Math Applying Problem Solving Strategies
Math APSS
The teacher:
• Poses a rich problem or task
• Asks probing question (not leading questions)
• Listens to students and groups
• Watches for different strategies and reasoning and misconceptions
• Selects students to share their strategies
• Allows students to struggle
• Conducts meaningful classroom discussions
The students:
• Spend time with problems
• Grapple with difficult problems
• Work with others to solve problems
• Discuss their thinking-strategies-confusions
• Listen and critique other students’ thinking-strategies-confusions
• Reflect on their thinking and revise if necessary-look for and try out different or more
efficient strategies
• Use math tools appropriately-give them up if not needed to solve a problem
• Make connections to other problems-topics-subjects
• Synthesize information to draw conclusions and complete tasks
• Use appropriate mathematical language and compute accurately
The product is the student thinking.
The answer is part of the process not the product. The product is the student thinking. Do not
stop with the answer. The classroom discussion is where the students are able to share their
thinking, listen to and critique other students’ thinking and demonstrate the practices.
Problem Solving Throughout the Math Block
At the beginning of a topic pose a problem and tell students that by the end of the topic they will
be able to answer that question.
Spiral concepts to develop critical areas of focus over time.
In Singapore they do not reteach concepts. They revisit topics at a higher level of complexity.
Students develop and strengthen their mathematical practices.
Students are given the opportunity to practice mathematics at a high but achievable level.
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Math Applying Problem Solving Strategies
Grade 3-Math APSS
Teacher Introduction
Math APSS is the first 20 minutes of your one hour block of math. To begin the first quarter you
will be using problems from the “Common multiplication and division situations” table found on
page 89 of the Common Core State Standards for Mathematics. The table is included in the
problem solving teacher introduction. The problems provided at the beginning of the Math
APSS document are meant to help you set up the procedures and routines for open-ended
problem solving in your classroom and to expose students to the different problem structures
required by the Common Core State Standards. It is important that students understand the
different problem situations. Your focus is student thinking and students sharing that thinking
with one another. After you have worked through the problems situations from the table, you
will choose other problems from problems provided or from other sources of rich problems.
The class discussion should center around what is known in each problem and what the student is
looking for in each problem. Students should gain an understanding of the different problem
situations possible in a multiplication or division problem. This will lead to a complete
understanding of multiplication and division problem situations. Students should be able to
recognize the number of groups, size of groups and product in these problems.
During the focus lessons you should introduce problems in which the wording is not as straight
forward as the wording in the problems given during the Math APSS. To help students
understand the structure of those problems, always bring them back to what they know (groups,
size of groups, total or product) and what the question is asking (groups, size of groups, total or
product).
In addition to the problem structures from the Common Core, more difficult problems will be
given during the Math APSS block. Students may work on a problem one day and the class
discussion will take place the second day. There may be some tasks that extend over several
days. Students must be given problems and tasks that take them time to solve so they will be
able to demonstrate the Mathematical Practices that includes persevering through a problem.
You will also use problems during the focus lesson and intervention blocks of your one hour
block of math. Open-ended problems are provided for each domain within the Curriculum
Guide.
The focus of any problem solving situation is the process students go through in order to solve a
problem. Phil Daro, a writer for the Common Core said that the answer is part of the process, not
the product. The product is the student thinking. Therefore, students must be given ample time
to discuss their thinking and listen to other students’ thinking. During this discussion time,
students are given the opportunity to develop proficiency in the Common Core Standards for
Mathematical Practice. This opportunity should not be limited to the Math APSS portion of your
math block.
When students listen to other student’s thinking and processing, they can and will adopt a
strategy if it makes sense to them. When students listen to the teacher’s strategy, they attempt to
adopt it even if it does not make sense. They stop their own struggle because they think the
“right” way is the teacher’s way.
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Math Applying Problem Solving Strategies
Grade 4-5-Math APSS
Teacher Introduction
Math APSS is the first 20 minutes of your one hour block of math. The problems provided at the
beginning of the Math APSS document are meant to help you set up the procedures and routines
for open-ended problem solving in your classroom. Your focus is student thinking and students
sharing that thinking with one another. After you have worked through these problems you will
choose other problems from the problems provided or from other sources of rich problems.
The class discussion should center around what is known in each problem, what the student is
looking for in each problem and their strategy for solving the problem. Students must justify
their thinking.
You will also use problems during the focus lesson and intervention blocks of your one hour
block of math. Open-ended problems are provided for each topic within the Curriculum Guide.
More difficult and time consuming problems will be given during the Math APSS block.
Students may work on a problem one day and the class discussion will take place the second day.
There may be some tasks that extend over several days. Students must be given problems and
tasks that take them time to solve so they will be able to demonstrate the Common Core
Standards for Mathematical Practice that includes persevering through a problem.
The focus of any problem solving situation is the process students go through in order to solve a
problem. Phil Daro, a writer for the Common Core said that the answer is part of the process, not
the product. The product is the student thinking. Therefore, students must be given ample time
to discuss their thinking and listen to other students’ thinking. During this discussion time,
students are given the opportunity to develop proficiency in the Common Core Standards for
Mathematical Practice. This opportunity should not be limited to the Math APSS portion of your
math block.
When students listen to other student’s thinking and processing, they can and will adopt a
strategy if it makes sense to them. When students listen to the teacher’s strategy, they attempt to
adopt it even if it does not make sense. They stop their own struggle because they think the
“right” way is the teacher’s way.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
3x6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Group Size Unknown
(“How many in each group?”
Division)
3 x ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
A x ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? x b = p, and p ÷ b = ?
Unknown Product
Equal
Groups
Arrays, 4
Area, 5
Compare
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
axb=?
Number of Groups Unknown (“How
many groups?” Division
? x 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
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Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
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Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
students used math tools to solve the problem, they should include a picture to represent how the
tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
•
•
•
•
•
•
Act it Out
Draw a Picture
Find a Pattern
Guess and Check
Make a List
Make a Table
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When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Standards for Mathematical Practice
The Mathematical Practices are student habits of mind that engage students in the processes of
problem solving, communicating mathematical ideas in meaningful discourse, reasoning and
justifying solutions, and creating and sharing multiple representations of mathematical concepts
and procedures. Experiences such as these will develop good thinkers so that they can persevere
through non routine problems. The 21st century learner is consistently questioning their own
thinking, taking risks and applying past knowledge to new situations.
In classrooms where the mathematical practices are evident, the teacher becomes the facilitator
of developing thinking skills. Teachers need to create and provide daily opportunities for
students to develop their mathematical practices so they can think, reason, and communicate
their mathematical understanding. Students need to be given opportunities to be creative,
collaborate with their peers, communicate, and develop critical thinking skills that are essential
to the success of developing their mathematical practices.
In the new Common Core State Standards, for both Mathematical Content and Standards for
Mathematical Practice, students are expected to have mastery by the end of their grade level.
Ongoing assessment to determine mastery and/or the need for intervention should occur
frequently. Included below are descriptions of behaviors observed in students using the eight
Mathematical Practices.
1. Make sense of problems and persevere in solving them.
Students solve problems in which solutions are not immediately apparent. Students solve
problems that they first need to struggle with themselves and then often work with other
students. When given real-world scenarios, first they figure out the right questions to ask, what
previous experiences they have had to help with the solution, what additional information they
need and then they determine an appropriate entry into solving the problem. Students explore
problems using objects or pictures and then learn to derive methods of solutions first before they
are formally taught the algorithms. Once they find a solution, they can use another method to
check their answer. This engagement of thinking leads to a deeper understanding of the
mathematics.
Lucas planted 22 corn plants and 14 tomato plants. Students determine what questions can be
answered or what could you figure out from that information about Lucas.
2. Reason abstractly and quantitatively.
Students understand abstract problems using manipulatives. They understand how to represent a
problem situation by using symbols and see connections between the number sentence and the
original problem. Students can interpret the answer and make sense of the units in problem
situations. For example, if students were determining how many cars are needed for a field trip
if they have 26 students in their class and each car can only hold 4 students. Students can
approach this problem in several different ways. They understand that 26 represents the number
of students in the class. There will be 6 cars with 4 children and they will need an additional car
to transport 2 more children. Students learn how to make sense of quantities and their
relationships in problem situations.
3. Construct viable arguments and critique the reason of others.
Students explain their solutions and the thinking that led them to that solution. They can listen
and ask questions about others’ thinking so that they can refine their own thinking. Students
determine whether solutions are correct, incorrect, or partially correct. Students draw from
previous experiences and conjectures to help them defend their arguments or critique the
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reasoning of others. They defend their thinking by using counterexamples, manipulatives, and/or
pictures. If students were asked to show how many different ways they can make a 5 cube tower
using 1 blue cube and 4 white cubes, they are able to explain all the towers they made and how
they know they have all the different towers possible.
4. Model with mathematics
Students use different representations, such as pictures, drawings, tables, graphs, number
sentences, etc. to explain their thinking and solutions to problem situations. Through modeling,
students record their ideas and reflect on their solutions to determine whether the results make
sense or if they need to revise their model. Students use modeling to clarify, support, and extend
their thinking. Students understand that some models are more helpful than others to represent a
problem situation.
5. Use appropriate tools strategically.
Students select from a collection of tools ones that would be most helpful to use and explain how
the tool was used to result in the solution. Diagrams and arrays are used as tools to make sense
of what otherwise might be complicated steps in a procedure or algorithm to solve problems
(e.g., ten frames, five frames, dot arrays, open number lines, associative property, branching,
etc). These tools help build mental images that are powerful in understanding the mathematics.
6. Attend to precision.
Students communicate accurately their mathematical thinking through discussions and written
explanations. As students share their ideas, their explanations have clearer descriptions and are
more refined and precise in their selection of mathematical vocabulary words to describe their
solutions, patterns, and representations.
7. Look for and make use of structure.
Students make connections from prior experiences to build conceptual understanding. Students
use this type of reasoning to derive rules that make sense to them. For example, when adding
8 + 9, they see both numbers are close to ten, so they add 10 + 10 = 20 then take away 1
(because 9 is 1 away from 10) = 19 and take 2 more away (because 8 is 2 away from 10) = 17.
8. Look for and express regularity in repeated reasoning.
Students use connections to solve new and non routine problems. They recognize structures or
patterns that can be applied to different situations. Students explain the reasonableness of
solutions. They see connections, use efficient strategies, and draw conclusions in problem
situations.
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Math Applying Problem Solving Strategies
Grade 3
Problem Type: Unknown Product-Equal Groups
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Geri gave each of her 5 friends 4 pieces of candy. How much candy did the friends get altogether? If she
kept the same amount as her friends, how much candy did she have before she gave it out?
• Kim learned how to make friendship bracelets in her art class. She needed 6 lengths of yarn, each piece of
yarn needed to be 9 inches long. How much yarn did Kim need for her bracelet? She also wanted to make a
bracelet for her best friend in Asia. How much yarn would she need for both bracelets?
• Maggie was planting tulips in her garden. She planted 4 rows with 6 tulips in each row. How many tulips
are there? If she planted 4 more rows with the same number of tulips in each row, how many tulips would
Maggie have planted?
Possible Strategies for Unknown Product-Equal Groups Problems
Students may use these or other strategies to solve this type of problem/task.
•
•
•
•
Memorized fact (i.e. 5 x 4 = 20)
Repeated addition (i.e.4 + 4 + 4 + 4 + 4 = 20)
Drawing a picture of the equal groups
Using math tools to model the equal groups
Classroom Discussion for Unknown Product-Equal Groups Problems
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated addition pose a question such as “How did you know to add 6 nine times?”
• If no students use repeated addition pose the question: “Could you use addition to solve this problem?”
• Explain what each number represents in the problem.
• Which number represents the number of groups in the problem?
• Why should another student use your strategy?
• How do you know your answer is reasonable? Accurate?
• Is there another way to solve this problem?
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Math Applying Problem Solving Strategies
Grade 3
Problem Type: Group Size Unknown-Equal Groups
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample Problems:
•
•
•
Shantel had 28 pieces of candy she shared her candy equally with herself and 6 friends. How much candy
did each friend get? How much candy did Shantel get?
Jamal had 72 inches of string. He cut it into 8 equal lengths to braid together to make a chain for his house
key. How long was each length of string he used for his chain? How much string would Jamal need to
make the same chain for his sister and brother? How much string would he need for all three chains?
Mrs. Jones has 27 students in her class. She wants to arrange the desks into 3 equal rows. How many desks
will there be in each row? If a student were removed from her class, how would this affect the desk
arrangement?
Possible Strategies for Group Size Unknown-Equal Groups Problems
Students may use these or other strategies to solve this type of problem:
•
•
•
•
•
Memorized fact (i.e.28 ÷ 7 = 4)
1
2
3
4
Repeated subtraction (i.e. 28 – 7 = 21; 21 – 7= 14; 14 – 7 = 7; 7 – 7 = 0; I took 7 away 4 times so they each
get 4 pieces.)
Drawing a picture
Use math tools to model the problem
Using the inverse operation (i.e. They each get 4 pieces of candy because I know that 7 times 4 is 28.)
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated subtraction pose a question such as “How did you know to subtract 7 four times?”
• If students do not use repeated subtraction pose the question: Could you use subtraction to solve this
problem?
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Math Applying Problem Solving Strategies
•
•
•
•
•
•
•
What represents the number of groups in this problem?
Which number represents the number in each group (size of the group) represent in the problem?
What does each number represent in this problem?
What happens if there are an odd number of items to be shared?
How do you know your answer is reasonable? Accurate?
Why should another student use your strategy?
Is there another way to solve this problem?
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Number of Groups Unknown-Equal Groups
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Harriet had 32 pieces of candy. She divided the candy with some friends. If Harriet and her friends each
got 4 pieces of candy, with how many friends did Harriet share her candy?
• Grover had 45 inches of rope. He was practicing making knots for a Boy Scout badge. He cut his rope into
pieces 9 inches long. He could make one knot with each piece of rope. How may knots could he make?
After he learned the knots, Grover cut the same number and size of pieces of rope to help his friend earn his
Knot Badge. How much rope did Grover use altogether? How many knots did he and his friend tie?
• Mr. Baker is taking his 30 third-graders to the zoo. Six students can ride in each van. If the maximum
number of students travels in each van, how many vans will Mr. Baker need to take to the zoo? If a new
student is added to the class, would this change the number of vans that Mr. Baker needs to order? What
would you suggest for the number of students in each van? Justify your answer.
Possible Strategies for Number of Groups Unknown-Equal Groups Problems
Students may use these or other strategies to solve this type of problem.
•
•
•
•
•
Memorized fact (i.e. 32 ÷ 4)
Repeated subtraction (
1
2
3
4
5
6
7
8
32 – 4 = 28; 28 – 4 = 24; 24 – 4 = 20; 20 – 4 = 16; 16– 4 =12; 12– 4 =8; 8– 4 =4; 4– 4 =0
o I took 4 away 8 times so there were 8 people)
Drawing a picture
Using math tools to model fair sharing
Using the inverse operation (i.e. There are 8 people including Harriet because I know that 4 times 8 is 32.
Harriet has 7 friends.)
Classroom Discussion for Number of Groups Unknown-Equal Groups Problems
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated subtraction pose a question such as “How did you know to subtract 4 six times?”
Math APSS
14 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
•
•
•
•
•
•
•
•
If students do not use repeated subtraction pose the question: “Could you use subtraction to solve this
problem?”
Which number represents the number of groups in the problem?
Which number represents the number in each group in the problem?
What does each number represent in the problem?
How is this type of problem different from a Group Size Unknown problem?
How do you know your answer is reasonable? Accurate?
Is your strategy efficient?
Why should another student use your strategy to solve this type of problem?
Math APSS
15 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Unknown Product- Arrays/Area
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Kim and her mother are building a holder for her 36 Nintendo DSi® games. Kim wants to build a holder
with 4 columns and 8 rows. Will this hold all of Kim’s games? Why or Why not? What are all of the
possible holders they could build for 36 games that have an equal number of columns and an equal number
of rows?
• What is the area of a 6 cm by 12 cm rectangle?
• There are 8 rows of chairs in the auditorium with 6 chairs in each row. How many chairs are there in the
auditorium?
Possible Strategies for Unknown Product- Arrays/Area
Students may use this or other strategies to solve this problem/task.
•
•
•
•
•
Guess and Check (i.e. 36 ÷ 4 = 9, 36 ÷ 8 = 4, remainder 4 OR 4 x ? = 36, 8 x 4 = 32, with 4 games that will
not fit. A holder with 4 columns and 8 rows will not hold all of the games because there
will be 4 games that would not fit in the holder)
Memorized facts (i.e. 1 x 36, 36 x1, 2 x 18, 18 x 2, 3 x 12, 12 x 3, 4 x 9, 9x 4, 6 x 6
Model of arrays (i.e. for 36)
Repeated addition
Subtraction strategies
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Is that a reasonable answer? Explain.
• What do each of the numbers in the problem represent?
Math APSS
16 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Group Size Unknown- Arrays/Area
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Mohamed has 42 CDs in his music collection. His mom bought a CD stand which has 7 shelves. If
Mohamed put the same number of CDs on each shelf, how many CDs would go on each shelf?
• Michelle’s backyard has an area of 120 square feet. If the length of the yard is 15 feet long, how long is the
width of Michelle’s yard?
• Lin bought a box of 60 chocolates. When he opened the box he noticed the chocolates were arranged into 5
equal rows. How many chocolates were in each row?
Possible Strategies for Group Size Unknown- Arrays/Area
Students may use this or other strategies to solve this problem/task.
•
•
Memorized fact (i.e. 42 ÷ 7 = 6)
Model of an array (i.e. 7 shelves with 6 cds on each shelf)
Some students may still use the repeated addition or subtraction strategies in place of multiplication and division.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Is that a reasonable answer? Explain.
• What do each of the numbers in the problem represent?
Math APSS
17 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Number of Groups Unknown- Arrays/Area
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• There were 63 marching band members marching in the parade. The members were arranged into equal
rows of 7 members. How many rows did they have in the parade?
• Jeff bought a crate for his new puppy. The area of the bottom crate is 12 square feet. If one dimension of
the crate is 3 feet long, how long is the other dimension?
• A rectangle has an area of 32 square centimeters. If one dimension of the rectangle is 8 centimeters long,
how long is the other dimension?
Possible Strategies for Number of Groups Unknown- Arrays/Area
Students may use this or other strategies to solve this problem/task.
•
•
Memorized fact (i.e. 63 ÷ 7 = 9)
Model of an array (i.e. 9 rows with 7 band members)
Some students may still use the repeated addition or subtraction strategies in place of multiplication and division.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Is that a reasonable answer? Explain.
• What do each of the numbers in the problem represent?
Math APSS
18 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Unknown Product- Compare
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Kija was cutting string for a project in art class. She had one piece of string that measured 7 cm in length.
She needed a piece of string that was 4 times longer than the 7 cm long piece of string. How long was the
piece of string she needed for her project?
• Oliver has 8 stickers in his journal. Alejandre has 6 times as many stickers as Oliver. How many stickers
does Alejandre have?
Possible Strategies for Unknown Product- Compare
Students may use this or other strategies to solve this problem/task.
Memorized fact (i.e. 4 × 7 = 28 cm. long)
Model 4 equal groups of 7
Repeated addition (i.e. 7 + 7 + 7 + 7 = 28)
7 cm
7 cm
7 cm
7 cm
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated addition pose the question: How did you know to add 8 six times?
• If no students use repeated addition pose the question: Could you use addition to solve this problem?
• Which number represents the number of groups?
• Which number represents the size of each group?
• What does the answer represent?
• After students show their solutions, select two different ways to represent the problem and ask students to
compare the two strategies.
• If no one represents the problem with a picture of 4 separate pieces of string, show that representation and
ask the students if it is an appropriate representation for the problem. Explain.
Math APSS
19 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Group Size Unknown- Compare
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Violet has 12 stickers in her journal. That is 4 times as many as the number of stickers that Stanley has.
How many stickers does Stanley have?
• Kyle was cutting string for a project in art class. The piece of string is 27 cm. long. That is three times
longer than the original piece of string. How long was Kyle’s original piece of string?
Possible Strategies for Group Size Unknown- Compare
Students may use this or other strategies to solve this problem/task.
•
4 × ? = 12 Stanley has 3 stickers because 4 times 3 equals 12.
Stanley has 3 stickers because I put 12 stickers
into 4 groups and got 3 in each group.
•
•
27 ÷ 3 = 9 cm. long
Students may show 27 fair shared into 3 equal groups of 9 cm.
?
27 cm
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated subtraction pose the question: how did you know to subtract 4 three times?
• What number represents the number of groups in this problem?
• Which number represents the size of the group in the problem?
Math APSS
20 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
•
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What does 12 represent in this problem?
What do we know in the problem?
What are we trying to find out in the problem?
Why would someone divide 12 stickers into 4 groups?
What would the problem say if we knew how many stickers Stanley had and wanted to know how many
stickers Violet had?
Math APSS
21 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 3
Problem Type: Number of Groups Unknown-Compare
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sample problems:
• Bailey has 24 stickers and Janet has 8 stickers. How many times as many stickers does Bailey have as
Janet?
• Korbett was cutting string for a project in art class. His first piece of string was 8 cm. long. The second
piece of string was 40 cm. long. How many times as long was the second piece of string than the first?
Possible Strategies for Number of Groups Unknown-Compare
Students may use this or other strategies to solve this problem/task.
•
24 ÷ 8 = 3
Bailey has 3 times more stickers than Janet.
1 2
3
8 + 8 = 16; 16 + 8 = 24 Bailey has 3 times as many stickers as Janet.
8 cm
24 cm
16 cm
? × 8= 40 cm. long
40 ÷ 8 = 5
2
3
1
•
32 cm
4
40 cm
5
40 cm
8 + 8 = 16; 16 + 8 = 24; 24 + 8 = 32; 32 + 8 = 40
I kept adding 8 until I got to 40. I added 8 five times so the answer is 5.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If students use repeated subtraction pose the question: how did you know to subtract 8 three times? Why
does this not really represent what is happening in the problem?
• If no one used repeated addition, ask students if they could use repeated addition to solve the problem.
Explain.
Math APSS
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
•
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Which number represents the number of groups in the problem? What does that number represent? (number
of times more)
What does 24 represent in the problem?
What number represents the size of the group? What does that number represent? (stickers)
Math APSS
23 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Tyree has a 48 inch length of ribbon. She is going to put it around 2 identical (congruent) rectangular picture
frames. What might be the dimensions of the frames, if she uses all of the ribbon? Could the frames be square?
Prove your claim.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
That would be 24 inches of ribbon for each frame, so the frames could be 5 inches by 7 inches.
5 + 5 + 7 + 7 = 24
•
They could be 4 inches by 8 inches because 4 + 4 = 8 and 8 + 8 = 16 and 16 + 8 = 24 and 24 + 24 = 48
inches.
•
Yes, because it could have 6 inch sides. 4 × 6 = 24 inches
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problem mean?
• What do we know in the problem?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• If the frames have 6 inch sides, would it still be a rectangle? Explain. (yes, a square is a special rectangle
with congruent side lengths)
• Could we find all of the possible answers if the side lengths are all whole numbers (no fractions)? What are
all of the possible whole number lengths?
Math APSS
24 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
If you earned $42 a week doing chores and saved all of it, how much money would you have in a month? Six
months? A year? If you earned the money mowing lawns and you make $14 for each lawn you mow, how many
lawns would you mow in a week to earn $42?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
$42 × 4 = $168 month (assuming 4 week months); $168 × 6 = $1,008 six months; $168 × 12 = $2,016 year
•
$42 × 52(weeks in a year) = $2,184 That is $168 more than the year when you multiply every month by the 4 week
per month total. $168 ÷ $42 = 4 So there must be 4 months that have 5 weeks in them.
•
$42 ÷ $14 = 3 lawns
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students should be
able to justify their own thinking as well as the thinking of others. This can be done through comparing strategies,
arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy to share
in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problem mean?
• What do we know in order to solve the problem?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• What does the $14 represent in the problem?
• If you earn the same amount every day and worked every day of the week, how much would you earn in one day?
What might you do to earn that amount?
Math APSS
25 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
The product of two numbers is greater than 100 but less than 200? One of the numbers is 20. What could the
other number be? If one number was 9, what could the other number be?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
Any number greater than 5 but less than 10 because 5 × 20 = 100 and 10 × 20 = 200
6, 7, 8, or 9 because 5 × 20 = 100 and 10 × 20 = 200
•
•
9 × 11 = 99 so it has to be at least 12 and 9 × 22 = 198 so it has to be less than 23.
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, or 22
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problem mean?
• What do we know in the problem?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• If no one includes all numbers between 5 and 10, ask “Are there any numbers between 5 and 6?”
• If we include fractions and decimals, how could we find the range for the second factor if 20 is one of the
factors? If 9 is one of the factors?
Math APSS
26 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Glidden paint says that a gallon of paint will cover 375 square feet to 400 square feet. I want to paint a square
room that is 12 feet long and 8 feet to the ceiling. How many gallons of paint will I need to buy to apply 2 coats of
paint? Justify your response.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
•
8 × 12 = 96 square feet so you only need 1 gallon for one coat and 2 gallons for 2 coats if it covers 400
square feet. 96 is less than 100 so 4 times 96 is less than 400 (4 × 4 = 16 square feet less or 400 – 16 = 384
sq.ft.) If it only covers 375 square feet it should still be okay because there would be a door and probably a
window. A door is about 3 ft. by 6 ft. That would be 18 sq. ft. off for the door. 384 – 18 = 366 sq. ft. to
cover.
4 × 96 = 384 sq. ft.; 384 – 18 = 366 sq. ft. and 366 is less than 375.
8 × 12 = 96 square feet and 4 × 96 = 384 square feet so it would probably take 2 gallons because if it only
covered 375 square feet for the first gallon the second coat wouldn’t take as much. Also, there would be a
door and maybe windows to take away.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problems mean?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• What strategy do you use to help you make sure you answer all of the parts of a problem?
• Did you need an exact answer in this problem? Explain.
• What if the ceilings were 9 feet high?
Math APSS
27 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
A baseball diamond is a square. Therefore, if a runner runs in a straight line from home plate to first, second and
third bases then back home, they have run the perimeter of the square. It is 90 feet from home plate to first base.
What is the perimeter of the baseball diamond? What is the area of the infield?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
The perimeter is 360 feet. 4 × 90 = 360 feet
90 + 90 + 90 + 90 = 360 feet
•
90 ft. × 90 ft. = 8,100 sq. ft. so the area of the infield is 8,100 sq. ft.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problems mean?
• What do we know in each problem?
• What do we need to know in each problem?
• What strategy do you use to help you make sure you answer all of the parts of a problem?
• If the pitcher covers one-third of the area of the infield, how much area would he cover?
Math APSS
28 of 113
Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
There were 3 large windows along one wall in Ms. Evans’ classroom. Each window had 4 large window sections.
Each window section had 4 panes. How many window panes were in Ms. Evans’ room? Mrs. Hatfield’s
classroom had two walls with windows just like the windows in Ms. Evans’ room. The south wall was just like the
wall of windows in Ms. Evans’ classroom. The west wall only had 2 large windows. How many window panes
were in Mrs. Hatfield’s classroom?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
3 × 4 × 4 = 48 window panes
Students may draw a picture (the picture may not look like a traditional window, however students must be
able to defend their drawing by referring to the problem). Make sure they record the answers.
1
1
2
3
4
sections of windows
1
2
3
Large windows
2
3
4
Window panes
There are 48 panes altogether, because 3 large windows times 4 sections on each window times 4 panes on
each section is 48 panes.
Mrs. Hatfield’s room
• There are 48 panes on the south wall because it is just like the windows in Ms. Evans’ room.
The west wall has 32 panes, because there are 2 windows, with 4 sections on each window and 4 panes on
each section.
2 × 4 × 4 = 32 window panes
That makes 48 + 32 = 80 window panes in Mrs. Hatfield’s room.
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problem mean?
• What do we know in the problem?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• Why might Mrs. Hatfield’s room have more windows?
• How do the windows in Ms. Evans’ room compare to the windows in our room?
• If each window section was separated into 6 window panes each, how many panes would there be in Ms.
Evans’ room? Mrs. Hatfield’s room?
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Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Ms. Quint’s class has 4 windows. Each window has the same number of panes. Her classroom has 32 window
panes. How many panes are on each window? Mr. Burke’s classroom has 6 windows just like the windows in Ms.
Quint’s room. How many window panes are in Mr. Burke’s room? What is the same about each room?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
4 × ? = 32 The answer is 8 because I know that 4 times 8 equals 32.
Students may draw a picture. Make sure they record the answers.
There are 8 panes on each window, because I kept dividing them until I had 32 window panes altogether.
Mr. Burke’s room
• There are 48 panes in Mr. Burke’s room because 6 × 8 = 48
• He has 2 more windows than Ms. Quint and since each window is the same there are 16 more window
panes (8 + 8). 32 + 16 = 48
• The number of panes (8) on each window is the same in Ms. Quint’s room and Mr. Burke’s room.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
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•
•
•
•
•
•
•
•
•
What does each number in the problem mean?
What do we know in the problem?
What do we need to know in the problem?
How did you know you answered every part of the problem?
Why might Mr. Burke’s room have more windows?
How do the windows in Ms. Quint’s room compare to the windows in our room?
If each window section was separated into 6 window panes each, how many panes would there be in Ms.
Quint’s room? Mr. Burke’s room?
If Mr. Burke’s room had 2 times as many windows as Ms. Quint’s room, how many windows and window
panes would there be in Mr. Burke’s room?
Compare how you determined the number of panes in Ms. Quint’s room as compared to how you
determined the number of panes in Mr. Burke’s room in the problem.
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Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
•
Miss Calhoun has some windows in her room. Each window is divided into 6 window panes. There are 72
window panes altogether. How many windows are in Miss Calhoun’s room? Ms. Jackson has
as many
windows as Miss Calhoun. How many windows and window panes are in Ms. Jackson’s classroom? Miss
Calhoun has 3 times as many windows as Mr. Brown has in his room. How many windows and window
panes are in Mr. Brown’s classroom?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
1
72 ÷ 6 = ? The answer is 12 because I know that 72 divided by 6 equals 12.
Students may draw a picture. Make sure they record the answers.
2
3
4
5
6
7
8
9
10
11
12
There are 12 windows in Miss Calhoun’s room, because I kept adding windows with 6 panes on each
window until I got 72.
•
10 would be 60 panes and 2 more windows would be 12 more panes to make 72 altogether, so there are
10 + 2 = 12 windows in Miss Calhoun’s room
Ms. Jackson’s room
• There are 6 windows and 36 window panes in Ms. Jackson’s room. Half of 72 is 36. Half of 12 is 6.
• There are 4 windows in Mr. Brown’s classroom. I know because 3 groups of 4 equals 12. There are 24
panes in his room because 4 × 6 = 24.
• I drew a model. So there are 4 windows in Mr. Brown’s classroom and 24 panes because 4 × 6 = 24.
Mr. Brown
Miss Calhoun
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problem mean?
• What do we know in the problem?
• What do we need to know in the problem?
• How did you know you answered every part of the problem?
• Which number represents the number of groups in the problem?
• What does 6 represent in the problem?
• What does 72 represent in the problem?
• If no one used repeated addition, ask if they could use repeated addition to find the answer and how.
• How do the windows in Ms. Quint’s room compare to the windows in our room?
• Miss Calhoun bought curtains to cover half of the panes on each window. How many panes are uncovered?
What could one of her windows look like after the curtain is hung?
• If each window is 60 inches high, about how long would the curtains Miss Calhoun bought be? Explain.
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Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Read the problems we did the last three days (only the first part of each question is here to compare). Write any
comparison statements about the three problems.
Be prepared to discuss their similarities and differences. You may look back where you solved the problems to
help you discuss them.
1. There were 3 large windows along one wall in Ms. Evans’ classroom. Each window had 4 large window
sections. Each window section had 4 panes. How many window panes were in Ms. Evans’ room?
2. Ms. Quint’s class has 4 windows. Each window has the same number of panes. Her classroom has 32
window panes. How many panes are on each window?
3. Miss Calhoun has some windows in her room. Each window is divided into 6 window panes. There are 72
window panes altogether. How many windows are in Miss Calhoun’s room?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Possible comparison statements
• Each problem is about windows and window panes.
• The first problem has large windows, sections of windows and window panes but the other two only talk
about windows and window panes.
• The first one you have to find out how many window panes there are in each window and in all of the
windows altogether. You know how many panes are in each section and how many sections are in each
window.
• The second one you know how many windows and how many panes there are altogether. You have to find
out how many panes are on each window.
• The third one you know how many panes there are altogether and how many are on each window. You
have to find out how many windows there are.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
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•
•
•
•
•
•
•
•
What have you already learned which helped you in solving this problem?
What does each number in the problems mean?
What do we know in each problem?
What do we need to know in each problem?
How did you know you answered every part of the problem?
What is the difference in how you determined each missing number?
Which problem was easiest to solve? Hardest? Why?
What strategy do you use to help you make sure you answer all of the parts of a problem?
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Grade 4
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Mr. Rodger’s classroom has 6 tables. There are 6 chairs at each table. How many chair legs are there? Every
other chair is blue. There are also grey chairs. How many blue chairs are there? The students wanted to arrange
the room so that there are no grey chairs at a table with blue chairs. Could they do this? Explain.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
•
6 × 6 × 4 = 36 × 4; 36 × 4 = 144 there are 144 chair legs
6 × (6 × 4) = 6 × 24 = 144 chair legs
I drew 6 tables and put 6 chairs at each table. I multiplied each chair by 4, because there are 4 legs on each
chair. Then I added 24 six times.
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
24 + 24 + 24 + 24 + 24 + 24 = 144 chair legs.
•
•
•
Half of the chairs are blue so there are 18 blue chairs.
36 ÷ 2 = 18 blue chairs
I colored in every other chair on my picture and got 18 blue chairs.
6 × 4 = 24
•
•
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
6 × 4 = 24
Yes they would have 3 tables with blue chairs and 3 tables with grey chairs.
I colored in all of the chairs at one table and no chairs at another table to show that they could have all blue
and all grey chairs at every other table.
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What does each number in the problems mean?
• What do we know in each problem?
• What do we need to know in each problem?
• What strategy do you use to help you make sure you answer all of the parts of a problem?
• Could the students get their wish to arrange the room so that there are no grey chairs at a table with blue
chairs, no matter how many chairs they had in the room? Can you prove a number of chairs that this is not
possible? Or can you prove that it could always be possible? Explain. If you assume that they always
have to have 6 chairs at a table. Does that change your answer? Explain.
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Grade 5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Mr. Hunter is ordering school supplies for Lee Elementary School. There are 300 students that attend Lee
Elementary School. Mr. Hunter is ordering enough pencils for every student to have three pencils. Pencils come in
boxes of 50. How many total boxes of pencils will Mr. Hunter need to order? If the pencils cost $0.15 each, how
much would a box of pencils cost?
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
• 300 × 3 = 900 pencils; 900 ÷ 50 = 18 boxes
• 50 × $0.10 = $5.00 and 50 × $0.05 = $2.50; $5.00 + $2.50 = $7.50; I knew that 50 times 10 cent equaled $5
so 50 times a nickel would be half of that which is $2.50. Then I added them together to get the cost of a
box.
• 50 × $0.15 = $7.50 a box of pencils
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• What does the number 50 represent?
• How much would Mr. Hunter spend on all of the pencils he bought?
• Suppose Mr. Hunter wanted to buy pencils with a picture of the school mascot. What might the pencils
cost? Explain.
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Grade 5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Chimer and his Dad were building a fence around their 25 ft. x 40 ft. yard. If fencing comes in 5 ft. sections, how
many sections of fencing does Chimer and his Dad need to completely enclose their yard?
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
25 + 25 = 50 and 40 + 40 = 80; 50 + 80 = 130 feet of fence. 130 ÷ 5 = 26 sections
25 + 40 = 65 and 65 × 2 = 130 feet perimeter; 130 ÷ 5 = 26 sections
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• If the fence cost $25 for each section, how much would Chimer and his dad have to pay for the fence?
• What about a door/gate?
• If a gate is 3 feet wide, what will Chimer and his dad need to buy to enclose the yard including a gate?
(They will still need to buy 26 sections of fence and a gate. They will only use 2 feet from one of the
sections of fence.)
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Grade 5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Brad was keeping track of the total miles he ran each week. On every Monday of the month Brad ran 3 miles. On
every Wednesday he ran twice as much as he ran on Monday. Every Friday he ran 5 miles. How many miles did
Brad run in a week? A month?
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
14 miles per week and 56 miles in one month, if there are 4 weeks in the month.
3 + (2 × 3) + 5 = 14 miles in one week; 14 × 4 = 56 miles in a month or 14 × 5 = 70 miles in a month if
there are 5 weeks in that month
•
3 + 6 + 5 = 14 miles in a week; 14 + 14 + 14 + 14 = 56 miles in a month
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• Is there a month that Brad would run farther than 56 miles? Explain
• Is there a month that Brad would run fewer miles than 56 miles, if he ran the same each week? Explain.
• How many miles would Brad run in a year if he stayed with this same pattern? Justify your answer.
• If Brad wanted to increase his total miles per week to 18 miles, how might he do this?
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Grade 5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Jerome is baking cookies for summer camp. Jerome has decided to bake 575 cookies for each group of campers. If
there are 15 groups of campers, how many total cookies will Jerome have to bake? How many campers might be
in a group? Justify your answer.
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
575 × 15 = 8625 cookies
575 × 10 = 5750; 575 × 5 = 2875; 5750 + 2875 = 8625 cookies
•
•
About 300 because that would be about 2 cookies for each person.
About 400 because most people would only eat one cookie but some would eat 2 cookies.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• What does the number 575 represent in the problem?
• What does the number 15 represent in the problem?
• If they packaged the cookies in packs of two, could they bake 575 cookies for each group? Explain your
answer?
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Grade 5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
A farmer grows 196 pounds of carrots. He sells them to a grocery store that places them in either 2 pound
containers or 5 pound containers. If the grocer uses the same number of 2 and 5 pound containers how many of
each did he use?
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
196 ÷ (2 + 5) = 28
2 × 28 = 56 and 5 × 28 = 140; 56 + 140 = 196
I knew there was the same number of 2 pound and 5 pound packages, so I added them together and divided
196 by the total which was 7 and got 28 packages of each size. I checked it by multiplying each number
times 28 and adding the products together.
•
I used guess and check. First I tried 20 of each and only got 140 pounds. So I tried 30 packages of each
and got 210. Because 210 was closer, I tried 27 and got 181. I knew it was between 27 and 30 so I tried 28
and it was right.
2 × 28 = 56 and 5 × 28 = 140; 56 + 140 = 196
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem? What do you need to find out? What is the question asking?
• How do you know you answered all of the parts of the question?
• If no one uses the distributive property show (28 × 2) + (28 × 5) and 28 × (2 + 5) and discuss the
relationship of the two problems. Why are they the same? Could that help them solve the original problem
more efficiently?
• If the store sells the 2 pound package for $1.50, how much do you think they should charge for the 5 pound
package? Justify your answer.
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Joleen’s class is going to plant a garden in the spring. They voted on 5 kinds of fruits and vegetables they want to
plant. They decided to plant tomatos, green beans, cucumbers, bell peppers and bib lettuce. Each row in their
garden will hold 8 tomato plants, 10 green bean plants, 6 cucumber plants, 12 bell pepper plants or 16 lettuce
plants. They will plant only one kind of plant in each row and they will fill each row they plant. They also
decided to make sure they have at least one row of each kind of plant. Could they have 53 total plants? If you
have 72 plants in the garden, how may rows would be in the garden? Explain your answers.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Could they have 53 total plants?
• No, they can not have 53 total plants. Referring to the given statement “they will plant only one kind of
plant in each row they plant and will fill each row they plant.” Students can add the number of plants that
each row can hold. (16 Lettuce plants + 12 Bell Pepper Plants + 10 Green Beans plants + 8 Tomato Plants +
6 Cucumber Plants = 52 total plants).
•
No because 53 is an odd number and all of the rows have an even number of plants. You cannot add all
even numbers and get an odd number.
•
Students may also create a list to show the rows needed.
Row 1 – 8 tomato plants
Row 2 – 10 green beans plants
Row 3 – 12 bell pepper plants
Row 4 – 16 lettuce bibs
Row 5 – 6 cucumber plants
Total Plants – 52 plants
If you have 72 plants in the garden, how may rows would be in the garden? Explain your answers.
Answer may vary. Students can create different combinations with 72 plants.
•
A possible solution is:
Row 1 – 8 tomato plants
Row 2 – 10 green beans plants
Row 3 – 12 bell pepper plants
Row 4 – 16 lettuce bibs
Row 5 – 6 cucumber plants
Row 6 – 8 tomato plants
Row 7 – 12 bell pepper plants
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Total plants – 72 plants
It took 7 rows to plant 72 different plants. Using previous knowledge, student knew that one row of each kind of
plant totals 52 plants. 20 more plants are needed to make 72.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If they did not need to have at least one row of each vegetable, could they have a garden with 65 plants?
Explain.
• If they did not need to have at least one row of each plant, could they have a garden of 46 plants?
• What could be in that garden?
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Class Garden
Joleen’s class is going to plant a garden in the spring. They voted on 5 kinds of fruits and
vegetables they want to plant. They decided to plant tomatos, green beans, cucumbers, bell
peppers and bib lettuce. Each row in their garden will hold 8 tomato plants, 10 green bean
plants, 6 cucumber plants, 12 bell pepper plants or 16 lettuce plants. They will plant only one
kind of plant in each row and they will fill each row they plant. They also decided to make sure
they have at least one row of each kind of plant. Could they have 53 total plants? If you have 72
plants in a garden. how may rows would be in the garden? Explain your answers.
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Use the Hundreds Chart to answer the questions.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Possible solutions to questions are as follows:
What do all the numbers shaded with dots have in common?
All of the numbers are multiples of 7
What do you notice about all the numbers shaded with lines?
All of the numbers are multiples of 5.
What is the relationship between the numbers shaded in dark gray?
Both numbers are multiples of 5 and 7. 35 is half of 70.
Can you explain the shading this time?
The numbers that are shaded with lines are even numbers and multiples of 2. The number shaded with dots (35) is
an odd number and a multiple of 5 and 7
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Here are some more parts of the 100 chart, each one shaded according to different rules. Can you work out what the
rules are for each?
Is there only one solution each time?
The numbers that are shaded with dots are multiples of 11.
The number shaded in dark gray is a multiple of 8 and 11
The number that is shaded with lines is a multiple of 2, 4, and 8.
The numbers that are shaded with dots is a multiple of 3 and 9.
The number that is shaded with dots is a multiple of 5 and 9.
The black numbers are a multiple of 6 and 9.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
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to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Show each small square one at a time and ask students if there could be other numbers on the hundred’s
chart shaded the same way as…?
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Patterns on a Hundreds Chart
Adapted from: http://nrich.maths.org
Here is a 100 grid with some numbers shaded:
What do all the numbers shaded with dots have in common?
What do you notice about all the numbers shaded with lines?
What is the relationship between the numbers shaded in dark gray and the numbers with dots?
(and the numbers with lines?)
Now, here is part of a 100 chart shaded in a different way:
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Can you explain the shading this time?
Here are some more parts of the 100 chart, each one shaded according to different rules. Can you
work out what the rules are for each?
Is there only one solution each time?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
There are exactly twelve children in Ryan’s class. Only four of the children are boys. The following questions refer
to a time when all the children are present in the class. There are no visitors in the class. There might be more than
one correct answer to a question.
1. Which of the lettered statements must be true?
2. Which of the lettered statements cannot be true?
3. Which of the lettered statements could be true or not true?
a. There are twice as many girls as boys in Ryan’s class.
b. There are eight more girls than boys in Ryan’s class.
c. There are four more girls than boys in Ryan’s class.
d. If Ryan is sitting at a table with all the girls, there are exactly nine children at that table.
e. If only three of the boys are standing on their heads, one of the boys is not standing on his head.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Students can make a list using the statements to determine the number of boys and girls.
boy boy boy boy girl girl girl
girl girl
girl girl girl
a. There are twice as many girls as boys in Ryan’s class.
This statement must be true. There are 12 total children in Ryan’s class. Four out of the 12 children are boys.
Multiplying the number of boys by 2 is 8 (which is the number of girls).
b. There are eight more girls than boys in Ryan’s class.
This statement cannot be true. There are 8 girls, not 8 more girls than boys. The total number of children is 12.
There are 4 boys. So there are only 4 more girls than boys.
c. There are four more girls than boys in Ryan’s class.
This statement must be true. Increase the number of boys by 4 to get the total number of girls. The number of girls
in the class is 8. 4 + 4 = 8 girls
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d. If Ryan is sitting at a table with all the girls, there are exactly nine children at that table.
This statement could be true or not true. Ryan is one of the four boys. The other statements have proved that there
are 8 girls in the class. 1 boy and 8 girls means there are 9 children at the table. However, the statement does not
mention if there are other boys sitting at the table. The statement could be false as well.
e. If only three of the boys are standing on their heads, one of the boys is not standing on his head.
This statement must be true. There are four boys in Ryan’s class. Three of the boys are standing on their heads,
which leaves one boy who is not.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Create an “if, then” statement that could not be true, one that must be true and one that could be true or not
true. Share one of your statements with a partner and have your partner tell which it is and why?
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Ryan’s Class
There are exactly twelve children in Ryan’s class. Only four of the children are boys. The
following questions refer to a time when all the children are present in the class. There are no
visitors in the class. There might be more than one correct answer to a question.
1. Which of the lettered statements must be true?
2. Which of the lettered statements cannot be true?
3. Which of the lettered statements could be true or not true?
a. There are twice as many girls as boys in Ryan’s class.
b. There are eight more girls than boys in Ryan’s class.
c. There are four more girls than boys in Ryan’s class.
d. If Ryan is sitting at a table with all the girls, there are exactly nine children at that table.
e. If only three of the boys are standing on their heads, one of the boys is not standing on his
head.
From: http://nrich.maths.org
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Double, double toil and trouble
Fire burn, and cauldron bubble.
From Macbeth by William Shakespeare
The witches are making a potion. They use gummy bears, licorice spiders, jelly beans and candy crows. They
want 32 legs in their brew. What could they use to make their potion? How many ways can you find?
Witch Hazel said she wanted to make a potion with 27 legs because that is her favorite number.
What combination could the witches use to make Hazel’s potion?
Possible Solutions
They want 32 legs in their brew. What could they use to make their potion? How many ways can you find?
Answers may vary. A possible solution is:
4 gummy bears
2 licorice spiders
2 jelly beans
Students may use this or other strategies to solve this problem/task.
Students could draw a picture to show the number of legs each item has. Then create an equation based off picture.
32 legs = gummy bear + gummy bear + gummy bear + gummy bear + licorice spider + licorice spider + jelly bean
+ jelly bean
32 legs = 4 (4 legs) + 2 (8 legs) + 2 (0 legs)
= 4 legs + 4 legs + 4 legs + 4 legs + 8 legs + 8 legs + 0 legs + 0 legs
Witch Hazel said she wanted to make a potion with 27 legs because that is her favorite number.
What combination could the witches use to make Hazel’s potion?
There are no possible combinations because each of the candies has an even number of legs.
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Could you find all of the possible combinations? Why not?
• What do you know about any combination of legs using these candies?
• Would the combination have to be a multiple of 2? 4? 8? Justify your reasoning.
• Why can’t Witch Hazel make her combination of 27 legs?
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Secret Potion
Double, double toil and trouble
Fire burn, and cauldron bubble.
From Macbeth by William Shakespeare
The witches are making a potion. They use gummy bears, licorice spiders, jelly beans and candy
crows. They want 32 legs in their brew. What could they use to make their potion? How many
ways can you find?
Witch Hazel said she wanted to make a potion with 27 legs because that is her favorite number.
What combination could the witches use to make Hazel’s potion? Explain your reasoning.
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Sophia rode her bike to her friend’s house. She stopped at the grocery store to get some candy when she was half
way to her friend’s house. When she was one-fourth of the way between the grocery store and her friend’s house,
she stopped at the bakery to get her friend a donut. Her friend lives 2 miles from Sophia’s house. How far is the
bakery from Sophia’s house? What fraction of the trip is left after Sophia gets to the bakery?
Possible Solutions
How far is the bakery from Sophia’s house?
The bakery is 1 1 mile away from Sophia’s house.
4
Students may use this or other strategies to solve this problem/task.
•
Students can create a model (number line) to determine the distance between Sophia’s house and the
bakery.
Grocery
Store
Sophia’s
house
1
mile
0
mile
Bakery
1¼
mile
Friend
1½
mile
1¾
mile
2
miles
•
The problem stated that the distance between Sophia’s house and her friend’s house is 2 miles. Half of 2
miles is 1 mile, which is the location of the grocery store. The problem stated that she stopped at the bakery
which was one-fourth of the way from the grocery to the friend’s house. Students will need to divide the
number line from the grocery to the friend’s house into intervals of 4 to determine the location of the
bakery.
•
Sophia still has 3 of the distance to go, because the grocery is half of the way and the bakery is 1 of the
8
8
5
way further so that is of the trip.
8
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If Sophia’s friend’s house was only one mile away and everything else in the problem was the same, how
far would the bakery be from Sophia’s house? (grocery half mile, bakery 5 mile)
8
• I had a student who said that Sophia had 3 of the trip left to go when she left the bakery. What would you
4
say to that student to help him understand his error?
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Sophia’s Ride
Sophia rode her bike to her friend’s house. She stopped at the grocery store to get some candy
when she was half way to her friend’s house. When she was one-fourth of the way between the
grocery store and her friend’s house, she stopped at the bakery to get her friend a donut. Her
friend lives 2 miles from Sophia’s house. How far is the bakery from Sophia’s house? What
fraction of the trip is left after Sophia got to the bakery?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Use the target to answer the questions.
You have 5 darts and your target score is 44.
You are an excellent player and every dart hits the board.
How many different ways could you score 44?
BUT
You can NOT hit the same 5 numbers in a different order. At least one number has to be different for each score of
44.
Try putting your own set of numbers onto the dart board. You can use 44 as your target number or choose a new
one.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Answers will vary. Possible solutions are as follows:
22 + 22+ 0 + 0 + 0 = 44
18 + 2 + 11 + 11 + 2 = 44
18 + 8 + 4 + 10 + 4 = 44
22 + 10 + 10 + 2 + 0 = 44
10 + 10 + 10 + 10 + 4 = 44
8 + 2 + 10 + 22 + 2 = 44
Emphasize that the same 5 numbers can NOT be hit in a different order.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Could you get an odd number using the same rules? Justify your reasoning.
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Target Number
You have 5 darts and your target score is 44.
You are an excellent player and every dart hits the board.
How many different ways could you score 44?
BUT
You can NOT hit the same 5 numbers even in a different order. At least one number has to be
different for each score of 44.
Try putting your own set of numbers onto the dart board. You can use 44 as your target number
or choose a new one.
From: http://nrich.maths.org
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Use the Deca Tree poster to find out how many leaves are left on the tree.
Possible Solutions
How many leaves were left on the tree?
•
There are 8,889 leaves left on the tree.
Students may use this or other strategies to solve this problem/task.
1 Deca Tree = 10 trunks = 100 branches = 1000 twigs = 10,000 leaves
1 trunk = 10 branches = 100 twigs = 1000 leaves
1 branch = 10 twigs = 100 leaves
1 twig = 10 leaves
Cut off one trunk: 10,000 leaves – 1000 leaves = 9,000 leaves left
Cut off one branch: 9,000 leaves – 100 leaves = 8,900 leaves left
Cut off one twig: 8,900 leaves – 10 leaves = 8,890 leaves left
Pull off one leaf: 8,890 leaves – 1 leaf = 8,889 leaves left
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• How does this tree relate to place value?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Jayvoni made this ruler to see how many different size lines he could draw by moving the different lengths around
the black dots. How many different lengths of lines can he make with his J-Ruler?
Possible Solutions
How many different lengths of lines can he make with his J-Ruler?
•
There are 13 different lengths shown below that can be made with his J-Ruler.
Students may use this or other strategies to solve this problem/task.
Students can create the J-Ruler using tagboard and brads to identify the different lengths.
1 inch
2 inches
3 inches is created with 1 inch and 2 inches
4 inches
5 inches is created by folding the 1 in. over the 2 in. to have a 1 in. line to add to 4 inch
6 inches is created with 4 inches and 2 inches
7 inches is created with 1 inch, 2 inches, and 4 inches
8 inches
9 inches is created with 7 inches and 2 inches
10 inches is created with 8 inches and 2 inches or with 1 inch, 2 inches, and 7 inches
11 inches is created with 1 inch, 2 inches, and 8 inches or with 4 inches and 7 inches.
12 inches is created with 4 inches and 8 inches.
15 inches is created with 8 inches and 7 inches.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
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•
•
•
What might be an advantage to this kind of ruler as opposed to a 12 inch ruler?
What is the advantage of the 12 inch ruler?
Could you create a similar ruler that could measure all of the whole inches from 1 to 15? 20?
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J-Ruler
Jayvoni made this ruler to see how many different size lines he could draw by moving the
different lengths around the black dot. How many different lengths of lines can he make with his
J-Ruler?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
A grocery store parking lot has space for 1000 vehicles. On Tuesday, mini vans were parked in two-fifths of the
spaces. The parking lot was
3
full. How many mini vans were in the parking lot? How many cars were in the
4
parking lot?
Possible Solutions
How many cars were in the parking lot?
•
There are 350 cars are in parking lot.
Students may use this or other strategies to solve this problem/task.
Students may draw a picture to represent the fractional amount of cars in the parking lot.
1000 vehicles
200 vehicles 200 vehicles 200 vehicles 200 vehicles 200 vehicles 2/5 of vehicles
are mini vans
•
2
of 1000 vehicles is equivalent to 400 vehicles. 400 parking spaces are occupied by mini vans.
5
250 vehicles 250 vehicles
250 vehicles
250 vehicles
3/4 of parking
lot is full
•
3
of 1000 parking spaces is equivalent to 750 spaces. 400 mini vans were parked in the parking lot. 400
4
subtracted from 750 is 350. There are 350 cars parked in the parking lot.
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If there were only 100 parking spaces and the rest of the information was the same, how many mini vans
would be in the parking lot and how many cars would be in the parking lot?
• What do you notice about the new numbers? Why does this happen?
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Park Here
A grocery store parking lot has space for 1000 vehicles. On Tuesday, mini vans were parked in
two-fifths of the spaces. The parking lot was
3
full. How many mini vans were in the parking
4
lot? How many cars were in the parking lot?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
The auditorium at West Side Elementary school has 40 seats in a row. There are 10 rows of seats. How many
seats are in the auditorium? The kindergarten students sit in the front, then the 1st graders, 2nd graders and so on to
the 5th graders in the back. The table shows the number of students in each classroom. The teachers sit with their
class. For these questions assume that there is 100% attendance.
Which row/s would the 3rd graders sit in? Would there be any rows with only one grade level? Which rows?
Would there be any rows with only one classroom? Which rows? Would there be any empty rows?
Possible Solutions
Which row/s would the 3rd graders sit in?
The 3rd grade classes will sit in Rows 5and 6.
Would there be any rows with only one grade level? Which rows?
Row 1 has only kindergarten, row 4 has only 2nd grade, Row 7 will contain only the 4th grade classes. Row 9 will
contain the 5th grade class.
Would there be any rows with only one classroom? Which rows?
Row 9 will have one class of 5th grade.
Would there be any empty rows?
Row 10 will be the empty row.
Students may use this or other strategies to solve this problem/task.
Students may draw a model of the seating arrangements and create simple equations that will equal the total
number of seats in each row. For organizational purposes, students can relabeled the grade levels. In this strategy,
the classrooms were identified by letter A or B
Teacher
Ms. Jarvis and Mr. Kale
Mrs. Harvey and Ms. Young
Mr. Bockey
Mr. King
Ms. Garner
Mr. Warner
Ms. Yankers
Mrs. Sanders
Ms. Travis
Ms. Harte
Mr. Zimmerman
Mrs. Isley
Math APSS
Grade level
KA
KB
1A
1B
2A
2B
3A
3B
4A
4B
5A
5B
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Number of students
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26
31
27
32
28
26
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Math Applying Problem Solving Strategies
Row 1 – 23 students (KA) + 2 teachers (KA) + 15 students (KB) = 40
Row 2 – 11 students (KB) + 2 teachers (KB) + 27 students (1A) = 40
Row 3 – 4 students (1A) + 1 teacher (1A) + 1 teacher (1B) + 27 students (1B) = 33 seats + 7 students (2A) = 40
Row 4 – 25 students (2A) + 1 teacher (2A) + 1 teacher (2B) + 13 students (2B) = 40 students
Row 5 – 15 students (2B) + 1 teacher (3A) + 24 students (3A) = 40
Row 6 – 2 students (3A) + 1 teacher (3B) + 29 students (3B) + 8 students (4A) = 40
Row 7 – 20 students (4A) + 1 teacher (4A) + 1 teacher (4B) + 18 students (4B) = 40
Row 8 – 12 students (4B) + 1 teacher (5A) + 27 students (5A) = 40
Row 9 – 25 students (5B) + 1 teacher (5B) = 26
Row 10
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• How many rows would be filled if there were only 20 seats in a row? Explain.
• How many seats would you estimate a movie theater might have? Why is that a reasonable estimate?
• Find out the number of seats, seats per row and number of rows in a movie theater.
• Was your estimate a good estimate? Justify your reasoning.
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Auditorium Seating
The auditorium at West Side Elementary school has 40 seats in a row. There are 10 rows of
seats. How many seats are in the auditorium? The kindergarten students sit in the front, then the
1st graders, 2nd graders and so on to the 5th graders in the back. The table shows the number of
students in each classroom. The teachers sit with their class. For these questions assume that
there is 100% attendance.
Teacher
Ms. Jarvis and Mr. Kale
Mrs. Harvey and Ms. Young
Mr. Bockey
Mr. King
Ms. Garner
Mr. Warner
Ms. Yankers
Mrs. Sanders
Ms. Travis
Ms. Harte
Mr. Zimmerman
Mrs. Isley
Grade level
K
K
1
1
2
2
3
3
4
4
5
5
Number of students
23
26
31
27
32
28
26
29
28
30
27
25
Which row/s would the 3rd graders sit in? Would there be any rows with only one grade level?
Which rows? Would there be any rows with only one classroom? Which rows? Would there be
any empty rows?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Complete the “Shape Challenge” worksheet.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
Students can use guess and check to solve expressions. Students will use factors to identify the value of
shape. Students will use numbers 0 thru 12 to solve equation.
•
The product for each multiplication problem is 12 or less than 12. In the expression below, students will
know that a single digit multiplied by itself (3 times) will produce an answer less than 12.
×
•
•
×
=
If students use the 1 as the value for the circles, then the hexagon would have a value of 1 as well; which
cannot be true. Each shape has it’s own value. If the circles have a value of 2, then the hexagon would
have a value of 8. If the circles have a value of 3, then the hexagon would have a value of 27. All shapes
must have a value of either 12 or less than 12. 27 is greater than 12. By identifying the value of the circle
and the hexagon, students can use this information to identify the values of the remaining shapes.
It is important that the students use their own reasoning skills to complete the worksheet.
=2
=9
=5
=1
=
=3
=6
=8
=
=0
=4
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2
2 ×
×
2 ×
=
4
=
2
8
8
3
×
2
×
=
3
5
9
=
10
3
×
4
=
12
3
×
1
=
3
3
×
2
=
6
1
×
10
=
10
2
=
12
2
×
0
=
0
4
0
×
6
2
×
× 2
=
8
=
0
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Explain how you got the number in the first shape you figured out? How did this help you with other
shapes?
• How did you figure out the triangle? Did it help you figure out the dodecagon or the square? Explain.
• How did you figure out the trapezoid? Did it help you figure out the circle or the hexagon? Explain.
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Shape Challenge
Each shape below represents a different number from 0 to 12. Find the value of each shape.
×
×
×
=
=
=
×
=
×
×
=
×
=
×
=
×
=
=
×
=
×
×
=
×
=
Challenges:
1. Create expressions that equal an odd or even number.
2. Use all of the symbols in one equation.
3. Create an expression that equals 100.
4. Use the fewest symbols to create an expression that equals 31.
5. Which two numbers could not be a value for a shape? Why?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
There are 44 people coming to a birthday dinner party. There are 15 square tables that seat 4 people - one at each
side as shown or the tables can be joined together to make groups as shown- And so on...
Find a way to seat the 44 people using all 15 tables, with no empty places.
Possible Solutions
Find a way to seat the 44 people using all 15 tables, with no empty places.
Answers may vary.
Students may use this or other strategies to solve this problem/task.
•
Students may draw a picture and create a chart to determine a way to sit each of the 44 people using all 15
tables.
Table 1 Seats 4 •
2 6 3 8 4 10 5 12 6 14 7 16 8 18 9 20 10 22 11 24 12 26 13 28 14 30 Using the chart, students can now select combination of tables to seat 44 people.
A possible solution is as follows:
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15 32 Math Applying Problem Solving Strategies
12 + 8 + 6 + 6 + 4 + 4 + 4 = 44
•
Challenge the students to create an expression to represent the picture.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Are there other combinations?
• What do you notice about all of the combinations?
• What would be the best arrangement for a birthday dinner party? Justify your reasoning.
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Table Arrangement
There are 44 people coming to a birthday dinner party. There are 15 square tables that seat 4
people - one at each side - like this:
Or the tables can be joined together to make groups like this:
And so on...
Find a way to seat the 44 people using all 15 tables, with no empty places.
From: http://nrich.maths.org
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the
information they are learning. Students should be able to summarize, analyze, organize, evaluate, predict,
design, create, innovate, etc.
Find the number on a Hundreds Chart.
Four of the clues below are true but do nothing to help in finding the number.
Four of the clues are necessary for finding it.
Here are eight clues to use:
1.
2.
3.
4.
5.
6.
7.
8.
The number is greater than 9.
The number is not a multiple of 10.
The number is a multiple of 7.
The number is odd.
The number is not a multiple of 11.
The number is less than 200.
Its ones digit is larger than its tens digit.
Its tens digit is odd.
What is the number?
Can you sort out the four clues that help and the four clues that do not help in finding it?
Possible Solutions
What is the number?
The answer is 35.
Can you sort out the four clues that help and the four clues that do not help in finding it?
Four clues that helped identify the number
• The number is a multiple of 7.
•
The number is odd.
•
Its ones digit is larger than its tens digit.
•
Its tens digit is odd.
Four clues that did not help identify the number.
• The number is greater than 9. The numbers 0 – 9 have zero tens.
•
The number is not a multiple of 10. The ones digit is larger than the tens digit. Multiples of 10 have zero
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in the ones place. Also multiples of ten are even numbers.
•
The number is not a multiple of 11. The ones digit is larger than the tens digit; not equal to each other.
•
The number is less than 200. All of the numbers on the Hundreds Chart are already less than 200.
Students may use this or other strategies to solve this problem/task.
Students do not have to follow the clues in order to identify the number. Students may start with statement #3 to
eliminate a majority of the numbers. Student may also start with statement #4.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Possible Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Could you change only two clues to make the secret number a different number?
• What clue eliminated the most numbers on the chart?
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Useful Clues
Find the number on a Hundreds Chart.
Four of the clues below are true but do nothing to help in finding the number.
Four of the clues are necessary for finding it.
Here are eight clues to use:
1.
2.
3.
4.
5.
6.
7.
8.
The number is greater than 9.
The number is not a multiple of 10.
The number is a multiple of 7.
The number is odd.
The number is not a multiple of 11.
The number is less than 200.
Its ones digit is larger than its tens digit.
Its tens digit is odd.
What is the number?
Can you sort out the four clues that help and the four clues that do not help in finding it?
From: http://nrich.maths.org
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
What do the digits in the number fifteen add up to?
How many other numbers have digits with the same total if we only include numbers without zeros?
Possible Solutions
What do the digits in the number fifteen add up to?
The digits add up to 6.
How many other numbers have digits with the same total if we only include numbers without zeros?
There are 31 possible solutions.
Students may use this or other strategies to solve this problem/task.
Students can create a list of digits combined together that add up to 6.
24
33
42
51
15
114
123
132
141
221
213
222
231
312
411
1,113
1,122
1,131
1,212
1,221
1,311
2,112
2,121
2,211
3,111
11,112
11,121
11,211
12,111
21,111
111,111
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Why do you think they said to not include zeros?
• Would there be fewer or more numbers that would add up to 7? Justify your reasoning.
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Adding Digits
What do the digits in the number fifteen add up to?
How many other numbers have digits with the same total if we only include numbers without
zeros? (Hint: There are more than 15 possible solutions.)
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
On Annie's ninth birthday her mom made her a cake which had the figures from 0 to 9 round the edge in red icing
instead of candles. Annie cut the cake into 3 pieces so that the numbers on each piece added to the same total.
Where were the cuts and what fraction of the whole cake was each piece?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
The fraction of each cut is 2/10 (or 1/5), 3/10, and 5/10 (or ½).
•
Students can divide the clock into equal pieces, one number in each piece. Each piece is 1/10 of the cake.
By adding all of the numbers on the cake. It equals 45. Referring back to the problem, the cake will be cut
with 3 cuts, so that the numbers on each piece added to the same total. Divide the number by 3 to get 15.
Students must now find a combination of numbers that will equal 15.
•
8 + 7 = 15; remaining numbers are 0, 1, 2, 3, 4, 5, 6, 9
6 + 5 + 4 = 15; remaining numbers are 0, 1, 2, 3, 9
0 + 1 + 2 + 3 + 9 = 15
•
The fraction of each cut is 2/10 (or 1/5), 3/10, and 5/10 (or ½).
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Could the cake be cut into a different number of pieces and still have the same total in each piece? Justify
your answer.
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Dividing a Cake
On Annie's ninth birthday her mom made her a cake which had the figures from 0 to 9 round the
edge in red icing instead of candles.
Annie cut the cake into 3 pieces so that the numbers on each piece added to the same total.
Where were the cuts and what fraction of the whole cake was each piece?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Fill in the table with the numbers listed. Each column should add up to the number beneath it and each row should
add up to the number beside it.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Fill in the table with numbers 1 to 6.
Even
6
2
3
11
a quart
5
1
10
10
7
4
4 cups in
•
Students must first determine the number of cups in a quart. Identifying that amount will reveal the
remaining numbers.
Fill in the table with numbers 1 to 9.
5
7 days in
a week
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1
5
of 5 = 1
2
6
12
8 pints in
17
a gallon
3
9
4
15
12
18
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1
of 5, and pints in a gallon). In the second row, the box in
5
the second column is empty. Combining 7 and 8 equals 15, the entire row is equal to 17. The empty box is 2.
Identifying that number will reveal the remaining numbers.
Students will solve the clues given (days in the week,
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What strategy did you use to start the grid?
• What decisions did you have to make to create your own grid>
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Number Puzzles
Fill in the table with the numbers listed. Each column should add up to the number beneath it
and each row should add up to the number beside it. For each puzzle, use each number only
once.
Fill in the table with numbers 1 to 6.
11
even
cups in a
quart
1
10
7
10
4
Fill in the table with numbers 1 to 9.
1
5
of 5
12
pints in
a gallon
days in a
week
17
16
15
12
18
Write your own to share.
Fill in the table with numbers _____________.
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
On the planet Vu there are two kinds of creatures Zios and Zepts. The Zios have three legs and the Zepts have 7
legs. The great space explorer Nicko, who first discovered the planet Vu, saw a crowd of Zios and Zepts. He
managed to see that there was more than one of each kind of creature before they saw him. As soon as they saw
Nicko, they all rolled over onto their heads and put their legs in the air. Nicko counted 52 legs. How many Zios
and how many Zepts were there?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
There are 8 Zios and 4 Zepts.
•
Students may create a number sentence and use the guess and check strategy to solve this problem. A
possible number sentence could be 3(p) + 7 (s) = 52 legs. First, substitute the digit 2 for p. If we subtract 6
legs from 52 legs, 46 legs are remaining.
3(p) + 7(s) = 52
3(2) + 7(s) = 52
6 + 7(s) = 52
52 – 6 = 46; There is no factor that can be multiplied by 7 that is equal to 46.
Continue to substitute in values for the variables.
•
Students can also solve the problem by listing the multiples of 3 and 7 in a table.
3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 7 14 21 28 35 42 49 •
Students know that by adding the legs of the Zios and the Zepts, 52 legs were seen. Students also know
that each Zio and Zept is seen more than once. Looking at the multiples of 3 and 7, students will be able
to combine three sets of numbers to get the sum of 52. Referring back to the question, there had to be more
than one of each kind of creature, so the possible answer is 8 Zios and 4 Zepts.
24 + 28 = 52
3(p) + 7(s) = 52
3(8) + 7(4) = 52
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Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Could there be 47 legs altogether? 71? 10? 55? 40? How could you quickly tell if a number of legs is or is
not possible?
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Planet Vu
On the planet Vu there are two kinds of creatures Zios and Zepts. The Zlos have three legs and
the Zepts have 7 legs.
The great space explorer Nicko, who first discovered the planet Vu, saw a crowd of Zios and
Zepts. He managed to see that there was more than one of each kind of creature before they saw
him. As soon as they saw Nicko they all rolled over onto their heads and put their legs in the air.
Nicko counted 52 legs. How many Zios and how many Zepts were there?
From: http://nrich.maths.org
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Traditional Tatami mats are 35.5” × 71” × 2.5”.
You may wish to use “friendlier” numbers in order to estimate the following arrangements of these mats. What
measurements might you use? What measurements would they use in Japan?
As stated above, from Shoji Styles, Tatami mats define the size of traditional Japanese rooms.
If you wanted to cover a 6 foot by 15 foot room, how many ways could you arrange the mats? How do you know
you have all of the possible arrangements?
Draw an arrangement of mats you think could cover the floor in a small, medium, or large bedroom.
At one retailer Tatami Mats cost $45.95 each tax included. How much would it cost for the mats for the bedroom
you drew?
Possible Solutions
Students may use this or other strategies to solve this problem/task.
Answers may vary. A possible solution may be as follows:
71
in.
71
in.
71
in.
35 ½ in.
~6
35 ½ in.
35 ½ in.
35 ½ in.
71
in.
35 ½ in.
71
in.
~15’
•
Students can create models/templates of the Tatami Mats to identify the possible arrangements within the 6
foot by 15 foot room.
Draw an arrangement of mats you think could cover the floor in a small, medium, or large bedroom.
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•
After students show an arrangement for a small, medium or large bedroom, have them discuss if the
dimensions are reasonable. What could they use as an example?
At one retailer Tatami Mats cost $45.95 each tax included. How much would it cost for the mats for the bedroom
you drew?
Answers will vary. A possible solution is as follows:
•
If you use the number of tatami mats from the 6’ by 15’ room. Students know that it takes five tatami mats
to cover the floor. Students can create an equation and use estimation to solve the problem.
$ 45.95 x 5 =
$ 46.00 x 5 = $230.00
(Estimate $45.95 to the nearest whole dollar, which is $46.00)
(Students should realize that it took an extra $0.05 per mat to get $46.00. Subtracting the
extra amount will give the exact amount.)
$230.00 - $0.25 = $229.75
It will cost $229.75 to purchase five tatami mats to cover a 6’ by 15’ room.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• Why did you take $0.25 away from your estimated total instead of $0.05?
• Why do you think the measurements for Tatami Mats are not a whole number of inches? (They are actually
sold in cm—Japanese do not use the US customary units of measurement. Students could research the mats
and find this out.)
• What is the relationship of the length and width of the mats? Why might they have this relationship?
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Tatami Mats
“Tatami mats are usually used to measure the dimensions of Japanese rooms and have become
the standard measurement of Japanese interior architecture.” They do not use square feet as a
measurement for area. They instead say that the room is so many tatami mats in size.
“Japan has a… floor focused culture, so many of life's most important social rituals take place on
floors thus making tatami mats of particular significance. It is very important in traditional
Japanese culture, for example, not to step on the mats with one's shoes on. Additionally one is
not supposed to talk (while) standing on tatami mats, conversation being restricted to when
sitting or kneeling. Eating on tatami mats, however, is acceptable.”
http://www.shojistyles.com/tatami/mats.html
Traditional Tatami mats are 35.5” × 71” × 2.5”.
You may wish to use “friendlier” numbers in order to estimate the following arrangements of
these mats. What measurements might you use? What measurements would they use in Japan?
As stated above, from Shoji Styles, Tatami mats define the size of traditional Japanese rooms.
If you wanted to cover a 6 foot by 15 foot room, how many ways could you arrange the mats?
How do you know you have all of the possible arrangements?
Draw an arrangement of mats you think could cover the floor in a small, medium, or large
bedroom.
At one retailer Tatami Mats cost $45.95 each tax included. How much would it cost for the mats
for the bedroom you drew?
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Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Ms. Carson’s class is making a class flag using 1 inch x 1 inch squares of four colors: yellow, green, red, and blue.
These 1 inch x 1 inch squares will be used to make a larger square flag measuring 4 inches x 4 inches. The class
decided to use the following quantities of each color: 1 yellow, 3 green, 3 red and the rest will be blue. What
8
8
16
fraction of the large square will be blue? Justify your response. How many 1 inch x 1 inch squares of EACH color
will be needed for the 4 inch x 4 inch square?
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
•
5 would be blue, because 1 + 3 + 3 + 5 = 1 All of the colors would have to add up to one , because
16
8 8 16 16
that would be the whole flag.
1 + 3 = 4 = 1 and 1 = 8 and 8 − 3 = 5 so the blue would be 5 . They all have to add up to 1.
8 8
8 2
2 16
16 16 16
16
One-half plus one-half equals one.
The whole flag has 16 squares. So I divided
Y G R B
it into 16 pieces and put the colors in for how many 16ths there were.
Y G R B
I had 5 left over, so there are 5 blue which is 5 of the flag.
16
G G R B
1=
8
3 =
8
2
16
6
16
3 = 3
16 16
5 = 5
16 16
•
•
G
so there are 2 yellow squares
G
B
B
so there are 6 green squares
so there are 3 red squares
so there are 3 red squares
If the flag is 4 in. by 4 in. it would have 16 squares. 1 are yellow so there would be 2 yellow squares; 3
8
8
are green so there would be 6 green squares; 3 are red so there would be 3 red squares; 5 are blue so
16
16
there would be 5 blue squares
My picture shows 2 yellow, 6 green, 3 red and 5 blue squares.
1 = 2 so there are 2 yellow squares
8 16
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Math Applying Problem Solving Strategies
3 = 6
8 16
so there are 6 green squares
3 = 3
16 16
5 = 5
16 16
so there are 3 red squares
so there are 3 red squares
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• Why did you have to convert the eights to sixteenths?
• What would happen if they wanted the flag to be a rectangle that was not square? What dimensions could
the rectangle be?
• What if they wanted a bigger flag? What would be an easy size of flag to make so that they could keep the
same fraction of each color?
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Classroom Flag
Ms. Carson’s class is making a class flag using 1 inch x 1 inch squares of four colors: yellow,
green, red, and blue. These 1 inch x 1 inch squares will be used to make a larger square flag
measuring 4 inches x 4 inches. The class decided to use the following quantities of each color:
1 yellow, 3 green, 3 red and the rest will be blue. What fraction of the large square will be
8
8
16
blue? Justify. How many 1 inch x 1 inch squares of EACH color will be needed for the 4 inch x
4 inch square?
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Math Applying Problem Solving Strategies
Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Clarissa planted a flower garden. 1 of the flowers were tulips, 3 were daffodils and the rest were irises. There
4
8
were 12 irises. How many of each kind of flower were in Clarissa's garden? How many flowers altogether?
Explain.
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
I found the common denominator for 1 and 3 which is 8. So there are 2 tulips and 3 daffodils.
4
8
8
8
2 + 3 = 5 The irises would be the rest of the whole, so there are 3 irises. That would mean that
8
8
8
8
there are the same number of daffodils as irises, 12. 12 + 12 = 24 If 3 is 12, then 2 would be 8.
8
8
There are 32 flowers altogether. 12 + 12 + 8 = 32 flowers in the garden.
•
I knew that 1 = 2 and 2 + 3 = 5 . So there are 3 irises. I took 12 counters and put them into 3
4
8
8
8
8
8
piles to see how many are in one-eighth and it was 4 flowers. So there are 12 irises, 12 daffodils and 8
tulips. That makes 32 flowers altogether.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• How do you know you answered all of the parts of the question?
• What if Clarissa wanted to double the size of her garden but keep the same proportion of each kind of
flower? How many of each flower would there be? What fraction of the garden would be tulips?
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Columbus City Schools 2012
Clarissa’s Garden
Clarissa planted a flower garden. 1 of the flowers were tulips, 3
4
8
were daffodils and the rest were irises. There were 12 irises. How
many of each kind of flower were in Clarissa's garden? How
many flowers altogether? Explain.
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Gina, Lola, Brigit and Gilda shared a pizza. Brigit and Lola each ate 1 of the pizza. If Gina and Gilda divided the
8
remaining pizza so that Gina ate twice as much as Gilda, what fraction of the pizza did Gina eat? How much did
Gilda eat? Explain your answer.
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
Brigit and Lola ate 2 ( 1 + 1 = 2 ) of the pizza, so there is 6 ( 8 − 2 = 6 ) left for Gina and
8 8
8
8
8
8
8
8
Gilda. If Gina ate twice as much as Gilda, she ate 4 of the pizza ( 4 + 2 = 6 ). 4 is twice as much as
8
8
8
8
8
2
8.
Students may draw a picture. They must include an explanation with their picture.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• How do you know you answered all of the parts of the question?
• Did all of the girls eat the same amount of pizza?
• Who ate the most? How much less pizza would she have to eat in order for all of the girls to eat the same
amount of pizza?
• Who ate the least? How much more would she get if all of the girls fair shared the pizza?
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Sharing Pizza
Gina, Lola, Brigit and Gilda shared a pizza. Brigit and Lola
each ate 1 of the pizza. If Gina and Gilda divided the
8
remaining pizza so that Gina ate twice as much as Gilda,
what fraction of the pizza did Gina eat? How much did
Gilda eat? Explain your answer.
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
You have invited 3 friends over for pizza. Two of your friends said they can come for sure. One friend might
come. How could you cut your pizza to make sure everyone could have an equal number of slices whether or not
the third friend comes? Explain your reasoning.
If two friends show up and one of your friends only ate half of their pizza, what fraction of the pizza would be left
over? Explain your reasoning.
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
•
I would cut the pizza into 12 equal sized pieces, because if there are only 3 of us we could each have 4
pieces and if there are 4 of us we could each have 3 pieces.
I found the common denominator for one-third and one-fourth and it is 12. I would cut the pizza into 12
pieces because it is the common denominator for thirds and fourths. 1 = 4 so if there are 3 people they
3 12
1
3
would get 4 pieces; =
so if there are 4 people they would each get 3 pieces.
4 12
Students may draw a picture. They must include an explanation with their picture.
2 friends and me
•
•
3 friends and me
3 people get 4 pieces or 4
people get 3 pieces
If 2 friends come we would all get 4 pieces or one-third of the pizza. If one person only eats half of that
there would be 2 pieces left which is one-sixth of the whole pizza.
Students may draw a picture, but they must tell the fraction of pizza left over.
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Math Applying Problem Solving Strategies
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• How do you know you answered all of the parts of the question?
• What if four people shared the pizza and one of the friends only ate half of their pizza? How much pizza
would be left over?
• Can you use the same strategy you used to find out how much was left over when there were 3 people
sharing the pizza? Expalin.
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Columbus City Schools 2012
Pizza with Friends
You have invited 3 friends over for pizza. Two of your friends said
they can come for sure. One friend might come. How could you cut
your pizza to make sure everyone could have an equal number of
slices whether or not the third friend comes? Explain your reasoning.
If two friends show up and one of your friends only ate half of their
pizza, what fraction of the pizza would be left over? Explain your
reasoning.
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Take any three consecutive numbers (less than 20) and add them together. Try several different sets of consecutive
numbers. Add them together and draw/make the representation of their sum.
What do you notice about the answer?
Look closely at your model.
Would it work in exactly the same way if you used different numbers?
Can you use your example to prove what will happen every time you add three consecutive numbers?
See if you can explain this to someone else. Are they convinced by your argument?
Once you can convince someone else, see if you can find a way to show the argument on paper. You might draw
something to prove that your result is always true from your example.
Possible Solutions and Possible Misconceptions
Students may use this or other strategies to solve this problem/task.
•
•
•
•
2+3+4=9
If you take one dot from the largest number and add it to the
smallest number, you get a rectangle. So you can multiply the
middle number 3 times to get the answer. 3 × 3 = 9
2 + 3 + 4 = (2 + 1) + 3 + (4 − 1)
The middle number will always be one more than the smallest number and one less than the largest number.
So you can add one to the smallest number (and get the middle number) and take one away from the largest
number (and get the middle number). The net change is zero. So the sum stays the same. Now you have 3
of the same number. So to get the sum of any 3 consecutive numbers multiply the middle number by 3.
29 + 30 + 31 = 3 × 30 easy
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Math Applying Problem Solving Strategies
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• What do you know in the problem?
• What do you need to find out? What is the question asking?
• How do you know you answered all of the parts of the question?
• Could this help you add 3 big consecutive numbers?
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Consecutive Numbers
Take any three consecutive numbers (less than 20) and add them together. Try several different
sets of consecutive numbers. Add them together and draw/make the representation of their sum.
What do you notice about the answer?
Look closely at your model.
Would it work in exactly the same way if you used different numbers?
Can you use your example to prove what will happen every time you add three consecutive
numbers?
See if you can explain this to someone else. Are they convinced by your argument?
Once you can convince someone else, see if you can find a way to show the argument on paper.
You might draw something to prove that your result is always true from your example.
Math APSS
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Columbus City Schools 2012
Math Applying Problem Solving Strategies
Grades 3-5
Problem or Task
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Use the Antonia’s Pizzeria menu to answer the following questions.
Which pizza is the best deal? Justify your response. Which slice is the best deal? If you were going to have your
pizza delivered which pizza would be the best deal?
Antonia is thinking about making her pizzas in the shape of a square instead of round. She would still sell 16" and
8" pizzas. The pizzas would cost the same as the same size round pizza. Would you advise her to switch to square
pizzas? Justify your reasoning.
Possible Solutions
Students may use this or other strategies to solve this problem/task.
•
•
The 16” pizza is the better deal. It is 4 times as big as the 8” pizza but only cost 2 times as much as the 8”
pizza.
The 16” pizza is the better deal because it is 2 times as much as the 8” pizza and more than 2 of the 8”
pizzas will fit in the 16” pizza.
8”
8”
•
1
1
slice is the best deal because it is 2 times the size of the slice but not 2 times the cost. It is only
4
8
$0.50 more.
•
I would buy the large pizza to be delivered because you get more than two times as much pizza for only two
times the price and I would buy one slice of pizza so that would make the total over $20 and it wouldn’t
1
cost anything for delivery. The
slice would only cost $0.50.
8
•
I would tell her not to make square pizzas because most people like round pizzas and because it would take
more dough and toppings to fill a square pizza that is the same width as a round pizza. The round pizza
would fit inside the square pizza and not cover the corners (like the picture).
The
1”
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Math Applying Problem Solving Strategies
•
I would tell her to switch to the square pizza and advertise that her pizza has more pizza for the same price.
She could show this picture to show people that they would get a better deal.
Classroom Discussion
The strength of problem solving lies in the rich discussion afterward. As the school year progresses students
should be able to justify their own thinking as well as the thinking of others. This can be done through
comparing strategies, arguing another student’s solution strategy or summarizing another student’s sharing.
Have several students share their strategies and responses and explain their reasoning.
Select students to share different strategies. You may choose a student with an incorrect solution or flawed strategy
to share in order for the class to reason and justify other’s thinking.
•
•
Discuss what students know and what they are solving for in each problem.
Do they know the product or total, the number of groups or the number in each group?
Questions:
• Why? How do you know?
• What have you already learned which helped you in solving this problem?
• If no one shows a picture with the 8” circle inside the 16” circle, ask the students to draw or cut out circles
one that has a diameter that is two times the size of the other and then put them together to justify their
thinking.
• Why is the 16” pizza not just 2 times as big as the 8” diameter pizza? (To find area you have to square the
dimensions πr2 = area; students may not know the formula for the area of a circle but they should know that
area is squared. They may use 16” and 8” squares to show the relationship.)
• Would you buy a small pizza, 4 slices of the small pizza or 8 slices of the small pizza? Why?
• Check the prices and sizes of pizzas where you buy pizza. Which is the best deal? Justify your reasoning.
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Antonia’s Pizzeria
Item
$2 delivery charge
for orders under $20
Size
Cost
16"
$19.98
8"
$9.98
Slices of Small Pizza
1
of pie
4
1
8 of pie
$3
$2.50
Which pizza is the best deal? Justify your response. Which slice is the best deal? If you were
going to have your pizza delivered which pizza would be the best deal?
Antonia is thinking about making her pizzas in the shape of a square instead of round. She
would still sell 16" and 8" pizzas. The pizzas would cost the same as the same size round pizza.
Would you advise her to switch to square pizzas? Justify your reasoning.
Math APSS
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Columbus City Schools 2012