Math Pre KK
Counting Cars
Task
Max counts cars going past his house. First, Max counts 3 cars. Next, Max counts 1 car. Last, Max counts 2 cars. How many cars did Max count going past his house? Show and tell how you know.
Alternative Versions
More Accessible Versions:
Max counts cars going past his house. First, Max counts 2 cars. Next, Max counts 1 car. How many cars did Max count going past his house? Show and tell how you know.
More Challenging:
Max counts cars going past his house. First, Max counts four cars. Next, Max counts three cars. Last, Max counts one car. How many cars did Max count going past his house? Show and tell how you know.
NCTM Content Standards and Evidence
Instructional programs from prekindergarten through grade 12 should enable all students to:
 Solve problems using informal counting strategies up to totals of 10.
 Solve problems which require the joining of two or more sets of objects in order to find the whole.
 In this task, learners must be able to combine three, one and two objects and determine the total of six.
Links
This task would link well to a unit on transportation. Instruction allowing children to combine sets of objects should have taken place prior to teaching this lesson.
Support
A child can be given manipulatives or toy cars to represent/model the six cars and can be encouraged to transfer the model to paper if s/he is comfortable doing so, or 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 there is a complete record of the child's reasoning.
Task Specific Assessment Notes
General Notes: Many children will solve this task by diagramming the cars and either count on or use addition to find a total of six cars.
Task Specific Rubric/Benchmark Descriptors
Click on a level for student example. 


Novice  The Novice will be unable to solve the task and could simply draw a “picture” of a car(s). No understanding of the underlying mathematics of the task will be evident. 
Apprentice  The Apprentice will be able to partially solve the task. S/he will understand that the task involves three cars, one car and two cars. The child may not be able to find the total number of cars. The child could also model/diagram six cars but not have the correct sets. 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  The Practitioner will be able to correctly solve the task by demonstrating a total of six cars. The Practitioner will use mathematical language and/or numbers. Terms could include, but are not limited to, total, more than, less than, 1, 2, 3 ..., first, next and last. The Practitioner will be able to construct an appropriate and accurate representation (usually a diagram, but could also use a model or table). The Practitioner will be able to make a mathematically relevant observation (connection) about her/his solution, such as three is the most cars that Max counted at one time. 
Expert  All the Practitioner criteria are evident and the Expert will be able to demonstrate a deeper understanding of the mathematical concept of sets or addition in the task. 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, table, first, next, last, second …, even, odd, pair, total, sum, equal, fair share, pattern and equation. The Expert will often use her/his representation to explore the underlying mathematical concepts in the task. The Expert could, but is not limited to, conclude that you need to add two cars to the second count and one car to the third count for Max to have seen the same number of cars each time; recreate the task with different numbers of cars at each counting; conclude that Max saw an even number of cars two times and an odd number of cars one time. The Expert could construct a new representation or make an equation to verify her/his answer or relate the “Counting Cars” task to a similar task and state the mathematical similarities. 
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