Structure Spotlight
Size Model : Next Steps
Exploring Number Relationships
The Size Model is a bar model that has equal-sized cells in each row. Each horizontal row represents the same total quantity. So in the model here, the sum of the cells in row C is the same for the sum of row B or A.
The Size model always has equal-sized groups. That is, each row will have the same number in each cell.
Whole-Half-Quarter Size Model
Whole-half models, continued
As you move to 2-D representations, continue to revisit the 3-D physical connection by modeling the quantities with cubes or other same-sized units, such as square tiles. Cubes work well to stack on top of each other to show the connections between 2-D and 3-D. You can use color here to show the different parts, or a marker line to show the delineation.
Focus on the relationship of double and half. This relationship is more complex than it may seem! For instance, when we discuss half of something, we often speak about it in the singular, as one thing. If we say, “find half of 28,” we are cognitively expecting one number, 14, although there are of course two halves. When operating, as well, when we divide by 2 the result is one number which is half of the original. These models can help show the two halves, even when we speak about half as one number.
Try it: Activity
Young students may also have a hard time realizing the amount that half is of its double. Have you ever asked a child to find half of a number and they come up with a number that is much larger than half? It can be hard to think about half being so small in comparison!
To illustrate the relationship between whole and half even more, performing an action along with the discussion will allow students to experience the proportion between them.
Try it: Give students strips of cut grid paper with a perfectly cut even number of squares, such as 12 or 20. The squares should be large enough so that the strip feels significant in their hands, such as 1-inch boxes. Ask students to hold their strips horizontally and vertically, getting a sense of the total number of squares. They can count them if they would like, but the specific number is unimportant.
Demonstrate with your strip. Hold the strip up and let it hang down vertically. Bring the top of the strip down to meet the bottom exactly and fold at the mid-point. Model several times the opening and closing of the strip by keeping the bottom half where it is and drawing the top half up.
You can also show this with the strip held horizontally. Bring one side over the meet other and note the difference. Here, the work is the visual intake of noticing the proportional difference as we see something’s surface area reduced by half, and then doubled back to the original. We also mimic the concept of halving or doubling number (or the operations of dividing by 2/multiplying by 2) as we repeatedly open and close the strips. Our minds are experiencing the relationship through our eyes and our hands as we imprint the idea of “half” into our brain. Open & close the strips many times.
You can note with the students how the squares match up; each has a partner. This is one definition of an even number: it can be broken evenly into groups of 2.
Next, relate the strips of paper to the Size model with their same number of total squares. For example, if they had a strip of 12 squares, they can now outline 12 squares on a piece of grid paper and draw another row under it, same size, but with a divider line in the middle. They can label each cell of the model with their specific numbers (ex: 12; 6, 6).
Note: we started with even numbers, so students will have whole-number halves (not like 2.5, for example) but the conversation about half of an odd number may come up. Ensure that they know that there is half of every number but right now we are working with whole-number halves.
Ask students to share their models. Some may even notice that their whole is another person’s half. This relationship will be explored more in the next post!
So much of math is noticing what changes and what stays the same...
