Standards-based assessment and Instruction

# Spanish Math 3-5

## Bulletin Board Border

Please help me. I would like to make a geometry bulletin board that has a border of circles, triangles and squares. I know that 20 shapes will fit across the board and that 12 shapes will fit down the board. If I start in the top left- hand corner with a circle followed by a triangle then a square and repeat this pattern all around the board, how many of each shape will I need?

Explain your solution using words and pictures.

#### More Accessible Version:

Please help me. I would like to make a geometry boarder on the wall across my classroom. I would like to use a pattern of circles, triangles and squares. I know that 21 shapes will fit across the wall. If I start with a circle, followed by a triangle, then a square and repeat this pattern, how many of each shape will I need? Explain your solution using words and pictures.

#### More Challenging Version:

Please help me. I would like to make a geometry bulletin board that is surrounded by a border of polygons. I will start in the top left-hand corner with a 4-sided shape, followed by a 5-sided shape, then a 6-sided and so on all the way up to a 10-sided shape. Once I get to a 10-sided shape I will start the pattern again with a 4-sided shape. I know that 20 shapes will fit across the board, and that 12 shapes will fit down the board. How many of each shape will I need to surround the bulletin board? Explain your solution using words and pictures.

### Context

This problem worked well because it allows students to diagram the problem and show all their work. Students first thought they could add all the numbers given and solve the problem without a diagram. When they became involved in the problem solving they realized why drawing clear diagrams is so useful.

This task allows students to explore a real-life problem using perimeter. It assesses their ability to take information given and apply it to a diagram.

### What Students Will Do

The task provides information that most students will need to diagram. Some students in my fourth-grade class had difficulty placing the corner pieces and counted them twice. The more accurate the diagram, the more accurate the solution.

Approximately 60 minutes

None, this is strictly a problem-solving task.

### Teaching Tips

Teachers tell students that a diagram will help them with problem solving, but we often get, “I did it in my head.” This problem allows students to draw a simple diagram to a challenging problem. It allows students the opportunity to actually make the border on a bulletin board.

### NCTM Standards

• Numbers and Operations
• Geometry and Measurement

### Concepts to be Assessed and Skills to be Developed

• Problem solving
• Reasoning
• Communication
• Patterns
• Perimeter

### Suggested Materials

• Paper
• Pencil
• Actual shapes may be used

### Possible Solutions

#### Original Version:

Students may conclude that 64 shapes are needed. They will add the numbers given in the task and draw a diagram that represents their interpretation. Hopefully by drawing a diagram, students will use the corner pieces correctly, and conclude that 60 shapes are needed, 20 of each shape.

#### More Accessible Version:

21 ÷ 3 shapes = 7 of each shape

#### More Challenging Version:

20 + 20 + 10 + 10= 60 shapes around board ÷ 7 different polygons = 8 polygons with 4 shapes having an extra one:

Trapezoids = 9
Hexagon = 9
Septagon = 9
Octagon = 9
Nonagon = 9
Decagon = 8

Click on a level for student example.
Novice The Novice will use inappropriate concepts and procedures to solve the problem (s/ he may have multiplied 20 by 12 to get 240). There will use little evidence in the explanation of a strategy or reasoning. The diagram will not relate to the problem (there will be no evidence of a border).
Apprentice The Apprentice will understand part of the problem and will show some mathematical reasoning, (using a pattern of shapes for the border) but will not use the corner pieces as a continuation of the pattern. There will be some use of a diagram and mathematical notation.
Practitioner The Practitioner will have an understanding of the problem and a strategy will be used that successfully solves all parts of the task. The student will use an accurate pattern of geometric shapes in the corners and a connection, observation or verification of the solution will be made.
Expert The Expert will have a clear understanding of the problem and all of the parameters. The pattern of geometric shapes will continue around the corners. Accurate mathematical representation will be shown, and mathematical reasoning will reflect refined reasoning skills. A connection, observation or verification of the solution will be made.

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"I have to admit that this workshop was PHENOMENAL... I feel so fortunate to have had the opportunity to attend this workshop with my fellow Atlanta Public Schools colleagues. Not only was it insightful and informative, but the presentation of the material covered was creatively done. She [the instructor] allowed us to make many of the activities she discussed. It seems as if both facilitators believe in the educational proverb: 'Tell me and I forget, teach me and I remember, involve me and I understand.'"

B. Cummings

Mathematics Instructional Coach

Atlanta Public Schools

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