Transcript Ratios_percents_fraction_grds_5and6

November 2013
5th and 6th Grade
Please make a name tent
Group Commonalities
On a blank sheet of paper, draw a vertical and
horizontal line, cutting the paper into quarters.
Number each one of the sections 1 - 4
1
3
2
4
Group Commonalities
•State something that you like
•Find out how many in your group also like it
•However many people in your group like it, write
what it is in that numbered spot
•Pass the paper to the next person and continue
the steps above for about 5 minutes
How Many Noses Are in Your Arm?
•The Statue of Liberty’s nose measures 4 feet 6
inches from the bridge to the tip.
•How long is her right arm?
•Use tape measures, calculators and what you
know about body parts to determine the length of
the Statue of Liberty’s right arm, the one holding
the torch.
•Decide on a problem solving strategies and to
record it.
•How long is her arm? (The actual length of
her arm is 42 feet.)
Share Solution Strategies
• Look through the handout of questioning
students
▫ What questions could you (have) asked
 Before
 During
 After
Ratios, Proportions, and
Percents
5th and 6th Grade
November 2013
Common Core Standards
• On a piece of paper:
▫ Write one thing going well
▫ Write one thing you are concerned about
Mathematical Practice
• Including in your lessons?
• Are the placemats working?
• How are you implementing them?
Standards
• Look through district and state standards being
addressed today
Work Through…
• Pattern Block handout AND
• Lesson 2 handout
• Compare the two handouts in terms of
questioning format
▫ What are strengths and weaknesses of each
handout?
Proportion Card Game
• Remove all face cards
• For the purposes of this game, aces = 1
• One player shuffles the deck of cards and deals out 4
cards to each team. The remaining cards should be
placed face down in the center of the table.
• Teams will take turns drawing a card from the deck and
discarding one from their hand.
• Each time a team finds a proportion in their hand, they
should call out “Prop!” Then, they lay down their cards
on the handout in the proper places to make a true
proportion statement.
• For each correct answer, teams earn a point.
• Each time a team wins a point, both teams should
discard their cards, the deck should be shuffled and both
teams should be dealt new hands of 4 cards.
I. Introduction to Ratio and Proportion
•The quantitative relation between two
What is a ratio? amounts showing the number of times
one value contains or is contained
within the other
•A statement of how two numbers
compare
What are some examples of ratios?
# of girls to # of boys in this room
# of teachers in Roosevelt to # of teachers in Arizona
# of tables in this room to # of chairs in this room
What is a proportion?
A statement about 2 equal ratios
Analogy:
A nest is to a bird as a den is to a fox.
Proportion:
2 is to 4 as 6 is to 12.
The proportion above can also be written:
2:4 as 6:12 2:4 = 6:12
2 6
=
4 12
According to Piaget’s theory of intellectual
development proportional reasoning is an
important skill for children to develop.
Proportional reasoning is something that is built
through an individuals’ experience and takes
time to develop. The following diagram
illustrates the concepts which are an important
components of proportional reasoning.
To assist students’ development of proportional
reasoning, it is not sufficient to teach a section on each of
these topics to develop proportional reasoning.
Relative Thinking
Relative thinking can be described as thinking
multiplicatively. It is important for students to
understand both absolute change and relative
change. Making the transition from absolute
change to relative change is an important step in
transitioning from additive to multiplicative
reasoning.
x
y
1
6
2
12
3
18
4
24
y = 6x
Partitioning
The partitioning of an object is the process of
dividing the object into a number of disjoint
parts that collectively make the whole. When
dealing with fractions, to determine fractional
parts, one partitions the object into parts of
equal size.
Unitizing
Unitizing is the cognitive process used to assign a
unit to a given quantity. For example, when
asked to think about a case of soda, do you
picture 24 cans, two 12- packs, or four 6-packs
[Lamon, 2006]? It is desirable for students to
build flexibility in determining the size of the
chunk of a quantity that they use for a unit, as
different situations may call for different sized
chunks. Asking students to explain their choices
can help encourage this flexibility.
UNIT RATE
If 3 cost $5 how much will 1 cost?
3 ÷ 5 = 0.6
Ratio Sense
Ratio Sense involves the ability to think flexibly in
problem situations involving ratios.
Rational Numbers
Rational Numbers are numbers of the form a/b ,
where b ≠ 0 and a and b are b integers. That is,
numbers that can be written as fractions. More
importantly, reasoning with rational numbers
requires the ability to reason flexibly with
fractions, ratios, rates, and percents, and the
operations on them.
Quantities and Change
Quantitative reasoning involves the ability to
interpret and operate with change. This may
require operating with constant or varying rates
of change.
y = 6x
6 is the constant
The ability of students (and people, in general) to
distinguish between relationships which are
proportional and relationships which are nonproportional is a key aspect of proportional
reasoning.
For this reason, students should be exposed to
situations which are proportional and nonproportional to learn when it is appropriate to
use a multiplicative solution strategy.
These types of experience are important because
in the absence of such experiences students tend
to inappropriately model non-proportional
problem situations with proportions.
One of the below questions involves
proportional reasoning the other does not,
which does and which does not? Discuss in
your groups
Question 1: Kris and Rich like to skate laps together
around an ice rink since they both skate at the same
constant rate. Today, Kris started skating first. By the
time that Kris had completed 9 laps Rich had completed
3 laps. How many laps will Kris complete by the time
that Rich completes 15 laps? Explain.
Question 2: If 3 bags of mulch weigh 21 pounds, how
many pounds will 8 bags of mulch weigh?
The temptation in Question 1 is to set up a proportion
to solve the problem, since there are three known
quantities and one unknown quantity, resulting in
45 for the answer as shown below.
9 = x
9 x 15 = 3 x x
3
15
45 = x
The thinking behind this approach is that 9 laps for
Kris is to 3 laps for Rich as x laps for Kris is to 15
laps for Rich. However, this particular problem can
not be represented by setting up a proportion, so
this approach is incorrect.
To see that the above approach will not work,
reflect on whether or not 45 laps is a
reasonable answer.
Let‘s take a closer look at this analysis of the
situation by making a table to represent the
information in the problem.
# of laps Rich completes
# of laps Kris completes
3
9
4
10
5
11
6
12
7
13
8
14
9
15
10
16
11
17
12
18
13
19
14
20
15
21
The
correct
answer, 21
laps, can
be seen by
making a
table.
In Question 2, it is implied that we are dealing
with the same size bags of mulch. Since 3 bags of
mulch weigh 21 pounds, 1 bag of mulch weighs 7
pounds. Hence, when determining the weight of
8 bags of mulch ones needs to only multiply the
number of pounds per bag by the number of
bags (i.e., 7 lb/bag × 8 bags = 56 lb). Thus, the
situation can be modeled by the linear function y
= 7x, where y represents the total weight of x
bags.
y = 21
8 3
y x 3 = 21 x 8
3y = 168
y = 56
Can proportions be used to solve
either of these problems? Why or
why not? Discuss in your groups
Question 3: Alisa is painting her living room. She
can paint the entire living room in 4 hours.
Assuming that Karen can complete the job in the
same amount of time as Alisa, how long will it take
Karen and Alisa to paint the living room together?
Question 4: If one player on a soccer team weighs
170 pounds, then what is the weight of 4 players?
Question 3: Alisa is painting her living room. She can paint the
entire living room in 4 hours. Assuming that Karen can complete the
job in the same amount of time as Alisa, how long will it take Karen
and Alisa to paint the living room together?
Common thing to do is set up the proportion:
1 person 2 people
=
4 hours ? hours
Which would give an answer of 8 hours. Is that
reasonable?
Can not use a proportion to solve this problem
Stopping to evaluate whether or not 8 hours is a
reasonable answer quickly leads one to conclude
that this problem can not be solved by setting up
a proportion. To see the correct answer, one
notices that both people can complete the job
alone in 4 hours so it should take half as much
time for the two people to complete the job
together, or 2 hours. Another way to analyze the
problem is to use a unit rate strategy – each
person can complete 1 job every 4 hours so let‘s
assume that each person can complete ¼ of a
job per hour.
Understanding Ratios and
Proportions
A. Consider the following situations and:
 Determine whether the problem being modeled
is a proportion.
 Explain your reasoning.
 If it is a proportion, solve the problem
1. Ray and Crystal invested money in a business
and will split the profit in a ratio of 2:3. If the
profit from the business is $1000, how much
money will each person receive?
2. Can Courtney enlarge a 3 inch by 5 inch photograph
proportionally to a 4 inch by 6 inch photograph? Is this
possible?
3. A fourth-grade class needs 5 leaves each day to feed its
2 caterpillars. How many leaves will they need each day
to feed 12 caterpillars?
4. If it takes 6 construction workers 4 days to complete a
job, how long will it take 2 construction workers to
complete the job? [Assume that each construction
worker can complete the entire job in the same amount
of time when working alone—they work at the same
rate.]
1. 2 = Ray
5 1000
3 = Crystal
5
1000
Ray: $400
Crystal: $600
2. No, a proportion can not be used as they are not
equal ratios
3≠ 4
5 6
3. 2 = 12
5 x
2x = 5 (12) x = 30 leaves
4. workers 6 = 2
days 4 x
x = 1 1/3 – Not Reasonable
A proportion cannot be used
Jeopardy
Percents
• Work through both sides of the handout with
your table
• What are the objectives being met with these
questions?
SECTION 1: THE MEANING OF PERCENT
In your own words, what is the meaning of
percent?
Realistic or Unrealistic?
1. George ate 130% of his breakfast. Unrealistic
2. Callie ran 2 miles per hour yesterday. Today she ran 50%
Realistic
faster.
3. Ming found a coat on sale. It was marked 100% off.
4. Matthew got a pay raise of 5%. Realistic
5. Nancy plays the state lottery. She figures she either wins
or loses, so she has a 50% chance of winning?Unrealistic
6. Mary went on a diet. She lost 80% of her previous weight.
Unrealistic
7. Tom’s boss gave him a pay cut of 50%. Then the boss gave
him a raise of 50%. Tom figures he now is making the same
Unrealistic
salary.
8. Carolyn baked a cake. She gave 40% to her sister, 30% to
her mother, and kept the other 30% for herself.
Realistic
What percent is the exact equivalent of 1/3?
Percent Can Make Sense
• This activity is designed to help students develop an
understanding of the relationships between
fractions and percent.
• The use of a particular visual representation that is
explicitly related to the meaning of per cent helps
not only to reinforce a conceptually oriented
understanding of percent but also suggests a mental
math method for the conversion of fractions to
percent. Now to examine a model of percent and its
connections to reasoning
What does ¼ mean?
1 out of every 4.
Use one of the hundreds tables to color one out
of every group of four squares
No Calculators!
• Use the other hundreds tables to represent the
following fractions:
1/5,
1/2,
3/4,
1/8
1/8 = ?
12 and 4 left over?
These 4 are
remaining.
What do we
do with
them?
12 4/8 = 12.5%
Still No Calculators
Now try:
1/9,
1/12,
1/3
1/9
11 squares shaded,
1 left over out of
the 9.
11 1/9 %
Examples
• Change 125% to a fraction.
125% means
125
, which simplifies to
100
• Change 0.005 to a percent.
0 . 005 =
5
1, 000
=
0 .5
100
, or 0 . 5 % .
5
4
Convert to Percents
Write each of the following as a percent.
1. 0.03
3%
2. 0.3
33.3%
3. 1.2
120%
4.0.00042
0.042%
5. 1
100%
6.
60%
7.
66.6%
8.
Convert to Decimals
Write each of the following percents as a decimal.
1. 5%
0.05
2. 6.3%
0.063
3. 100% 1
4. 250%
2.5
5.
6.
0.3
0.006
SECTION 3: CONVERTING BETWEEN
FRACTIONS-, PERCENTS, AND
DECIMALS—STUDENT RESOURCES
http://www.khanacademy.org/search?page_searc
h_query=fractions+decimals+percents
Ratio and Proportion Questions
• Questions come from Questioning book
discussed last session
Georgia lessons
• Look through the lessons/activities from the
state of Georgia
Feedback
• Please complete the ½ sheet