Mental Math: "How Many to 100?" on the 100-Dot Array

I mentioned in an earlier post that one of my favourite tools to use with students is the 100-dot array. I want to share one way that I use it to help students develop some mental math skills. I have used this with students as young as grade two, but if older students have developed few mental math skills, this is a great way to add to their strategies. The fact that this mental math is grounded in the use of a visual tool allows every student to have success with the method. I have traditionally done this strategy using an overhead and a transparency of the 100-chart, but if you have access to a document camera, you can easily use that. The other thing I usually do while doing this activity with students is have them write their responses on some kind of response boards (e.g., small chalk boards, small white boards, etc). It is a great way to incorporate formative assessment in the math classroom. As students write their responses, you can see at a glance who is “getting it” quickly and who needs a bit more help. I should mention, too, that the first time I use the 100-dot array with any group of students I always spend 10-15 minutes having the kids notice things about the array and talk about it. We pay attention to the 10’s, the 5’s the 25’s and the 50’s that are displayed on the chart. It is an important step if you are going to use the tool for any activity. So, now to the actual activity. I begin by displaying the 100-dot array on the overhead, the I cover some rows with a piece of paper. I ask two questions: How many do you see? and How many to 100? I look for the answers in two forms: the number of tens the students see, and the number of dots they see. For instance, if I have covered 3 rows, I want students to say they see 7 rows of 10 dots, or 70 dots. The “how many to 100?” is answered by 3 rows of 10 dots or 30 dots. Each answer uses the relationship of 10 as an anchor, which is one of the foundational number relationships that students need to develop. Depending on the age and quickness of the students, I spend the first number of days doing this part of the activity. The second stage is to use two pieces of paper to cover dots. With the first, I cover full rows as before. With the second I cover part of the last exposed row. So, from the previous example, if I were covering 3 full rows, I would go on and cover part of the next row, say 8 more dots. Now when I ask, “How many do you see?” the answer is 62 (6 full rows and 2 more dots). When I ask, “How many to 100?” students must “complete” two types of tens: in the individual row and in the number of rows. So, completing the row in the example there are 8, and then there are 3 full rows covered, for a total of 38 needed to make 100. The year I taught grade 2 I did this activity regularly throughout the year. By late spring I could just say to my students (without showing the 100-dot array), if you start at 57, how many do you need to make 100? Across the whole classroom, almost as one, the eyes of every student would close as they would visualize the array. Heads would swivel and bob as students were completing the 10’s, and they all could find the needed amount to make 100. It’s an easy way to build some mental math skills with your students. I recommend doing it fairly regularly as a warm-up activity. Build these skills, and then build some more! Below  are some links to download some 100 dot arrays in different sizes. (Students use the small ones all the time in my class. When they are problem solving, if their strategy makes use of the array, they grab a small one from the basket at the back of the classroom and glue it onto the sheet they so they can represent their thinking. Choose a size that works for your students!) one large 100 dot array 4 small 100 dot arrays 6 small 100 dot arrays 12 small 100 dot arrays Mathematically yours, Carollee PS The link for the 100 dot arrays was not working for a while (thanks, Pat, for pointing out the glitch!), but I think I have it fixed now! — at least it downloaded OK for me. Let me know if you have more trouble.