D691 – Elementary Mathematics Curriculum

Explanation:

Favorite colors are categories, not values along a continuous scale or timeline. Bar graphs are designed to compare counts or frequencies across categories, making them ideal here. Line graphs connect points to show change across an ordered, typically continuous variable (often time), which does not apply to color categories.

Correct Answer:

A bar graph is suitable for categorical data, while a line graph is used for continuous data.

Why Other Options Are Wrong:

A bar graph requires numerical data, while a line graph does not.

Both graphs plot numerical values; the difference is the nature of the x-axis (categorical for bars vs. ordered/continuous for lines), not whether numbers are required.

A bar graph is easier to read than a line graph.

“Easier” is subjective and not the key criterion. The suitability depends on data type, not general readability.

A bar graph can show trends over time, while a line graph cannot.

This reverses the roles. Line graphs are typically used to show trends over time or another continuous variable.

Explanation:

From the prior estimate, 2015 was about 550 books. The increase to 2016 is therefore 700 − 550 = 150 books, which falls in the 150–200 range.

Correct Answer:

About 150 to 200 books

Why Other Options Are Wrong:

About 50 to 100 books

Too small; it would imply 2015 was roughly 600–650, not matching the earlier estimate of ~550.

About 100 to 150 books

The exact increase computes to about 150, and the best matching range provided is 150–200, not this lower band.

About 200 to 250 books

Too large; it would require 2015 to be around 450–500, which doesn’t align with the prior estimate.

Explanation:

The conservation of number principle means that the number of objects in a set stays the same even if their spatial arrangement changes, as long as no objects are added or removed and none are hidden. Rearranging them closer together or farther apart does not alter the set’s cardinality. This idea supports stable counting and one-to-one correspondence in early math.

Correct Answer:

Quantity does not change with physical arrangement

Why Other Options Are Wrong:

Numbers can be rearranged without changing their value

This statement is vague and can be misread as moving digits in numerals or reordering terms in an equation. Conservation is specifically about the count of concrete objects being invariant under spatial rearrangement, not about symbolic manipulation of numbers.

Physical arrangement affects the total quantity

This is the opposite of conservation. Spreading objects out or clustering them does not change how many there are; only adding or removing objects changes quantity.

The sum of numbers remains constant regardless of order

This describes the commutative property of addition, not conservation. Commutativity concerns arithmetic with symbols; conservation concerns the cardinality of a set of objects under rearrangement.

Explanation:

This describes matching each object to exactly one number word while proceeding through the stable counting sequence—ensuring you don’t skip or double-count any item. That practice is called one-to-one correspondence and is foundational for accurate counting and for understanding that the final count reflects the quantity present.

Correct Answer:

One-to-one correspondence

Why Other Options Are Wrong:

Number names

This refers to knowing or saying the words “one, two, three…,” not the act of pairing each object with a single count word during counting.

Count sequence and counting

Reciting the sequence is necessary, but the key idea in the prompt is assigning one number word to each item. That specific matching is one-to-one correspondence, not merely knowing the sequence.

Cardinality

Cardinality is the understanding that the last number said tells “how many” are in the set. It depends on correct counting, but it is not the act of one-by-one matching described.

Explanation:

Teaching patterns means establishing and following a rule (e.g., ABAB, AAB, ABC) and asking students to continue or create sequences that fit that rule. Having students repeat a color sequence directly targets pattern recognition, description, and extension. It also supports language like “unit of repeat,” prediction of “what comes next,” and transition from simple repeating to growing patterns—all central goals in early algebraic thinking.

Correct Answer:

Introduce a sequence of colors to be repeated in the stacking process.

Why Other Options Are Wrong:

Focus solely on the height of the stacks without considering color.

Height comparisons shift the task to measurement, not patterning. While you could form growing patterns by height, ignoring color removes the clear, categorical cue that helps young learners perceive the repeating unit. The prompt is about extending a color block activity; dropping color weakens the connection to pattern rules.

Limit the colors used to only primary colors.

Choosing only primary colors constrains materials but does not introduce or reinforce a pattern rule by itself. Without an explicit sequence to follow (e.g., red–blue–red–blue), students are not practicing recognition or extension of patterns. Limiting color choice is a materials decision, not a pedagogical move toward patterning.

Encourage students to create random stacks without any specific order.

Random arrangements lack a governing rule, so they neither model nor practice pattern recognition. Students need purposeful sequences to analyze, continue, and generalize. Randomness may encourage creativity, but it does not build the algebraic reasoning targeted in a lesson on patterns.

Correct Answer:

Cube B has 8 times the mass of cube A.

Explanation:

With equal densities, mass is proportional to volume. The volume of a cube is the cube of its edge length. Cube A has volume 1³=1 cm³, while cube B has volume 2³ = 8cm³. Therefore, cube B’s mass is 8 times cube A’s mass.

Why Other Options Are Wrong:

Cube B has 2 times the mass of cube A.

This would be true only if mass scaled linearly with edge length. However, for three-dimensional objects, volume—and therefore mass at constant density—scales with the cube of the linear dimension. Doubling the edge does not double the mass; it increases volume by 2³. Thus, the mass becomes eight times, not two times.

Cube B has 4 times the mass of cube A.

A factor of four suggests squaring the scale factor, which applies to surface area, not volume. Mass at constant density follows volume, not area. Since the edge length doubles, volume scales by 2³ = 8, not 2² = 4. Therefore, four times the mass underestimates the correct ratio.

Cube B has the same mass as cube A.

Equal mass would require equal volume at the same density. The cubes have different edge lengths (2 cm vs. 1 cm), so their volumes differ significantly. Cube B’s volume is eight times larger, so its mass cannot be the same. Keeping density constant rules out this possibility.

Cube B has a smaller mass than that of cube A.

A smaller mass would imply a smaller volume at the same density. But cube B has a longer edge, making its volume greater, not smaller. Specifically, the volume increases by a factor of eight when the edge doubles. Consequently, cube B must be heavier, not lighter.

Correct Answer:

The activity helps students develop their skills in collecting and representing data through graphing.

Explanation:

When students survey classmates about favorite shapes, they generate categorical data, practice tallying, and organize results into tables or simple graphs. They interpret counts, compare frequencies, and draw conclusions from the displays. This work strengthens vocabulary such as “most,” “least,” and “equal to,” while connecting real-world questions to mathematical representations. It directly targets early data collection and graphing standards.

Why Other Options Are Wrong:

The activity teaches students about the properties of geometric figures.

While shapes are mentioned in the survey, the task does not analyze attributes like side lengths, angles, or symmetry. Students are choosing preferences, not classifying by properties. Any discussion of why someone prefers a shape is incidental to the data goal. Therefore, it does not systematically build knowledge of geometric properties.

The activity introduces students to basic probability concepts.

A preference survey records what students like; it does not model random events or sample spaces. Without random selection or chance processes, the results do not represent probabilities. Treating the class’s favorites as probabilities confuses frequency with likelihood in a chance context. Thus, probability is not the mathematical focus of this activity.

The activity focuses on understanding spatial relationships between shapes.

Spatial reasoning involves position, orientation, and relationships like “above,” “inside,” or “congruent.” The survey does not require manipulating shapes in space or analyzing how they relate. Students are counting responses, not exploring transformations or spatial arrangements. Consequently, spatial relationships are not the primary learning outcome here.

Explanation:

In base four, the digits are 0, 1, 2, and 3. The number 100₄ equals 1·4² + 0·4 + 0 = 16 in base ten. The number just before it must therefore be 16 − 1 = 15 in base ten. Converting 15 to base four gives 3·4 + 3 = 33₄, which is the immediate predecessor of 100₄; equivalently, 100₄ − 1 = 033₄ → 33₄ after borrowing.

Correct Answer:

33₄

Why Other Options Are Wrong:

32₄

This equals 3·4 + 2 = 14 in base ten. That is two less than 16, not one less, so it cannot be the immediate predecessor of 100₄. If you counted up in base four, you would say 32₄, then 33₄, and only then 100₄. Therefore, 32₄ is too small to be the number just before 100₄.

333₄

This equals 3·16 + 3·4 + 3 = 48 + 12 + 3 = 63 in base ten. It is far larger than 100₄ (which is 16 in base ten), so it cannot come immediately before 100₄ in counting order. The confusion often comes from thinking of base ten where 99 is before 100, but here we are comparing a three-digit base-four number to a three-digit base-four benchmark. Thus, 333₄ is not adjacent to 100₄ and is much greater.

323₄

This equals 3·16 + 2·4 + 3 = 48 + 8 + 3 = 59 in base ten. Like 333₄, it is much larger than 100₄, so it cannot be the number just before 100₄. The presence of a leading 3 in the highest place value makes it exceed any number starting with 0 or 1 in that position. Hence, 323₄ is not the immediate predecessor of 100₄.

Explanation:

This activity centers on spatial language and following/giving simple movement directions. Words like left, right, forward, backward, above, and below build students’ understanding of position and movement in space. Practicing these terms supports early geometry and navigation skills and prepares students for later map work. The focus is immediate, kinesthetic use of directional vocabulary.

Correct Answer:

Using directional terms

Why Other Options Are Wrong:

Reading and using a compass rose

A compass rose involves cardinal directions (north, south, east, west) and is typically introduced later. The activity uses relative directions tied to the child’s body and classroom landmarks, not global orientation with N/S/E/W. Thus, it doesn’t directly teach compass-rose skills.

Understanding chronological order

Chronology is about time sequencing (first, next, then, last), not spatial movement. While the search may have steps, the instructional target here is spatial language, not temporal ordering. Therefore, this option misidentifies the skill being practiced.

Reading and using a map

Map reading requires interpreting symbols, scale, and spatial correspondences on a representation. In this activity, students navigate the real environment using verbal cues, not a map. Although it can be a precursor to mapping, map skills are not explicitly taught here.