PECT Math Study Guide

1. Which of the following statements best describes the purpose of the PA Core Math Content Standards?

A. To establish a minimum skill set for students entering kindergarten.

B. To provide a detailed, step-by-step curriculum for teaching math at each grade level.

C. To outline the knowledge and skills students should achieve at each grade level to be college and career ready.

D. To assess teacher effectiveness in delivering math instruction.

C. To outline the knowledge and skills students should achieve at each grade level to be college and career ready.

2. A teacher asks her students to analyze a word problem, identify key information, and decide how to solve it. This activity most closely aligns with which Standard for Mathematical Practice?

A. Attend to precision.

B. Reason abstractly and quantitatively.

C. Make sense of problems and persevere in solving them.

D. Model with mathematics.

C. Make sense of problems and persevere in solving them.

3. The Math Process Standards emphasize that effective mathematics instruction should include opportunities for students to:

A. Memorize formulas and use them in real-world applications.

B. Explore concepts through hands-on activities, communication, and reasoning.

C. Practice repetitive problem-solving strategies to build fluency.

D. Rely solely on technology for accurate computation.

B. Explore concepts through hands-on activities, communication, and reasoning.

4. Which of the following classroom practices demonstrates the Standard for Mathematical Practice, “Construct viable arguments and critique the reasoning of others”?

A. Students solve math problems independently and write their solutions in a journal.

B. Students work in pairs to explain their solutions and challenge each other’s thinking respectfully.

C. The teacher provides students with a list of formulas to use when solving problems.

D. Students participate in timed math fact drills to improve fluency.

B. Students work in pairs to explain their solutions and challenge each other’s thinking respectfully.

Which of the following teacher actions would best support the development of a productive disposition in students?

A. Rewarding students only for getting the correct answer on a math test. B. Encouraging students to view mistakes as opportunities for learning and growth.

C. Assigning repetitive drills to reinforce procedural skills.

D. Providing immediate solutions to challenging problems to prevent frustration.

B. Encouraging students to view mistakes as opportunities for learning and growth.

A teacher notices that a student can solve addition and subtraction problems accurately but struggles to explain why their solution works. To help the student develop a well-rounded mathematical proficiency, the teacher plans activities that focus on which of the following strands?

A. Adaptive reasoning and conceptual understanding

B. Productive disposition and algorithmic accuracy

C. Procedural fluency and productive disposition

D. Algorithmic accuracy and procedural fluency

A. Adaptive reasoning and conceptual understanding

A third-grade teacher notices that students are able to solve multiplication problems using a standard algorithm but make frequent errors when solving multi-step word problems involving multiplication. To improve their procedural fluency, the teacher should:

A. Provide timed multiplication drills to improve speed and accuracy.

B. Assign more word problems for students to practice solving independently.

C. Engage students in activities that connect multiplication algorithms to visual models and real-world contexts.

D. Require students to memorize the steps of the multiplication algorithm.

C. Engage students in activities that connect multiplication algorithms to visual models and real-world contexts.

A second-grade teacher is having students work on adding double digit numbers. The teacher writes the problem on the board, asks students to estimate the answer, and then asks them to explain how they arrived at their answers. As students listen to their peers’ explanations, some students are able to revise their answers correctly while others are not. Which of the following follow-up activities would be most appropriate?

a. Working with small groups to help students at different ability levels construct their understanding of the concept of addition.

b. Providing more challenging problems such as triple digit addition for students to practice their estimation and reasoning skills.

c. Drawing on students’ understanding of double-digit numbers to introduce other arithmetic concepts involving double-digit numbers.

d. Having students take a short test to find out who has grasped the concept and who may need more individual focused instruction

a. Working with small groups to help students at different ability levels construct their understanding of the concept of addition.

A teacher provides mathematics instruction using manipulatives, worksheets, games, and class discussion. The primary benefit of providing instruction in multiple formats is that it:

a. Helps students engage prior knowledge

b. Encourage the development of social skills through group participation

c. Addresses a variety of learning styles

d. Allows students to see diferent ways to solve the same problem

Addresses a variety of learning styles

A teacher wants to help students develop their understanding of relative magnitude of numbers. Which of the following types of mathematics learning materials would be most effective for the teacher to use for this purpose?

a. Number lines

b. Plastic coins

c. Pattern blocks

d. Number cubes

d. number cubes

A preschool teacher wants to promote children’s skills related to counting and comparing numbers. Which of the following activities is most appropriate for this purpose?

a. Asking the children to look around the classroom for the largest numeral they can find

b. Letting the class vote on which book the teacher should read next and then asking the class which book won based on the votes

c. Having the class count the number of times a flipped coin comes up tails and predicting whether heads will come up on the next flip

d. Selecting one child each day to count the number of children in the class and set the appropriate numbers of cups out for juice at snack time

b. Letting the class vote on which book the teacher should read next and then asking the class which book won based on the votes

Which of the following strategies is most appropriate for solving the problem below?

One week Liam baked some cookies and gave half of them away. The next week Liam baked twice as many cookies and gave half of them away. The third week Liam again baked twice as many cookies as the week before and gave half of them away. If Liam gave away 24 cookies the third week, how many did he give away altogether?

c. Working backward

To correctly use the ordinal numbers first through fifth with a group of five objects placed in a row, a young child must first be able to:

a. Recognize and write numerals one through five

b. Understand that order of the objects in the group does not affect the total number

c. Recognize that the last number counted is the number of objects in the group

d. Understand that the number five is made up of smaller numbers

b. Understand that order of the objects in the group does not affect the total number

Which of the following strategies is most effective for helping kindergarten students develop fluency with basic addition facts?

A. Encouraging students to memorize all sums up to 10 through repetition

B. Teaching students to use counting-on and number relationships to find sums

C. Having students practice long addition with two-digit numbers

D. Providing students with calculators to verify their answers

B. Teaching students to use counting-on and number relationships to find sums

A first-grade teacher wants to improve students' fluency with subtraction facts within 10. Which of the following activities would best support this goal?

A. Using timed tests to assess how quickly students can recall subtraction facts

B. Introducing fact families to help students see the relationship between addition and subtraction

C. Encouraging students to solve multi-step word problems with subtraction

D. Having students repeatedly write subtraction facts in a math journal

B. Introducing fact families to help students see the relationship between addition and subtraction

A teacher introduces ten-frames and rekenreks to students when teaching basic addition and subtraction facts. What is the primary benefit of using these manipulatives?

A. They provide students with a visual representation of number relationships.

B. They encourage students to rely on rote memorization of facts.

C. They help students understand long division concepts.

D. They allow students to focus only on even-numbered sums.

A. They provide students with a visual representation of number relationships.

A first-grade teacher wants to help students understand the value of digits in two-digit numbers. Which of the following activities would best support this goal?

A. Asking students to memorize the names of all two-digit numbers

B. Having students decompose numbers into tens and ones using base-10 blocks

C. Teaching students to count by ones up to 100 without grouping numbers

D. Asking students to recite numbers backward from 50 to 0

B. Having students decompose numbers into tens and ones using base-10 blocks

Which of the following student activities best demonstrates an understanding of place value in a three-digit number?

A. Identifying the largest and smallest three-digit numbers

B. Writing the number 342 and explaining that the "3" represents 300, the "4" represents 40, and the "2" represents 2

C. Listing all possible ways to write a number using different digits

D. Memorizing a song that lists numbers from 100 to 900

B. Writing the number 342 and explaining that the "3" represents 300, the "4" represents 40, and the "2" represents 2

A kindergarten teacher asks students to count out 25 cubes and then group them into tens and ones. This activity primarily supports students in developing which of the following concepts?

A. Skip counting by twos and fives

B. The ability to add two-digit numbers

C. An understanding of the base-10 number system

D. The concept of odd and even numbers

C. An understanding of the base-10 number system

A third-grade teacher notices that some students struggle to determine the value of digits in four-digit numbers. Which of the following instructional strategies would best support these students?

A. Using expanded form to break numbers into thousands, hundreds, tens, and ones

B. Having students write numbers in order from least to greatest

C. Teaching students to count by ones to 1,000

D. Asking students to round all four-digit numbers to the nearest thousand without explanation

A. Using expanded form to break numbers into thousands, hundreds, tens, and ones

A first-grade teacher is introducing addition using direct modeling. Which of the following activities best aligns with this approach?

A. Asking students to memorize basic addition facts up to 20

B. Having students use counters to physically represent and combine two groups of objects

C. Teaching students to write addition equations using only numerals

D. Encouraging students to use mental math without visual aids

B. Having students use counters to physically represent and combine two groups of objects

A second-grade teacher introduces an invented strategy for subtraction called "Counting Up." Which of the following student responses best illustrates this strategy when solving 63 - 58?

A. "I subtracted 50 from 63 to get 13, then subtracted 8 more to get 5." B. "I started at 58 and counted up to 63: 58 to 60 is 2, then 60 to 63 is 3, so the answer is 5."

C. "I lined up the numbers and borrowed from the tens place before subtracting."

D. "I wrote the problem as 60 - 50 = 10, then adjusted for the extra 3 and 8."

B. "I started at 58 and counted up to 63: 58 to 60 is 2, then 60 to 63 is 3, so the answer is 5."

Which of the following instructional strategies best helps students transition from direct modeling to using more efficient addition strategies?

A. Encouraging students to memorize the addition table first

B. Teaching students to use strategies such as making a ten and doubles facts

C. Requiring students to use only the standard algorithm for all addition problems

D. Providing students with only word problems instead of equations

B. Teaching students to use strategies such as making a ten and doubles facts

4. A third-grade student is using an invented strategy to solve 46 × 3 by breaking apart the numbers. Which of the following student responses best represents this approach?

A. "I multiplied 40 × 3 to get 120, then 6 × 3 to get 18, and added 120 + 18 to get 138."

B. "I wrote the numbers in columns and multiplied starting with the ones place."

C. "I counted by threes forty-six times to get the answer."

D. "I used a calculator to find the answer quickly."

A. "I multiplied 40 × 3 to get 120, then 6 × 3 to get 18, and added 120 + 18 to get 138."

A fourth-grade teacher introduces the standard algorithm for multiplication. Which of the following should students understand before using this method?

A. The order of operations in multi-step problems

B. How to break apart numbers using place value and partial products

C. How to add and subtract fractions

D. The properties of exponents and square roots

B. How to break apart numbers using place value and partial products

A student is using direct modeling to solve 24 ÷ 4. Which of the following strategies best demonstrates direct modeling for this problem?

A. Drawing 24 objects and circling groups of 4 to count how many groups there are

B. Writing a division equation and solving it mentally

C. Using a calculator to find the answer

D. Relying on memorized division facts without visual representation

A. Drawing 24 objects and circling groups of 4 to count how many groups there are

Which of the following best explains why it is beneficial for students to use invented strategies before learning standard algorithms for operations?

A. Invented strategies promote number sense and flexible thinking before students memorize procedural steps.

B. Invented strategies are more efficient than standard algorithms in all cases.

C. Standard algorithms are too complex for elementary students to learn. D. Using invented strategies prevents students from ever needing standard algorithms.

A. Invented strategies promote number sense and flexible thinking before students memorize procedural steps.

A second-grade teacher introduces students to the concept of equality using a balance scale. Which of the following activities best supports students' understanding of the equal sign?

A. Asking students to solve equations only by memorizing number facts B. Using a balance scale to show that both sides of an equation must have the same total value

C. Telling students that the equal sign means "the answer comes next" D. Having students solve problems without using visual models

B. Using a balance scale to show that both sides of an equation must have the same total value

A teacher gives students the equation 5 + __ = 12. Which of the following strategies best supports students in understanding how to find the missing number?

A. Encouraging students to guess numbers until they find the correct one B. Having students count on from 5 to 12 using fingers or a number line C. Teaching students that the missing number is always the larger number in the equation

D. Asking students to solve the problem using a calculator

B. Having students count on from 5 to 12 using fingers or a number line

Which of the following represents an early algebraic thinking skill that a kindergarten student might demonstrate?

A. Using a variable to write an equation

B. Recognizing and continuing a repeating pattern with shapes or colors C. Graphing ordered pairs on a coordinate plane

D. Solving two-step word problems involving multiplication

B. Recognizing and continuing a repeating pattern with shapes or colors

A fourth-grade teacher asks students to describe the pattern in the number sequence: 2, 5, 8, 11, 14, __. Which of the following responses best demonstrates algebraic reasoning?

A. "The next number is 25 because 11+14 is 25”

B. "The pattern adds 3 each time, so the next number is 17."'

C. "The numbers are random, but they get larger"

D. "The pattern alternates between odd and even numbers, so the next number is even."

B. "The pattern adds 3 each time, so the next number is 17."

A teacher asks students to represent 3/4 using a length model. Which of the following would be the most appropriate representation?

A. Using a number line to mark off three out of four equal sections between 0 and 1

B. Drawing a rectangle and shading three out of four equal parts

C. Grouping 3 out of 4 objects in a collection

D. Stacking fraction strips to compare 3/4 to ½

A. Using a number line to mark off three out of four equal sections between 0 and 1

A teacher introduces the set model for fractions by showing a collection of 12 counters and asking students to find 1/3 of the set. What should students do to determine the answer?

A. Divide the 12 counters into three equal groups and count the number in one group

B. Draw a rectangle and shade one-third of it

C. Use a number line to mark one-third of the distance from 0 to 12

D. Subtract 1 from 12 until they reach three groups

A. Divide the 12 counters into three equal groups and count the number in one group

A third-grade teacher introduces the concept of iterating fractions by asking students to find how many 1/4-sized pieces fit into a whole. Which of the following best explains this process?

A. Dividing a whole into four equal parts and considering just one of those parts

B. Using a number line to find where 1/4 is located

C. Repeatedly adding 1/4 until reaching the whole, showing that four copies of 1/4 make 1

D. Arranging different-sized fractions to compare them visually

C. Repeatedly adding 1/4 until reaching the whole, showing that four copies of 1/4 make 1

A teacher is introducing decimals to fourth-grade students using base-ten blocks. Which of the following representations best helps students understand the place value of 0.3?

A. Using ten unit cubes to represent 1 and showing 3 of them as 0.3

B. Drawing a number line from 0 to 10 and marking 3 on it

C. Showing 3 tens blocks and calling it 0.3

D. Counting by threes to reach 30

A. Using ten unit cubes to represent 1 and showing 3 of them as 0.3

A teacher asks students to compare 0.45 and 0.5. Which of the following strategies best supports students’ conceptual understanding?

A. Telling students to ignore the zero in 0.45 and compare 45 to 5

B. Converting both decimals to fractions and comparing 45/100 to 50/100

C. Asking students to round both numbers to the nearest whole number D. Rewriting both numbers as whole numbers by multiplying by 1000

B. Converting both decimals to fractions and comparing 45/100 to 50/100

A student is solving 2.4 × 3 using an area model. Which of the following representations correctly models the problem?

A. Drawing a rectangle with one side labeled 2.4 and the other side labeled 3, then dividing it into 2 whole units and 0.4, multiplying each part by 3, and summing the partial products

B. Using a number line and skip-counting by 2.4 three times

C. Rounding 2.4 to 3 and multiplying by 3

D. Writing the problem as 24 × 3 and moving the decimal after solving

A. Drawing a rectangle with one side labeled 2.4 and the other side labeled 3, then dividing it into 2 whole units and 0.4, multiplying each part by 3, and summing the partial products

A teacher is introducing ratios using real-life examples. Which of the following best illustrates a part-to-part ratio?

A. "For every 2 red marbles, there are 3 blue marbles."

B. "The car travels 60 miles in 1 hour."

C. "There are 8 boys in a class of 20 students."

D. "The price of 4 apples is $2."

A. "For every 2 red marbles, there are 3 blue marbles.

A student is trying to determine if the ratios 4:6 and 6:9 are equivalent. Which method would best help the student confirm equivalence?

A. Adding the numbers in each ratio to compare their sums

B. Converting each ratio to a decimal and comparing the values

C. Cross multiplying to see if 4 × 9 equals 6 × 6

D. Dividing each number in the ratios by 2 to simplify

B. Converting each ratio to a decimal and comparing the values

A teacher presents the problem: "If 3 pencils cost $1.50, how much do 10 pencils cost?" Which of the following strategies would best help students solve this proportion problem?

A. Dividing $1.50 by 3 to find the unit rate, then multiplying by 10

B. Estimating the total cost by rounding $1.50 to $2

C. Multiplying 1.50 by 10 directly

D. Adding $1.50 repeatedly until reaching 10 pencils

A. Dividing $1.50 by 3 to find the unit rate, then multiplying by 10

A preschool teacher places several containers of different sizes and shapes at a water table, along with measuring cups and scoops. The teacher observes that some children are filling one container with several scoops and saying, “This one takes more scoops than that one.”

Which of the following teacher actions best supports the development of foundational measurement concepts?

A. Encouraging students to compare the weight of the containers when full and empty

B. Asking students to estimate how many scoops it will take to fill each container before trying

C. Explaining to students that they need to use standard measuring cups to be accurate

D. Asking students to record their findings in a data table and make a bar graph

B. Asking students to estimate how many scoops it will take to fill each container before trying

A second-grade class is learning about using rulers to measure the length of objects. A student measures a pencil by placing it in the middle of the ruler, starting at the 3-inch mark instead of 0, and says it is 7 inches long.

What should the teacher do to best address this student's misunderstanding?

A. Show the student a video that explains how to use a ruler correctly

B. Review that all measurements should begin at the “zero” mark on the ruler

C. Tell the student the correct length and move on to the next object

D. Have the student measure longer objects to develop more accuracy

B. Review that all measurements should begin at the “zero” mark on the ruler

A fourth-grade student is solving a problem involving the perimeter and area of a rectangle. The student correctly calculates the perimeter as 24 units but writes that the area is also 24 square units. The dimensions of the rectangle are 4 units by 8 units.

Which teacher response would most effectively guide the student toward conceptual understanding of the difference between perimeter and area?

A. Remind the student to double-check their arithmetic for the area formula

B. Give the student a formula chart to reference for perimeter and area

C. Have the student use square tiles to build the rectangle and count the area

D. Explain that perimeter and area always have different values

C. Have the student use square tiles to build the rectangle and count the area

A first-grade teacher gives students a set of pattern blocks and asks them to sort the blocks into groups. One student groups all shapes with straight sides together, but does not distinguish between triangles, squares, and rectangles.

What does this student’s thinking indicate about their geometric development?

A. The student is reasoning at a level where properties of shapes are used for classification

B. The student is at the visual level of geometric thinking, focusing on appearance C. The student is able to identify shapes by name but not by number of sides

D. The student is confusing two-dimensional and three-dimensional shapes

B. The student is at the visual level of geometric thinking, focusing on appearance

A third-grade teacher asks students to investigate quadrilaterals and identify which ones have pairs of parallel sides. One student identifies a rectangle but not a rhombus, saying, “This one is tilted, so the sides don’t look straight.”

What instructional approach would best help the student advance in geometric reasoning?

A. Have the student use a ruler to measure the sides of both shapes

B. Provide names and definitions of all types of quadrilaterals to memorize

C. Use tracing paper or a geoboard to manipulate shapes and explore properties D. Tell the student that parallel lines always stay the same distance apart

C. Use tracing paper or a geoboard to manipulate shapes and explore properties

A second-grade teacher asks students to survey their classmates about their favorite type of fruit and record the results using tally marks. Afterward, the teacher asks the class to use the tally chart to create a bar graph. Which of the following learning goals is best supported by this activity?

A. Developing an understanding of mean, median, and mode

B. Learning to draw graphs using a ruler and accurate scales

C. Building skills in organizing, representing, and interpreting categorical data

D. Practicing multiplication facts in a real-world context

C. Building skills in organizing, representing, and interpreting categorical data

A fourth-grade class is analyzing a line plot that shows the number of books students read over the summer. One student notices that most students read between 4 and 6 books, but one student read 20 books. What mathematical concept is best introduced through this student’s observation?

A. The concept of variability and outliers in a data set

B. The use of double bar graphs for comparison

C. The process of random sampling in data collection

D. The importance of collecting data using consistent units

A. The concept of variability and outliers in a data set

A second-grade teacher shows students a spinner divided into four equal sections: 1 red, 1 blue, and 2 green. The teacher asks, “Which color do you think the spinner will land on the most if we spin it many times?” Which follow-up activity would best support students’ understanding of probability through hands-on learning?

A. Giving students a worksheet to write the fraction of each color

B. Asking students to predict the next five spins in a row

C. Having students spin the spinner multiple times and tally the results

D. Explaining that each spin is completely random and unpredictable

C. Having students spin the spinner multiple times and tally the results

A fourth-grade student designs a game using a number cube labeled 1 to 6. The student says, “If you roll a 6, you win a prize. All the other numbers mean you lose.” Another student says, “That’s not fair!” What instructional approach would best help students evaluate whether the game is fair?

A. Explain that in games, the rules are always made by the creator

B. Have students compare the number of winning outcomes to total outcomes

C. Ask students to play the game and decide how they feel about the results

D. Change the rules so that you win on a 5 or 6 instead

B. Have students compare the number of winning outcomes to total outcomes

A teacher introduces exponents by writing expressions such as 32 on the board. One student says, “I think it means 3 times 2, so it’s 6.”

What teacher response would most effectively support conceptual understanding of exponents?

A. Remind the student to memorize that exponents mean repeated multiplication B. Tell the student the correct answer and ask them to copy it into their notes

C. Use base-ten blocks or arrays to show how it represents 3 groups of 3

D. Explain that exponents follow a different rule than multiplication

C. Use base-ten blocks or arrays to show how it represents 3 groups of 3