TExES Core Subjects EC-6 Math (902) practice test

Students who can build and explain concrete models are ready for a pictorial stage that links the models to written numbers. Drawing each array as an area model and writing the partial products in its parts (20 × 4 = 80 and 3 × 4 = 12, so 23 × 4 = 92) connects the blocks to the numbers the standard algorithm records. Moving from concrete to pictorial to symbolic representations keeps the meaning of each step visible.

A

Jumps to abstract practice and skips the pictorial step that connects the blocks to the written numbers.

B

Larger products keep the students at the concrete stage, while the stated goal is to move toward the written algorithm.

C

Memorizing steps teaches the procedure without meaning and does not build on what the students already understand from their models.

In a lesson built on problem solving, students work on a problem before any method is shown, draw on what they already know to invent strategies, and then compare solutions so that connections and more efficient strategies come out of the discussion. That structure matches the goal of students developing their own strategies before a standard method is taught. Modeling the method, a video lesson and a homework set all supply a method first or presume one.

A

Modeling followed by guided and independent practice is sound for teaching a known procedure, but showing the standard method first defeats the goal of students developing their own strategies.

B

A video lesson presents a method for students to copy, so it supplies the strategy instead of letting students develop one.

D

A homework set of similar problems checks answers rather than developing strategies, and with no method taught students have no support for a new type of problem.

In its measurement meaning, division asks how many groups of the divisor fit in the dividend, so 3/4 ÷ 1/8 asks how many eighths are in three-fourths. On a number line marked in eighths, 3/4 falls at 6/8, so six jumps of 1/8 reach it and the quotient is 6. The model shows the meaning of the division, not just its result.

B

An overlapping area model shows the product 3/4 × 1/8 = 3/32, so it represents multiplication, not division.

C

Strips placed end to end show the sum 3/4 + 1/8 = 7/8, not how many eighths fit in 3/4.

D

Circling 3/4 of 8 counters shows 3/4 × 8 = 6, the result of the invert-and-multiply rule, without showing why dividing by 1/8 gives that result.

Subtraction is modeled by taking counters away, and 2 − (−3) asks to take 3 negative counters from a set that has none. Adding 3 zero pairs does not change the value of the set, and removing the 3 negative counters leaves the 3 added positive counters with the original 2, a value of 5. The model shows that taking away −3 has the same effect as adding 3.

A

Adding 3 negative counters models 2 + (−3) = −1, the misconception that subtracting a negative number means adding a negative number.

B

Removing 3 positive counters models 2 − 3 = −1, so the sign of the number being subtracted is lost.

C

Joining 3 positive counters shows only that 2 + 3 = 5, and because nothing is taken away, it cannot show why the subtraction gives the same result.

Connecting plotted points with a line shows that every value between them is possible, which fits only an input that can vary continuously. Time passes continuously and a burning candle has a height at every moment, so its graph is a line segment. Tickets, books and rows come only in whole numbers, so their graphs are separate points.

A

The cost is in dollars, but it depends on a whole number of tickets, so no points exist between the plotted ones: half a ticket cannot be bought.

B

Weight is a measured quantity, but this weight changes only when a whole book is added, so the graph is a set of separate points, one for each number of books.

D

Rows of chairs come in whole numbers, so only the plotted points have meaning, even though the number of chairs increases at a constant rate.

Starting at 7 and adding 4 is a recursive description: it gives each term from the one before it. An explicit rule gives a term directly from its position, so a question about position turns students to the pairs (1, 7), (2, 11), (3, 15) and (4, 19), where each term is 4 times its position plus 3. With that rule the 100th term is 4 × 100 + 3 = 403, found without listing the terms before it.

A

Listing the next numbers extends the pattern with the add-4 rule the students already use, so their thinking stays recursive.

B

The students have already found the increase of 4, and the size of each step does not by itself connect a term to its position.

D

Creating a new pattern is a worthwhile task, but it asks for another recursive pattern instead of an explicit rule for this one.

Describing where objects are located depends on positional words such as under, behind, beside and between. Hiding a toy and giving a partner clues makes kindergarten students choose those words to tell exactly where the toy is, and the partner's search shows whether the description was clear. This everyday language of position is the foundation for later ways of describing location, such as maps and coordinate grids.

A

Sorting and naming shapes builds knowledge of shape attributes, a sound kindergarten goal, but not the language of location.

B

Comparing the heights of towers builds the measurement idea of taller and shorter rather than words that tell where an object is.

C

Ordered pairs describe location with symbols that build on positional language, so plotting them is a later, more abstract step than kindergarten students are ready for.

A ray has one endpoint and extends without end in one direction. A flashlight beam starts at the flashlight and travels outward in a single direction, so it models a ray. A segment has two endpoints, a point marks a single location, and a plane is a flat surface that extends in every direction.

B

A string held between two nails has two endpoints, so it models a line segment, not a ray.

C

A flat desktop models part of a plane, a flat surface, rather than a figure that extends in one direction.

D

A distant star models a point, a location with no length.

A stem-and-leaf plot groups the heights by their tens digits, the stems 0 through 4, so the lengths of the rows show the shape of the distribution, while each leaf keeps one plant's height, as 3 | 7 records 37 centimeters. A histogram and a box-and-whisker plot also show how data are distributed, but they summarize the values in intervals or in five numbers, so the individual heights are lost. Choosing a display that fits both the data and the purpose is a central decision in organizing data.

A

A box-and-whisker plot shows the spread through five summary values, so it does not keep the individual measurements visible.

B

A histogram shows how many heights fall in each interval, but it does not keep the individual measurements visible.

D

A bar graph compares categories; one bar per plant would keep each height but would not group the values to show how they are distributed.

Statements that hold only for whole numbers become misconceptions that students must later unlearn. Multiplying by 10 shifts each digit one place to the left, so each digit's value becomes ten times as great: in 2.5 × 10 = 25, the 2 moves from the ones place to the tens place and the 5 from the tenths place to the ones place. The same is true of negative numbers: −2.5 × 10 = −25, and each digit again moves one place to the left, so its place value becomes ten times as great. This place-value description is true for whole numbers, decimals and negative numbers alike.

A

Misconception, a rule that expires with decimals: 0.5 × 8 = 4, which is less than 8.

B

Misconception, a rule that expires with negative numbers: 3 − 5 = −2.

D

Misconception, a rule that expires with decimals: 2.5 × 10 is 25, not 2.50; the shortcut works only for whole numbers.