A cart's position on a track is s(t) = t³ − 6t² + 9t meters after t seconds. At which times is the cart at rest?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0001
A school raffle ticket wins a $10 prize with probability 0.1 and a $2 prize with probability 0.3; otherwise it wins nothing. What is the expected prize of one ticket?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0002
A shop's cost to make x lamps is C(x) = 0.02x² + 30x + 500 dollars. What is the marginal cost when x = 100?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0001
Scores on a placement test are normally distributed with mean 70 and standard deviation 8. About what percent of scores are above 86?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0002
Show all 50 questions (46 more)
1. Core content: calculus and statistics, continued
The weights of apples from an orchard have standard deviation 18 grams. For random samples of 36 apples, what is the standard deviation of the sample mean?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0002
What is the area under the line y = 2x + 1 from x = 0 to x = 3?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0001
What is the limit of (x² − 2x − 15) / (x − 5) as x approaches 5?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0001
2. Fractions, ratios and the number system 10
2.5 gallons of paint cover 875 square feet. At that rate, how many gallons cover 1,400 square feet?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0003
A bag holds 4½ cups of birdseed. How many ¾-cup scoops does it hold?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0003
A coach has 84 granola bars and 126 juice boxes. She wants the greatest number of identical snack bags with nothing left over. How many bags can she make?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0004
A lawn sprinkler starts a cycle every 20 minutes and a drip line every 36 minutes. Both start at 6:00 a.m. How many minutes pass before they next start together?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0004
A printer uses 2.5 × 10⁻³ liters of ink for each page. How many pages does a cartridge holding 4 × 10⁻¹ liters print?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0004
A student finds 25 × 48 mentally by thinking “25 × 4 is 100, and 100 × 12 is 1,200.” Which property justifies regrouping 25 × (4 × 12) as (25 × 4) × 12?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0004
A student writes 2/3 + 1/4 = 3/7. Which task would best lead the student to see the error without being told the rule?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0003
A tape diagram for a trail mix shows 4 equal parts of peanuts and 7 equal parts of raisins. The mix holds 45 more grams of raisins than peanuts. How much does the mix weigh?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0003
What is the value of 32 raised to the power 3/5?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0004
Which fraction is equal to a terminating decimal?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0003
3. Relations, functions and algebra 12
A battery's charge is recorded every hour: 96, 48, 24 and 12 units at hours 0, 1, 2 and 3. Which kind of function models the data?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0006
A diver's height above the water, in meters, is h = −5t² + 5t + 10 after t seconds. When does the diver reach the water?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0008
A farm stand sells ears of sweet corn for $0.75 and pumpkins for $4. A customer buys 22 items for $36. How many pumpkins did the customer buy?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0007
A plumber charges a $60 visit fee plus $85 an hour. Why is the bill a linear function of hours worked but not proportional to hours?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0007
A removal program cuts a lake's population of 1,600 invasive carp by 25% each year. How many carp remain after 3 years?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0008
A storage unit costs a $45 deposit plus $22 a month. Marta has $265 to spend. For how many months, at most, can she rent it?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0005
A student graphs f(x) = √x and g(x) = √(x + 9) and says that g is f moved 9 units to the right. What should the teacher point out?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0006
A student solves 5 − 2x ≤ 11 and writes x ≤ −3. Testing x = 0 gives 5 ≤ 11, a true statement. What does the test show?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0005
A student solves x² + 6x − 5 = 0 by completing the square. Which equation is a correct step?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0008
The area of a trapezoid is A = ½h(b₁ + b₂). Which equation solves the formula for b₁?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0005
In which equation is y not a function of x?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0006
Which equation is the line through (6, 1) that is perpendicular to y = 3x − 4?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0007
4. Measurement and geometry 7
A blueprint is drawn to a scale of 1 : 50. A patio measures 12 cm by 9 cm on the blueprint. What is the patio's actual area?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0009
A carpenter marks 20 inches along one edge of a doorframe corner and 21 inches along the other. The marks are 29 inches apart. What does this show about the corner?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0009
A paper cup is a cone 8 cm across at the top and 9 cm deep. To the nearest cubic centimeter, how much water does it hold?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0009
A photo is enlarged on a copier set to 250%. By what factor does the photo's area grow?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0010
A quilt pattern has a patch corner at (−3, 4). The pattern is reflected across the x-axis and then across the y-axis. Where does the corner end, and which single transformation gives the same result?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0010
A sprinter runs 100 meters in 12.5 seconds. What is her average speed in kilometers per hour?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0009
On a coordinate map of a campus, the gym is at (−6, 1) and the library at (6, 6). Each unit is 10 meters. How far apart are the two buildings in a straight line?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0010
5. Probability and statistics 7
A café's lunch special is one of 3 soups, one of 4 sandwiches and one of 2 drinks, each picked at random. How many different lunches are in the sample space, and how likely is any one of them?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0011
A district wants to estimate what share of its 2,400 seventh graders walk to school. Which plan is most likely to give a representative sample?Reasoned from the definition of a random sample. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0012
A drawer holds 6 calculators that work and 4 that do not. A teacher takes two at random without putting the first back. What is the probability that both work?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0011
A teacher picks 3 of her 8 mathletes for a committee in which every member has the same role. How many different committees are possible?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0011
A two-way table of 50 seventh graders shows that 30 ride a bus and that 12 of those riders bring lunch from home; 20 students in all bring lunch. Given that a student rides a bus, what is the probability that the student brings lunch?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0011
The mean of five quiz scores is 82. A sixth score of 94 is added. What is the new mean?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0012
Nine students read 2, 3, 5, 6, 8, 9, 11, 14 and 20 books this year. What is the interquartile range of the data?Worked from the numbers in the question. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0012
6. Disciplinary literacy in mathematics 7
A teacher introduces the word “coefficient” on Monday. By the IES practice guide on adolescent literacy, what should follow so that students keep the word?Improving Adolescent Literacy: Effective Classroom and Intervention Practices (Institute of Education Sciences, 2008), Recommendation 1. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0014
A teacher is choosing words for explicit study in a unit on linear functions. By the framework's criteria, which word is the best choice?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0014
Before a unit on proportional reasoning, a teacher models how to mark up a word problem: circle the question, underline each quantity, and note its units in the margin. What is the teacher providing?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0015
Each Friday, students in a mathematics class rate their own written explanations against a class checklist and set a goal for the next week. Which use of assessment is this?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0013
Students compare a bar graph with the paragraph printed beside it and find that the paragraph reports a different year's totals. Which skill from the framework are they using?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0015
Students write a paragraph arguing which of two phone plans costs less over a year. By the framework, what should a strong written argument rest on?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0016
Which routine best builds the supportive language and literacy environment the framework describes for a mathematics classroom?The wording of the objective in the test framework. Written to the ILTS Middle Grades (5–8) Mathematics (202) framework, objective 0013
How this list was compiled: Our own questions, written to the test design and framework of Illinois Licensure Testing System field 202, Middle Grades (5–8) Mathematics, in the study guide on the ILTS website (https://www.il.nesinc.com/Content/STUDYGUIDE/IL_SG_OBJ_202.htm, copyright 2016 Pearson Education, Inc.; test page https://www.il.nesinc.com/TestView.aspx?f=HTML_FRAG/IL202_TestPage.html, read 2026-10-12): 100 multiple-choice questions, 80 scorable and 20 not, in 3 hours 15 minutes, passing score 240; Core Content (objectives 0001 and 0002) 14%, Fractions, Ratios, and the Number System (0003 and 0004) 20%, Relations, Functions, and Algebra (0005 to 0008) 24%, Measurement and Geometry (0009 and 0010) 14%, Probability and Statistics (0011 and 0012) 14% and Disciplinary Literacy in Mathematics (0013 to 0016) 14%. The bank holds 50: 7, 10, 12, 7, 7 and 7. Each computed answer can be worked again from the numbers in the question; the seven literacy questions rest on the wording of the framework or on the Institute of Education Sciences practice guide named under them. Licence: Questions, choices and explanations are original work of this site.
How it works
This free ILTS math 202 practice test follows Middle Grades (5–8) Mathematics, the Illinois content test numbered 202: 50 questions of our own, each worked out. It is unofficial and not affiliated with the Illinois State Board of Education or Pearson.
Read the full guide
ILTS 202 is Middle Grades (5–8) Mathematics
The ILTS test list has both Middle Grades (5–8) Mathematics, field 202, and Mathematics, field 208. This page follows 202 only. The two differ on test day too: the 208 test page tells candidates to bring their own graphing calculator.
What the 202 framework weighs
Core content, which is calculus and college statistics, 14%; fractions, ratios and the number system 20%; relations, functions and algebra 24%; measurement and geometry 14%; probability and statistics 14%; disciplinary literacy in mathematics 14%, as read October 12, 2026. The 50 questions split 7, 10, 12, 7, 7 and 7. Exam mode gives 98 minutes, the real pace.
The test page says an on-screen scientific calculator and a formulas page are provided with test 202.
Subarea 6 is about teaching: vocabulary, reading, writing and discussion in a mathematics classroom, with nothing to compute.
Questions and answers
How many questions are on the ILTS Middle Grades Mathematics (202) test?
100 multiple-choice questions in 3 hours 15 minutes of testing, after a 15-minute tutorial. By the framework, 80 of them are scorable and 20 are not.
What is the passing score for ILTS 202?
The test page and the framework both give 240. A count of right answers on these 50 questions is not that score, so the result here carries no pass verdict.
Is there calculus on the ILTS middle grades math test?
Yes. Objective 0001 is calculus: limits, continuity, derivatives as rates of change, and integration as area under a curve. College statistics is objective 0002. Together they form Subarea 1, 14% of the score.