Question

part a
which of the following could be used to help find the product of 48 × 36 using an area model? select all that apply.
48
24
180
240
1,200
120

Explanation:

Step1: Recall Area Model for Multiplication

The area model for \(48\times36\) involves breaking down the numbers into tens and ones. We can write \(48 = 40 + 8\) and \(36=30 + 6\). Then the area model is a rectangle with length \(48\) and width \(36\), and we can divide it into four smaller rectangles: \(40\times30\), \(40\times6\), \(8\times30\), and \(8\times6\). We can also break the numbers in other ways, for example, \(48 = 24\times2\) and \(36 = 18\times2\), but let's check the given numbers.

First, let's calculate the products from the area model breakdown:
- \(40\times30=1200\)
- \(40\times6 = 240\)
- \(8\times30=240\) (wait, no, \(8\times30 = 240\)? Wait \(8\times30=240\), \(40\times6 = 240\), \(40\times30 = 1200\), \(8\times6 = 48\). Also, if we break \(48\) as \(48\) (the original number) and maybe other breakdowns. Let's check each option:
- \(48\): The original number, in the area model, the length or width can be \(48\), so it can be used.
- \(24\): If we break \(48\) into \(24\times2\) and \(36\) into \(18\times2\), but more directly, if we consider \(48\times36=(24\times2)\times36 = 24\times(2\times36)=24\times72\), but maybe not. Wait, another breakdown: \(48 = 48\), \(36 = 30+6\), but let's check the products from the standard breakdown:
- \(40\times30 = 1200\) (so \(1200\) is a product in the area model)
- \(40\times6=240\) (so \(240\) is a product in the area model)
- \(8\times30 = 240\) (same as above)
- \(8\times6 = 48\) (so \(48\) is a product in the area model)
- What about \(180\)? Let's see, if we break \(48\) as \(24\times2\) and \(36\) as \(15\times2.4\) no, that's not integer. Wait, maybe another breakdown: \(48 = 12\times4\), \(36 = 15\times2.4\) no. Wait, let's check the numbers:
- \(48\): Yes, because the area model uses the two factors \(48\) and \(36\) as the length and width, so \(48\) is part of the model.
- \(24\): If we consider \(48 = 24\times2\), then when calculating \(48\times36\), we can write it as \(24\times2\times36=24\times72\), but maybe not directly. Wait, no, let's go back to the standard area model with \(48 = 40 + 8\) and \(36=30 + 6\):
- The four sub - rectangles have areas: \(40\times30 = 1200\), \(40\times6=240\), \(8\times30 = 240\), \(8\times6 = 48\).
- So \(48\) (from \(8\times6\)), \(240\) (from \(40\times6\) or \(8\times30\)), \(1200\) (from \(40\times30\)) are products in the area model.
- \(180\): Let's see if \(180\) can be a product. \(48\times3.75 = 180\), but \(3.75\) is not an integer. So \(180\) is not a product from the standard area model breakdown.
- \(24\): Let's see, \(48\times3.75 = 180\), no. Wait, maybe I made a mistake. Wait \(48\times36 = 1728\). Let's check the options:
- \(48\): Yes, as one of the factors.
- \(24\): If we do \(48\times36=(24\times2)\times36 = 24\times72\), but \(72\) is not a given number. Wait, no, maybe another way: \(36\times5 = 180\), but \(48\times5 = 240\). Wait, no, the correct products from the area model (breaking into tens and ones) are \(40\times30 = 1200\), \(40\times6=240\), \(8\times30 = 240\), \(8\times6 = 48\). So the numbers that are products in the area model are \(48\), \(240\), \(1200\). Let's check each option:
- \(48\): Yes, because \(8\times6 = 48\) (a sub - product in the area model).
- \(24\): Is \(24\) a sub - product? Let's see, \(48\times0.5 = 24\), but \(0.5\) is not a factor in the area model (which uses whole numbers for breaking into tens and ones). So \(24\) is not a sub - product. Wait, maybe I was wrong. Wait \(48\times36=(24\times2)\times(18\times2)=24\times18\times4\), no. Wait, maybe the problem is not just sub - products but also the factors or intermediate values. Wait, the area model can also be thought of as a rectangle with length \(48\) and width \(36\), so \(48\) is a side length, so it can be used. \(1200\) is \(40\times30\), a sub - rectangle area. \(240\) is \(40\times6\) or \(8\times30\), a sub - rectangle area. \(48\) is \(8\times6\), a sub - rectangle area. So the correct options are \(48\), \(240\), \(1200\). Wait, let's re - calculate:
- \(40\times30 = 1200\) (so \(1200\) is valid)
- \(40\times6=240\) (so \(240\) is valid)
- \(8\times30 = 240\) (same as above)
- \(8\times6 = 48\) (so \(48\) is valid)
- \(24\): Let's see, \(48\div2 = 24\), \(36\div1.5 = 24\), but \(1.5\) is not an integer, so \(24\) is not a sub - product or a factor in the standard area model breakdown with whole numbers.
- \(180\): \(48\times3.75 = 180\), not an integer factor, so not valid.
- \(120\): \(48\times2.5 = 120\), not an integer factor, so not valid.

Step2: Conclusion on Each Option

- \(48\): Yes, as \(8\times6 = 48\) (sub - product) or as a side length.
- \(24\): No, not a sub - product or a factor in the area model breakdown with whole numbers.
- \(180\): No, not a sub - product or a factor.
- \(240\): Yes, as \(40\times6 = 240\) or \(8\times30 = 240\) (sub - products).
- \(1200\): Yes, as \(40\times30 = 1200\) (sub - product).
- \(120\): No, not a sub - product or a factor.

Answer:

The options that apply are \(48\), \(240\), \(1200\). So the checkboxes for \(48\), \(240\), and \(1200\) should be selected.