Jill’s teacher asked her to construct a rectangular prism from unit cubes. She has built the structure about. In order for Jill to complete her assignment, how many unit cubes will Jill need to add to the below structure? Explain how you were able to figure out how many more cubes will be needed. Is there a different way to solve this situation?

Explanation:

The first way to figure out how many more cubes Jill will need to add to complete the structure until forming a rectangular prism  is to add the cubes from column to colum,

The first column is complete: it has four rows each with 4 cubes.

The second column (the one next to the right of the first column) has three rows. Then Jill has to add one more row. This is 4 cubes.

The third column has two rows. Then, Jill will need to add two rows with 4 cubes each, for a total of 8 cubes. So far, the count is 4 + 8 = 12 cubes.

The fourth and fifth columns have just one row. Then, Jill will need 3 rows for each column, making it 2 × 3 × 4 = 24 cubes.

The total final count is 12 + 24 = 36 cubes. ← answer

There are different ways to figure out the number of cubes needed to complete the rectangular prism. This is another one:

How many cubes shall the complete pyramid have? : 4 × 4 × 5 = 80 (length × width × height).

How many cubes does the shown incomplete structure have?: 4×4 + 3×4 + 2×4 + 4 + 4 = 16 + 12 + 8 + 4 + 4 = 44.

Difference: 80 - 44 = 36 cubes. ← answer

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.

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