Strategy 1: Explore together.
The first time you expose students to a particular tool, take some time as a class to explore this tool with a mathematical eye. Imagine putting the tool out for students to explore independently but alongside others who are exploring, too. After a few minutes, have students turn and talk, or get into small groups to talk about how they see the tool helping them with math. As a large group, discuss and brainstorm some of the uses of the tool.
SAMPLE ACTIVITY - Grade 2+ Geoboards
Give students the opportunity to look at the geoboard without an elastic. Ask them to look for as much information as they can on the geoboard (number of pegs, number of squares, size, layout, etc) There is a great article by Marilyn Burns about the first time she introduced geoboards to students at the link below. In fact, it's a link to a publication called "Hands on help from Marilyn Burns". For interest's sake, note that the publication date is 1996!
Once the students have had some time to explore the geoboard and notice the details, have a large group discussion about their findings. Then, ask them to look at their findings and think about how they could use this as a math tool. A few years ago, I had the opportunity to do this exploration with over 90 classrooms. The thing that blew my mind was that in each case (in classrooms from Grades 2 to 6), students came up with great ways to use the geoboard that related to all 5 strands of math. Often, their teachers were totally surprised at the ideas they came up with. Usually, we used their thinking as a springboard to come up with a follow-up lesson. You'll see an example of such an activity, below.
SAMPLE ACTIVITY - Grade 1 - Cuisenaire Rods
One of my colleagues introduced cuisenaire rods to her students with an exploration activity. During their exploration, they noticed that the rods could be organized by length, and that there was a relationship in their lengths (each one is 1 cm longer than the other). Because of this exploration, the students were able to assign values from 1 to 10 to the rods. The students were looking at number bonds, and they began to use cuisenaire rods to find all of the possible ways to make 7, for example. Not only were they able to model the 7s using the rods, but they were also able to represent their rods as numerical expressions, and to engage in rich discussion as one combination led to another.
SAMPLE ACTIVITY - Grade 3/4 - Geoboards
As students begin to formalize their understanding of fractions, it can be helpful for them to see that a geoboard can be divided into 2 halves, and that the two halves may not look alike, but each half will have the same area. One way to explore this concept is to ask students to show one half on the geoboard. Students can record the different ways to show one half either on dot paper, or by taking photos of their geoboard and posting it on an electronic bulletin board, or by using a single device to photograph their work. There's a great geoboard app (it's free!) that can also be really handy for this, as students can screen shot their work and then compile a slideshow of examples, or find another way to compare their different solutions.