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During a lesson on physical and chemical changes, a teacher asks the students to come up with examples of changes in matter that can be observed in everyday life. Which of the following is an example of a chemical change? A. The cooking of an egg B. The mixing of salt and pepper C. The breaking in half of a pencil D. The freezing of water in a plastic tray

Option (A) is correct. A chemical change occurs when one substance turns into a new substance, as is the case when an egg is cooked.

A student found an incorrect answer to the problem 19.9 plus 1.319.9+1.3. The student's answer is represented in the following work. 19.9 +. 1.3 ________ 20.12 Which of the following incorrect answers to the problem 12.94 plus 1.0812.94+1.08 is the result of an error that is most similar to the error made in the preceding work? A. 13.102 B. 13.92 C. 14.74 D. 23.74

Option (A) is correct. Both this answer and the given student's answer are the result of treating the fractional part and the integer part of the decimal as two separate numbers and then adding the fractional parts and integer parts separately.

Students completed a simulation examining the erosion of soil by water and made several conclusions. Which of the following conclusions presents a misconception? A. "Erosion is always harmful to the environment." B. "Erosion involves the movement of material from one place to another." C. "The slope of the land affects erosion." D. "The amount of vegetation affects erosion."

Option (A) is correct. Erosion has several environmental benefits, such as the creation of new soil, the formation of new habitats, and the transport of carbon to wetland areas.

Ms. Carswell asks her students to order five eighths, seven twelfths, and two thirds from least to greatest. One student, Jacob, correctly answers that seven twelfths is less than five eighths and five eighths is less than two thirds, and Ms. Carswell asks him to explain his reasoning. Jacob says, "The number on the bottom tells you how many pieces there are, and the number on top tells you how many of them are shaded. The smallest fraction is the one that has the biggest number of its pieces not shaded." Ms. Carswell wants to provide a counterexample for Jacob to help him realize his method does not always work. Which of the following pairs of fractions provides a counterexample to Jacob's method? A. 1/3 and 2/5 B. 1/4 and 1/3 C. 3/10 and 2/5 D. 7/20 and 5/12

Option (A) is correct. For a counterexample to Jacob's method, the smallest fraction is the one that does not have the biggest number of its pieces unshaded. In this option, the smallest fraction, one third13, has 2 pieces unshaded, and the largest fraction, two fifths25, has 3 pieces unshaded.

Students in a classroom worked with a partner to solve the following problem. One-fourth of Anne's hot chocolate is made of milk. What is the amount of milk in 8 cups of Anne's hot chocolate? Devon explained how he found the answer to the problem, saying, "If one fourth14 is milk, it's like the hot chocolate is 4 cups and 1 of them is the milk. Then if you double the hot chocolate, you have to double the milk, so in 8 cups there are 2 cups of milk." Devon's partner, Milena, represented Devon's strategy using cubes as represented in the following figure. The figure shows two sets of cubes. Each set consists of four cubes. In each set, there is one unshaded cube and three shaded cubes. End figure description. Which of the following statements best characterizes how Milena's work represents Devon's strategy? A. Milena's work accurately represents Devon's strategy because it shows 14 of 4 and how that result was doubled. B. Milena's work accurately represents Devon's strategy, but an area model for multiplication would better represent the different units of measure involved in the problem. C. Milena's work does not accurately represent Devon's strategy because unit cubes cannot be used to represent parts of a whole. D. Milena's work does not accurately represent Devon's strategy because Devon's strategy used a different whole than the whole Milena used in her work.

Option (A) is correct. In Milena's work, the first set of 4 cubes represents 4 cups of hot chocolate, and the 1 unshaded cube represents the 1 cup of milk. The set of cubes is then doubled, just as Devon explained in his strategy. Milena's work makes use of a set model of fractions in which 4 cubes represent the whole, which is a valid way to represent the fractions in the problem.

A teacher presents the students in a class with the characteristics of a certain animal. The teacher describes the animal as having a backbone and soft, moist skin covered by a slippery layer of mucus. The teacher also says that the animal goes through complete metamorphosis in its life cycle and that it lives part of its life in water and part of its life on land. The teacher then asks the students to classify the animal on the basis of its characteristics. Based on the information provided, the animal is best classified as a member of which of the following groups? A. Amphibians B. Insects C. Mammals D. Reptiles

Option (A) is correct. The characteristics presented are typical of frogs. Frogs belong to the class Amphibia, which also includes toads and salamanders.

Ms. Stewart asked her students to find the quotient 5/9÷5/7. One student's answer is represented in the following work. The figure shows a student's work. The work is as follows. 5/9 divided by 5/7, equals 35/63 divided by 45/63, which equals 35/45. End figure description. Which of the following statements is true of the student's method? A. The method is valid for dividing any two fractions. B. The method is valid for dividing fractions only when the quotient is less than one. C. The method is valid for dividing fractions only when the fractions in the original problem have the same numerator. D. The method is not valid for dividing fractions because division of fractions requires multiplying the first fraction by the reciprocal of the second fraction.

Option (A) is correct. The first step of the student's method replaces the original two fractions with two equivalent fractions that have common denominators, and the quotient is unchanged. The second step of the student's method treats the unit fraction with the common denominator as the "whole," and the quotient is equal to the quotient of the two numerators. In the second step of the given work, the fraction 1 over 63163 is treated as the "whole," and the division problem becomes "35 wholes divided by 45 wholes" with quotient the fraction 35 over 453545.

Ms. Sengupta is working with her students to name fractions represented by area models. In particular, she wants her students to focus on the importance of equal parts. She asks her students to write their answers to the following problem. The figure shows a problem, which consists of a question and a figure. The question reads: What fraction of the big square is shaded? The figure is a big square. A vertical line is drawn through the center of the square, which first divides the square into two parts. The left part of the big square is further divided by a horizontal line that extends from the midpoint of the left side of the square to the center of the square. The big square is divided into two small squares and one small rectangle. One of the small squares is shaded. End figure description. Ms. Sengupta notices that students wrote several different incorrect answers. She wants to choose an incorrect answer to discuss as a class that will highlight the importance of attending to equal parts when naming a fraction. Which of the following incorrect answers best supports Ms. Sengupta's mathematical goal? A. one third B. one half C. three fourths D. 1 and one half

Option (A) is correct. The fraction one third13 is likely produced by counting the number of shaded pieces and the total number of pieces, without attending to equal parts.

Which of the following describes a student-created model that best represents the movement of matter through an ecosystem? A. An interconnected web showing the movement of matter between plants, animals, decomposers, and the environment B. A pyramid model showing the relative biomass at each trophic level C. A linear model showing the movement of matter from plants to animals to decomposers to the environment D. A concept map diagramming how a plant uses water, soil, and decomposed materials to grow and reproduce

Option (A) is correct. The movement of matter through an ecosystem is best represented by a model that shows connections between all the organisms in the ecosystem and between the organisms and their environment.

Ms. Aguilar's class is learning about word problems involving division. Ms. Aguilar's goal is to assess whether her students recognize when the remainder forces the answer to the next-highest whole number when solving a word problem. Which of the following word problems is aligned with Ms. Aguilar's goal? A. 15 quarts of orange juice will be poured into a certain number of 2-quart containers. What is the least number of containers needed for the orange juice? B. 16 markers will be put into a certain number of bags. There will be 5 markers in each full bag. How many full bags of markers can be made? C. 3 children will share a chocolate bar equally. How much of the chocolate bar will each child get? D. 10 yards of fabric will be cut into 3 equal pieces. What is the length, in yards, of each piece?

Option (A) is correct. The problem in option (A) can be solved by dividing 15 by 2, with quotient 7 and remainder 1, which forces the answer to the next-highest whole number, 8. The least number of containers needed is 8, where 7 of the containers are full and the remaining container contains 1 quart of orange juice.

A student is asked to compare three pairs of decimal numbers. The student correctly states that 0.59 is less than 0.721. The student correctly states that 0.96 is greater than 0.008. The student incorrectly states that 0.7 is less than 0.021. Which of the following misconceptions most likely explains the student's answers? A. The student is comparing decimals as if they are whole numbers. B. The student is assuming the longer decimal will always be the greater decimal. C. The student thinks that the decimal with more zeros will always be the greater decimal. D. The student thinks that the decimal with the larger digit in the rightmost place will always be the greater decimal.

Option (A) is correct. The student's answers likely arise from ignoring the decimal point and all zeroes to the left of the leftmost nonzero digit.

In a classroom activity using images of the night sky, a teacher helps the students observe that the positions of stars seen in the summer at 11 P.M. are different than those seen in the winter at the same time. Which of the following statements best helps explain the observation? A. Earth revolves around the Sun. B. The Moon revolves around Earth. C. Constellations are moving away from Earth. D. Constellations revolve around Earth.

Option (A) is correct. The view of stars from Earth changes as Earth revolves around the Sun.

Students are studying various aspects of magnetism. Which of the following questions should upper-elementary students be able to answer experimentally with classroom resources? A. What would cause a compass to not point to the north? B. Why do all magnets have a north and south pole? C. Is navigation using a compass more accurate than using the Sun and stars? D. Why are magnets made of particular materials?

Option (A) is correct. This question can be investigated empirically inside the classroom with compasses, magnets, and other objects.

In an activity, Mr. Shope's students mix one cup of warm water, one tablespoon of borax powder, and one‑fourth cup of glue. After a period of thorough mixing, the students are left with a puttylike substance commonly referred to as slime. The students determine the properties of their substance and record their observations. Mr. Shope then asks the students to state whether they created a mixture or a new substance and to support their claim with evidence. Which of the following students made the best use of evidence to support his or her claim? A. Alyssa, who wrote that they had created a new substance because the slime has properties that are different from the properties of the three ingredients B. Max, who wrote that they had created a mixture because all three ingredients were mixed together C. DeShawn, who wrote that they had created both a mixture and a new substance because they mixed the ingredients together and that by mixing them in water, they could not get the ingredients back again D. Alessandro, who wrote that they had created a mixture because they could use filtration to separate the slime into the original ingredients

Option (A) is correct. This question tests your ability to critique students' work on the basis of how well the students used evidence to support a scientific claim. The student described in option (A) correctly cited the properties of the slime that are different from those of the original ingredients as evidence that a new substance had been formed in the activity.

A teacher poses the following problem to the class. Helena has 20 red beads for making bracelets. She has 12 more red beads than yellow beads. How many yellow beads does she have? Which of the following problems has the same mathematical structure as the problem the teacher poses? A. Alex has 25 toy cars and 12 more stuffed animals than toy cars. How many stuffed animals does he have? B. Chanelle has 13 crayons. Chanelle has 7 more crayons than Chuck has. How many crayons does Chuck have? C. Britney has 7 more packs of snacks than Bella has. Bella has 5 packs of snacks. How many packs of snacks does Britney have? D. Daryl has some white shirts. After buying 3 new white shirts, he has 8 white shirts. How many shirts did Daryl have before buying the 3 new white shirts?

Option (B) is correct. Both the problem in option (B) and the problem the teacher poses have a "comparison/start unknown" structure, in which a larger quantity is given, the amount by which the larger quantity differs from a smaller quantity is also given, and the problem asks for the smaller quantity.

Ms. Hammonds asked one of her students, John, to write the number five hundred. John wrote "500" in his workbook. Next, Ms. Hammonds asked John to write the number that is ten more than five hundred. John immediately said, "Five hundred ten," and wrote "50010" in his workbook. Which of the following mathematical skills is most evident in John's response? A. Reading any three-digit number B. Writing any three-digit number C. Mentally adding ten to a three-digit number D. Understanding the value of the digits in a three-digit number

Option (C) is correct. John's spoken response, "five hundred ten," provides evidence of understanding how to mentally add ten to a three-digit number.

Mr. Walters asked his students to order 89, 708, 37, and 93 from least to greatest, and to be ready to explain the process they used to order the numbers. One student, Brianna, ordered the numbers correctly, and when Mr. Walters asked her to explain her process, she said, "The numbers 89, 37, and 93 are less than 100, so they are all less than 708, since that is greater than 100. Also, 37 is the least because it comes before 50 and the other two numbers are close to 100. Then 89 is less than 90, but 93 is greater than 90." Which of the following best describes the strategy on which Brianna's explanation is based? A. A counting strategy B. A benchmarking strategy C. An estimation strategy D. A place-value strategy

Option (B) is correct. Brianna first indicates that 708 is the greatest number because it is greater than 100, while 37, 89, and 93 are all less than 100. Next, Brianna indicates that 37 is the least number because it is less than 50, while 89 and 93 are greater than 50. Finally, Brianna recognizes that 89 is less than 93 because 89 is less than 90, while 93 is greater than 90. Thus, over the course of her explanation, Brianna used 100, then 50, and then 90 as points of reference for comparisons, which is exactly what benchmark numbers are: points of reference for comparison.

Ms. Roderick asked her lunch helper in her kindergarten class to get one paper plate for each student in the class. Which of the following counting tasks assesses the same mathematical counting work as this task? A. Having students line up according to the number of the day of the month in which they were born B. Showing students 10 pencils and asking them to get enough erasers for all the pencils C. Showing students a row of 12 buttons and asking them to make a pile of 8 buttons D. Asking students to count the number of triangles printed on the classroom rug

Option (B) is correct. Getting one paper plate for each student in the class assesses whether students can determine when the number of objects in one set is equal to the number of objects in another set, and the task described in option (B) involves a similar determination.

Ms. Simeone is working with her first-grade students on writing two-digit numerals. She wants to use an activity to assess whether her students are attending to the left-to-right directionality of the number system. Which of the following activities is best aligned with Ms. Simeone's purpose? A. Asking students to read the numbers 20 through 29 B. Asking students to represent the numbers 35 and 53 using base-10 blocks C. Asking students how many tens and how many ones are in the number 33 D. Showing students 23 cubes and 32 cubes and asking them which quantity is greater

Option (B) is correct. Having the students represent 35 and 53 using base-10 blocks will help Ms. Simeone assess whether students know which place is the tens place and which place is the ones place or whether students have reversed the ones place and the tens place, thinking that the ones place is on the left and the tens place is on the right. Representing the numbers provides more information about students' understanding of place value than just reading numbers does.

In word problems that have a multiplicative comparison problem structure, two different sets are compared, and one of the sets consists of multiple copies of the other set. Which of the following best illustrates a word problem that has a multiplicative comparison problem structure? A. There are 4 shelves in Joaquin's bookcase, and there are 28 books on each shelf. How many books are in Joaquin's bookcase? B. Marcus drives 3 times as many miles to get to work as Hannah does. Hannah drives 16 miles to get to work. How many miles does Marcus drive to get to work? C. A football field is 360 feet long and 160 feet wide. A soccer field is 300 feet long and 150 feet wide. The area of the football field is how many square feet greater than the area of the soccer field? D. An ice cream parlor sells 29 different flavors of ice cream and 4 different types of cones. How many different combinations consisting of an ice cream flavor and a type of cone are available at the ice cream parlor?

Option (B) is correct. In the problem in option (B), the two values being compared are the number of miles that Marcus drives to get to work and the number of miles that Hannah drives to get to work, and the number of miles that Marcus drives is 3 times the number of miles that Hannah drives.

Ms. Lee is teaching her first graders addition and subtraction within 20 by counting. She has her students work individually on finding the answer to 7 plus 57+5. During the whole-class discussion of the answer to the problem, James says, "I think 7 plus 5 equals 11, because I counted 7, 8, 9, 10, 11." James puts up a finger each time he says a number in the sequence. Which of the following statements most likely explains the reason behind James's error? A. James does not know the count sequence. B. James did not start his count on the correct number. C. James does not understand that the last count word indicates the amount of the set. D. James does not know how to decompose numbers greater than 10 into tens and ones.

Option (B) is correct. James mistakenly starts his count on the first addend, 7, instead of on the next whole number, 8.

Mr. Wilson wrote the numbers 657, 756, and 576 on the board and asked the students to state which number is the greatest and explain why. One student, Lily, answered, "756 is the greatest. The first number has 6 hundreds, the second number has 7 hundreds, and the third number has 5 hundreds, so 756 is the greatest." Which of the following statements best describes the evidence of understanding place value that Lily's explanation provides? A. The value of the digit in each place of a three-digit number is a multiple of a power of 10. B. The leftmost digit in a three-digit number represents the number of hundreds in the number. C. A digit in a certain place has ten times the value the digit would have in the next place to the right. D. The greatest three-digit number that can be made with three different digits is the number that has the digits in order from greatest to least.

Option (B) is correct. Lily's answer is based on comparing the leftmost digit in each of the three-digit numbers and states that the leftmost digit represents the number of hundreds.

Students in an upper elementary classroom are studying fossils. Which resource would best support the idea that fossils provide evidence that ecosystems can change over time? A. Actual samples of fern fossils found in a swamp B. Pictures of seashell fossils found in a desert C. A map showing locations in a forest where scientists have discovered woolly mammoth fossils D. A chart listing the number of extinct shark teeth found at different shorelines around the world

Option (B) is correct. Marine life cannot exist in a desert. Showing students pictures will enable them to see that the shells look out of place and that the desert environment must have been a sea many years ago.

A student investigated whether matter is conserved during a change of state. The student weighed an empty glass beaker and then filled it with water before weighing it again to determine the weight of the water. The student then poured the water from the beaker into an ice-cube tray and placed it in a freezer. The student later removed the ice cubes from the tray and weighed them. After repeating the procedure five times, the student found that the weight of the ice cubes was slightly less than the initial weight of the water. The student concluded that freezing the water reduced its mass. Which of the following modifications to the procedure would best remove the error that affected the student's data? A. Removing the water from the beaker and the ice cubes from the tray using a spatula B. Weighing the water after it is placed in the tray and the ice cubes before they are removed from the tray C. Repeating the experiment three times instead of five times D. Drying the beaker and ice-cube tray with a paper towel before water is added

Option (B) is correct. Modifying the procedure by weighing the water after it is placed in the tray and weighing the ice cubes before they are removed from the tray would most likely eliminate the error. Transferring the water twice for weighing probably led to a systematic loss of mass.

During a lesson in her second-grade class, Ms. Costa draws two squares of the same size, each representing the same whole. She then divides and shades the squares as represented in the following figures. The figure shows two squares. In the first square, two diagonals are drawn, dividing the square into four triangles of the same size. One of the triangles is shaded. In the second square, a vertical line and a horizontal line are drawn through the center of the square, dividing the square into four smaller squares of the same size. One of the squares is shaded. End figure description. Her students consistently identify the area of each shaded region as one-fourth, but when they are asked if the areas are equal, some students say no. Which of the following statements most likely explains why the students interpret the areas as being unequal? A. The students think that the areas are unequal because the wholes are different sizes. B. The students think that the areas are unequal because the shaded regions are different shapes. C. The students have difficulty determining the size of geometric figures that include diagonal lines. D. The students have difficulty determining the part-to-whole relationship when they are working with visual models of fractions.

Option (B) is correct. One misconception that students often have when they first begin to work with area models of fractions is that the parts of the whole must be congruent for the areas of the parts to be equal, and this misconception explains the responses described in the question.

When Sean, a third-grade student, solved the following problem, he gave an incorrect answer. Adele is baking cakes. Each cake needs 3 eggs. How many cakes can she make with 22 eggs? Sean said the answer was "7 remainder 1 cakes." Which of the following statements describes the error Sean made to arrive at his incorrect answer? A. Sean divided incorrectly. B. Sean interpreted the remainder incorrectly. C. Sean did not use the correct units in his answer. D. Sean used a strategy that works only with certain numbers.

Option (B) is correct. Sean answered in terms of the number of whole groups and the remainder, but the question asked only for the number of whole groups. He included the remainder 1, which is the number of eggs that were not needed to make the cakes. Therefore, Sean did not interpret the remainder correctly.

Ms. Marley is working with her students on fraction addition. She wants to highlight how the commutative property of addition can be used to make landmark numbers when adding fractions. Which of the following expressions best highlights the use of the commutative property of addition for this purpose? A. 1/3+1/4+1/3 B. 1/3+1/4+2/3 C. 2/3+2/3+2/3 D. 3/5+2/5+2/3

Option (B) is correct. The commutative property can be used to rewrite 1/3+1/4+2/3 as 1/3+2/3+1/4 to make the landmark number 1+1/4.

On a class field trip, students look at a profile of a local canyon wall. Sandy says, "I didn't realize this area was once an ocean." The teacher asks the other students to explain why they think Sandy may have come to this conclusion. Which student response indicates a lack of understanding about the evidence and geologic history? A. "I can see a layer of rock with all kinds of shells in it." B. "The layers of rock are in different colors in the shape of an ocean wave." C. "Two of the layers are sand, with a layer of shells between them." D. "The fossils we can see are from species that lived in the ocean."

Option (B) is correct. The student lacks an understanding of the evidence because the appearance of different-colored layers of rocks is not evidence that the area was once covered by an ocean.

A student presents a project to the class describing the relationship between flowering plants and bees. The student concludes the presentation by stating, "Flowers have bright colors so they can trick bees into helping them reproduce." Which of the following responses by the teacher would best help the student develop a more accurate view of the relationship between bees and flowers? A. Have you considered the role of other animals in helping plants reproduce? B. Have you considered a benefit for bees when they help flowers reproduce? C. What benefits do humans receive from this bee-flowering plant relationship? D. Do flowering plants produce more pollen when bees are present?

Option (B) is correct. This question will direct students toward a more accurate view of the relationship between bees and flowers by adding a missing component to the model and to the explanation.

Ms. Stine is doing an activity with her students in which she says a decimal number out loud and her students respond by writing the numeral. When Ms. Stine says, "one hundred eight and eight‑hundredths," a student named Roy writes the numeral "108.800." The students are then asked to write the numeral corresponding to "three hundred twenty‑four and seven‑thousandths." Which of the following responses shows an error that is most similar to the error Roy made? A. 324.0007 B. 324.7000 C. 30024.0007 D. 30024.7000

Option (B) is correct. This response shows "seven thousandths" as 0.7000, which is similar to the error Roy made when he wrote "eight hundredths" as 0.800.

Ms. Rodriguez is working with her kindergarten students to develop the skill of counting on. Which of the following tasks is best aligned with the goal of having students count on? A. The teacher gives each student a number book with a different number on each page. The students must count out and glue the same number of pictures to match the given number on each page. B. The teacher gives each student a 10-piece puzzle, disassembled with a single number written on each piece. The students must put the puzzle together with the numbers in order. C. The teacher gives each student a shuffled deck of 10 cards, each with a single number from 1 to 10. When the students draw a number card, they must count to 20, starting from the number on the card they drew. D. The teacher gives each student 8 blocks and a number cube, with the sides of the number cube numbered from 3 to 8. When the students roll the number cube, they must count out the same number of blocks as the number rolled and create a tower with that number of blocks.

Option (C) is correct. A student would begin with the number drawn and count on from that number until 20 is reached. For example, if the student draws a card with 15 on it, the student would count on from 15, saying, "15, 16, 17, 18, 19, 20." The other tasks described do not require students to count on.

Ms. Robinson is working with her students on mixed-number subtraction, and she wonders how well her students understand that each whole in a mixed number is equivalent to the fraction n/n equal parts. She wants to construct a subtraction problem for her students that will assess their understanding of the concept. Which of the following problems will give Ms. Robinson the most information about her students' understanding that each whole in a mixed number is equivalent to the fraction n/n equal parts? A. 5 3/4−3 1/2 B. 6 4/5−2 2/7 C. 4 1/3−1 5/6 D. 5 2/5−4 2/5

Option (C) is correct. Because one third13 is less than five sixths56, students can use regrouping to rewrite 4 and one third413 as 3 plus three thirds, plus one third, equals 3 plus four thirds3+33+13=3+43 and then subtract 4 and one third, minus 1 and five sixths413−156 as open parenthesis, 3 minus 1, close parenthesis, plus, open parenthesis, four thirds minus five sixths, close parenthesis(3−1)+(43−56).

Students are conducting an investigation on dissolving solids in a liquid. After mixing a spoonful of salt into a large container of water, a student says, "I can no longer see the salt in the water because the salt is gone." What can the teacher do to clarify this misconception? A. Add another teaspoon of salt to the water and stir B. Show the students that other substances also can dissolve by mixing sugar and water together C. Evaporate the water and show that the salt still remains D. Give the students a magnifying glass and allow them to inspect the sample

Option (C) is correct. Evaporating the water will show that the salt still remains in the container.

Mr. Bennett's class was discussing strategies for adding whole numbers. One student, Katie, said, "When I add two numbers, I get the same answer as when the numbers switch places." Mr. Bennett asked his students to explain why Katie's claim is true when adding any two whole numbers. After giving the class time to work, he asked another student, Joel, to present his explanation. Joel said, "If the answer is a little number like 4, then 2 plus 2 is the same either way. And if the answer is a big number like 400, then 200 plus 200 is the same either way." Which of the following statements best characterizes Joel's explanation? A. It clearly explains why Katie's claim is true when adding any two whole numbers. B. It assumes that Katie's claim is true, but it does not establish in general why her claim is true. C. It neither gives useful examples for showing that Katie's claim is true, nor does it establish in general why her claim is true. D. It gives only two examples that show that Katie's claim is true, but more examples are needed to establish in general why her claim is true.

Option (C) is correct. In each of Joel's examples, he chose two addends that are equal. Joel's examples are not useful for showing that Katie's claim is true because changing the order of the addends does not change the appearance of the sums. Joel's explanation does not establish in general why Katie's claim is true because the examples cannot establish that something is true in general. In particular, Joel's examples do not address any case in which the addends are two different numbers.

One of Mr. Spilker's students, Vanessa, incorrectly answered the addition problem 457+138 as represented in the following work. 457 +138 _______ 585 Mr. Spilker wants to give Vanessa another problem to check whether she misunderstands the standard addition algorithm or whether she simply made a careless error. Which of the following problems will be most useful for Mr. Spilker's purpose? A. 784+214 B. 555+134 C. 394+182 D. 871+225

Option (C) is correct. In the work shown, after adding the ones and recording the 5 in the ones place, Vanessa did not record that the additional 10 ones were 1 ten, nor did she add the regrouped ten in the tens place. The problem in option (C) will be most useful for Mr. Spilker's purpose because it requires regrouping from the tens place to the hundreds place.

A third-grade teacher wants to assess students' mastery of providing explanations. The teacher asks the class to select a species that lives in groups and to write an explanation of how certain interactions within a group will help increase the likelihood of an individual's survival, growth, and reproduction. Which of the following explanations best meets the objective? A. "Whales live in pods. Whales migrate to warmer waters to survive." B. "Bees live in colonies. Each bee has a specific duty to maintain the colony." C. "Fish live in schools. By being one of a large number of fish, each individual has less chance of being eaten." D. "Elephants travel in herds. The calves run slower and are surrounded by the older elephants."

Option (C) is correct. In this explanation, the student describes a specific behavior to support the claim and makes a connection to how the school provides protection for all its members.

A teacher shows students a short video of a ball being thrown vertically upward, momentarily coming to rest, and falling back down toward the ground. When asked about the force of gravity on the ball throughout its motion, a student says that gravity pulls down on the ball at all times except for when the ball stops briefly before reversing direction. Which of the following responses by the teacher will best help the student overcome the misconception about gravity? A. Showing the video again but in slow motion B. Posing a follow-up question: How does a parachute change the motion of a skydiver in free fall? C. Posing a follow-up question: If gravity is not pulling on the ball when it stops, why does it start to fall back down? D. Showing a video highlighting variations in gravity between Earth and the Moon and at different altitudes on Earth

Option (C) is correct. It will help the student overcome the misconception by knowing that an object can change direction only if there is a force acting on it. This response will show the student why gravity is still pulling on the ball even when the ball is momentarily at rest.

While Mr. Lynch's students were working on finding the answer to the problem 5/6÷1/3, one student, Maggie, said the following. "I think that fraction division works just like fraction multiplication. You can divide the numerators and the denominators to find the answer. Like in this problem, 5 divided by 1 is 5, and 6 divided by 3 is 2, so the answer is 5 over 2." Which of the following is true of Maggie's conjecture about fraction division? A. Maggie's conjecture is not true, because her method did not produce the correct answer to the problem. B. Maggie's conjecture is not true, because division of fractions requires multiplying by the reciprocal of the second fraction. C. Maggie's conjecture is true for all fraction division problems, but in some problems, the resulting quotients from the numerators and denominators will not be whole numbers. D. Maggie's conjecture is partially true. Her method gave the correct answer in this problem because the denominators and numerators divided evenly, but the method will not give the correct answer in other problems.

Option (C) is correct. Maggie's method works for all fraction division problems but in some problems the result will be a "complex fraction" where one or both of the numerator and the denominator are fractions; as an example, one half divided by three fourths, equals the fraction with numerator one third, and denominator two fourths12÷34=(13)(24) is a true equation but the fraction on the right side of the equation is not written in a form where both the numerator and the denominator are whole numbers.

A teacher helps the students in a class create a poster showing a food web in a prairie ecosystem. The teacher asks the students to label the organisms in the food web according to each organism's role in the ecosystem. Which of the following terms should the students use to label an animal that grazes on grass as its primary source of food? A. Producer B. Carnivore C. Primary consumer D. Secondary consumer

Option (C) is correct. Primary consumers are animals that eat the producers in an ecosystem. An animal that eats plants, such as a herbivore, should be labeled as a primary consumer.

Mr. Keller presents the following equations to his class and asks his students to decide whether each equation is true or false. 2+3=4+1 2+4=6−1 5=8−3 7=7 5=6 Which of the following is most clearly highlighted by the activity? A. Fluently adding and subtracting within 10 B. Determining the value of unknown numbers C. Understanding the meaning of the equal sign D. Using properties of operations to add and subtract

Option (C) is correct. Students must decide whether each equation is true or false based on whether the two sides of the equation represent the same value. To complete the activity, students must interpret the equal sign to mean "is the same as."

Mr. Aslanian's class watched a video about a species of mosquito that is found only at relatively high altitudes on a particular mountain in Africa. Mr. Aslanian then asked the students to use the information presented in the video to propose an explanation for the mosquitoes' limited geographical range. Which of the following responses indicates the most accurate understanding of the factors that influence the natural range of a species? A. Farran wrote that the mosquitoes probably migrate up and down the mountain every day, but that scientists study the mosquitoes only when the mosquitoes are high up on the mountain. B. Nash wrote that the mosquitoes live at high altitudes on the mountain because they need air, sunlight, water, and a source of food to live. C. LaShawn wrote that the mosquitoes are better adapted to the cooler, damper climate at higher altitudes on the mountain than to the hotter, dryer climate at lower altitudes. D. Cole wrote that mosquitoes that like warmer climates probably want to live where there is clear sunlight, and clear sunlight is often found at the top of a mountain.

Option (C) is correct. The question tests your ability to critique students' explanations of natural phenomena on the basis of whether the explanations are consistent with existing knowledge. The response presented in option (C) provides the most accurate explanation for the limited geographical range of the mosquito species.

Ms. Karp gives each of her students 24 counters. She asks the students to count the number of counters each of them has and to write the number on their worksheets. All the students write "24" on their worksheets. Next, she asks the students to explain the meaning of the 2 and the 4 in "24." Which of the following student explanations provides the most evidence of place-value understanding? A. The 2 is in the tens place and the 4 is in the ones place. B. The 2 is twenty counters and the 4 is four single counters. C. The 2 is two groups of ten counters and the 4 is four counters. D. The 2 is two counters and the 4 is four counters, and so it's twenty-four.

Option (C) is correct. The student explanation in option (C) provides the most evidence of place-value understanding because it connects the 2 to two groups of ten counters and the 4 to four individual counters.

During a class demonstration, a teacher floats a small toy boat in the center of a long, narrow rectangular tank filled with water. Initially, the water is still and the boat is not moving. The teacher asks students to predict what will happen if they use their hand to create a wave crest that travels toward the other end of the tank. A student predicts that the boat will be carried by the wave to the other end of the tank. Which of the following statements indicates a misconception that the student most likely has about water waves? A. Water waves get smaller as they get further away. B. Water waves transfer energy from one location to another. C. Water particles at the surface move along the surface with the wave. D. Water waves slow down when they encounter an obstacle.

Option (C) is correct. This statement indicates a misconception that water moves along with waves and will take the boat with it.

Ms. Lussier asked one of her students, Matthew, to explain how he found the answer to the subtraction problem 54−37. Matthew explained his answer as follows. "I didn't want to have to trade, so I added 3 to 54 so I would have 7 like in 37. Then I added 3 to 37 to keep the problem the same, but now I have 57 minus 40 so I just do 7 minus 0 and 5 minus 4." Which of the following statements best characterizes Matthew's strategy for finding the answer to two-digit subtraction problems? A. Matthew's strategy is not valid because he solved a different subtraction problem than the one given. B. Matthew's strategy is not valid because it can only be used when the ones digit of the number to be subtracted is greater than the ones digit of the number from which to subtract. C. Matthew's strategy is valid, but it will not always provide an equivalent subtraction problem where the ones digit of the number to be subtracted is less than the ones digit of the number from which to subtract. D. Matthew's strategy is valid, and it will always provide an equivalent subtraction problem where the ones digit of the number to be subtracted is less than the ones digit of the number from which to subtract.

Option (C) is correct. This strategy is valid and is similar to the strategy known as shifting the problem. The strategy will not always provide an equivalent subtraction problem, but the method works as long as the difference between the values in the ones place of the subtrahend and the minuend is greater than the difference between 10 and the value of the ones place of the subtrahend.

Marina explained how she found the difference 35−18, saying, "I knew that 18 plus 2 is 20, and 35 plus 2 is 37, so 35 minus 18 is the same as 37 minus 20, which is 17. So 35 minus 18 is 17." Marina's partner, Jeremy, represented Marina's strategy using a number line, as shown in the following figure. The figure shows a number line. There are four tick marks drawn on the number line. The first tick mark is labeled 18, the second tick mark is labeled 20, the third tick mark has no label, and the fourth tick mark is labeled 35. Between the numbers 18 and 20 is a curved arrow labeled 2. Between the number 20 and the third tick mark is a curved arrow labeled 10. Between the third tick mark and the number 35 is a curved arrow labeled 5. End figure description. Which of the following statements best characterizes how Jeremy's work represents Marina's strategy? A. Jeremy's work accurately represents Marina's strategy because it shows that she correctly found the difference between 35 and 18. B. Jeremy's work accurately represents the part of Marina's strategy in which she considered 20 instead of 18 as the subtrahend, but it does not accurately represent how she took 20 away from 37. C. Jeremy's work does not accurately represent Marina's strategy because Marina's strategy involved shifting the problem, but Jeremy's work shows a counting-up strategy. D. Jeremy's work does not accurately represent Marina's strategy because Marina used a comparison interpretation of subtraction, but Jeremy's work shows a takeaway interpretation of subtraction.

Option (C) is correct. When Marina adds 2 to 18 and 2 to 35, she shifts the problem from 35−18 35 minus 18 to 37−2037 minus 20, which has the same difference, and then she subtracts. In contrast, Jeremy's work shows how a student would start from 18 and count up 2 to get to 20, then count up 10 more to get to 30, and then count up 5 more to get to 35, after which the 2, 10, and 5 would be added to find the difference of 17. Marina's strategy uses a takeaway interpretation of subtraction because she subtracted 20 from 37 to find the difference, but Jeremy's work shows a comparison interpretation of subtraction because he found the distance between 18 and 35 on the number line.

One of Mr. Terry's students, Yvonne, found the answer to the problem 3/8×2/9 as represented in the following work. 3/8x2/9 = 2/8x3/9 = 1/4x1/3 = 1/12 When Mr. Terry asked Yvonne to explain her work, she said, "It's easier to just switch the numerators to make simpler fractions." Which of the following statements is true of Yvonne's strategy? A. Yvonne's strategy can only be used to rewrite products of fractions where the difference between the numerators is 1 and the difference between the denominators is 1. B. Yvonne's strategy can only be used to rewrite products of fractions where both fractions are less than 1. C. Yvonne's strategy can be used to rewrite any product of two fractions, but it will not always result in fractions that can be simplified. D. Yvonne's strategy can be used to rewrite any product of two fractions, and it will always result in fractions that can be simplified.

Option (C) is correct. Yvonne's strategy is valid because it is based on the definition of the product of two fractions and the commutative property of multiplication. For any two fractions the fraction a, over bab and the fraction c over dcd, the product of the fractions is defined as the fraction a, over b, end fraction, times the fraction c over d, equals the fraction with numerator a times c, and denominator b times d, end fractionab×cd=a×cb×d. Based on the commutative property of multiplication, the fraction with numerator a, times c, and denominator b times d, end fraction, equals the fraction with numerator c times a, and denominator b times d, end fractiona×cb×d=c×ab×d, which means that the fraction a, over b, end fraction, times the fraction c over d, equals the fraction c over b, end fraction, times the fraction a, over dab×cd=cb×ad. However, Yvonne's strategy does not always result in fractions that can be simplified. For example, four fifths times six sevenths45×67 is equivalent to six fifths times four sevenths65×47, but none of these fractions can be simplified, and the product the fraction 24 over 352435 cannot be simplified either.

Mr. Marzotto asks his students the following question. "If you have a 2 and a 5, what is the greatest two-digit number you can make using the 2 and the 5 ?" One student, Addy, responds, "With those two digits, I can make 25 and 52. The greatest number of the two is 52 because it has 5 tens, and 25 only has 2 tens, and 5 is greater than 2." Addy has demonstrated evidence of understanding which of the following mathematical ideas or skills? A. Using numerals to describe quantities B. Recognizing that 1 ten is made of 10 ones C. Mentally calculating 10 more than a given number D. Knowing the value of each place in a two-digit number

Option (D) is correct. Addy has demonstrated evidence of understanding that 52 is made up of 5 tens and 2 ones.

Ms. White's students investigate why some stars appear to be brighter than others in the night sky. They take observational notes about the apparent brightness of two different-sized flashlights that they shine from various distances onto the same wall. After the investigation, the students conclude that brighter stars must be larger than dimmer stars. Which of the following activities will provide the best data for directing the students toward a more accurate conclusion? A. Repeating the investigation, aiming the flashlights at black construction paper taped to the wall B. Combining the observations with those made by the students in another class C. Repeating the investigation, measuring the diameter of the circles of light produced at various distances D. Repeating the investigation, using two flashlights of the same size

Option (D) is correct. The students focused on the size of the flashlights rather than the change in apparent brightness as the distance changed. If the flashlights are the same size, that variable will be removed and the students will see that the apparent brightness changes with the distance of the flashlight from the wall.

Ms. Shaughnessy is working with her class on measuring area using nonstandard units. While the students are finding the area of the surface of their desks using rectangular note cards, one student says, "I can just measure the long side of the desk with the long side of the card, then measure the short side of the desk with the short side of the card, and multiply them." Which of the following best describes the validity of the student's strategy? A. The strategy is not valid because the same unit must be used to measure each side of the desk. B. The strategy is valid only if the note cards are squares. C. The strategy is valid and the unit of measurement is square units. D. The strategy is valid and the unit of measurement is note cards.

Option (D) is correct. Area can be measured using any two-dimensional unit that covers a surface, but the label of the area must reflect that unit. In this case the student has used note cards as the unit to measure the area of the desk. When using square units, one counts how many times the side of the square unit fits on each side of the rectangle whose area is to be measured. When using a unit that is not a square, like a note card, it is important to keep the orientation of the unit constant to cover the area without overlapping. This method results in one dimension of the rectangle being measured with the long side of the note card and the other dimension of the rectangle being measured with the short side of the note card.

Ms. Carter shows one of her students, Brandon, a set of cubes. She tells Brandon that there are 13 cubes in the set and asks him to take 1 cube away from the set. Ms. Carter then asks Brandon, "How many cubes do you think are in the set now?" Brandon quickly answers, "Twelve." Brandon has demonstrated evidence of understanding which of the following mathematical ideas or skills? A. Using numerals to describe quantities B. Counting with one-to-one correspondence C. Recognizing a small quantity without counting D. Knowing that each previous number name refers to a quantity which is one less

Option (D) is correct. In the scenario, Ms. Carter shows Brandon a set of cubes, explicitly tells him how many cubes are in the set, and asks him to take one cube away from the set. This process allows Ms. Carter to ensure that Brandon knows that there is now one less cube in the set. When Ms. Carter asks how many cubes are in the set after one cube is removed, Brandon readily states, without counting the cubes, that there are 12 cubes. This provides evidence that Brandon knows that 12 is the number name that precedes 13 and that 12 refers to a quantity that is one less than 13; it can also be assumed that Brandon has the same understanding for other whole numbers.

An elementary school class has been learning about how animals can change the environment to meet their needs. After viewing a video of a beaver building a dam, the students are asked to explain how changing its environment helps the beaver to support its needs. Their explanation needs to be supported by evidence. Which explanation responds to the teacher's direction and is most likely to be supported by evidence from a video about beavers? A. "When the dam is built, the flow of water in the river changes, and a lake is formed." B. "Dams are good for beavers, and beavers that build dams are better off than beavers without dams." C. "Beavers are very good swimmers and can stay underwater for long periods of time." D. "Because the beaver lives in a lodge in the lake formed by the dam, it is protected from predators."

Option (D) is correct. In this explanation, the student provides a claim and links the claim to a scientific reason.

Ms. Lawson asks one of her students, Jen, to explain how to use base-10 blocks to add 67 and 40. Jen lays out some base-10 blocks as shown in the following figure. Jen says, "I used 6 rods for the 6 tens in 67 and 7 cubes for the ones. Then I used 4 rods for 4 tens because I have to add 40. I counted all the rods and cubes and got 10 rods and 7 cubes, so it makes 10 tens and 7 ones altogether." Jen has demonstrated evidence of understanding which of the following mathematical ideas or skills? A. Understanding that 100 is equivalent to 10 tens B. Using base-10 written numerals to describe quantities C. Mentally calculating ten more or ten less than a number without counting D. Understanding that the digits of a two-digit number represent amounts of tens and ones

Option (D) is correct. Jen's description of how to use rods and cubes has demonstrated evidence of understanding that 67 represents 6 tens and 7 ones and that 40 represents 4 tens.

Which of the following activities would best introduce students to a method of erosion control? A. Students form mounds of soil that vary in height. After pouring water on the mounds, they compare the amount of erosion in each sample. B. Students blow through a straw on a tray of sand. They blow with varying force and compare the amount of sand that is blown away after each test. C. Students pour a different amount of water onto each of three identical trays of soil set at various slope angles, graphing the relationship between the amount of runoff and the slope angle. D. Students are presented with two trays of soil set at a slight angle, one containing growing grass. The students pour water on both samples and compare the amounts of soil lost.

Option (D) is correct. Planting grass or other plants is one of the best ways to prevent erosion. Students will notice that less soil was lost in the sample with vegetation.

In Ms. Werner's classroom, students are measuring the length of objects around the room. Shaun measures the length of the rug using a ruler and gets 108 inches. Tim measures the length of the same rug using a yardstick and gets 3 yards. Ms. Werner overhears them discuss their results. Shaun says that inches are bigger than yards because there are more inches in the length of the rug, but Tim says that yards are bigger than inches because there are fewer yards. Which of the following statements best expresses Shaun's misconception about measurement? A. The final measurement does not depend on the length of the unit. B. The size of the unit of measure is dependent on the distance it covers. C. The same measurement cannot be obtained when using different measurement tools. D. The number of units of measure increases as the size of the unit of measure increases.

Option (D) is correct. Shaun's misconception is that the number of units of measure increases with the size of the unit, so 108 inches is bigger than 3 yards because 108 is bigger than 3. Shaun may be overgeneralizing his conception of counting numbers to his understanding of measurement units.

Consider the following incomplete statement about rounding a decimal number. When 390.396 is correctly rounded to the nearest ____________, the result is 390. Which of the following words make the preceding statement true when inserted into the blank in the statement? Select ALL that apply. A. Hundredth B. Tenth C. Unit D. Ten E. Hundred

Options (C) and (D) are correct. When 390.396 is correctly rounded to the nearest unit, the result is 390, and when 390.396 is correctly rounded to the nearest ten, the result is 390.

Mr. Marquez and his students are working on naming fractions with denominators of 10 or 100 using decimal notation. Mr. Marquez finds an activity online that involves having students shade some parts of a tenths grid and some parts of a hundredths grid, name the fractions that have been shaded, and write the fractions in decimal notation. The following figure shows an example from the activity. The figure shows an example from the activity. The example shows various representations of two values. Each representation consists of a shaded grid and an equation. First representation. The first representation consists of a 10-unit grid, which has 7 of the 10 units shaded. Below the grid is the following equation: the fraction 7 over 10, equals 0.7. Second representation. The second representation consists of a 100-unit grid, which has 22 of the 100 units shaded. Below the grid is the following equation: the fraction 22 over 100, equals 0.22. End figure description. Which of the following is the most significant limitation of using the grids in this way? A. It does not make the equivalency of a fractional part of a whole and a fractional part of a set clear. B. It does not make the equivalency between decimal notation and fraction notation clear. C. It makes it difficult to determine what constitutes the unit fraction in each representation. D. It does not make the relative value of the wholes in the two representations clear.

Option (D) is correct. The activity assumes that both the tenths grid and the hundredths grid represent a whole, but some students may interpret the tenths grid as representing 10 hundredths because the tenths grid looks like a single column of the hundredths grid. In the example, some students may misinterpret the 7 shaded squares in the tenths grid as seven one hundredths7100, which would be the correct interpretation of 7 shaded squares in a single column of the hundredths grid.

Which of the following activities provides fourth-grade students with the best opportunity to make and test predictions about the transfer of kinetic energy between objects? A. Dropping a soccer ball from the roof of a building and measuring the height of the rebound B. Placing two soccer balls touching each other on the floor with both balls at rest C. Measuring the distance traveled by a soccer ball when it is allowed to roll down a ramp and onto a smooth, flat surface D. Rolling one soccer ball along the floor into another that is not moving

Option (D) is correct. The activity involves the transfer of kinetic energy between soccer balls. Because the activity is easy to observe and easy to repeat, it offers fourth-grade students an excellent opportunity to test their predictions about the transfer of kinetic energy between objects.

A teacher provides students with several rectangular samples of flexible materials, with the same length and width but with varying thickness. The students are asked what causes the difference in stiffness among the materials. A student picks up a thin sample that is very flexible and a thick sample that is much stiffer. The student concludes that the thick sample must be made of a stiffer material since it is harder to bend. Which of the following sets of samples should the teacher provide to allow the student to determine that both the type of material and the thickness of the material can affect stiffness? A. Two samples of the same material of equal thickness B. Samples of the same material but of varying thickness C. Thick samples of one material that can be peeled in layers D. Samples of two different materials with two thicknesses each

Option (D) is correct. The four samples will allow the student to determine that both the thickness of the material and the type of material the sample is made of can affect stiffness.

1, 9, 17, 25, 33, 41, 49, 57 Which of the following statements best describes the numbers in the preceding list? A. All whole numbers that are not divisible by 4 and are less than 60 B. All whole numbers that are not divisible by 8 and are less than 60 C. All whole numbers that are less than 60 and have a remainder of 1 when divided by 4 D. All whole numbers that are less than 60 and have a remainder of 1 when divided by 8

Option (D) is correct. The list consists precisely of the whole numbers that are both 1 greater than an integer multiple of 8 and less than 60.

In an activity using an illustration of a plant, a teacher guides students through a lesson about the different parts of a plant. When discussing the structure and function of the roots, the teacher points to small projections that are shown on the plant's roots. The teacher tells the students that the projections are called root hairs and asks the students to predict what would happen if all the root hairs were suddenly gone. Which of the following is likely to be the most direct effect on the plant of the sudden loss of all its root hairs? A. Excess water will build up in the stem. B. Transport of sugars throughout the plant will stop. C. Carbon dioxide will not be able to reach the leaves. D. Water and minerals will not be absorbed from the soil.

Option (D) is correct. The root hairs provide the roots of plants with the surface area needed to absorb water and minerals.

Ms. Chambers writes the following equation on the board. 5×19−40=40−5×19 Ms. Chambers asks her students to determine if the equation is true. One student says, "The equation is true because on both sides of the equation, you subtract after you multiply 5 times 19." Which of the following statements most likely gives the reason underlying the student's error? A. The student disregarded the order of operations. B. The student incorrectly applied the associative property. C. The student does not understand the meaning of the equal sign. D. The student has overgeneralized by applying the commutative property to subtraction.

Option (D) is correct. The student applied the order of operations to determine that multiplication is performed before subtraction in the expression on either side of the equation but misapplied the commutative property, which works only with addition or multiplication. Using the student's reasoning, the equation would simplify to 95 minus 40, equals 40 minus 9595−40=40−95 or 55 equals negative 5555=−55.

In an investigation, students kicked soccer balls along thick grass, short grass, and smooth pavement. The students' teacher then asked the students to use ideas about forces to explain the observed changes in the movement of the soccer balls. One student responded by saying, "The ball rolled after I kicked it. The force of my leg made the ball move. When that force was gone, the ball stopped rolling." Based on the student's response, the teacher should focus on which of the following concepts to correct the student's misunderstanding? A. A force can change the direction in which an object is moving. B. A push or pull can cause an object to start moving. C. A heavier object can require a larger force to start it moving. D. An object will continue moving at a constant velocity unless a force is exerted on it.

Option (D) is correct. The student may be confusing the concepts of energy and force. The student indicates understanding that the kick force affects the ball only when there is contact with the ball. But the student indicates lack of understanding of the concept that motion continues unchanged unless there is a force, such as friction, that slows down and stops motion.

Ms. Keane had her students plant seeds in pots of soil. The students then placed some of the pots on a sunny windowsill and some on a table that was out of direct sunlight. As the seeds began to sprout, the students cared for them and observed their growth. After a weeklong break, the students found that some of their plants were wilted and some were dead. Ms. Keane asked the students to suggest a way to investigate why some of the plants had died and others had not. Which of the following plans will best allow the students to collect data that can be used to determine why some of the plants died? A. Move all of the dead plants to the windowsill and water them the same amount each day. B. Move all of the wilted plants to the dark and water half of them daily and half of them weekly. C. Examine the pots with the wilted plants to determine whether the soil is dry and water each plant when the soil feels dry. D. Place new groups of plants in the original locations and water half of the plants in each group daily and water the other half weekly.

Option (D) is correct. This plan will allow students to test only the water variable while keeping the plants alive and allowing a comparison between groups of plants in different locations.

Ms. Aggarwal tells her students that the x in the equation x plus 4, equals 7x+4=7 is called a variable. She then asks her students to write down what they think the word "variable" means in this context. One student wrote, "The variable is a letter that stands for something we don't know." Which of the following revisions to the student's definition most improves the precision of the definition as it relates to this use of the word "variable"? A. The variable is a letter that stands for any number that is unknown. B. The variable is a letter that stands for a number that is unknown in the equation. C. The variable is a letter that stands for an unknown number that can take on different values. D. The variable is a letter that stands for the unknown number that will make the equation true.

Option (D) is correct. This revision of the definition revoices the student's use of "something we don't know" as an unknown and states that the unknown stands for the single unknown number that makes the equation true.

Ms. Pellegrino divides her class into four groups and gives each group a dozen bean seedlings to monitor for the semester. She asks each group to use the seedlings to investigate whether plants need sunlight to grow. Although she wants each group to develop its own procedure, she wants to prevent errors that will result in the collection of unusable data. Which of the following is a procedural error that Ms. Pellegrino should correct to prevent the collection of unusable data? A. Measuring plant height in inches instead of centimeters B. Not having a backup plan if all the students in the group are absent for one day C. Recording data on loose-leaf paper instead of in a bound laboratory notebook D. Measuring the final heights of the plants but not the initial heights

Option (D) is correct. To rely on changes in plant height as an indicator of plant growth, the students will need to measure both the initial height and final height of each plant in their investigation.

A teacher leads a discussion about the impact of environmental change on an ecosystem. The teacher asks the students in the class to explain how the death of a tree can be beneficial. To help the students, the teacher suggests that they consider how a fallen tree can affect the species diversity of the surrounding forest. In which of the following ways can a tree-fall gap result in increased species diversity in a mature forest community? A. By reducing the amount of carbon dioxide gas in the atmosphere B. By increasing competition between birds for limited resources C. By creating a barrier to the dispersal of seeds produced by mature trees D. By providing a habitat in which shade-intolerant plants can grow

Option (D) is correct. When a tree falls in a forest, a gap in the forest canopy is created, allowing more sunlight to reach the forest floor. Plant species that prefer sunny spots can colonize the area until trees regrow and create a full canopy.

Ms. Vargas asked her students to write an expression equivalent to 4(x−y). A student named Andrew tried two substitutions of values for the ordered pair (x,y) in the expression, then incorrectly wrote that 4x−y is equivalent to 4(x−y), based on the substitutions he tried. Which TWO of the following substitutions of values for the ordered pair x comma y(x,y)could be ones that Andrew tried? A. x=0 and y=0 B. x=0 and y=1 C. x=1 and y=0 D. x=1 and y=1 E. x=1 and y=−1

Options (A) and (C) are correct. Since 4 times, open parenthesis, x minus y, close parenthesis, equals 4 x minus 4 y4(x−y)=4x−4y, the expression 4 x minus y4x−y is equivalent to 4 times, open parenthesis, x minus y, close parenthesis4(x−y) only when 4 x minus y, equals 4 x minus 4 y4x−y=4x−4y. The last equation can be rewritten as 3 y equals 03y=0, or more simply, y equals 0y=0. The two substitutions Andrew tried must have used the value 0 for the variable y.

Mr. Varela asked his students to define a square in terms of other two-dimensional geometric figures. Which TWO of the following student definitions precisely define a square? A. A square is a rectangle that has 4 sides of equal length. B. A square is a parallelogram that has 4 angles of equal measure. C. A square is a parallelogram that has 4 sides of equal length. D. A square is a rhombus that is also a rectangle. E. A square is a rectangle that is not a rhombus.

Options (A) and (D) are correct. A square is a quadrilateral with 4 sides of equal length and 4 angles of equal measure, whereas a rectangle is a quadrilateral with 4 angles of equal measure, a rhombus is a quadrilateral with 4 sides of equal length, and a parallelogram is a quadrilateral where opposite sides are parallel. Therefore, a rectangle that has 4 sides of equal length is a square, and a rhombus that is also a rectangle is a square, so options (A) and (D) are both precise definitions of a square.

Ms. Egbuniwe wants her students to analyze information and provide evidence that individuals inherit traits from their parents but with variation between the individuals and their parents. She would like to provide them with resources that contain appropriate evidence for an upper-elementary student. Of the following, which TWO would provide the best evidence? A. A picture of a female cocker spaniel with her litter of eight puppies, some black, some golden, some brown B. An illustration that shows the evolution of the modern horse over geologic time, including representations of older, extinct species in the horse family C. A picture of a large tree, surrounded by shoots sprouting from its roots D. A diagram that shows a corn plant with red kernels and a corn plant with yellow kernels that have produced offspring containing both red and yellow kernels

Options (A) and (D) are correct. These two resources clearly show a parent, its offspring from sexual reproduction, and both similarities between and variation among all the individuals.

Which THREE of the following actions will result in reduced energy consumption? A. Commuting to work using public transportation instead of driving alone in a gasoline-powered car B. Setting the thermostat for a home cooling system to a higher temperature in warm weather C. Leaving the engine of a gasoline-powered car running when the car will be parked for more than five minutes D. Preventing electrical appliances from drawing power in standby mode by unplugging them when they are not in use

Options (A), (B), and (D) are correct. Commuting to work by public transportation will reduce the miles traveled by private, gasoline-powered vehicles, which will reduce gasoline consumption. Setting a thermostat to a higher temperature in warm weather will reduce the operation of a home cooling system, which will reduce electric energy consumption. Unplugging electrical appliances when they are not in use will reduce the amount of electric power used by the appliances, which will reduce electric energy consumption.

Ms. O'Neil asked her students to find the sum 5/6+7/10+2/15. One student found the sum as represented in the following work. 5/6+7/10+2/15 = 25+21+4/30 = 50/30 Which THREE of the following mathematical strategies were used in the student's process? A. Finding equivalent fractions B. Finding a common denominator C. Finding the simplest forms of fractions D. Finding the least common multiple of the denominators E. Finding the greatest common factor of the denominators

Options (A), (B), and (D) are correct. In order to write the fraction 25+21+430with numerator 25 plus 21, plus 4, and denominator 30, the student used three of the strategies listed. First, the student found a common denominator of 30 for the three fractions in the given sum, where 30 is the least common multiple of the denominators of those three fractions. Next, the student found an equivalent fraction with denominator 30 for each of the three fractions in the given sum to determine the three addends in the expression 25 plus 21, plus 425+21+4.

Mr. French's students are working on finding numbers less than 100 that are multiples of given one-digit numbers. When Mr. French asks them how they know when a number is a multiple of 6, one student, Crystal, says, "Even numbers are multiples of 6!" Mr. French wants to use two numbers to show Crystal that her description of multiples of 6 is incomplete and needs to be refined. Which of the following numbers are best for Mr. French to use for this purpose? Select TWO numbers. A. 15 B. 16 C. 20 D. 24 E. 27 F. 30

Options (B) and (C) are correct. The best numbers for Mr. French's purpose are even numbers that are not multiples of 6, and 16 and 20 are both even numbers, but they are not multiples of 6.

Which THREE of the following phenomena are related to the relative positions of the Sun, the Moon, and Earth? A. Solar flares B. Eclipses C. Phases of the Moon D. Tides

Options (B), (C), and (D) are correct. The relative positions of the Sun, the Moon, and Earth affect eclipses, tides, and the phases of the Moon.

Mr. Benner places a row of 5 cubes on a student's desk and asks the student, Chanel, how many cubes are on the desk. As Chanel points at the cubes one by one from left to right, she counts, saying, "One, two, three, four, five." Then she says, "There are five cubes!" Mr. Benner then asks Chanel to pick up the third cube in the row. As Chanel points at three cubes one by one from left to right, she counts, saying, "One, two, three." She stops, then picks up the three cubes, and gives them to Mr. Benner. Chanel has demonstrated evidence of understanding which TWO of the following mathematical ideas or skills? A. Using numerals to describe quantities B. Recognizing a small quantity by sight C. Counting out a particular quantity from a larger set D. Understanding that the last word count indicates the amount of objects in the set E. Understanding that ordinal numbers refer to the position of an object in an ordered set

Options (C) and (D) are correct. Chanel first counts the cubes one by one and then she states that there are 5 cubes. Her work demonstrates that she understands that when counting a set of objects, the last word count indicates the cardinality of the set (that is, the number of objects in the set), so (D) is correct. When she is asked to pick the third cube in the row, she is able to count and stop at three, thus demonstrating that she can count out a quantity from a larger set, so (C) is correct.

Which TWO of the following word problems can be answered by dividing 4 by 1/5 ? A. A quilt requires 1/5 square meter of purple cloth. How many square meters of purple cloth are needed for 4 of these quilts? B. A machine working at a constant rate takes an hour to sort 4 bins of mail. How many bins of mail does the machine sort in 1/5 hour? C. Each person at a dinner was served 1/5 of a fish fillet. A total of 4 fish fillets were served. How many people were served at the dinner? D. A roll of copper wire that was 4 meters long was cut into sections that were each 1/5 meter long. How many sections were made from the roll of copper wire? E. Each of 4 people in a study group read 1/5 of the total chapters in a book. No student read the same chapter as any other student. What fraction of the total chapters in the book did the study group read?

Options (C) and (D) are correct. Problem (C) can be answered by dividing the number of fillets, 4, by the fraction of a fillet served per person, 15 fillet per person, to determine that 20 people were served at the dinner. Problem (D) can be answered by dividing the length of the roll of wire, 4 meters, by the length per section, one fifth15 meter per section, to determine that 20 sections were made from the roll.

Mr. Wilson's upper elementary class is researching the positive and negative effects of pesticide and herbicide use in local agriculture. The class has already spent many hours searching on the Internet but has collected very little information. He would like to provide two additional opportunities for the students to collect information that would be the most useful. Which TWOactivities would provide the best opportunities to collect useful information? A. Mr. Wilson can schedule a trip to a local soybean farm. The students will tour the farm and interview the farmer for information on his farming practices. B. Mr. Wilson can schedule a trip to a pesticide production plant. The students will have a chance to interview the scientists employed by the company and receive materials that may not be present on the Internet. C. Mr. Wilson can meet with the school librarian, outlining his objectives and requirements. He can then bring his class to the library for several research periods after the librarian has had a chance to collect and lay out relevant journals, newspapers, and books. D. Mr. Wilson can schedule a panel discussion with representatives from the Environmental Protection Agency, United States Department of Agriculture, and the local extension service. They can provide materials that may not be available on the Internet, provide suggestions for additional resources, and answer student questions.

Options (C) and (D) are correct. They describe good sources of unbiased, authoritative information. The other options would present limited, possibly biased, information.

An athlete runs 3 laps around an oval track. The distance the athlete runs in one lap is 400 meters. What is the total distance, in kilometers, that the athlete runs?

The correct answer is 1.2 kilometers. The total distance that the athlete runs in 3 laps is 3 times 400 meters 3×400 meters, which is equal to 1,200 meters. Since 1 kilometer is equivalent to 1,000 meters, the athlete runs 1,200 meters times the fraction 1 kilometer over 1,000 meters1,200 meters×1 kilometer1,000 meters, which is equal to 1.2 kilometers.

What value of x makes the equation 8 x equals 1368x=136 true?

The correct answer is 17. The value of x that makes the equation true can be determined by dividing 136 by 8, which gives the quotient 17.

A student uses cubes to build a rectangular solid. Each of the cubes has a side length of 1/2 inch. The rectangular solid has a length of 4 inches, a width of 3 inches, and a height of 2 1/2 inches. How many cubes does the student use to build the rectangular solid? _____________cubes

The correct answer is 240 cubes. The number of cubes placed along the length of the solid is 4÷1/2 , which equals 8. The number of cubes placed along the width of the solid is 3÷1/2 , which equals 6. The number of cubes placed along the height of the solid is 2 1/2÷1/2 , which equals 5. The total number of cubes the student uses is 8×6×5 , which equals 240.

A student's incorrect solutions to two equations are represented in the following work. The figure shows a student's work. The work shows the student solving the following two equations. Each solution consists of two steps. Equation 1. Step 1: 7 x equals 70. Step 2: x equals 0. Equation 2. Step 1: 4 x equals 44. Step 2: x equals 4. End figure description. If the student continues to use the same strategy, what will be the student's solution to the equation 2x=24 ?

The correct answer is 4. The work shown indicates that the student likely thinks that x is a placeholder for the digit in the units place and that the coefficient of x represents the digit in the tens place. The student likely does not understand that 7 x 7x means 7 times x or that 4 x 4x means 4 times x . Therefore, if the student continues to use the same strategy, the student will indicate that the solution to the equation 2 x equals 24 2x=24 is x equals 4 x=4 since 4 is the digit in the units place in 24.

A chef uses one eighth 1/8 pound of shredded potatoes to make a serving of hashed brown potatoes. How many servings of hashed brown potatoes can the chef make with 5 pounds of shredded potatoes?

The correct answer is 40 servings. Since the chef uses one eighth18 pound of shredded potatoes to make a serving of hashed brown potatoes, the number of servings the chef can make with 5 pounds of shredded potatoes is 5 divided by one eighth5÷18 , or 40.

A worker made stacks of either 6 chairs or 4 chairs with the chairs used during a band practice session. The worker put 3/4 of the chairs in stacks of 6 and the rest of the chairs in stacks of 4. The worker stacked 48 chairs in all. What was the total number of stacks of chairs that the worker made?

The correct answer is 9. The worker stacked 48 chairs in all, and put three fourths34 of the chairs in stacks of 6 and the rest of the chairs in stacks of 4. The number of chairs that the worker put in stacks of 6 is three fourths times 4834×48 , or 36. The number of chairs that the worker put in stacks of 4 is 48 minus 3648−36 , or 12. The number of stacks of 6 chairs is 36 divided by 636÷6 , or 6. The number of stacks of 4 chairs is 12 divided by 412÷4 , or 3. The total number of stacks of chairs that the worker made is 6 plus 36+3 , or 9.