiply a
fraction by a whole number.
Direct the class’ attention to meanings of multiplication. Explain that you would like them to
list, show or draw all ideas they can recall for what 5 • 3 means or how to find the product. A
few possibilities should include: a) 5 + 5 + 5; b) 3 + 3 + 3 + 3 + 3; c) 5 groups of 3 (with perhaps
5 circles and 3 objects in each); d) 3 groups of 5 (3 circles with 5 objects in each); e) 5 “jumps”
of 3 on a number line; f) 3 “jumps” of 5 on a number line; g) an array with 5 rows of 3 or 3 rows
of 5; h) a rectangle with dimensions 3 by 5 or 5 by 3.
Give the students a few minutes to list all ideas. Put the class in groups of 4 and use roundtable
to have them share. Choose one person to start in each group and have each students share 1
new idea and then rotate until all ideas are shared. While students are sharing. select a student
who used repeated addition and one who used “groups of” to come share with the class. Note
that we will deal with area and number line in future lessons!
Once the class recalls methods a, b, c and/or d, have them turn to page 2 of the activity sheet.
Have a volunteer read the directions aloud and walk through the example problem with the class.
Encourage students to use any type of drawing they would like for the “groups of”. Encourage
students to be thinking about how they add fractions with like denominators when simplifying
IMP Activity: Multiplying Fractions with Whole Numbers
5
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the repeated addition portion. Make sure students record the product as both a fraction greater
than 1 and a mixed number (when applicable). Give students about 15 minutes to complete
problems 1-6 on their own. Have early finishers check solutions with a partner and then begin
the analysis. If the class was struggling at all, choose students to come show and explain their
work for the class. If most the class was able to complete the 6 problems without much struggle,
then turn to the analysis as a group. Note that the analysis also serves as an answer key. Use
think-pair-share to have students answer the 6 questions. Below are some possible answers.
b a• b
Note: while it is great for students to leave this lesson understanding that a • =
, it is more
c
c
important they understand what they are doing (and they will have two more opportunities to
make this generalization in the unit!).
Possible Answers
1. What part of the factors stays the same in the product? _denominator
2. What part of the factors changes to get the product? _numerator/whole number
3. How do you get the numerator for the product? multiply the whole number by the numerator
4. What other patterns or short-cuts did you notice for multiplying a fraction by a whole
number?
When multiplying a whole number by a unit fraction, the whole number becomes the numerator
of the fraction in the product.
5
In the case of #5, the “5’s” “cancel” out (the becomes 1 and 1 multiplied by any number is the
5
number)
For any of the problems, you can first “divide” the whole number by the denominator and then
1 10
multiply (i.e., 10 • = •1
2 2
5. Predict: Based upon what you discovered above, find the product of the following fractions:
6 48
4
9 90
7
b a• b
a) 8 • =
b) 10 • =
c) (challenge!) a • =
=4
=1
11 11
11
83 83
83
c
c
6. How is multiplying a fraction by a whole number similar to and different from multiplying
two whole numbers?
similar as you multiply the whole number by the numerator; different as you have the
denominator which remains constant.
IMP Activity: Multiplying Fractions with Whole Numbers
6
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Teacher Directions
Materials:
• 2 competition number lines- one marked from 0 to 5 in fourths and eighths and one
marked from 0 to 5 in thirds and sixths.
• Competition problems written on index cards.
• Optional: Chalk for each pair to draw their own number line and practice
• Chart paper on which you can record group times outside.
• Stop watch or cell phone timer
Objective: Students will use a number line (on paper and with their bodies) to represent
multiplication of a whole number with a fraction.
Directions:
Pass out the activity sheet and have a student read the opening scenario. Give students about 3-5
minutes to list all methods they can think of (with an illustration if possible) to show how to find
this product (which they can also represent as a sum!). Put students in groups of 4 and use
roundtable to have students share. To do this, have one person in each group share 1 idea and
then rotate around the group, having each student share 1 new idea until all ideas have been
shared. Make a class list of the methods that have been used.
Direct the class’ attention to Multiplying Fractions on a Number Line. Give students a minute
to try to figure out how to represent this on the number line. Have them share ideas with a
partner and then select a student to share with the class.
Explain to the class that they will be competing to simplify multiplication of fractions with
whole number problems by hopping on a number line. Direct the class’ attention to the
Competition Practice. Have a student read the directions and walk through the example with
the class. Make sure they understand that they must take hops of equal length and that they must
represent the final location using both a mixed number and a fraction greater than 1. Ask the
1
students how many hops of would get them to the same location (answer 12). Once the class
4
understands, put them in teams of 2 and give them about 8-10 minutes to complete the practice
problems.
Have students read the Hopping Along the Number Line Competition Rules. Model what the
competition will look like using a number line under the document camera if this would help
your class (but if they have been using the chalk number lines throughout, they should be fine!)
Optional: Once outside, you can give each team of 2 a piece of chalk and let them draw and then
practice the same 5 problems on a number line.
Bring the class outside to the two competition number lines. Have each team of two seated
around the competition lines. Explain that you will have problems written out on an index card.
You’ll call up a team and hand them the card and the time will start. The team must figure out
which number line to use and then they must model the problem on the number line. Time stops
IMP Activity: Hopping Along the Number Line
4
FD_T15
once the runner is on the final location and can state the name of the point as a mixed number
and a fraction greater than 1. As soon as one group finishes, have them sit back down and have
the next group come up. Tell the recorder the time and be ready to hand out the next problem
and begin the time again. This competition should go quickly!
Take out the chart paper and a marker. Choose a student to record times. Have the students not
competing try to figure out the product in their heads before the runner gets there. Ask the
students how many unit fractions (of the same denominator as the fraction in the factor) they
could have hopped to arrive at the same location. Do as many rounds as you have time for and
then announce the team with the lowest total score!
Round 1
4
5•
8
2
3•
6
2
7•
3
3
9•
8
3
6•
4
5
8•
6
1
10 •
2
2
6•
4
7
5•
8
1
7•
2
1
14 •
6
2
11•
8
6
6•
8
4
5•
6
7
3•
8
Competition Problems: Write out on index cards
Round 2
5
3•
4
7
2•
6
6
3•
4
10
3•
8
5
2•
2
4
3•
3
7
2•
4
3
3•
2
9
3•
8
5
3•
3
13
2•
8
9
2•
6
8
2•
4
8
3•
6
7
2•
3
IMP Activity: Hopping Along the Number Line
5
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Name: ________________________________________ Date:________________
The Factor Game-Fraction Multiplication
Rules
• The object of the game is to be the first team to get 4 in a row. This
can be horizontal (
), vertical (
) or diagonal (
) or (
).
• Each team/person selects one color of counters to represent them on
the game board.
• The first team or player to go chooses two (2) factors from the bottom
of the board and calls out the product. The team or player will place
a paperclip on each of the factors at the bottom of the page and then
place their counter on the game board in the box representing the
product. If the product shows up twice on the game board, the team
or player must choose which one they want.
• The next team or player now must chose one (and only one) of the
paperclips to move to another factor and call out the product. They
then place their counter in the box representing the product.
• Note: Both paperclips may be placed on the same factor (e.g., 2 x 2
=4).
• Note: Recall equivalent fractions when looking for the product you
would like!
IMP Activity The Factor Game- Fraction Multiplication
1
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Fraction Factor Game
1
8
3
4
1
1
4
1
1
2
1
3
2
1
12
1
6
1
2
1
1
3
3
8
2
3
5
8
6
9
1
5
8
2
5
6
1
2
2
1
4
16
1
1
4
10
4
2
5
1
3
3
4
5
1
2
3
4
2
3
3
1
2
3
4
5
6
8
9
1
5
1
6
1
8
2
3
3
4
1
2
1
3
1
4
IMP Activity The Factor Game- Fraction Multiplication
2
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Teacher Directions
Materials
2- color counters (about 20 per student or team)
Paperclips- 2 per team
Overview:
Students will play a game to practice multiplication of whole numbers and fractions by choosing
two factors and placing a counter on a square in an attempt to cover 4 squares in a row.
Directions
This is a game for two players or two teams and can easily be played with the whole class by
dividing the class into two groups and placing the game board on the document camera.
The object of the game is to be the first team to get 4 in a row.
The first team to go chooses two factors from the bottom of the board and calls out the product.
Place a paperclip on each of the factors and place a counter on the square representing the
product (note that the group or person choosing the factor must explain which square to cover,
demonstrating their understanding of multiplication as well as fraction equivalence). If the
product shows up twice on the game board, the team must chose which one they want.
The next team now must choose one of the paperclips to move to another factor and call out the
product. They may only move one of the paperclips.
Note: Both paperclips may be placed on the same factor (e.g., 2 x 2 =4).
Teams should quickly realize that they must not only think about how to get 4 in a row for
themselves, but also be careful to block the other team and avoid leaving the paperclips on
factors that the other team needs to complete their 4 in a row.
IMP Activity The Factor Game- Fraction Multiplication
3
FD_T19
Teacher Directions
Materials:
◊ Grade 4 Decimal Place Value Cards (See Template in Unit Folder)
Objective
Students will look at fractions shaded to represent tenths and hundredths and record their fraction
name and the name in words. Students will then be introduced to decimal fractions and extend
the place value chart to learn how to record tenths and hundredths in decimal notation. Students
will use place value cards to build decimal fractions that involve both tenths and hundredths.
Directions:
Pass out the activity sheet and have a student read the directions. Give the students 1-2 minutes
to complete number 1, leaving “Place Value” blank for now. Have students share the fraction
and words they recorded with their neighbor and then use random selection to have students
share. Students should record the fraction
and the words “one-tenth”. Once students
understand, have them work independently on problems 2-9. When most students have tried
numbers 5 and 6, bring the class back together to discuss what they wrote. The goal here is to
record a number as _____ and ____. See answer key below.
1.
, one tenth
2.
, three tenths
3.
, one hundredth
4.
, seven hundredths
5.
, 1 and 5 tenths
6.
, two tenths and seven hundredths
7.
, six tenths and nine hundredths
8.
, four tenths and four hundredths
9.
, one tenth and five hundredths
Part 2
Have a student read the top of the page under part 2. Ask each student to point to the word
“tenth” on the place value chart. Ask them to locate the ones place as well as the tens, hundreds,
etc. Ask them to point to the hundredths place too. Pass out the place value cards to each
student or pair. Show the class .1 (call it one tenth) and ask them to hold this up and say the
name of this card with you, “One tenth.” Repeat the same process for one hundredth. Next,
explain you will be playing a game where you say the name of a decimal fraction and they try to
be first to hold up that card. As you say the name, look around to see who is holding up the
correct card.
1. Show me three tenths.
2. Show me three hundredths.
IMP Activity: Fractions and New Place Value Cards
5
FD_T20
3.
4.
5.
6.
7.
8.
Show me six tenths.
Show me two tenths.
Show me seven hundredths.
Show me nine tenths.
Show me eight hundredths.
Show me eight tenths.
If students are struggling, play for a longer time. If students are doing well, move on to part 3.
For this, students need to go back to problems 1-9 and record the name of the decimal fraction
using place value notation. Model number 1 with the students, asking them to show you the card
that represents one tenth. When you see the card that say .1, record this number on the place
value line. Let the students work alone on 2-5 and then pause to see what students recorded for
number 5 (let them share and debate). (Answers are below). Do the same for number 6 and then
allow students to finish before coming together to check again. Note, for problems 6-9, it is
vitally important that students model EACH number with the cards and then place the cards one
on top of another to see how to record the number using place value.
1)
2)
3)
4)
5)
6)
7)
8)
9)
.1
.3
.01
.07
1.5
.27
.69
.44
.15
IMP Activity: Fractions and New Place Value Cards
6
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Teacher Directions
Materials:
◊ Colored Pencils
◊ Place Value Cards (to the hundredths)
Objective
Students will shade given fractions or decimal fractions on ten and hundred grids and record the
name the number represents both as a fraction and a decimal fraction. Students will explain how
fractions and decimal fractions are both valid ways of expressing tenths and hundredths.
Directions:
Pass out the activity sheet, place value cards and colored pencils to each student. Go over the
example by asking the students to count how many boxes are shaded and how many boxes there
are in the one whole box. Ask the class how to record this as a fraction and then ask them to
show the number with the place value cards. Finally, have the students record the number using
place value. Once students understand, allow them to work alone or with partners to complete
problems 1-3. Have students who finish early put their work up on the board and then explain
3
30
their thinking. Note: for problems like #1, students can see this as
. Find students who
or
10 100
thought about it each of these ways to share and question students to ensure they all agree these
are both correct answers (and thus .3 is equivalent to .30).
If needed, bring the class back together to discuss the directions for part 2. Here the number to
shade is given, and students must shade and then give the fraction and decimal fraction names.
There is not one write way to shade; a student can see 42 out of 100 as 4 tenths and then 2
hundredths or just any 42 boxes. Bring the class together to discuss Analysis Questions 1 and 2.
Assign half the class question #1 to study and the other half of the class #2. Let pairs work
together on their answer and then use inside outside line to have students teach each other what
they found. Have students complete problems 7 and 8. Come back together to discuss problem
8. To be correct, students must show 1 whole 100 grid and then 10 out of the next WHOLE
hundred grid, as each grid represents 1. Adding a column of 10 to the existing 100 grid just
makes the fraction 110 out of 110 and adding a column of 10 as a new box and shading those 10
now makes it 2 wholes. (See below for a correct representation).
IMP Activity Fractions & Decimal Fractions
6
FD_T22
Have the students work on part 3. Encourage students to use their place value cards to first
build the number and then shade and record the fraction name. Choose some early finishers to
put their work up on the board and then explain their thinking once most students have
completed this section. Ask for alternate methods to shade or think about each problem.
Finally, have students complete part 4. Students will need to draw addition grids for the
fractions greater than 1 (numbers 18 and 20). If time remains, give students a few minutes to
think about and share ideas to the final analysis question. (Note there is not a correct answer for
this.)
IMP Activity Fractions & Decimal Fractions
7
FD_T23
Teacher Directions
Materials
◊ Strips of butcher paper 1 meter long by about 3 cm wide for each person ◊ Index cards or paper cut to be 10cm long by 1in, 3 per table ◊ Index cards or paper cut to be 10cm long by 1in, marked in 1 cm increments, 3 per table ◊ Choose 3 objects, ranging from about 10 cm to about 280 cm long for display. Objective: By building and using a meter stick to measure various objects, students will develop
an understanding of tenths and hundredths. Students will explain the meaning of tenths and
hundredths by writing out numbers using expanded notation.
Directions
Give everyone a strip of paper and ask them to compare the length of their strip to other people's
strip around them. When students have agreed that they all have the same length strip, declare
that this length will be called a length of 1. From the front of the room, show the class the three
objects you have selected. Pass out the activity sheet. Ask each table to estimate from where
they are (in their seats), how long each object is as a fraction of 1 and record this on their
activity sheet.
Record the estimates on a class table (by either asking groups to share out or having a group
representative some record their estimates on the board). Expect there to be many different
denominators in the fractions used. Identify some of the denominators and draw attention to
what they mean - for example if someone estimated an object as 3/4 then it means the whole was
divided into 4 equal parts and the object is the length of 3 of those 4 equal parts. Give the
students a minute to think about and record their ideas to question number 2. The goal here is for
students to see that it is not easy to compare fractions with different numerators and
denominators without doing some additional math.
Ask the students what would make it easier to compare the measurements. Lead them to
answering that it is important to have the same denominator and have them record their ideas on
question 3. Announce that everyone will use tenths, and that to do this, we will all fold our strip
of paper into ten parts. Instruct the students to fold their paper into 10 equal parts. You might
want to model this by folding the paper in half and then the halves in fifths.
Pass out the index cards that measure 10cm (or one-tenth of a meter) (without the 1 cm
markings) and direct the participants to check their folds by using the index card. Instruct the
students to mark their strips at the tenths markings.
On the board list the decimal representations of the fractions now marked on their strip and ask
1
them to mark each mark of line they drew with the correct decimal representations. i.e.
= 0.1,
2
= 0.2,
10
10
etc.
IMP Activity: Meter Stick Decimals
3
FD_T24
When students have their strips marked, ask that a representative go and measure the same
original three objects to the nearest tenth. Collect the group data again so that all students can
see the results. Ask if any group thought any of the objects were between tenth markings. Have
that person identify one that was and ask which two measurements it was between. Let the
group know that you are going to draw another person to ask this next question of, but will first
give everyone a chance to think and discuss with a neighbor. Ask what number is in between 0.3
and 0.4 (if the object was between 3 tenths and 4 tenths - otherwise use whichever is appropriate
from the first person's response.) Give them a moment to think discuss with a partner and record
their thoughts on #6. Ask if anyone else had another object between two tenths and repeat the
same process.
Build off of this discussion to say that if we cut each tenth into ten more parts, we would have
100 parts and could label them. Give each table two index cards marked with 1 cm increments
and have them mark their strips.
Have groups again measure the objects, this time to the nearest hundredth, and document their
measurements by writing it in the table under number 9.
Once each group has recorded the measurement to the nearest hundredth, show the class how to
write this using expanded notation. Refer the strip of paper to explain as you expand. (Ex: tissue
3
box 2.3 = 2 + 0.3 and 2 +
10
99
9
9
Ex:#10 397
= 300 + 90 + 7 + +
or300 + 90 + 7 + 0.9 + .09
100
10 100
Have the students work with a partner or alone to complete the rest of the table, encouraging the
use of their strip to answer the questions.
Adding: Ask the students to explain to a partner the meaning of 0.3+ .02 (#13) or 3/10 plus
2/100. Lead their answers to drawing a sketch of a strip and marking the location and the parts
that lead to it. Have students complete the remaining practice problems (#14-18). Have early
finishers come show their work up front and explain how they thought about these 5 problems.
IMP Activity: Meter Stick Decimals
4
FD_T25
Teacher Directions
Materials: None Objective: Students will study a number line to determine how it was partitioned and use this information to determine the value of a certain point. Directions: Pass out the activity sheet. Explain to the students that their task is to determine the value of the “?” for each number line. Point out the directions, noting that the tick marks are equally spaced but not necessarily 1 unit apart. Set the timer for 4 minutes and have students work alone, silently. Then give them 4 minutes to work with a partner. As they work, circulate and ask questions to guide the students, such as, “Do you think each line represents 1 whole?” “How many spaces are between the 0 and the 1? How many pieces is the whole divided into?” Use the last 2 minutes to have students share what label they gave the point and why (use random selection to do this). Note: some of these will be challenging to the students; it is okay to let them struggle and have others explain their reasoning. IMP Activity What’s the Point Decimals?
2
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Teacher Directions Materials: 6-sided dice (3 per pair)
Post-it flags (a pack per pair)
Number Line Paper (butcher paper cut about 6 inches by 3-4
ft.)- 1 per pair
Objective: Students will roll dice to generate decimal fractions and reason to determine where to
place the decimal fraction on a number line. Directions: Using a number line you draw on the document camera or board, begin by
demonstrating how to play the game. To do this, make the points 0 and 3 on
the ends of the number line. Roll three dice and explain to the class that you
need to decide which digit will represent ones, which will represent tenths
and which will represent hundredths. Note that the number needs to be in
between 0 and 3. (For example, if you roll a 2, 5 and 1, you can make 1.25,
1.52, 2.15 or 2.51). Ask a few students to share where they would place that
point and why. Once you agree on a general location, write the decimal
fraction on a post-it flag and place the flag on the number line. Then, record
the decimal fraction in the middle column of the chart on the bottom of the
activity sheet and record a decimal fraction that was less than this and one
that was greater, as suggested by the class in trying to reason where to place
the fraction. Explain that now it would be the next student’s turn to roll and
decide where to place their decimal fraction.
The game is played in pairs, with each student deciding which decimal
fraction to use and where to place it, and the other student agreeing or
disagreeing with the placement before taking their turn. Both students place
their flags on the same number line. Students continue taking turns rolling,
recording and placing the flags while you circulate to assess and question.
Once the class understands, put the class in pairs. Have each pair send a
person up to get one number line paper, three dice and two sets of post-it
flags for the group. IMP Activity On The Line Game-­‐ Decimal Fractions 2 FD_T27
Teacher Directions Materials: None Objective: Students will study a number line to determine how it was partitioned and use this information to determine the value of a certain point. Directions: Note: There are two “what’s the point” activities here. You can choose to do one on each of two days or do them both in a single day. Pass out the activity sheet. Explain to the students that their task is to determine the value of the “?” for each number line. Point out the directions, noting that the tick marks are equally spaced but not necessarily 1 unit apart. Set the timer for 4 minutes and have students work alone, silently. Then give them 4 minutes to work with a partner. As they work, circulate and ask questions to guide the students, such as, “Do you think each line represents 1 whole?” “How many spaces are between the 0 and the 1? How many pieces is the whole divided into?” Use the last 2 minutes to have students share what label they gave the point and why (use random selection to do this). Note: some of these will be challenging to the students; it is okay to let them struggle and have others explain their reasoning. IMP Activity: What’s the Point?
3
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