It is no secret that Quarter the Cross is one of my favourite tasks. I’ve written about it twice before: as a Day 1 activity and in connection with Fraction Talks. The original source is apparently T. Dekker & N. Querelle, 2002, Great Assessment Problems (www.fi.uu.nl/catch). It has proliferated in recent years, including with an active Twitter hashtag: #QuarterTheCross.
Quarter the Cross promotes creative thinking, encourages students to find multiple ways of achieving the same goal, and compels students to justify their reasoning.
In the task, students verbally justify their visual representation of one quarter of the cross. This connects two representations together: verbal and visual. So far, students have been considering the cross as one whole.
We can incorporate a richer symbolic representation by considering the cross as composed of five smaller squares. (I was originally inspired by David Butler’s mention of this in a blog post.) In this second method, if one square is a unit, then what does it mean to ‘quarter the cross’? We can colour 1/4 of one unit and do this five times. This is represented symbolically as 5/4 = 5×(1/4). One way to represent this visually is shown below.
We can now explore different ways of expressing 5/4, and connect them to corresponding visual representations. For example, 5/4 = 1 + 1/4. Colour in one whole unit and a quarter of another. One way to represent this visually is shown below.
In the past, I’ve shown students a few different ways to symbolically express 5/4 and their corresponding visual representations. Occasionally I’ll pose one as a challenge. (‘Show me a visual representation of 5/4 = 2×(5/8)’.)
This year I wanted to try having students actively connect the representations. Inspired by Maureen Hegarty, I decided to have students try a card sort, with the following instructions:
- Cut out the cards.
- Match each visual representation to the symbolic representation. (You can write your own card, if needed.)
- When you are finished, try writing alternative expressions using different operations.
The pages are shown below. You can also download the entire PDF here.
What I found interesting is that because all representations show 5/4, students are focused on which symbolic representation best suits a particular cross. They also naturally debate whether more than one symbolic expression could be used. Next time I might include a few easier representations (students found some of these to be very challenging). In summary, I loved this way the discussion was mathematically rich, focused on connecting representations, and encouraged students to justify their reasoning to one another.