Math 902 practice test 1 and 2

Teachers are using a combination of formative and summative assessments in their classrooms to assess their students' understanding of the mathematical concept they are learning. Which of the following lessons is an example of a summative assessment?

Giving a test at the end of the unit. [A unit test is a summative assessment. Summative assessment provides a gauge at a particular point in time to measure student learning relative to content standards.]

A school administrator has 45 erasers, 75 pencils, and 105 stickers to distribute to each homeroom teacher in the school to use as prizes. The administrator wants to give the same number of each item to each homeroom teacher. What is the greatest number of each item that can be given to each homeroom teacher?

15 [The greatest common divisor of 45, 75, and 105 is 15.]

Molly is going to roll a die 120 times. About how many times can you expect her to get a number other than 5 or 6?

80. [This is the correct answer. She will roll a 5 or a 6 approximately 20 times. There are 4 numbers other than 5 and 6: 1, 2, 3, and 4. Each of these numbers is expected to happen 20 times, since the product of 4 and 20 is 80.]

A teacher gives the following word problem to students.

Mike has 24 candy bars that he gives to his friends. How many friends does he give the candy bars to if each friend gets 6?

Which of the following methods would be acceptable ways for the teacher to help students solve the problem?

- Write the equation 24 ÷ x = 6 and then solve.

-Make circles with 6 candy bars in each circle until reaching 24 candy bars. Count the number of circles.

-Subtract 6 from 24 until reaching 0. Count the number of times 6 was subtracted.

-Create an array with 6 dots in each row until reaching 24 dots. Count the number of rows.

Henry is meeting a friend for dinner at a restaurant that is 15 miles from his house. Two different cab companies are available for hire. Cab Company A charges a $3 base rate in addition to $0.75 for every mile. Cab Company B charges a $5 base rate in addition to $0.50 for every mile. Using slope-intercept form, which cab company would charge Henry a lesser amount for his 15-mile trip?

Cab Company B: 0.50(15) + 5 = 12.50 [The equation 0.50(15) + 5 = 12.50 correctly shows the total cost that Henry paid using Cab Company B. In the equation, the slope represents the cost per mile, which is $0.50. Since Henry traveled 15 miles at a rate of $0.50 per mile, these numbers are multiplied to determine how much it costs to go 15 miles, which is $7.50. The y-intercept represents the base rate of $5, which needs to be added to $7.50 (the total cost for 15 miles) to determine the total cost of the trip, which is $12.50.]

The map of a city is drawn in the first quadrant on a coordinate plane. The location of the library can be found at (4,5). A line drawn from the library to the police station can be represented by the equation y = 3/4x + 2, and the straight-line distance from the library to the police station is 10 units. What coordinate represents the location of the police station?

(12,11) [The point (12,11) is on the line y = 34x + 2. Using the Pythagorean theorem with side lengths of 6 and 10, the distance between (4,5) and (12,11) is 10.]

Taylor is redecorating his room. At the store, he purchased paint brushes for $23, paint for $61, painter's tape for $15, and a tarp for $12. What is the estimated amount of money that Taylor spent on his items, rounding the price of each item to the nearest $10?

$110 [All numbers were correctly rounded to the nearest $10: $23 was rounded down to $20, $61 was rounded down to $60, $15 was rounded up to $20, and $12 was rounded down to $10. So, 20 + 60 + 20 + 10 = $110]

Morgan and her family just moved to a new town, where she will be attending a new school. Her math class is currently learning how to multiply multi-digit numbers. Since arriving, Morgan has shown difficulty with the topic. What assessments can her math teacher use now to determine what is preventing Morgan from performing well on her assignments? Select ALL that apply.

Error analysis

Constructed response

Anecdotal records

[An error analysis is a type of formative assessment the teacher creates that contains specific problems that focus on a certain skill. This analysis will allow the teacher to identify any common errors Morgan is making, The teacher could use a constructed response with the whole class that requires students to first solve a multiplication problem and then write the steps they followed. The written portion of the response will help the teacher identify any errors in the multiplication process that students are making, Anecdotal records are a type of formative assessment that use direct observation and written notes to document student achievement. With these records, the teacher could observe Morgan when she is completing specific problems and document any errors or misconceptions that are occurring.]

In Ms. Lori's math class, students are working in groups to identify shapes by observing the properties of each shape. One group of students receives a shape that is described as a parallelogram, with both pairs of opposite sides parallel, two sets of opposite congruent angles, and sides that are of equal length. Which shape do these properties describe?

Rhombus. [A rhombus is a shape that is identified by its two pairs of opposite, parallel sides. All four sides in a rhombus are congruent, and opposite angles are congruent.]

Simplify the expression 2(11 − 2) ÷ (−2). Which of the following classifications correctly describe the result? Select ALL that apply.

-9, Integer. [The answer −9 is classified as an integer because it is the negative of a natural number.]

Rational Number. [The answer -9 is classified as a rational number because it can be expressed as the fraction −91.]

Which of the following concepts would a teacher best illustrate to students using a geoboard?

Area of triangles. [A geoboard is a mathematical manipulative used to teach plane geometry. It is a physical board with nails half driven in neat rows and columns. Rubber bands are wrapped around the nails to make triangles and other polygons. When a triangle is made on the geoboard, a student can count the height and base to calculate the area or imagine the rectangle that could be used to determine the area of the triangle.]

A teacher put the following expression up on the board for students to simplify. Which strategies correctly simplify the expression?

8(7 − 2) + 6(2 + 4)

8(5) + 6(6)

8(7) − 8(2) + 6(2) + 6(4)

[This strategy shows that you can simplify what is in the parentheses first, This strategy applies the Distributive Property to each set of parentheses first.]

Ryan and Jamal are competing in a swim-a-thon. Ryan has $40 of donations and pledges to donate another $0.50 for every lap he swims. Jamal has $25 in donations and pledges to donate another $1.00 for every lap he swims. Which graph correctly models the number of laps that Ryan and Jamal must swim to reach the same amount of money?

(30,55) [The linear function that represents the amount of money Ryan will raise has a y-intercept of 40, which represents base donations. The graph has a slope of 0.5, which represents $0.50 raised for each lap. The equation is y = 0.5x + 40. The linear function that represents the amount of money Jamal will raise has a y-intercept of 25, which represents the base donations. The graph has a slope of 1, which represents $1 raised for each lap. The equation is y = x + 25.]

A teacher gives a formative assessment to evaluate student learning halfway through a particularly challenging instructional unit. In which of the following ways could the teacher best use such an assessment to decide whether to make any modifications to teaching methods for the remainder of the unit?

Compare the results to a set of learning goals set at the beginning of the unit. [Comparing the results to a set of learning goals would help the teacher determine whether any modifications should be made. If the goals have been met, likely few or no modifications needs to be made; if the goals have not been met, modifications would likely be beneficial.]

Which fraction is shown on the 100-grid? (24 boxes shaded out of 100)

6/25. The 100-grid shows 24 of the 100 squares shaded, 24/100 = 6/25.

Which of the following methods can be used to correctly find the area of the trapezoid? (A=8, B=14, H=5)

Cut the trapezoid into a rectangle in the middle and into two right triangles on the sides. Find the areas of the rectangle and both triangles. Add the areas of the rectangle and both triangles. [The trapezoid can be broken down into two right triangles and a rectangle. Adding the areas of all the figures, the two right triangles and the rectangle, will equal the area of the trapezoid.]

Consider x = 2y+8/6. Which equation is rewritten as a function of y in terms of x?

y = 3x-4 [The equation is correctly solved for y by first multiplying by 6, then subtracting 8, and finally dividing by 2.]

Which measure of central tendency is most influenced by outliers?

Median [The mean is the average of the values in a set of data. Outliers in a set of data affect the total, which is used to calculate the mean.]

survey asked people how often they brush their teeth in the morning. The resulting data is shown in the table.

What is the relative probability of always brushing in the morning for people 30 and under compared to people over 30?

0.80 [The probability of the individual age groups is 0.40 for individuals 30 and under and 0.50 for individuals over 30, which, when compared, produces the result of 0.80.]

The length of segment AB is represented by the expression −4a − 9. Which of the following is a reasonable solution for a?

-5 [If a = −5, then AB = −4(−5) − 9 = 11. Any positive length is reasonable; therefore, −5 is a reasonable value of a.]

To choose a line leader each day, a teacher puts each student's name in a bucket and draws a name. In a class of 25 students, what is the probability that the same student will be chosen two days in a row?

0.0016 [The probability of a student being chosen for one day is 125 = 0.004. The probability of a student being chosen two days in a row is 0.04(0.04) = 0.016.]

After reviewing students' homework, the teacher realizes most of the class is having difficulty with solving multi-step equations. Which of the following activities would be the most effective strategy for the teacher to use to help the students understand the process of solving multi-step equations?

Have students work individually on small whiteboards to solve equations while the teacher circulates to provide feedback. [Providing immediate and consistent feedback gives students an opportunity to recognize their mistakes right away and build confidence as they get more answers correct.]

To estimate the population of fish in a small lake, researchers caught, tagged, and released 50 fish. They then caught another sample of fish and counted 29 tags in the 300 fish in their sample. Which of the following is the best estimate for the number of fish in the lake using this data?

517 [To estimate the population of fish, first write the proportion 29300 = 50x. Next multiply 300(50) = 15,000, and finally divide 15,000 by 29, which is approximately 517.]

A town's population data for the previous ten years is published in a teacher's local paper. The data suggests a linear relationship. Which of the following would be the best activity for the teacher to use to demonstrate the connection that mathematics has to the real world?

Predict the population of the town in five years. [The data can be used to create a linear model that allows for future population predictions. This option best shows the power of mathematics in the real world.]

Six friends are sitting on a bench in the park. Tyler is sitting between Sam and Paul, and Heather is sitting between Colin and Barb. If Barb is sitting on the left of Sam, who is sitting between Heather and Sam?

Barb. [Drawing a picture and placing the six friends on the bench based on the clues provided in the problem will help determine who was sitting between Heather and Sam. The first clue puts Tyler between Sam and Paul, indicating an order of Sam, Tyler, and Paul. The second clue places Heather between Colin and Barb, indicating an order of Colin, Heather, and Barb. The third clue states that Barb is sitting on the left of Sam, indicating an order of Colin, Heather, Barb, Sam, Tyler, and Paul. Therefore, Barb is sitting between Heather and Sam.]

What expressions are equal to (2/3 × 1 2/4) × 5? Select ALL that apply.

5/1, 5, 60/12. [multiply 12/12 × 51 = 60/12]

The band students at two different high schools sold shirts for one week to raise money to buy new instruments. The line graph represents the amount of money each high school raised per day. What percent of the total money High School A raised was made on Wednesday?

22.5 [To determine what percent of the total amount High School A earned was made on Wednesday, use the equation, 45/390 = x1/00. The fraction represents how much money was made on Wednesday over the total amount High School A earned. The total is calculated by adding the amount of money made on each day of the week, 20 + 25 + 45 + 60 + 10 + 5 + 35 = 200. The fraction x100 represents the percent of money made out of 100. To find x, cross-multiply to yield 45(100) = 200x and divide both sides by 200 to isolate x, which yields x = 22.5.]

The first-grade class is learning about money. Which activities would help strengthen the students' understanding of the name and value of different coins? Select ALL that apply.

The students match a picture of a coin to its name and value. [This activity will require students to recall the name and value of coins to match them.]

The students learn a song that incorporates visuals to teach them the name and value of each coin. [Teaching the students a song with visuals meets the needs of a variety of multiple intelligences, including visual and auditory. It is also a tool the students can use when they are trying to recall the name and value of a coin.]

A bush plane can carry less than 1,200 pounds of freight and luggage. A grocery store has shipped an order on the plane that weighs 600 pounds, and each passenger aboard the plane brought 80 pounds of luggage. If each passenger on the plane brought 80 pounds of luggage, how many passengers could bring luggage on the plane?

7. [The inequality 1,200 > 600 + 80x correctly represents the variables in the problem needed to find the total number of passengers that could bring luggage on the plane. Combining like terms yields 600 > 80x, which yields to x < 7.5. This value must be rounded down to 7 since rounding up would put the plane over its weight limit for freight and luggage.]

During weekly collaboration, the second-grade team is working with a math specialist to write a clear learning goal for its upcoming unit on shapes. What learning goal would be the most appropriate for a unit on shapes?

Students will be able to identify two-dimensional and three-dimensional shapes. [This learning goal focuses on the instruction students will receive to be successful in the unit. If students are able to identify two-dimensional and three-dimensional shapes, they will be able to complete the activities that are part of the unit.]

What is the eighth term in the following sequence? 2, 7, 17, 37, 77, ___, ___, ___

637. [The correct pattern for the sequence is n(2) + 3, where n represents the previous number in the sequence. The eighth term is found by first identifying the sixth and seventh terms. The sixth term is 157, and the seventh term is 317. The eighth term is determined using the expression 317(2) + 3, which yields 637.]

The fourth-grade students are working on multiplying three-digit by two-digit numbers using different strategies. For today's lesson, the students were asked to independently solve problems on their own before discussing the problems with the whole class. During the class discussion, the teacher observed that the conversations lacked mathematical discourse. Which questions can the teacher write on the board to help increase mathematical discourse among her students? Select ALL that apply.

Can you walk the class through the steps you took to solve the problem?

Can you solve this problem using an alternative strategy?

If you answered the question incorrectly, can you explain what you need to correct?

A concession stand at a high school football game sells hotdogs, h, and cheeseburgers, c. A hotdog sells for $2, and a cheeseburger sells for $3. In the course of one football game, the stand received $510 from these items. A total of 195 hotdogs and cheeseburgers are sold altogether. Which two equations are needed to solve for the number of hotdogs and cheeseburgers sold? Select ALL that apply.

2h + 3c = 510. [The equation 2h + 3c = 510 is used in a system of equations, with c + h = 195, to solve for the number of hotdogs and cheeseburgers sold. The equation is written using the information about the costs of each hotdog and cheeseburger and total money received from these items.]

- c + h = 195

Which of the following phrases can be used to express 4 ÷ x = 12? Select ALL that apply.

-The quotient of 4 and x is 12. ["The quotient of 4 and x is 12" correctly describes the equation because 4 is the dividend, x is the divisor, and 12 is the quotient.]

-4 divided by x equals 12. [The statement "4 divided by x equals 12" correctly describes the equation because 4 is the dividend, x is the divisor, and 12 is the quotient.]

What expression represents the area of the polygon?

1/2(4)(6) + (6 × 6) [This is the correct answer. To find the area of the polygon, find the areas of the triangle and rectangle and add them. The area of the triangle is 1/2bh = 1/2(4)(6) and the area of the rectangle is bh = 6 x 6. Adding these areas yields 1/2(4)(9) + (6 × 6).]

Select all of the scenarios that would be a relevant application of tangrams. Select ALL that apply.

Introducing geometric figures to a group of students. [Tangrams would give students a tangible model when introducing geometric figures to students.]

Decomposing shapes to find area. [Decomposing shapes to find area is easy to visualize with tangrams.]

Developing an understanding for fraction algorithms. [Tangrams can be used when developing an understanding for fraction algorithms. A figure composed of four triangles would help show 1/4, 1/2, 3/4, and 1 whole.]

Gloriana deposits $550 into a savings account and does not plan to deposit or withdraw any money from the account for 8 years. The bank offers her four options for interest. Which option will accumulate the most money in 8 years?

5% interest compounded annually. [To find the total after 8 years of 5% compound interest, use the compound-interest formula A = P(1 + r/n)nt, where P represents the principal, r represents the interest rate, n represents the number of times interest is applied each year, and t represents the number of years. This formula yields an account balance after 8 years of A = 550(1.05)8 = $812.60.]

Which of the following are ways to increase a diverse group of students' understanding of mathematical language regarding polygons in introductory geometry? Select ALL that apply.

- Encourage English-language learners to support each other, and ask them to describe figures in their native language.

-Provide pictures, graphic representations, and real and touchable examples of a mathematical term. Draw each figure and hand students physical models of the shapes.

-Show that vocabulary can have multiple meanings. Students most likely have heard of "The Pentagon." Explain the difference and the connection between the shape and building.

-Identify and pre-teach key vocabulary. Make sure that students know what angles, sides, vertices, and lines mean before starting the lesson.

In the figure above, ∠A ≅ ∠N and AB ≅ No. Which additional piece of information would be required to prove that triangles ABC and NOP are congruent?

AC ≅ NP [Given that these sides are congruent, it can be proven that these two triangles are congruent because of the Side-Angle-Side Congruence Theorem. This theorem states that if the two sets of corresponding sides and included angles are congruent, then the triangles are congruent.]

A gumball machine has 15 red gumballs, 20 blue gumballs, 10 yellow gumballs, and 25 green gumballs. If Mateo has enough money to get one gumball, what is the probability of getting a gumball that is not red or yellow?

9/14. This is the probability of not getting a red or yellow gumball. The sample space includes a total of 70 gumballs, of which 45 are not red or yellow, 45/70 = 9/14.

Which of the followings examples could a teacher best use to show how mathematics is used in a variety of careers?

-A farmer uses a ratio to determine how many lettuce plants he could fit in a given-size field.

-A store owner finds the profit on each pumpkin using the price that the seeds cost and the price at which he plans to sell them.

-A warehouse supervisor determines the percentage of product damaged in a flood.

What is the nth term in the sequence?

0, 6, 24, 60, 120, 210....

n³ − n. [The pattern is n³ − n. The first number in the sequence is 1³ = 1 − 1 = 0. The second number in the sequence is 2³ = 8 − 2 = 6. The nth term in the sequence is n³ − n.]

In which of the following fields of mathematics is the Pythagorean theorem most utilized?

Geometry. [The Pythagorean theorem is a geometric theorem that states that the sum of the squares of the measures of the legs of a right triangle is equal to the square of the measure of the hypotenuse.]

Janelle, Michael, and Corey went fishing. Janelle caught 24 fish. Janelle caught twice as many fish as Michael, and Michael caught three times as many fish as Corey. Which of the following approaches can be used to model the number of fish that Corey caught?

Put 24 number tiles into two equal groups, and then create three equal groups from each of those groups. [Dividing 24 tiles into two groups finds the number of fish Michael caught, which is half as many as Janelle. Dividing those two groups into three equal groups determines the number of fish Corey caught, since he would have caught a third of the number of fish Michael caught.]

An elementary school is selling discount cards as a fundraiser. The team pays $5 for each card and sells them for $25. One donor contributed $100 to begin the fundraiser. Which of a following correctly models the profit the team made, y, as a function of the number of cards sold, x?

- y = 20x + 100

-graph where top is 360. [The correct y-intercept is the amount of money they start with, which is the $100 donation. The slope is the amount of money they earn on each card. Since they pay $5 for each card and sell the cards for $25, the amount of money they make on each card is $20 − $5 = $20.]

Which of the following tools are used to find a unit of measure? Select ALL that apply.

Ruler, Stopwatch, Protractor.

If x + y = 12 and y = x + 9, which of the following statements is a valid conclusion?

x + (x + 9) = 12. [This choice substitutes x + 9 for y, which yields (x + 9) + x = 12. The second given equation shows these expressions to be equal, so this is a valid substitution.]

A fish tank in the shape of a cylinder is completely filled with water. The tank is 3 feet high, and the base has a diameter of 4 feet. Which of the following statements explain an approach to finding the volume of the water in the tank?

- Use the formula V = πr²h, and substitute 2 for r and 3 for h.

- Find the area of the base of the tank by squaring the radius and multiplying by π. Then multiply the area of the base by the height.

When reading notes in music, different notes have specific lengths. For example, a whole note has four beats, a half note has two beats, and a quarter note has one beat. What is the best approach to connect the concept of reading notes to mathematics?

Have students practice part- to whole-fraction relationships by finding different combinations of notes equivalent to whole notes. [Different combinations of notes can be used to create the same number of beats as a whole note. This models the part to whole relationship of fractions perfectly.]

A restaurant has a total of $1,800 in sales for dinner and $650 in sales for the rest of the day. The cost of labor for the day is $1,100. The cost of food and all other expenses is estimated to be $445. Which of the following is the gross income for the restaurant?

$2,450. [The gross income is the amount of profits before subtracting expenses. In this case, it is the total sales for the restaurant for the day, 1,800 + 650 = 2,450.]

Molly is driving to visit a friend for the weekend, and she notices that her gas tank is getting low. She knows that her gas tank holds 24 gallons of gas and her car can go 15 miles for every gallon of gas. The diagram above depicts Molly's gas tank. How far can Molly drive before she will need to stop for gas?. [graph shows three out of 8 bars being shaded.]

135 miles. [Use 3/8 = x/24 to determine how many gallons of gas are left in the car. In the equation, 3/8 represents how full the gas tank is, 24 represents how much gas the tank holds, and x represents how many gallons are left in the tank. Cross multiply to have 8x = 3(24), which yields x = 9. The car has 9 gallons of gas. To determine how many miles the car can go until empty, multiply 9(15) = 135, where 9 represents how many gallons of gas are left in the car and 15 represents how many miles can be driven on each gallon. This indicates that the car can go 135 more miles before the tank is empty.]

Jeff wants to start a business selling the fish he catches. He has secured a deal to sell a specific amount of fish to a local store each day. Below is information he has gathered from his previous 10 days of fishing.

10, 15, 20, 22, 30, 32, 50, 99, 99, 100

When determining the amount of fish Jeff will be able to sell to the store each day, which central tendency should he use

The median. [The median is the middle number in a set of numbers when ordered from least to greatest. This measure would be the most appropriate one to use because it is the most representative of the numbers in the population. It ensures that Jeff does not overcommit himself to selling more or less fish than he can catch any one day.]

Ms. Ellen is grading her students' quizzes from the first part of their unit on shapes. She wants to use the results of the assessment to guide the rest of her instruction in the unit. When reviewing the results, which of the following questions should she consider to develop relevant content that fits her students' needs?

-Was there a certain type of problem or strategy that a majority of the students missed that I should cover again? [Identifying which problem types or strategies students missed will help the teacher determine if a review lesson is needed before moving on to new content.]

-Should I change the format of my lessons to incorporate additional time for group or individual practice? [Giving students adequate time to practice the content they have learned is an important part of instruction. It allows the teacher to see where students are showing difficulty, and it allows students to master the process and skills they need to be successful.]

-What additional resources or strategies could I use to help students grasp the concepts we are learning in class? [If the teacher identified in advance additional strategies or methods for teaching the content, she would be able to use them in upcoming lessons if students were having difficulty grasping the main strategy given. Having multiple strategies available will allow her to differentiate her lessons to meet the needs of all of her students.]

[regular simple pentagon] which of the following words correctly describes the polygon shown above?

regular, convex, pentagon

During a visit to California, Karen bought her friend dinner. When she received the bill, the total was $90 before tax. The sales tax for California is 7.25%. Karen wanted to leave a 15% tip on the amount that includes the cost of the meals and the sales tax. What is the total amount of money Karen will pay for dinner, the sales tax, and the tip?

$111.01. [90 x .0725 = 6.525, 90+6.525=96.525, 96.525 x .15 = 14.47875, 96.525+14.47875 = 111.00375]

[The graph above represents information on 10 people and their travel time to work in minutes on a typical workday. Using the normal distribution graph and a standard deviation of 11.33 minutes for this population, which of the following statements are true? [diagram of a regular curve on a x-axis, with 30 marked as the mean.]

- the mean of the data is 30

-68% of the time traveled is between 18.67 minutes and 41.33 minutes.

-34% of the time traveled is between 18.67 minutes and 30 minutes

-Since the data is symmetric, the mean best represents the data.

Of which of the following models can be used to solve the equation y = 2 (3x + 1) for when x = 2?

- y = 6x+2

- -1,4 0,2 1,8, etc etc

- graph with intercept at 2

Triangle ABC is shown on the coordinate plane. Triangle ABC is reflected across the y-axis and then rotated 90 degrees clockwise about the origin, resulting in triangle DEF. Which graph shows the correct placement of triangle DEF given the transformations?

F (1,1) / D (6,1) / E (1,4)

A teacher has an upcoming unit assessing proportionality. Which of the following actions would best aid instruction based on this standard?

Define clear learning goals for students about the standard. [It is very important for all students and teachers to understand what they are expected to achieve for a given TEKS.]

Which of the following responses would be a proper application of visual media used to aid instruction in the mathematics classroom?

- A circle graph to show the percent of students that prefer each subject

-A stem-and-leaf plot to show the heights of the students on the basketball team

-A table to teach graphing two-step equations

-An animation of rotations, translations, and reflections of a figure on the coordinate plane

x = -1, 0, 2, 6

y = -1/2, 1, 2,4

Which of the following rules could be used to represent the pattern shown in the table?

y = x/2 + 1

Each student in a group reported the number of letters in his or her first name.

11, 7, 8, 4, 3, 5, 6, 10, 10

Which statement states and explains the meaning of the median with respect to the data?

The median is 7, and this value represents the middle number of letters in the students' names when ordered from least to greatest.

Line x and line y are perpendicular lines. Which of the following are properties of line x and line y?

- Line x and line y intersect to form a right angle. [Perpendicular lines must intersect to form a right angle.]

-The slopes of line x and line y are the negative reciprocal of each other. [The slopes of perpendicular lines always are the negative reciprocal of each other.]

A teacher writes the problem 357 − 291 = ? on the board. Some students solve the problem using various learned procedures, and others begin by using manipulatives to represent the problem. Which of the following responses explains a procedure or representation that can be used to evaluate this subtraction problem?

-Use an algorithm for borrowing and regrouping.

-Work backward to determine the number that can be added to 291 to reach 357.

-Model the problem with base 10 blocks to highlight each place value and eliminate common blocks from both numbers.

Which of the following are instructional methods that a teacher could use when teaching subtraction of multidigit numbers?

-Use visuals and manipulatives to represent the numbers.

-Use frequent formative assessments to assess for understanding.

-Connect subtraction of multidigit numbers to addition of multidigit numbers.

A teacher asks her class to complete the problem 500 − 237 = ? using mental math. Which of the following responses explains a strategy that would help students complete the problem most efficiently?

Subtract one from each number and then subtract the resulting numbers. [Subtracting one from each number creates a problem that can be completed without borrowing: 499 − 236. This problem can be completed quickly using mental math.]

[circle inside of a square, with sqaure lines being 10 by 10.] Two darts are randomly thrown at the target. Which response shows the probability that one dart will hit the shaded region and the other dart will hit the unshaded region?

0.17. [The probability of hitting the shaded area is found by the ratio 5^2π/100 = 25π/100 = 0.79, and the probability of hitting the unshaded region is found by the ratio 100−25π/100 = 0.21. To determine the probability of both happening, multiply the two probabilities: 0.79 × 0.21 = 0.17

A teacher notices many students are having difficulty adding fractions with unlike denominators after they have been taught to find a least common denominator. She realizes that even though they are able to determine a least common denominator, they cannot consistently change the fractions correctly. Which of the following approaches would be appropriate activities the teacher could use to help her students?

- Ask students to use a rectangle to model each fraction in the addition problem. Then divide the rectangles into the same number of parts. Use the new rectangles to rewrite the addition problem with a common denominator.

- Have students model each fraction in the addition problem with a pie chart. Divide the pie charts until they have the same number of pieces, and rewrite the fractions that represent the new pie charts.

Which of the following is an accurate description of the evolution of mathematics instruction from the 1990s to the present?

This is the correct answer. Before introducing an algorithm, the teacher uses lessons and activities to create a conceptual understanding of the process and procedures.

Prior to introducing the algorithm for long division, a teacher uses strategies to develop the concept. Which of the following methods can she use to build a conceptual understanding of a long division problem similar to 360 ÷ 4?

A. Using multiplication to divide by using partial products

C. Using base-ten blocks to model place value in a long division problem

D. Using a number line to show the change in the number during division.

E. Using an area model to give a visual representation of the divisor and dividend in a long division problem

The table shows the number of daily downloads of a song.

Days/X = 1, 2, 3, 4

Streams/Y = 14, 12, 10 ,8

Which equation accurately represents the data in the table?

y = −2x + 16

An animal shelter is selling homemade dog biscuits in batches of three at a flea market to raise money for the animals. The shelter paid $50 to rent the booth, and it costs the shelter $4 to make each batch of dog biscuits. The shelter sells each batch of dog biscuits for $8. What is the minimum amount of dog biscuits the shelter must sell to make a profit of $200?

63. [profit for each dog biscut is $4, 63 x 4 = 252, - the $50, its $202.]

Equivalent to 3x−9/8x−24

- 3/8, .375, 6/16

The fourth-grade math teacher is introducing new manipulatives to her class to help the students understand place value. Which of the following activities would be appropriate for the teacher to do to ensure that students use the manipulatives appropriately?

-Demonstrate how the manipulatives are used.

-Assign activities with the manipulatives for students to do at home.

-Assign activities with the manipulatives for students to do at home.

Tyler works at a game store. The store just received a new shipment of merchandise. The shipment includes 10 childrens' games, 5 travel-size games, 3 card games, and 12 video games. What is the probability that Tyler selects a travel-size game and then, when replacing the travel game, a card game?

1/85. [Determine the outcome of the first event, selecting a travel-size game. The probability is 5/30 since the total number of travel-size games is 5 and the total number of games is 10 + 5 + 3 + 12 = 30. Then determine the outcome of the second event, selecting a card game, 3/29. The bottom number is changed from 30 to 29 because a game was taken out and not replaced. Then multiply 5/30 × 3/29 = 15/870. This fraction reduces, as 15÷15/870÷15 = 1/85.