Nikki Romano - Factors Task

My students have been working on multiplication, especially factors and the multiplication families. The students have to list in numerical order the factors for each number listed on the worksheet. The students are given colored squares, shown in the image above to help them find the factors. For example, if the students are trying to find the factors for the number 10, the students have to use 10 colored squares. With the 10 squares they have to make equal rows and lines with no extra squares hanging off the sides. For example, the easiest factor to make is one row of 10 and then making two rows of 5 squares. The students continue checking factors by making more rows so for making 3 rows, they will discover that there will be extra squares and can't make an even row line ratio. The purpose of the this task is to turn the concept of factors into something concrete. By using the colored squares the students can physically create a number using the different numbers (factors) and discover for themselves what doesn't work. 

One possible effective way to approach this problem is for the teacher to model the activity and practice with the students before letting them accomplish it by themselves. I had to explain a lot and give a lot of guidance during this task because the students didn't understand the role the colored squares represented. Another possible way to approach this problem is to along with writing down the factors for each number draw a picture for each factor that the squares look like in each example. This way the students are practicing writing the factors and drawing what they look like; various way to represent something. One anticipated approach my students took to this problem was writing the factors as multiplication problems. For example for 32, the students would write 8x3 3x8. This is correct but the students were unable to come up with the maximum amount of factors this way. They thought of all the different ways they could remember but without double checking using the colored square method they  missed many factors. Another anticipated student approach to this problem is to get tired of moving the colored squares every time to double check if the answer is correct. This could lead to many mistakes because of lack of motivation and the students wanting to complete this as fast as possible. 

With much guidance and support this student was able to accomplish this task. With this case this student waited and I had to make sure to check each square formation she made instead of having confidence and moving on. This student was able to make rows that represent one of the factors and had to make a even square formation. When I ask the student questions like can you count by 5s to get to 32? She was able to answer no and was able to skip checking with rows of 5 by answering and thinking of mathematical questions like that. This student also demonstrates that she can order the factors in numerical order which shows she listens to directions and can list the maximum amount of factors. She also was able to create the correct square formation which shows she is understands her task and when she gets an extra square that is not the correct answer. One way to advance the students thinking is to give them the list of factors and have the students identify what number belongs to those specific factors. Another way to advance this students' thinking is to give numbers that have many factors and maybe that the students have not been exposed to in order to figure out the factors. This is a great way to introduce higher numbers with more difficult factor families. 

3 questions about this artifact:

1. Can this student produce the complete list of factors for each number without the squares?

- can use this method to practice and after much practice, give an assessment by asking the students to list factors to numbers from practice and similar numbers.

2. Does every student in the class understand the purpose of the squares and how it can help?

- the teacher can express during the introduction of this method why the squares are being used. The students can discuss what the squares represent and why they think it is important and useful to use. Then the students can be assessed by asking them to write a sentence explaining their thoughts of what the squares mean and their purpose in this task.

3. Can the student use this method to find factors for numbers they are not so familiar with?

- after practice and assessments with frequently used numbers, use an assessment with bigger numbers and see how they do. But during this assessment give them the squares in order to assist them with discovering the factors.