Tony Gardiner did some hunting and found a document containing PISA’s released mathematics items for 2022. In a comment on our PISA non-post, Tony made some remarks about a triple of triangle questions contained in the document. The questions, each with the same triangle picture prompt, are below. The questions are followed by PISA’s discussion. The detailed marking rubric can also be found in the items document (pp 9-14). Note that PISA is a test for 15-year olds; as with all such tests, the questions are intended to range from easy to difficult.
1. For the first item, students are asked to compute the percentage of blue triangles shown in the first four rows of the pattern. There are six blue triangles and 16 total triangles, so the percentage of blue triangles is 37.5% (6 ÷ 16 = 0.375). This is an easy item (Level 1a) and is intended to get students thinking about the pattern by employing a simple algorithm with all information shown.
2. The second item in the unit builds off the first item by again asking students to compute the percentage of blue triangles, but this time it is based on five rows of the pattern. Since the fifth row is not shown, students have to extend the pattern by one row to determine new values for the number of blue triangles and the total number of triangles. With five rows, the percentage of blue triangles is 40.0% (10 blue triangles ÷ 25 total triangles).
This item is intended to be easy and to get students thinking about extending the pattern beyond what is shown, but not extending the pattern so that it requires generalising. This is a Level 2 item, so it is slightly more difficult than the first item in the unit, possibly because it requires working with a part of the pattern that is not shown but is still an overall easy item for students.
3. [The third item] is the final item in this unit, and it builds off the previous two items to now generalise with the pattern. The task for the students is to evaluate a claim that the percentage of blue triangles in the pattern will always be less than 50% as more rows are added. Students have to select either “Yes” or “No” to indicate if the claim is or is not true, but then they also have to provide an explanation to support their selection. This is a reasoning item that requires students to analyse the pattern to recognise a relationship between the number of red and the number of blue triangles in each row, and then use that relationship to support their selection.
The correct selection is “Yes,” that the claim is true, and an acceptable explanation recognises that the number of red triangles in each row will always be greater than the number of blue triangles in each row. Note that students can phrase their response in terms of either the number of blue triangles being fewer or the number of red triangles being greater, as long as there is some language indicating that this relationship is true for every row. Partial-credit responses to this item generally either focus on just the first row, which contains only a red triangle, or do not clearly communicate that the relationship between the number of each color triangle applies to every row.