#### 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 test | Grade 8 assessment test | If you would rather print this page out, click here.

### Sample Test Items

**Standard**: 7.EE.A.1, 7.EE.A.2

**Item Type**: Multiple choice

1. A charity has collected $*x*. They collected $x/2 from a car wash and $x/3 from a bake sale. The rest of the money was directly donated to them. What fraction of the money was directly donated?

A. ⅓

B. ¼

C. ⅕

D. ⅙

**Standard**: 7.EE.B.3

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

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

Caroline will use the proportions in this recipe to make as many cookies as she can with ingredients in her kitchen.

She has 4 tablespoons of butter, ¼ cup baking cocoa, and more than enough of the remaining ingredients.

Caroline will make {18/27/36} cookies. She will use {1.5/2.25/3} cups of oats.

**Standard**: 7.EE.B.4a, 7.EE.B.3

**Item Type**: Constructed response

- 2 cups sugar
- 8 tablespoons butter
- ½ cup milk
- ⅓ cup baking cocoa
- 3 cups oats

Makes about 36 cookies

She has 4 tablespoons of butter, ¼ cup baking cocoa, and more than enough of the remaining ingredients.

Caroline will make {18/27/36} cookies. She will use {1.5/2.25/3} cups of oats.

- A carpenter installed 210 inches of wood trim around a large window. After installing trim around five identical smaller windows, he had installed a total of 860 inches of trim.

The cost of the trim was $4 per foot.

What was the cost, to the nearest cent, of the wooden trim the carpenter used on one of the smaller windows?

*Show your work.*

**Standard**: 7.G.A.1

**Item Type**: Multiple choice

- Hiro is building a porch swing. The scale drawing below represents the base of the swing.

What is the actual area of the swing?

- A. 6 square feet
- B. 8 square feet
- C. 12 square feet
- D. 16 square feet

**Standard**: 7.G.A.2

**Item Type**: Multiple select

- Which of these sets of conditions determine more than one triangle? Select
**all**that apply.

- A. Side lengths of 3 cm and 4 cm, and a 30° angle
- B. Side length of 3 cm, a 30° angle, and a 45° angle
- C. Side lengths of 5 cm and 5 cm, and a 60° angle
- D. Side lengths of 6 cm, 8 cm, and 10 cm
- E. Side lengths of 8 cm and 8 cm, and two 45° angles

**Standard**: 7.G.A.3

**Item Type**: Multiple select

- Which shapes can be the result of slicing a right rectangular pyramid with a plane that is parallel to one of the sides? Select
**all**that apply.

- A. triangle
- B. rectangle
- C. trapezoid
- D. pentagon
- E. oblique parallelogram

**Standard**: 7.G.B.4

**Item Type**: Multiple choice

- What is the area of a circle with a circumference of 8π units?

- A. 16π square units
- B. 32π square units
- C. 36π square units
- D. 64π square units

**Standard**: 7.G.B.5

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

- Select from the drop-down menus to correctly complete the sentences.

Two adjacent angles are {sometimes/always/never} supplementary.

Three angles are adjacent and have the same measure.

The greatest possible measure they could each have is {30°/60°/100°/120°}.

**Standard**: 7.G.B.6

**Item Type**: Constructed response

- The floor plan of a basement is shown below.

The basement ceiling is 2.5 meters above the floor.

- A. Partition the floor plan into a rectangle using a single line segment and another shape. Find the area of the rectangle. Show or explain how you got your answer.
- B. Find the area of the basement floor. Show or explain how you got your answer.
- C. Find the amount of water, in cubic meters, that would flood the basement from the floor to the ceiling. Show or explain how you got your answer.

*Show your work.*

**Standard**: 7.NS.A.1a

**Item Type**: Multiple choice

- Which results in a value of 0?

- A. an increase in temperature of 43 degrees from 43 degrees
- B. a decrease in elevation of 55 feet from 55 feet below sea level
- C. a deposit of $132 into a new bank account followed by writing a check for $132
- D. a score of –12 added to a score of 0

**Standard**: 7.NS.A.2a

**Item Type**: Multiple select

- Which of the following expressions is equal to 5 × (–3)? Select
**all**that apply.

- A. 5 + 5 + 5
- B. (–3) + (–3) + (–3) + (–3) + (–3)
- C. 3 + 3 + 3 + 3 + 3
- D. (–5) + (–5) + (–5)
- E. 2 + 2 + 2 + 2 + 2

**Standard**: 7.NS.A.2b

**Item Type**: Drag and drop

- Drag and drop a number into the blank to correctly complete the equation.

________÷ –15 = –3

**OPTIONS**

–45 –18 –12 –5 5 12 18 45

**Standard**: 7.NS.A.3, 7.RP.A.3

**Item Type**: Constructed response

- An elevator in an underground mine descended from station A at a depth of –250.25 feet relative to ground level to station D at a depth of –570 feet.

- A. What was the total distance, to the nearest hundredth of a mile, that the elevator descended?

The elevator moved at a steady rate of 4.5 miles per hour.

- B. How long did it take, to the nearest tenth of a minute, for the elevator to descend from station A to station D?

*Show your work.*

**Standard**: 7.RP.A.2c, 7.RP.A.2b

**Item Type**: Constructed response

- A recipe uses 3 cups of flour for every 2 cups of water. It also uses 4 cups of water for every 5 cups of milk.

Write an equation to determine the number of cups of flour, *f*, required for *m* cups of milk.

*Show your work.*

**Standard**: 7.RP.A.3

**Item Type**: Multiple choice

- Mr. Corwin buys a box of crayons for $3.74. When he bought a box a few weeks ago, the price was $3.52.

What is the percent increase in the price of one box of these crayons?

- A. 5.88%
- B. 6.25%
- C. 16%
- D. 17%

**Standard**: 7.SP.A.2

**Item Type**: Multiple choice

- A movie theater surveyed 80 randomly chosen people who saw a movie on a Saturday. They were asked to name their favorite type of movie from four choices. The data is shown in the table.

Favorite types of movie | |
---|---|

Type of movie | Number of votes |

Comedy | 30 |

Drama | 24 |

Musical | 6 |

Thriller | 20 |

Based on the data, how many of the 2,000 people who visited the theater that Saturday can be expected to say they like comedies best?

- A. 500
- B. 600
- C. 750
- D. 1,200

**Standard**: 7.SP.B.4

**Item Type**: Multiple select

- Two sets of data are shown. Each bar represents an equal number of children.

Which conclusions are supported by these two sets of data? Select **all** that apply.

- A. Before age 5, girls grow faster than boys.
- B. The average height for boys under age 5 is just over 27 inches, while the average height for girls this age is 30 inches.
- C. After age 5, boys grow faster than girls.
- D. The average height for boys under age 5 is 29 inches, while the average height for girls this age is 33 inches.
- E. After age 5, boys and girls grow at about the same rate.

**Standard**: 7.SP.C.8

**Item Type**: Multiple choice

- Two contestants are each given a box with 1 gold ball and 3 black balls. Each person is blindfolded and has to pull out one ball. They win if both balls are gold. What is the probability that they win?

- A.
^{ 1}⁄_{16} - B.
^{ 1}⁄_{10} - C. ⅛
- D. ¼

**Standard**: 7.SP.C

**Item Type**: Constructed response

- Part A.

Lin was given the table below, which lists the results of spinning two spinners. One spinner had three sections colored red, green, or blue. The other spinner had two sections numbered 1 or 2.

Result of spin | Frequency | Probability |
---|---|---|

Red and 1 | 26 | |

Red and 2 | 21 | |

Blue and 1 | 55 | |

Blue and 2 | 49 | |

Green and 1 | 25 | |

Green and 2 | 24 |

Complete the table, listing the experimental probability of each outcome as a decimal rounded to the nearest hundredth. Explain how you got your answer.

Part B.

Based on the data, explain how you would determine the number of “Red and 1” results you would expect after any number of spins.

Part C.

Lin knows that the numbered spinner is divided into two equal sections. She wants to determine the sizes of the colored sections on the other spinner. Using the data, explain how Lin can deduce the relative sizes of the colored sections.

### Solutions and Explanations

- A charity has collected $
*x*. They collected from a car wash and from a bake sale. The rest of the money was directly donated to them. What fraction of the money was directly donated?

**Key: **D

**Solution:**

** x – ^{x} ⁄ _{2} ** – x(1 – ½ – ⅓) = x(

^{6}⁄

_{6}–

^{3}⁄

_{6}–

^{2}⁄

_{6) = x(⅙ )}

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

The first step uses the Distributive Property in “reverse”: ab + ac = a(b+c). Another name for this is “factoring.” The property is stated with two terms and addition but works for 3 or more terms and with subtraction.

- Caroline will use the proportions in this recipe to make as many cookies as she can with ingredients in her kitchen.

- 2 cups sugar
- 8 tablespoons butter
- ½ cup milk
- ⅓ cup baking cocoa
- 3 cups oats

Makes about 36 cookies

She has 4 tablespoons of butter, ¼ cup baking cocoa, and more than enough of the remaining ingredients.

**Key: **Caroline will make __18__ cookies. She will use __1.5__ cups of oats.

**Solution:**

She has enough butter for ½ the recipe and enough cocoa for ¾ of the recipe. The most she can make is half of the 36 cookies. In this case, she will use half of the 3 cups of oats.

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

In general, to find proportional amounts that match the recipe, you can multiply every amount by the same positive constant *k*, and this will produce 36*k* cookies. In this particular problem, k = ½ was used.

- A carpenter installed 210 inches of wood trim around a large window. After installing trim around five identical smaller windows, he had installed a total of 860 inches of trim.

The cost of the trim was $4 per foot.

What was the cost, to the nearest cent, of the wooden trim the carpenter used on one of the smaller windows?

**Key: **$43.33

**Solution:**

Let *x* be the amount of trim, in inches, used on each smaller window. Then

210 + 5x = 860

5x = 650

x = 130

Each smaller window uses 130 inches of trim.

The cost for this was ^{130} ⁄ _{12}feet x $4 per foot = $^{130} ⁄ _{3} ~ $43.33.

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

Pay attention to units. Length is given in inches, and cost is given in dollars per foot. Notice in the last line how the 130 inches is converted to feet (by dividing by 12).

The main point of this problem is the solution of a linear equation (210+5x = 860). You can solve/rewrite/simplify any equation by performing the same arithmetic operation on each side. In the first step, 210 is subtracted from both sides. In the second step, both sides are divided by 5.

- Hiro is building a porch swing. The scale drawing below represents the base of the swing.

What is the actual area of the swing?

**Key: **C

**Solution:**

The actual dimensions are 6 feet by 2 feet, so the area is 6 • 2 = 12 square feet.

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

It would be incorrect to find the area of the scale drawing (3 square inches) and then multiply by the scale factor 2 feet/inch. The scale factor is applied to both dimensions, so the area of the scale drawing is multiplied by 4.

- Which of these sets of conditions determine more than one triangle?

**Key: **A, B

**Solution:**

- Side lengths of 3 cm and 4 cm, and a 30° angle: The 30° angle may or may not be the included angle between the given side lengths. (There are actually 4 possible triangles, though the student need not determine this.)
- Side length of 3 cm, a 30° angle and a 45° angle: The triangle will have 30°, 45°, and 105° angles, and the side of length 3 can be opposite any one of these. (There are 3 possible triangles.)
- Side lengths of 5 cm, and 5 cm, and a 60° angle: If 60° is a base angle of this isosceles triangle, then both base angles are 60° and so is the included angle. If 60° is the included angle, then the equal base angles are each half of 120°. So this can only be an equilateral triangle with all sides equal to 5 cm.
- Side lengths of 6 cm, 8 cm, and 10 cm: Three side lengths uniquely determine a triangle (as long as the triangle inequality is satisfied).
- Side lengths of 8 cm and 8 cm, and two 45° angles: The third angle must be 90°. The legs of the right triangle must be 8 cm each, and the hypotenuse is uniquely determined.

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

The triangle inequality states that any two side lengths in a triangle must have a sum that is greater than the third side length. In general, given three side lengths with the sum of the shorter two greater than the longest, they form a unique triangle.

If the triangle inequality was not satisfied for option D, then there would be 0 such triangles (e.g., side lengths 6, 8, and 15)

How many pieces of information about side lengths and angles determine a unique triangle?

- Two pieces: never unique
- Three pieces: may or may not be unique
- Four pieces: triangle is unique or impossible

- Which shapes can be the result of slicing a right rectangular pyramid with a plane that is parallel to one of the sides?

**Key: **B, C

**Solution:**

If the plane is parallel to the base, it will cut the four triangular sides and form a rectangle. If the plane is parallel to one of the triangular sides, it will cut three triangular sides and the base and form a trapezoid.

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

If a plane were to cut through the upper vertex of the pyramid, it would form a triangle.

- What is the area of a circle with a circumference of 8π units?

**Key: **A

**Solution:**

2πr = 8π → r = 4 → Area = 8π x 4^{2} = 168π square units

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

If the radius of a circle is *r*, then the circumference is 28πr and the area is 8π x r^{2}.

**Key:**Two adjacent angles are__sometimes__supplementary.

Three angles are adjacent and have the same measure.

The greatest possible measure they could each have is __120°__.

**Solution:**

Two adjacent angles may or may not add up to 180°, so they may or may not be supplementary.

The three angles may sum to at most 360°. So each is at most 360°/3 = 120°.** **

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

Supplementary angles together form a straight line, which has a measure of 180°.

Complementary angles together form a right angle, which has a measure of 90°.

- The floor plan of a basement is shown.

The basement ceiling is 2.5 meters above the floor.

- A. Partition the floor plan, using a single line segment, into a rectangle and another shape. Find the area of the rectangle. Show or explain how you got your answer.
- B. Find the area of the basement floor. Show or explain how you got your answer.
- C. Find the amount of water, in cubic meters, that would flood the basement from the floor to the ceiling. Show or explain how you got your answer.

**Key: **155.52 m^{2} OR 51.84 m^{2} ; 190.08 m^{2}; 475.2 m^{3}

**Solution:**

- For a horizontal line: 14.4 x 10.8 = 155.52 square meters

For a vertical line: 14.4 x 3.6 = 51.84 square meters

- For a horizontal line: The triangle has leg lengths 9.6 meters and 7.2 meters. Its area is ½(9.6)(7.2) = 34.56 square meters.

The total area is 155.52 + 34.56 = 190.08 square meters.

For a vertical line: the area of the trapezoid is ^{14.4 + 24} ⁄ _{2} x 7.2 = 138.24. The total area is the same.

- The basement is a prism. To find its volume, multiply the area of the floor times the height: 190.08 x 2.5 = 475.2 cubic meters.

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

In general, a prism is a three-dimensional object with identical cross sections. The cross section can be any polygon. For example, a cube is a simple prism with a cross section that is a square. The basement is a prism with a cross section that is a pentagon.

- Which results in a value of 0?

**Key: **C

**Solution:**

The deposit puts +132 into the account and the check or withdrawal “puts” –132 into the account. These amounts “cancel” and produce a result of 0.

Option A describes +43 combined with +43, which produces +86.

Option B describes a decrease or –55 combined with –55 (feet below sea level), which produces –110.

Option D combines –12 with 0, which produces –12.

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

Negative numbers are a convention used to indicate opposites. If you are standing at sea level, for example, you need some way to distinguish moving 10 feet up from 10 feet down. So we have decided to call the first +10 feet and the second –10 feet. You can hold +10 dollars in your hand, and you cannot hold –10 dollars, but it is convenient when doing bookkeeping to describe a debt with negative numbers.

- Which of the following expressions is equal to 5 × (–3)?

**Key: **B, D

**Solution:**

Multiplication is repeated addition, so 5 times any number is that number added 5 times, Namely, 5 x (-3) = (-3) + (-3) + (-3) + (-3).

Also, 5 x (-3) = -5 x 3 = 3 x (-5) = (-5) + (-5) + (-5).

Adding negative numbers produces a negative number, so the value of the expression is negative. Options A, C, and E all have positive values, so cannot be correct.

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

A positive times a negative number is *defined* to be negative. But this makes sense: if you borrow $3 (“have –$3”) on 5 occasions, then you owe $15 (“have –$15”). This is represented by the equation 5 x (-3) = -15.

The rules for multiplying signed numbers were put in place so that properties of operations you learned for positive numbers are still valid for all numbers. In particular, using the Distributive Property: 0 = 5 x 0 = 5 x (3 + -3) = 5 x 3 + 5 x (-3) = 15 + 5 x (-3)

In order for the last expression to be equal to 0, we must have 5 x (-3) = -15. Now 0 = -5 x 0 = -5 x (3+ -3) = -5 x 3 + (-5) x (-3) = -15 + (-5) x (-3)

In order for the last expression to be equal to 0, we must have (-5) x (-3) = 15.

**Key:**__45__÷ –15 = –3

**Solution:**

Use the relationship between division and multiplication.

☐ + -15 = -3 → ☐ = -15 x -3 = +45

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

The commentary for the previous item discussed the multiplication of signed numbers. In particular, it showed that the product of two negatives is a positive number.

- An elevator in an underground mine descended from station A at a depth of –250.25 feet relative to ground level to station D at a depth of –570 feet.

- A. What was the total distance, to the nearest hundredth of a mile that the elevator descended?

The elevator moved at a steady rate of 4.5 miles per hour.

- B. How long did it take, to the nearest tenth of a minute, for the elevator to descend from station A to station D?

**Key: **0.06 mile; 0.8 minute

**Solution:**

- A. Let
*x*be the change in elevation in feet. Then -250.25 + x = -570 → x = -570 + 250.25 = -319.75

The elevator descended 319.75 feet. This must be converted to miles:

^{? mile} ⁄ _{319.75 feet} = ^{1 mile} ⁄ _{5280 feet} → ? = ^{319.75} ⁄ _{5280} ~ 0.06

The elevator descended 0.06 mile.

- B. The time it took to descend this distance is time =
^{distance}⁄_{rate}=^{0.06 mile}⁄_{4.5 mile/hour}= 0.013 hour

This must be converted to minutes: 0.013 x 60 minutes = 0.8 minute.

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

It is important to note the units that you are given and apply the correct conversions. You are given distance in feet and asked for distance in miles. Then you are given speed in miles per hour and asked for time in minutes.

For an object moving at a constant rate, rate x time = distance, so time = distance/rate.

As noted previously, distance below ground is defined to be negative as a matter of convenience. The sum of the negative and positive number was negative because –570 has a greater absolute value (as opposed to 570 + -250.25 = 319.75, for example).

- A recipe uses 3 cups of flour for every 2 cups of water. It also uses 4 cups of water for every 5 cups of milk. Write an equation to determine the number of cups of flour,
*f*, required for*m*cups of milk.

**Key:** f = ^{6} ⁄ _{5}m

**Solution:**

The first sentence implies that 6 cups of flour are used for 4 cups of water. So 6 cups of flour are used for 5 cups of milk. This can be written as a proportional relationship:

^{f} ⁄ _{m} = ^{6} ⁄ _{5} → f ^{6} ⁄ _{5}m .

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

This item gives you two proportional relationships: between flour and water, and between water and milk. It concludes with a proportional relationship between flour and milk. In general: if *x* and *y* are in a proportional relationship, and *y* and *z* are in a proportional relationship, then *x* and *z* are in a proportional relationship. (See if you can prove this.)

- Mr. Corwin buys a box of crayons for $3.74. When he bought a box a few weeks ago, the price was $3.52. What is the percent increase in the price of one box of these crayons?

**Key: **B

**Solution:**

The percent increase is ^{3.74 – 3.52} ⁄ _{3.52} x 100% = ^{0.22} ⁄ _{3.52} x 100% = 0.0625 x 100% = 6,25%.

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

Remember to divide by the original amount (not the new amount) whenever you are computing a percent increase or decrease. Precisely,

percent increase (decrease) = ^{new amount – original amount } ⁄ _{original amount} x 100%

If the amount goes down, then the numerator is negative, so you obtain a negative percent. That signifies a decrease.

- A movie theater surveyed 80 randomly chosen people who saw a movie on a Saturday. They were asked to name their favorite type of movie from four choices. The data is shown in the table.

Based on the data, how many of the 2,000 people who visited the theater that Saturday can be expected to say they like comedies best?

Favorite types of movie | |
---|---|

Type of movie | Number of votes |

Comedy | 30 |

Drama | 24 |

Musical | 6 |

Thriller | 20 |

**Key: **C

**Solution:**

Set up a proportion using the ratio “Comedy to Total Number”:

^{30} ⁄ _{80} = ^{x} ⁄ _{2,000} → x ^{30} ⁄ _{80} x 2000 = 750

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

The proportion being set up here is not quite like a proportion in an ordinary ratio problem (one covered by the standards 7.RP.2 and 7.RP.3). Those are precise relationships. The proportions set up in statistics items such as this are (1) based on a belief that you have a representative sample, and (2) yield an approximate answer. So what the above solution says is that the data yields a belief that close to 3/8 of the people going to this theater prefer comedy and that somewhere around 750 of the 2000 people prefer comedy.

- Two sets of data are shown. Each bar represents an equal number of children.

Which conclusions are supported by these two sets of data?

**Key: **A, B

**Solution:**

The bars are higher for girls age 2, 3, 4, and 5, so girls grow faster during these ages.

The bars represent equal numbers of children, so you can find the average height (in inches) by averaging the heights of the bars:

No information is given beyond age 5, so options C and E are not based on the data. Option D does not use the average, but rather the heights for age 3.

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

If the bars did not represent equal numbers of children, you could not simply average their heights. To give a very simple counter-example, suppose that the bars for girls age 2 through 5 each represented one girl and the bar for age 1 represented 2 girls. The average height for those 6 girls would be

^{18 + 18 + 23 + 33 + 36 + 40} ⁄ _{6} ^{168} ⁄ _{6} = 28

Note also that answers to questions such as this must be based on the given data, and not on things you may “know” that fall outside that data (so option C may be true, but it is not a correct answer here).

- Two contestants are each given a box with 1 gold ball and 3 black balls. Each person is blindfolded and has to pull out one ball. They win if both balls are gold. What is the probability that they win?

**Key: **A

**Solution:**

There is 1 way to win and a total of possibilities. Therefore, the probability is 1/16.

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

The total number of possibilities follows from the “multiplication principle.” If one set has *n* objects and another set has *m* objects and you chose one object from each set, the total number of possibilities is n x m. In this problem, each set is a box with 4 balls, so n = m = 4.

You may have thought there was a total of 4 possibilities: gold-gold, gold-black, black-gold, and black-black. Even though you cannot distinguish between one black ball and another, picking one black ball versus another is a different result.

- Part A.

Lin was given the table below, which lists the results of spinning two spinners. One spinner had three sections colored red, green, and blue. The other spinner had two sections numbered 1 and 2. Complete the table, listing the probability of each outcome as a decimal rounded to the nearest hundredth.

**Solution:**

Result of spin | Frequency | Probability |
---|---|---|

Red and 1 | 26 | |

Red and 2 | 21 | |

Blue and 1 | 55 | |

Blue and 2 | 49 | |

Green and 1 | 25 | |

Green and 2 | 24 |

*There were 200 spins all together. Divide each frequency by 200 to get the experimental probability.*

Part B.

Based on the data, explain how you would determine the number of “Red and 1” results you would expect after any number of spins.

*Because the probability of spinning “red and 1” is 0.13, I multiply 0.13 times the number of spins. For example, in 700 spins, I would expect about 0.13 × 700 = 91 results of “Red and 1.” *

Part C.

Lin knows that the numbered spinner is divided into two equal sections. She wants to determine the sizes of the colored sections on the other spinner. Using the data, explain how Lin can deduce the relative sizes of the colored sections.

*The red and green probabilities are about the same, so their sections should be the same fraction f of the spinner. The blue probabilities are about as much as the red and green combined, so its fraction of the spinner should be 2f. So together, f + f + 2f =1 (the whole spinner), and it follows that f = ¼. Thus, red and green are each ¼ of the spinner, and blue is ½ of the spinner.*

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

This problem is about experimental probability, as opposed to theoretical probability. If ¼ of the colored spinner is red, then theoretically ¼ of the time you will obtain red. If ½ of the numbered spinner is 2, then theoretically ½ of the time you obtain 2. This means that theoretically ½ of the time you got red you also get 2. That is ½ of ¼ of the time, or 1/8 = 0.125 of the time you “should” get red and 2. Compare this to the experimental probability (0.11) of what was actually obtained. If the spinners are truly unbiased, then after performing a large number of experiments, the experimental probability should be close to the theoretical probability.

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