Math Pre KK
Justin's Placemats
Task
Justin makes a placemat from a piece of paper just like the piece of paper you have. Justin draws a different patternblock shape on each corner of the placemat. What could Justin's placemat look like? Show and tell how you know.
Alternative Versions
More Accessible Versions:
Justin makes a placemat from a piece of paper just like the piece of paper you have. Justin draws a patternblock shape on each corner of the placement. What could Justin's placemat look like? Show and tell how you know.
More Challenging:
Justin makes two placemats from two pieces of paper just like the two pieces of paper you have. Justin draws a patternblock shape on each corner of a placemat. Justin makes each placemat the same. What could Justin's two placemats look like? Show and tell how you know.
NCTM Content Standards and Evidence
Instructional programs from prekindergarten through grade 12 should enable all students to:
 Recognize and name some variations of the circle, square, triangle and rectangle
 Understand ideas such as over, under, corner and locate these positions
 In this task, learners must be able to draw and name four different shapes, as well as identify the corners of the given rectangle.
Links
This task would link well to a celebration of Thanksgiving in which the children have worked to design placemats. Instruction in naming, describing and drawing shapes needs to have taken place.
Support
A child may need pattern blocks, patternblock paper shapes, or stamps to represent/model the shapes to put on each corner of the placemat. The teacher can take a picture of the child's model. (Many children will select paper, pencil, crayon, etc. to show their solution.) A teacher, older student, paraprofessional, volunteer, etc. should scribe the child's solution so the teacher will have a complete record of the child's reasoning.
Task Specific Assessment Notes
General Notes: Many children will solve this task by placing patternblock shapes on the corners of the paper provided and then trace the patternblock shape. Other children may glue paper patternblock shapes on the corners or use patternblock stamps. Please note that all pattern blocks are not colored the same. Some companies will color the triangle green and others red. A child could decide to color the patternblock shapes on her/his paper, and the color choice does not matter.
Task Specific Rubric/Benchmark Descriptors
Click on a level for student example. 


Novice  The Novice will be unable to solve the task and could draw patternblock shapes all over her/his paper or could draw shapes other than patternblock shapes. 
Apprentice  The Apprentice will be able to partially solve the task. S/he will understand that the task involves patternblock shapes but not understand the property of corner (vertex, angle). The Apprentice could demonstrate understanding of corner but not be able to place a different patternblock shape in each corner. The Apprentice will attempt to communicate her/his reasoning by using a mathematical language term and/or number. The Apprentice will also attempt to make an appropriate representation. A connection may be attempted but it will not be mathematically relevant to the task. 
Practitioner  All the Practitioner criteria are evident and the Expert will be able to demonstrate a deeper understanding of the mathematical concept of geometric properties of patternblock shapes. The Expert will also bring more mathematical language and/or numbers to the task than the Practitioner. Terms could include, but are not limited to  diagram, key, model, triangle, square, rhombus, trapezoid, hexagon, sides, angle, vertex, even, odd, more than, less than. The Expert will often use her/his representation to explore the underlying mathematical concepts in the problem. The Expert could, but is not limited to, conclude that the triangle has the least number of sides of all the pattern blocks and the hexagon has the most; that the hexagon is the "largest" shape to place on the placemat; that there are two remaining shapes (if you count the two rhombus shapes separately) for the next placemat and then you have to repeat shapes; and that each shape on the placemat will have four sides if you use the square, trapezoid, and two rhombi. 
Expert  The Practitioner will be able to correctly solve the task by demonstrating that a different patternblock shape is on each corner of the paper. This would indicate a "correct answer." The Practitioner will use mathematical language and/or numbers. Terms could include but, are not limited to  diagram, square, triangle, rhombus, trapezoid, hexagon. The Practitioner will be able to construct an appropriate and accurate representation (many children will begin with a model and then represent the shapes with paper, tracing, stamps). The Practitioner will make a mathematically relevant observation (connection about her/his solution) such as there are four different shapes because there are four corners, the triangle only has three sides. 
Novice
Problem Solving/Reasoning/Proof
The child's placemat showing a hexagon patternblock shape near a corner ("This is the sun."), flowers and rain would not work to solve this problem and demonstrates incorrect reasoning.
(Novice)
Communication/Representation
The child uses no mathematical language or numbers.
(Novice)
The child's drawing cannot be considered a representation because the diagram does not support any mathematical reasoning. The one pattern block (hexagon) is not considered for a corner but to show the "sun."
(Apprentice)
Connections
The child is unable to make a mathematically relevant observation because s/he demonstrates no understanding of the underlying mathematical concept of "corner" or different patternblock shapes for each corner.
(Novice)
*Overall assessment
Apprentice
Problem Solving/Reasoning/Proof
The child's strategy of diagramming patternblock shapes on each corner indicates correct reasoning for "corner." The child does not place a different patternblock shape in each corner even when the task was reread using a new term for "different." The child is not able to arrive at a complete correct answer.
(Apprentice)
Communication/Representation
The child correctly uses the mathematical terms  square, triangle, rectangle and orally counts the shapes from one to four correctly.
(Practitioner)
The child diagrams a patternblock shape in each corner of her/his paper. This representation is appropriate but not accurate. The child does not indicate a different patternblock shape in each corner. Two corners have a square pattern block.
(Apprentice)
Connections
The child solved the problem and made a mathematically relevant observation about her/his solution. The child states, "Justin's paper is a rectangle. That is not a patternblock shape."
(Practitioner)
*Overall assessment
Practitioner
Problem Solving/Reasoning/Proof
The child's strategy of diagramming a square, trapezoid, triangle and rhombus on the placemat would work to solve this problem. The child's answer is correct because s/he used four different patternblock shapes.
(Practitioner)
Communication/Representation
The child correctly uses the mathematical terms  rectangle, square, shapes, triangle and sides. The child's placement of four different patternblock shapes on the placemat and tracing them to complete the diagram is appropriate and accurate to the task.
(Practitioner)
Connections
The child makes mathematically relevant observations about the properties of the patternblock shapes. "They are all different but three shapes got four sides. The triangle only has three sides," and, "This one (the hexagon) has six sides."
(Practitioner)
*Overall assessment
Expert
Problem Solving/Reasoning/Proof
The child's strategy of putting patternblock shapes in each corner of the paper placemat to finish the diagram would work to solve the problem and the answer is correct. The child uses four different shapes. The child then uses the geometric properties of the shapes to refl ect how the placemat could look as well as determining the surface area of each patternblock shape.
(Expert)
Communication/Representation
The child correctly uses the mathematical terms  right, square, left, rhombus, triangle, corner, shapes, sides, rectangle, circle and row. The child correctly notates each shape from one to four. The child's ability to use so many correct geometric terms earns an Expert Level. The child completes the diagram by correctly placing four different patternblock shapes in each corner. The child then makes a diagram to show which shapes have more than four sides and which shapes have four sides. The child makes another diagram to show the order of some patternblock shapes from "smallest" to "biggest."
(Expert)
Connections
The child demonstrates a strong understanding of the geometric properties of patternblock shapes as well as correctly naming the square, rhombus, triangle, as well as rectangle and circle. The child explores the geometric property of side and arranges the pattern blocks correctly in her/his diagram. Using this information the child states, "Justin can't just use four sides because the placemat has four corners. He needs four of them. Justin can do 13 side (one, threesided) 16 side (one, sixsided) and 24 sides (two, foursided) if he wants."
The child comments that there are no rectangle and circle pattern blocks and decides that there are more missing patternblock shapes. The child then determines and places the patternblock shapes in order according to the surface area of each. "Now I put them in a row from smallest to biggest." The child shares her/his thinking. "Put the triangle on the square. You still see orange. I can put three triangles on the red one and lots on the yellow."
(The rhombus does have more surface area than the square. Older children would find the area of a triangle and double the area to determine the area of the rhombus. Children would find the area of the square and compare the two shapes.) Visually comparing the area of a rhombus and square pattern block is difficult, but this young mathematician is beginning her/his geometric journey.
(Expert)
*Overall assessment