#### By Michael Avidon, math editor

## Knowing the format of the questions will help students succeed

A good part of the work we do here is writing and editing items (questions) for high-stakes statewide exams. Those items cannot be released to the public, so the items here are not from actual exams, but are written in a similar style and align with Common Core standards. Solutions, and additional useful information, are found in the last section.

In the past, all items on standardized tests were multiple choice, and many of the items in this documents will be in that format. With the advent of online exams, other formats are also used. Some of these are known as technology enhanced items (TEIs). Such items cannot be properly presented in a document, but some are presented in a mock format. These are described below.

**Multiple select/multiple response questions** appear to have the same format as multiple choice, but have more than one correct answer out of four or more options. The test-taker is asked to “select all that apply.” Sometimes these items state how many options to select and sometimes they do not.

**Multiple-choice drop-down/Inline choice items** will have any number of drop-down menus with multiple options from which the test-taker chooses to correctly fill in missing parts of sentences, equations, diagrams, or other components.

In this document, the existence of a drop-down menu is indicated with braces:

Start of sentence {option 1/option 2/option 3} end of sentence.

**Drag and drop items** provide blank spaces or boxes in sentences, equations, etc., into which test-takers drag and drop correct options (e.g., numbers, words) that appear below or to the side of the item. In this document, underlined blank spaces will generally indicate where to drop the correct answers from a list of options below the item.

**Fill in the blank items** usually require the test-taker to provide a whole number or decimal.

**Constructed response items** require complete, hand-written solutions.

One type of TEI that you will not find here is a **hot spot** item. In such an item, a student must point and click to indicate a correct answer (e.g., select a point on a number line).

**MORE ON THIS SERIES: **Grade 6 assessment exam | Grade 7 assessment exam | If you would rather print this page out, click here.

### Sample Test Items

**Standard:** 8.EE.A.1, 8.EE.A.2

**Item Type:** Drag and drop

1. Drag and drop the options from below into the blanks to correctly complete the equations.

If 2^{3}x^{2} = (2^{4})^{2} and x > 0, then x^{2} = 2 ^{☐} = _____, and x = _____.

OPTIONS

1 2 3 4 5 6 7 8 9 10 11 12

16 32 64 128 256 512 1024 2048

√2 √8 √32 √128 √512 √2048

**Standard: **8.EE.A.4

**Item Type:** Multiple choice

2. The average distance between Earth and the moon is 2.4 x 10^{5}miles. The first manned mission to the moon traveled at an approximate average speed of 3 x 10^{3}miles per hour. Which calculation will determine the number of hours it took to get to the moon?

A. Divide 2.4 by 3, then multiply by 10^{2}.

B. Divide 2.4 by 3, then multiply by 10^{5/3}.

C. Divide 3 by 2.4, then multiply by 10<sup)2.

D. Divide 3 by 2.4, then multiply by 10^{5/3}.

**Standard: **8.EE.B.5

**Item Type:** Multiple choice

3. The graph to the right shows the relationship between elapsed time and distance traveled by Airplane A after it reached its cruising speed.

Airplane B was cruising at a slower speed. Which representation could show the relationship between the elapsed time and distance traveled of Airplane B after it reached its cruising speed?

A. y = 5x, where y is distance traveled in miles and x is time in minutes

Standard: 8.EE.B.6

Item Type: Multiple select

4. The graph of a line is shown below.

Which of the following statements concerning this diagram are true? Select all that apply.

A. △ABC is similar to △DEF.

B. The slope of the line is 2.

C. The slope of the line is ½.

D. ^{AB} ⁄ _{BC} = ^{DE} ⁄ _{EF}

E. ^{AB} ⁄ _{BC} < ^{DE} ⁄ _{EF}

F. m < DFE > m < ACB

**Standard:** 8.F.A.2

**Item Type:** Multiple choice

5. A function is represented by the graph below.

Which of the following descriptions of functions will always have a greater output than the one shown?

A. Initial distance is 100 miles and rate of change is 50 m/h.

B. Initial distance is 200 miles and rate of change is 30 m/h.

C. Initial distance is –50 miles and rate of change is 50 m/h.

D. Initial distance is –50 miles and rate of change is 60 m/h.

**Standard: **8.F.A.3

**Item Type:** Multiple choice

6. Which equation does not represent a linear function?

A. xy = 9

B. y = 2x + 13/3

C. 8y = –9x

D. 5x + 10y = 80

**Standard:** 8.F.B.4, 8.EE.C.8

**Item Type:** Constructed response

7. Cindy made 11 identical centerpieces yesterday. This morning, after 1.5 hours, she had made 6 centerpieces. Cindy continued to work at the same rate from noon onward. Let h be the number of hours she worked after noon.

a. Write an equation for the total number of centerpieces, c, Cindy will have made in terms of h.

Her friend is also making centerpieces, and the number she has produced is modeled by the equation c = 14 + 5h.

b. At what time will they have produced the same total number of centerpieces, and what will be the grand total that they have produced?

Show your work.

**Standard: **8.F.B.5

**Item Type: **Multiple choice

8. The distance that Dana jogs during a 30-second interval is shown in the graph below.

Which explanation correctly interprets the graph?

A. Dana walked at a constant speed for 15 seconds, then stood still for 6 seconds, then walked at a greater constant speed over the next 9 seconds.

B. Dana walked at a constant speed for 15 seconds, then stood still for 6 seconds, then kept increasing his speed over the next 9 seconds.

C. Dana kept increasing his speed for 15 seconds, then walked at a constant speed for 6 seconds, then kept increasing his speed over the next 9 seconds.

D. Dana kept increasing his speed for 15 seconds, then walked at a constant speed for 6 seconds, then walked at a greater constant speed over the next 9 seconds.

**Standard:** 8.G.A.2

**Item Type:** Multiple choice

9. Triangles ABC and DEF are shown on the grid below.

Which sequence of transformations can be used on triangle DEF to prove that it is congruent to triangle ABC?

A. Rotation 180° counterclockwise, then translation 3 units right

B. Rotation 180° counterclockwise, then translation 3 units up

C. Rotation 90° clockwise around the origin, then reflection across the x-axis

D. Rotation 90° clockwise around the origin, then reflection across the y-axis

**Standard:** 8.G.A.3

**Item Type:** Multiple choice

10. A quadrilateral is shown on the grid below.

The quadrilateral undergoes a dilation with a scale factor of ½ , centered at the origin, to produce quadrilateral A’B’C’D’. What are the coordinates of A’, the image of A?

A. (–12, 8)

B. (–6, 2)

C. (–3, 2)

D. (–3, 4)

**Standard:** 8.G.A.4

**Item Type:** Multiple-choice drop-down

11. Select from the drop-down menus to make the statements true.

Two figures are shown on a coordinate plane below.

The two figures {are/are not} similar.

In general, any two figures are similar when there exists a transformation from one to the other using {only/dilations,} rotations, reflections, and translations.

For these two figures, there {does/does not} exist such a transformation.

**Standard:** 8.G.A.5

**Item Type:** Multiple choice

12. Angles A and B each measure 65°.

What is the measure of angle ACD?

A. 115°

B. 120°

C. 125°

D. 130°

**Standard: **8.G.B.6

**Item Type:** Multiple choice

13. A triangle has side lengths 5 inches and 10 inches. What measures, if any, could the third side have that would make it a right triangle?

A. It cannot be a right triangle regardless of the third side length.

B. It can be a right triangle, but only if the third side measures √75 inches.

C. It can be a right triangle, but only if the third side measures √125 inches.

D. It can be a right triangle if the third side measures √75 or √125 inches.

**Standard: **8.G.B.7

**Item Type:** Multiple choice

14. A teacher receives a pointer packed at an angle in a shipping tube, as shown below.

The shipping tube is a right circular cylinder, with a length of 30 inches and a radius of 2 inches. What is the greatest possible length, in inches, of the pointer?

A. √904

B. √916

C. 32

D. 34

**Standard:** 8.G.B.8

**Item Type:** Constructed response

15. A triangle has vertices at the following three points.

M (–4, –1) N (1, 5) P (4, 1)

Determine the longest side of the triangle. Show your work.

**Standard:** 8.NS.A.1

**Item Type:** Multiple select.

16. Which of the following numbers are irrational? Select all that apply.

A. √1

B. √2

C. 0.3817

D. ^{π} ⁄ _{2}

E. –2.5

**Standard: **8.SP.A.1

**Item Type:** x

17. The scatter plot below shows the number of 8-track players sold annually by two companies over a 15-year period.

Which pattern of association is shown between time and sales in this graph?

A. no association

B. positive association

C. negative association

D. nonlinear association

**
Standard:** 8.SP.A.2

**Item Type:**Multiple choice

18. The scatter plot below represents the relationship between time spent doing math homework (x) and the grade on a math exam (y).

Which equation is the best fit for these data?

A. y = 70

B. y = 75

C. y = 4x + 46

D. y = 4x + 55

**Standard: **8.SP.A.4

**Item Type:** Multiple choice

19. Julie took a survey of likely voters to see which of two candidates each preferred. The table below shows the data converted into percentages of the total number of voters surveyed.

Which conclusion is best supported by the data in the table?

A. 25% of all men who were likely voters preferred candidate X.

B. 67% of all women who were likely voters preferred candidate Y.

C. 40% of all men who were likely voters would vote in the election.

D. 20% of all likely voters who preferred candidate X were women.

### Solutions and Explanations

1. Key: If 2^{3}x^{2} = (2^{4})^{2} and x^{2} > 0, then 2^{5 } = 32 and x = √32.

**Solution: 2 ^{3}x^{2} = 2^{4×2} = 2^{8} → x^{2} = 2^{8} / 2^{3} = 2 ^{8-3} = 2^{5 = 32 → x = √32.
}**

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Two rules of exponents are used here:

• (x^{a})^{b} = x ^{ axb}(Did you add the exponents 4 and 2 instead?)

• x^{a} + x^{b} = x^{a-b}(Did you divide the exponent 8 by 3?)

Two properties of equality are used here:

• Dividing both sides (of 2^{3x2 = 28) by the same number (23 )
• Taking square roots of both sides (of x2 = 32)
You could solve this slightly differently by simplifying the exponents before dividing:
23x2 = 28→ 8x2 = 256 → x2 = 256 + 8
2. The average distance between Earth and the moon is 2.4 x 105miles. The first manned mission to the moon traveled at an approximate average speed of 3 x 103miles per hour. Which calculation will determine the number of hours it took to get to the moon?}

**Key: A**

Solution:

Time (in hours) equals distance (in miles) over rate (in miles per hour). So the number of hours is ^{2.4 x 105} ⁄ _{3 x 10<sup3} = ^{2.4} ⁄ _{3} x ^{105} ⁄ _{103} x 10^{5×3} = (2.4 / 3) x 10^{2}.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

This uses one rule of exponents: x^{a} / x^{b}.

Make sure that you pay attention to units. These need to cancel correctly. The distance units divided by the rate units given is:

(If, for example, you were given rate in miles per minute, you would have needed to convert it first to mph to get the correct answer.)

3. The graph to the right shows the relationship between elapsed time and distance traveled by Airplane A after it reached its cruising speed.

Airplane B was cruising at a slower speed. Which representation could show the relationship between the elapsed time and distance traveled of Airplane B after it reached its cruising speed?

**Key: C**

Solution:

Constant speed is given by distance over time. This is the same as the change in y over the change in x, or the slope of the ray. The ray passes through (6, 30) and (0, 0), so Airplane A has speed 30/6 = 5 miles per minute.

Examine the options to find the one with a slower speed (lesser slope).

First, y = 5x represents a line with slope 5, which is the same as the given graph.

The ray in option B passes through (4, 24) and (0, 0), so its slope is 24/4 = 6.

The ray in option C passes through (1, 4) and (0, 0), so its slope is 4/1 = 4.

The table in option D has distance increasing by 7 miles for each 1-minute increase.

So option C represents the only airplane traveling at a slower speed.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

You may have chosen option B because the ray appeared to have a lesser slope than the one in the problem. However, the y-axis has a different scale, and that accounts for the fact that the slope in B is actually greater.

The slope-intercept form of the equation of a line is y = mx + b, where m is the slope of the line. Slope can also be described as the “rate of change” of the output per unit of the input.

4. The graph of a line is shown. Which of the following statements concerning this diagram are true?

**Key: A, C, D**

**Solution:**

The measures of ∠DFE and ∠ACB are equal because they are formed by parallel (horizontal) lines cut by a transversal. So F is false. The two triangles have right angles, so they are similar by AA. Thus A is true. The ratio ^{AB} ⁄ _{BC} = ^{2} ⁄ _{4} = ½ and ^{DE} ⁄ _{BC} = ^{3} ⁄ _{6} = ½. Therefore, D is true and E is false. Either ratio is the slope of the line. So C is true and B is false.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

The first part of the solution uses the general fact that if a transversal (line) cuts two parallel lines, then “corresponding angles” are congruent.

The main point of the item is to confirm that the slope of a line is constant. That is, no matter which two points you use to calculate the slope (here, A and C vs. D and F), you will obtain the same ratio. That this is true in general follows from the fact that the ratios are quotients of corresponding sides in similar triangles, and such ratios must be equal.

Remember that slope is defined as ^{△y} ⁄ _{△x} (options B and C were checking that you did not confuse the numerator and denominator). The side lengths of the top triangle are smaller than those of the bottom triangle, but the ratio of the side lengths is not smaller (options D and E), and the angles are not smaller (option F).

5. A function is represented by the graph.

Which of the following descriptions of functions will always have a greater output than the one shown?

**Key:** A

**Solution:**

The chosen function must have a greater initial value than the initial value of 0 in the graph, so it is option A or B. Its graph must stay above the given ray, so its slope must be the same (50) or greater. Only option A matches this description.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Any two lines with different slopes will intersect somewhere. The graph of option B is a ray (y = 200 / 30x) that starts above the given ray (y = 50x ), but has a lesser slope, so it will intersect the given ray in the first quadrant (when 200/ 30x = 50x → x = 10). This tells you that it has the same output at the intersection point, and will have a lesser output farther to the right.

6. Which equation does not represent a linear function?

**Key:** A

**Solution:**

A linear function has an equation in the form y = mx + b. Solve each equation for y if it is not already isolated.

Equation A: xy = 9 → y = ^{9} ⁄ _{x}. The variable in the denominator implies that this is not linear. Here is the confirmation that the other options are all linear:

Equation B: y = 2x + 13/3. This is already in the correct form.

Equation C: 8y = -9x → y = –^{9} ⁄ _{8}x

Equation D: 5x + 10y = 80 → 10y = 80 -5x → y = 8 – 0.5x

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Any equation with two variables represents a relationship between those two variables. That relationship may be expressed explicitly (e.g., y = ^{9} ⁄ _{x} ) or it may be expressed implicitly (e.g., xy = 9). Explicitly means “Here it is!” (If the input is x, then the output is .^{9} ⁄ _{x} ) Implicitly means “Well, I am indirectly telling you.” (If the input is x, then the output is – well, you have to figure it out from the equation.) If you are given xy = 9and an input of 4, then you have 4y = 9 → y = ^{9} ⁄ _{4}. Now you know the output is ^{9} ⁄ _{4}.

If you “change” an equation by performing the same arithmetic operation on both sides (addition, subtraction, multiplication, division), then you are not changing the relationship. You are just writing it in another form. If you are given y = ^{9} ⁄ _{x}and an input of 4, then you have y = ^{9} ⁄ _{4}. As stated in the previous paragraph, an input of 4 will have an output of 9/4.

7. Cindy made 11 identical centerpieces yesterday. This morning, after 1.5 hours, she had made 6 centerpieces. Cindy continued to work at the same rate from noon onward. Let h be the number of hours she worked after noon.

a. Write an equation for the total number of centerpieces, c, Cindy will have made in terms of h.

Her friend is also making centerpieces, and the number she has produced is modeled by the equation c = 14 + 5h.

b. At what time will they have produced the same total number of centerpieces, and what will be the grand total that they have produced?

**Key:** c = 4h + 17; 3 p.m., 58

**Sample Response:**

a. Cindy’s rate today is 6 + 1½ = ^{6} ⁄ _{1} / ^{3} ⁄ _{2} = ^{6} ⁄ _{1} x ^{2} ⁄ _{3} = ^{12} ⁄ _{3} = 4 centerpieces per hour. As of noon, she had made 17 centerpieces. So c = 17 + 4h.

b. The question is to find out when h and c are the same in the two equations, so you need to solve the system c = 17 + 4h and c = 14 + 5h:

17 + 4h + 14 + 5h → 3 = h(3 hours after noon, or 3 p.m.).

At that time, c = 17 + 4(3) = 17 + 12 = 29. The grand total is 2 x 29 = 58.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

If something is being done at a constant rate, then the relationship between time and the total amount done will be a linear function. In particular,

total amount done = initial amount done + new amount done

= initial amount done + rate × time

The initial amount and rate are constants, while time is a variable. This is precisely the description of a linear function.

The commentary for item 5 is applicable to part b. The two equations in the solution (c = 17 + 4h and c = 14 + 5h) have graphs that are lines: the first has a greater initial value (17 > 14 ) and a lesser slope (4 < 5). The first line will start above the second, but the second is steeper and will intersect the first (at (3,29)).

8. The distance that Dana jogs during a 30-second interval is shown in the graph below.

Which explanation correctly interprets the graph?

**Key:** B

**Solution:**

A straight-line segment on a grid has a constant slope. On a graph of time versus distance, this is a constant speed (distance over time).

In the graph above, the segment for 0 < x <15 represents a constant positive speed.

The segment for 15 < x <21 is horizontal, so it has a slope of 0. It represents a constant speed of 0 (standing still) over 6 seconds.

The curved segment at the end goes up, so distance is increasing and speed is positive.

For 21 < x < 24 , the distance increases from 60 to about 63 yards; for 24 < x < 27, the distance increases from about 63 to about 72 yards; for 27 < x <30, the distance increases from about 72 to 100 yards. This would be speeds of about 1 yard per second, 3 yards per second, and 9 yards per second. That is, the speed is increasing over 9 seconds.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

In general, when a distance-time graph is curving upwards, as the last part of the above graph, the object in motion is accelerating (speed is increasing). When a distance-time graph looks runs up to the right, the object in motion is decelerating (speed is decreasing). In grade 8, you learn that the slope of a line in this context is rate of change. If you eventually study calculus, you will learn how to calculate rate of change at any instant when a curve represents the motion of an object.

9. Triangles ABC and DEF are shown on the grid.

Which sequence of transformations can be used on triangle DEF to prove that it is congruent to triangle ABC?

**Key:** C

**Solution:**

Transforming one figure to the other with rotations, reflections and translations will prove they are congruent.

When a horizontal segment is rotated 90° (in either direction), it becomes a vertical segment. Likewise, a vertical segment becomes horizontal. Rotating 180° is like rotating 90° twice. Consider what happens just to DE, the longer leg of the right triangle. Rotating 180° and translating in any direction will yield the longer leg in a horizontal position, so A and B must be incorrect. Rotating 90° clockwise will make it vertical, moving it from y = 8 to x = 8. In particular, the corner D will transform to (8, 5). It will also transform from a vertical segment to a horizontal along . Reflecting over the y-axis would put it back into Quadrant II. Reflecting over the x-axis keeps the longer leg vertical and the shorter leg horizontal. It transforms the corner vertex from (8, 5) to (8, –5), so everything “matches up.”

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Sometimes it may seem confusing to figure out what happens to a figure under rotations (or other transformations). It might be simpler or more helpful to consider what happens to just one side or individual vertices. Then piece those results together to determine what happens to the whole.

In mathematics (and in real life), sometimes the order of things can be switched with no effect (multiplying two numbers) and sometimes the order matters (subtracting numbers; putting on your socks and shoes). This is true when you combine two or more transformations: sometimes any order gives the same result, and sometimes different orders yield different results. It is probably not helpful to come up with a list of cases; just carefully examine each combination as it is presented.

10. A quadrilateral is shown on the grid.

The quadrilateral undergoes a dilation with a scale factor of ½ , centered at the origin, to produce quadrilateral A’B’C’D’. What are the coordinates of A’, the image of A?

**Key: **C

Solution:

The dilation transforms A = (-6,4) to A’ = ½(-6, 4) = (-3,2).

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

For a dilation centered at the origin with scale factor k, you simply multiply the original coordinates by k to find the new coordinates.

If the center of the dilation was not the origin, but was point B, for example, then A would move toward B and the distance would be halved. Thus, its image would be (-4.5, 4). If the center was the point A itself, then A’ = A.

11. Two figures are shown on a coordinate plane.

**Key: **The two figures are not similar.

In general, any two figures are similar when there exists a transformation from one to the other using dilations, rotations, reflections, and translations.

For these two figures, there does not exist such a transformation.

**Solution:**

Compare corresponding parts of these L-shapes. The ratio of the long sides is 4:6. The “bottoms of the L” has the same ratio, 2:3. But the “thickness of the bottom of the L” has the ratio 1:2, which is different. So the figures are not similar.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Rotations, reflections, and translations do not change the shape of an object; they just change the position, so they create congruent figures. Dilations expand or contract an object, operating proportionally on all sides. (In ordinary English, a dilation is only an expansion.) They create similar figures (the sides are in a proportional relationship).

12. Angles A and B each measure 65°.

What is the measure of angle ACD?

**Key:** D

Solution:

An exterior angle is the sum of the opposite interior angles, so this is 65° + 65° = 130°.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

ACD is called an exterior angle because it is the angle formed by the extension of one side of the triangle with another side. If you did not know the theorem cited, you could have calculated this as follows: the sum of the angles in the triangle is 180°, so the measure of angle BCA is 180° – 130° = 50°. Now, angle ACD is supplementary to BCA, so its measure is 180° – 50° = 130°.

13. A triangle has side lengths 5 inches and 10 inches. What measures, if any, could the third side have that would make it a right triangle?

**Key:** D

**Solution:**

If 10 is the length of the hypotenuse, and x is the length of the missing leg, then we need x^{2} = 5^{2} = 10^{2} → x^{2} + 25 = 100 → x^{2} = 75 → x = √75.

If 5 and 10 are the legs, and x is the length of the hypotenuse, then we need x^{2} = 10^{2} + 5^{2} = 100 + 25 = 125 → x = √125.

The hypotenuse cannot be 5, because the hypotenuse is the longest side.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

This uses the Pythagorean theorem and its converse: the three sides of a triangle satisfy a^{2} + b^{2} = c^{2} if and only if it is a right triangle.

Note that, if desired, you could rewrite the radicals (some would say simplify, but it is a matter of perspective as to which form is simpler): √75 = √25 x √3 = 5√3 and √125 = √25 x √5 = 5√5.

14. A teacher receives a pointer packed at an angle in a shipping tube, as shown below.

The shipping tube is a right circular cylinder, with a length of 30 inches and a radius of 2 inches. What is the greatest possible length, in inches, of the pointer?

**Key:** B

**Solution:**

The greatest possible length L is the hypotenuse of a right triangle (in the cross section, going the long way, of the tube). The legs of this right triangle are 30 inches (the length of the tube) and 4 inches (the diameter of the tube). So L^{2} = 30^{2} + 4^{2} = 900 + 16 = 916, and thus L √916.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

To clarify precisely why this is the greatest length, any pointer that extended across but did not touch “opposite corners” (there are no corners in a tube, but I mean corners of the cross section) would be the hypotenuse of a right triangle with one leg that is 30 inches and another leg that is less than 4 inches (draw a picture to verify this). If it did not extend all the way across, it would be the hypotenuse of a right triangle with a leg that is less than 30 inches and another leg that is at most 4 inches. Either case would yield a smaller value of L.

15. A triangle has vertices at the following three points.

M (–4, –1) N (1, 5) P (4, 1)

Determine the longest side of the triangle.

**Key:** MP

**Sample Response:**

Substitute the coordinates into the distance formula for each side.

Clearly √68 > √61 > √5.

So MP is the longest side.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

The distance from (x_{1}, y_{1} to (x_{2}, y_{2</sub) is given by the formula √(x2 – x1)2 + (y2 – y1)2.
Make sure that when you subtract a negative, you get a positive. For example, part of the calculation for MP was and 1 – (-4) = 1 + 4 = 5 and 5 1 (-1) = 5 + 1 = 6.}

16. Which of the following numbers are irrational?

**Key:** B, D

**Solution:**

The following numbers are rational, because they can be expressed as the quotient of two integers: √1 = ^{1} ⁄ _{1}; 0.3817 = ^{9817} ⁄ _{10,000}; -2.5 = ^{-5} ⁄ _{2}.

The square root of a number that is not a perfect square is irrational, so √2 is irrational. Because π is irrational, π/2 must also be irrational.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

The point of option A is that a radical symbol does not automatically imply that you have an irrational number. The point of option D is that even though this is a fraction, it is not the quotient of two integers, so it is not rational. More precisely: if π/2 were rational and you multiplied it by 2, you would get another rational number. But you get π, which is irrational. So π/2 must be irrational.

17. The scatter plot shows the number of 8-track players sold annually by two companies over a 15-year period.

Which pattern of association is shown between time and sales in this graph?

**Key:** D

**Solution:**

There is clearly a pattern here, but it is not a positive association. That would mean as the years increase, the sales roughly also increase. But this happens only from 1965 to 1974. Likewise, it is not a negative association, because the overall pattern would have to be decreasing, and that happens only from 1974 to 1980. It is not linear, because that would be roughly increasing or decreasing or constant.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

In grade 8, your study is essentially restricted to linear (1st degree) functions; so in statistics you would be asked whether a line fits a scatter plot or not. In grade 9, you will study quadratic functions, or those in the form y = ax^{2} + bx + c. These have a U-shape for a < 0, and an upside-down U-shape for . The latter type roughly fits this pattern.

18. The scatter plot below represents the relationship between time spent doing math homework (x) and the grade on a math exam (y).

Which equation is the best fit for these data?

**Key: **D

**Solution:**

The two constant functions would have graphs that are horizontal lines. That would not fit the increasing pattern of the points from left to right. The equation y = 4x + 46 follows the increasing pattern, but its graph has all the points above the line. The equation y = 4x + 55 has a graph that is a line from (0, 55) to (10, 95). This will run right through the middle of the data points, with 6 points above and 6 points below the line. That would be a reasonably good fit for the data.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

In grade 8, you are expected to just “eyeball” the data points and find a line that roughly fits the pattern. You should be able to find the equation of the line that you chose to draw. In high school, you will probably use a graphing calculator (or an online linear regression calculator) to input the points and find the exact “line of best fit.” There is a complicated formula that you could directly plug the numbers into, but it would be really unpleasant. The basic idea behind it is that the sum of the squares of the vertical distances to the line is minimized.

19. Julie took a survey of likely voters to see which of two candidates each preferred. The table below shows the data converted into percentages of the total number of voters surveyed.

Which conclusion is best supported by the data in the table?

**Key: **B

**Solution:**

All the percentages given are computed relative to P, the total number of people surveyed. The number of women surveyed is 0.6P and the number of women who preferred candidate Y is 0.4P. The fraction of all women surveyed (representing likely voters) who preferred candidate Y is ^{0.4P} ⁄ _{0.6P} = ^{0.4} ⁄ _{0.6} = ^{4} ⁄ _{6} = 0.66 or 67%.

**Commentary (formulas, ideas, mistakes, misunderstandings, advice):**

Whenever percentages are being computed, remember that you can take one number and find out what percent it is of different things. Suppose that 1000 people were surveyed. Then 60%, or 600, of these people are women, and 55%, or 550, of these people are candidate Y supporters. The 40%, or 400, of these people who are women who prefer candidate Y are a subset of three groups: (1) women, (2) candidate Y supporters, and (3) everyone surveyed. The correct option contained the percentage of group (1). You could compute the percentage of group (2): ^{400} ⁄ _{500} = 0.72, or about 73%. The original figure in the table, 40%, is the percentage of group (3).

The incorrect options show mistakes based on confusing which group you are finding the percentage of. Option A uses a percent of everyone, but talks about a percent of men who are likely voters (so the correct fraction is 25/40 or 62.5%). Option D uses a percent of everyone, but talks about a percent of voters who prefer candidate X (so the correct fraction is 20/45 or 44%). Option C completely confuses the issue: the information in the table does not give you any idea what percentage of people would actually vote. The 40% in the table simply refers to the fact of the people surveyed, 40% are men.

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