Primary PoW Packet
Up, Up, and Away
http://mathforum.org/pows/
Welcome
Welcome!
This packet contains a copy of the problem, the “answer check,” sample solutions and some teaching
suggestions. This is Library Problem #3791. The text of the problem is included below. A print-friendly
version is available using the “Print” link from the blue-shaded box on the problem page.
Standards
In Up, Up, and Away students are asked to find out how many of the balloons are yellow. If your state
has adopted the Common Core State Standards, this alignment might be helpful:
Grade K: Operations & Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking from.
Grade 1: Operations & Algebraic Thinking
Use addition and subtraction within 20 to solve word problems involving situations of adding to,
taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g.,
by using objects, drawings, and equations with a symbol for the unknown number to represent
the problem.
Grade 2: Operations & Algebraic Thinking
Represent and solve problems involving addition and subtraction.
Mathematical Practices
1.
2.
3.
4.
7.
The Problem
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Look for and make use of structure.
Up, Up, and Away!
A circus clown has 16 balloons. One half of the balloons are red. Half of the
remaining balloons are blue. The rest are yellow.
How many balloons are yellow? How do you know?
Extra: What fraction of the balloons is blue? What fraction of the balloons is
yellow?
Answer Check
After students submit their solution, they can choose to “check” their work by looking at the answer
that we provide. Along with the answer itself (which never explains how to actually get the answer) we
provide hints and tips for those whose answer doesn’t agree with ours, as well as for those whose
answer does. You might use these as prompts in the classroom to help students who are stuck and
also to encourage those who are correct to improve their explanation.
Four of the balloons are yellow.
If your answer does not match ours, did you
• draw a picture?
• use red, yellow, and blue counters to act it out?
• talk in your group?
If your answer does match ours, did you
• explain?
• write a number sentence?
• help anyone in your group?
Our Solutions
Method 1: Make a Diagram
We noticed:
There are 16 balloons.
Half of the 16 balloons are red.
Half of the balloons that are not red are blue.
The balloons that are not red or blue are yellow.
We made a diagram and labeled the colors:
We could see that 4 balloons are yellow!
Method 2: Use Manipulatives
We used counters to think about the balloons. We knew there were 16 balloons:
If half of them are red then we have:
If half of the balloons that aren’t red, are blue we have:
And so those last four are yellow:
Method 3: Draw a Picture
We drew a picture of the balloons.
We colored half of them red.
We colored half of the rest blue.
We colored what was left yellow. 4 of the balloons are yellow.
Method 4: Write a Number Sentence
There are 16 balloons. Some of the balloons are red, some are blue and some are yellow. Half of the
balloons are red and half of 16 is 8:
© 2011 Drexel University
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? + ? = 16
8 + 8 = 16
We know half of the remaining balloons are blue:
?+?=8
4+4=8
There are 4 blue balloons and 4 yellow balloons.
Extra: 4 out of the 16 balloons are blue and 4 out of the 16 balloons are yellow. 4/16 = 1/4
Method 5: Use Algebra (We would not expect this strategy from primary students but if older students
try this problem they might be encouraged to think about it algebraically.)
Let
r = the number of red balloons
b = the number of blue balloons
y = the number of yellow balloons
Then, r + b + y = 16
I also know that:
1
)16 = r which means that 8 = r
2
1
( )16 = b + y which means that 8 = b + y
2
1
1
( )(b + y) = b which means that ( )(8) = b which means that 4 = b
2
2
(
So,
8 + 4 + y = 16
12 + y = 16
y=4
There are 4 yellow balloons.
Teaching
Suggestions
This problem is one that was discussed during an online course offered to a group of primary level
teachers from Montgomery County, Maryland. Here are some thoughts that were shared in the course
about this particular problem:
Terrie’s Thoughts
The problem I chose to do in class this week was problem #3791. It’s called Up, Up, and Away! I
chose this problem because it focuses on fractions of a group and that is what we have been focusing
on in class. This problem had a clown with 16 balloons and 2nd graders were to come up with how
many were red, yellow, and blue knowing that 1/2 of the balloons were red, 1/2 of the remaining
balloons were blue, and the rest were yellow.
I went through a 5-day process to complete the teaching of this problem: reading the problem, notice
and wonder, brainstorm strategies to solve, computation using different strategies, response from
teacher, review and reflect on strategy and explanation. Out of a class of 18, 3 students were able to
explain and accurately compute the solution to the problem. Most students could show 1/2 of 16 or
the red balloons, but had a difficult time understanding how to come up with a strategy for the
remaining balloons: yellow and blue. Instead of having my students submit online I have handwritten
what I notice and what I wonder on their written submissions. After student discussions and reviewing
written responses from me, most of my 2nd graders were able to explain their solutions using words
and numbers. It will be interesting to see if they are able to apply these strategies to a different
situation.
We hope this information is useful in helping you make the most of the Primary Problems of the Week
in the Library. Please let me know if you have ideas for making them more useful.
~ Suzanne
© 2011 Drexel University
<[email protected]>
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