8th grade math: what your child learns this year

The correct choice is Added the exponents because $\frac{2^4}{2^2}$ simplifies to $2^{4-2} = 2^2$, not $2^6$.

1. Look at Riley's first line. Riley wrote $2^{4+2}$, which combines the exponents by addition rather than subtraction.
2. The quotient rule says to subtract the denominator exponent from the numerator exponent: $4 - 2 = 2$, so the correct result is $2^2$.
3. Riley wrote $2^{4+2} = 2^6$ instead of $2^{4-2} = 2^2$. The mistake is adding the exponents. The answer is Added the exponents.

Select (1, 3), (3, 5), and (5, 2) because those are the three plotted points on the graph.

1. Look at each point on the graph. Read the x-value (left/right) first, then the y-value (up/down).
2. The three plotted points are (1, 3), (3, 5), and (5, 2).
3. Compare to the choices and select only (1, 3), (3, 5), and (5, 2).

The answer is Jake. The image is a rotation of the pre-image: a quarter turn counterclockwise around the origin sends every point $(x, y)$ to $(-y, x)$. Jake names a rotation and gives a reason that proves it.

1. Look at the grid. The blue figure is the pre-image and the red figure is the image. The pre-image is in the upper right of the grid and the image is in the upper left of the grid. Mia says, 'This is a translation because the figure moved to another part of the grid.' Jake says, 'This is a rotation because the figure turned around the origin.' To be correct, a student must name the right transformation and give a reason that proves it. Test each claim.
2. Is Mia correct? Mia names a translation. In a translation, the figure slides: every point moves the same distance in the same direction. The figure does not turn and does not flip, so its leftmost corner stays its leftmost corner.
3. Test Mia's claim. The leftmost corner of the pre-image is at $(1, 1)$ and the leftmost corner of the image is at $(-3, 2)$. To get from one to the other, you move 4 units left and 1 unit up. If every corner moved that way, the corner at $(4, 1)$ would land on $(0, 2)$. The image has no corner at $(0, 2)$. So the image is not a translation of the pre-image. Mia's reason is true, but a reflection or a rotation can also move a figure to another part of the grid, so it cannot prove a translation. So Mia is not correct.
4. Is Jake correct? Jake names a rotation. In a rotation, the figure turns around a fixed point: every point turns by the same angle around that point. Jake's reason names the point: the origin, $(0, 0)$.
5. Test Jake's claim. A quarter turn counterclockwise ($90^{\circ}$) around the origin sends $(x, y)$ to $(-y, x)$. Check every corner: $(1, 1)$ lands on $(-1, 1)$, $(4, 1)$ on $(-1, 4)$, $(4, 2)$ on $(-2, 4)$, and $(2, 3)$ on $(-3, 2)$. All 4 corners land on corners of the image. So the image is a rotation of the pre-image: a quarter turn counterclockwise around the origin. The figure turned around the origin, as Jake says. Only a rotation turns a figure around a point, so the reason proves a rotation. So Jake is correct.

The correct statement is that the second power of 10 does not line up the repeating block 17. After $10x$, the next aligned equation should use $1000x$, not $100x$.

1. Liam's first shift is correct. Multiplying by 10 moves the decimal one place past the nonrepeating digit 3, giving $10x = 3.\overline{17}$.
2. To cancel the repeating block, it must start in the same decimal position in both equations. In Liam's $100x$ line there is still one extra fixed digit, 7, before the repeat begins, so the two tails do not line up.
3. After $10x$, two more decimal places are needed so the block 17 starts in the same position. The correct aligned pair is:
4. Now the repeating parts cancel cleanly.
5. Solve for $x$ to get the correct fraction.

The answer is scatter plot because each student has two numerical measurements, Height (cm) and Arm Span (cm), that can be plotted together as one point.

1. Look at the table. Each row is one student. The first column gives that student's ID number — it is just a label, not a measurement on any axis.
2. The two measurement columns are Height (cm) and Arm Span (cm). For example, student 101 has height 138 and arm span 140. Those two numbers are recorded together for the same person, so that row could become the point (138, 140) on a graph.
3. A scatter plot is designed for paired numerical data — each pair becomes one point. A bar graph compares category totals, a dot plot shows one list of numbers, and a line graph fits data ordered over time. None of those match two paired measurements per case.
4. The IDs 101–107 may look numerical, but they do not belong on either axis. Only Height (cm) and Arm Span (cm) are the true measurements, so a scatter plot is the correct choice.