Standards-based assessment and Instruction

# Math Pre K-K

## Counting Cars

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 pre-kindergarten 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.

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.

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## Here's What People Are Saying

The Southwest Vermont Supervisory Union (SVSU) in Bennington was trying to help its local community gain a better understanding of what students need to demonstrate in order to be successful on their math assessments. One of the most difficult sections of the exam is problem solving. Here, students are not only expected to determine the correct strategy, but also to explain their answers in writing, using math language and representation. For four years SVSU has used Exemplars as a curriculum tool in teaching its students how to solve these types of problems...

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