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Picture Frames
Meg is making picture frames by gluing red and blue square
tiles on to wooden frames.
The 1st frame uses 8 tiles.
The 2nd frame uses 12 tiles.
The 3rd frame uses 16 tiles.
Picture Frames
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If this pattern continues, how many tiles will Meg need to make
the 10th frame? How many tiles will Meg need to make the
20th frame?
Meg wants 1/2 of her 20th frame to be blue tiles and 1/2 of the
frame to be red tiles. Bags of 50 red tiles sell for $1.00. Bags of
25 blue tiles sell for 50 cents. A mixed bag of 50 red and 50
blue tiles sells for $2.25. Meg wants to spend as little money as
possible on tiles. What should Meg buy? Show all your math
thinking.
Picture Frames
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Picture Frames
Suggested Grade Span
3–5
Grade(s) in Which Task Was Piloted
4 and 5
Task
Meg is making picture frames by gluing red and blue square tiles on to wooden frames.
The 1st frame uses 8 tiles.
The 2nd frame uses 12 tiles.
The 3rd frame uses 16 tiles.
If this pattern continues, how many tiles will Meg need to make the 10th frame? How many tiles
will Meg need to make the 20th frame?
Meg wants 1/2 of her 20th frame to be blue tiles and 1/2 of the frame to be red tiles. Bags of 50
red tiles sell for $1.00. Bags of 25 blue tiles sell for 50 cents. A mixed bag of 50 red and 50 blue
Picture Frames
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tiles sells for $2.25. Meg wants to spend as little money as possible on tiles. What should Meg
buy? Show all your math thinking.
Alternative Versions of Task
More Accessible Version:
Refer to the original version of the task, and substitute:
If this pattern continues, how many tiles will Meg need to make the 4th frame?
If this pattern continues, how many tiles will Meg need to make the 5th frame?
If this pattern continues, how many tiles will Meg need to make the 10th frame?
Teacher Note:
See page 8 of the PDF to print a complete worksheet with graphics.
More Challenging Version:
Refer to the original version of the task, and substitute:
If this pattern continues, how many tiles will Meg need to make the 10th frame?
If this pattern continues, how many tiles will Meg need to make the 20th frame?
Write a rule for determining the number of tiles need to make any size frame.
Meg wants 1/2 of her 20th frame to be blue tiles and 1/2 of the frame to be red tiles. Bags of 50
red tiles sell for $1.00. Bags of 25 blue tiles sell for 50 cents. A mixed bag of 50 red and 50 blue
tiles sells for $2.25. Meg wants to spend as little money as possible on tiles. What should Meg
buy? Show all your math thinking.
Teacher Note:
See page 9 of the PDF to print a complete worksheet with graphics.
NCTM Content Standards and Evidence
Algebra Standard for Grades 3–5: Instructional programs from pre-kindergarten through grade
12 should enable students to ...
Understand patterns, relations and functions.
• NCTM Evidence: Describe, extend and make generalizations about geometric and
numeric patterns.
• Exemplars Task-Specific Evidence: This task requires students to identify how the
pattern in tiles is increasing and then to extend the pattern for the tenth and twentieth
frames.
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Number and Operations Standard for Grades 3–5: Instructional programs from prekindergarten through grade 12 should enable students to ...
Compute fluently and make reasonable estimates.
• NCTM Evidence: Develop fluency in adding, subtracting, multiplying and dividing whole
numbers.
• Exemplars Task-Specific Evidence: This task requires students to use computation to
determine which bags are the better deal.
Time/Context/Qualifiers/Tip(s) From Piloting Teacher
This is a medium-length task.
To print a copy of the more accessible version of the task with the graphic image, refer to page
8. To print a copy of the more challenging version of the task with the graphic image, refer to
page 9.
Links
This task would link well to craft projects and photography or could be given around school
picture time.
Common Strategies Used to Solve This Task
Most students make a chart which to record data. They analyze the data to identify a pattern
and then extend the pattern to draw conclusions. Students then use a computational approach
to determine the better deal on tiles.
Possible Solutions
Frame
Number
Number
of Tiles
1
8
2
12
3
16
10
44
20
84
Picture Frames
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Red bag: $1.00 ÷ 50 = $0.02 per tile
Blue bag: $0.50 ÷ 25 = $0.02 per tile
Mixed bag: $2.25 ÷ 100 = $0.0225 per tile
If a red bag and 2 blue bags are purchased, it will cost a total of $2.00. This is the better deal.
The mixed bag is $0.0025 more expensive per tile.
More Accessible Version Solution:
Frame
Number
Number
of Tiles
1
8
2
12
3
16
4
20
5
24
10
44
More Challenging Version Solution:
Frame
Number
Number
of Tiles
1
8
2
12
3
16
10
44
20
84
n
4(n + 1)
Red bag: $1.00 ÷ 50 = $0.02 per tile
Blue bag: $0.50 ÷ 25 = $0.02 per tile
Mixed bag: $2.25 ÷ 100 = $0.0225 per tile
If a red bag and 2 blue bags are purchased, it will only cost a total of $2.00. This is the better
deal. The mixed bag is $0.0025 more expensive per tile.
Task-Specific Assessment Notes
General Notes
This task lends itself well to students identifying a pattern and then generalizing it. Students
who do so accurately will achieve an Expert status.
Picture Frames
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Novice
The Novice will demonstrate little or no understanding of the task. Some attempt may be made
at drawing one or more picture frames, but the diagrams need to show evidence of leading to a
partially correct solution to move beyond Novice level. Reasoning errors in initial understanding
limit the student’s response.
Apprentice
The Apprentice will have a partially correct solution with a strategy that will work for solving the
task. The Apprentice may draw some correct picture frames that are free from reasoning errors.
The Apprentice may also correctly solve one part of the task, but not all.
Practitioner
The Practitioner will achieve a correct solution to the task and show supporting work. Math
representations such as charts or diagrams will be used as strategy and communication tools.
Expert
The Expert will not only achieve a correct solution to both parts of the task but will also utilize
other good problem-solving strategies, such as creating a rule to solve the task (4x + 4, with x
equal to the frame number) or verifying the solution by showing a comparison between rules
and diagrams.
Picture Frames
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7 of 14
More Accessible Version Worksheet
Meg is making picture frames by gluing red and blue square tiles on to wooden frames.
The 1st frame uses 8 tiles.
The 2nd frame uses 12 tiles.
The 3rd frame uses 16 tiles.
If this pattern continues, how many tiles will Meg need to make the 4th frame?
If this pattern continues, how many tiles will Meg need to make the 5th frame?
If this pattern continues, how many tiles will Meg need to make the 10th frame?
Picture Frames
Copyright 2008, Exemplars, Inc. All rights reserved.
8 of 14
More Challenging Version Worksheet
Meg is making picture frames by gluing red and blue square tiles on to wooden frames.
The 1st frame uses 8 tiles.
The 2nd frame uses 12 tiles.
The 3rd frame uses 16 tiles.
If this pattern continues, how many tiles will Meg need to make the 10th frame?
If this pattern continues, how many tiles will Meg need to make the 20th frame?
Write a rule for determining the number of tiles need to make any size frame.
Meg wants 1/2 of her 20th frame to be blue tiles and 1/2 of the frame to be red tiles. Bags of 50
red tiles sell for $1.00. Bags of 25 blue tiles sell for 50 cents. A mixed bag of 50 red and 50 blue
tiles sells for $2.25. Meg wants to spend as little money as possible on tiles. What should Meg
buy? Show all your math thinking.
Picture Frames
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Novice
Picture Frames
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Apprentice
Picture Frames
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Practitioner
Picture Frames
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Expert
Picture Frames
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Expert
Picture Frames
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