And then if you were to do a percent, well, this is 44 per 100, or 44/100, but even here I like lookĪt it as 44 per 100 or 44%. Spatial visualization involves the ability to image objects and pictures in the minds eye and to be able to mentally transform the positions and examine. Now, what about as a decimal? Well, 44/100, you could say, well, you have your ones place, and then this is the same thing. We could divide the numeratorĪnd the denominator by four, in which case you would get 11 over 25. Row is 10, 20, 30, 40, and then one, two, three, four. And how many of them are there? Well, let's see, this This is a 10 by 10 grid, so there's 100 equal sections here. So see if you can represent this as the part that's shaded Or another way of thinking about it, 60 per, instead of 100 you could say cent. You can actually pace it out on a life-sized number line or draw a picture. Multiply the numerator and the denominator by 10, that's the same thing as 60 per 100. In a measurement model, you have to pick a basic unit. Now, what about a percentage? Well, percent means per 100, so one way to thinkĪbout it is six over 10 is the same thing as what per 100? That is equal to, if we And so we have 6/10, so you could just put it right over there. What decimal would it be? Pause the video again and If you divide the numeratorĪnd the denominator by two, that's the same thing as three over five. So the blue represents 6/10 of a whole, or it represents, youĬould just say, 6/10. Nine, 10 equal sections, and six of them are filled in. Into one, two, three, four, five, six, seven, eight, Well, let's first thinkĪbout it as a fraction. ![]() Part that is shaded in blue as a fraction, as a decimal, and as a percent. Picture examples of some of the manipulatives for each math concept area can. What we're going to do in this video is try to represent the For students who have math learning problems, explicit teacher modeling of. Here are some of my favorite ways to incorporate the CRA model into a variety of math tasks.Assume that this entire square represents a whole. It will also mean that less intervention and re-teaching is needed. Once students can “see” the math in their heads, the abstract phase will be a natural, simple progression. Change your thinking – if the goal is flexible thinking, then the bulk of time should be spent with manipulatives.Remember that not every student thinks the same. During a whole class math talk, represent thinking in a variety of ways.Make manipulatives available in students’ table groups so that they are easily accessible for those who need them.Don’t store your manipulatives in a drawer and only bring them out for special occasions! These should be a regular part of your math instruction.Here are some tips to seamlessly incorporate the CRA model into your lessons: It helps to think about the CRA model as a Venn diagram rather than a sequential series of steps. This way you can be certain that you are differentiating for all your students, regardless of where they are in their understanding. When you teach a math lesson, make it your goal to incorporate concrete, representational, and abstract into the same lesson. We can help bridge this gap by giving our students opportunities with concrete materials so they construct the understanding that is essential to their future success in math.ĭESIGNING LESSONS WITH THE CRA MODEL IN MIND ![]() Instead of seeing 25 as two tens and five ones, they see it as literally a “2” and a “5.” This makes it very hard for them to make connections and see relationships. If you have students who are struggling with math, I encourage you to consider that the reason for their weakness could simply be that they don’t “see” the math in their heads. Have you? It is easy to see the abstract phase as that end goal that we rush to get to – but is it really our end goal? Or is the goal to help our students construct their understanding and become flexible thinkers? I know that I have been guilty of rushing through concrete activities to get to abstract activities faster. I believe that the reason for this gap is a lack of focus on concrete learning. Why do some students just seem to “get” math and some never do? It’s no secret that there is a huge gap in a lot of our students’ mathematical understanding and fluency. For example, the base ten blocks could now be represented as an equation. In the abstract phase, we represent our thinking with digits and symbols. For example, we could represent the base tens from the previous picture with a drawing of base ten blocks. In the representational phase, we draw representations. An example of this might be base ten blocks to represent an addition expression. Students should be able to move and manipulate 3D objects to represent their thinking. In the concrete phase, we focus on using hands-on manipulatives. The CRA Model is an instructional approach for teaching math.
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