Math Test 2

Which of the following best describes how to teach multiplication involving a whole number and a fraction?

A "fraction times a whole number" and a "whole number times a fraction" are conceptually different, so they should be taught separately.

One of the following explanations is flawed because the name of the strategy and the description of the strategy do not match. Which one is flawed?

A double number line is created on a coordinate axis; points on the graph can be used to solve problems A coordinate graph can be used to solve proportional situations, but a double number line is a one-dimensional line in which one unit is labeled above the line and the other unit is labeled below the line. This helps determine how to find the missing value.

Which of the following is a good explanation for how to add fractions?

Add equal-sized parts—finding a common denominator can help to solve the problem.

Which of the following is important to do before students learn the formal algorithms?

Address misconceptions.

One of the best approaches for teaching elapsed time is which?

An empty number line The empty number line is ideal for linking the computation required for elapsed time problems to the task of adding or comparing times. A number line is also suggested by the Common Core State Standards for student use on this type of problem.

A student says, "My answer must be wrong—my answer got bigger." Which of the following responses will best help the student understand why the answer got bigger?

Ask the to explain the meaning of 8 ÷ 2, using cutting ribbon as a context, and then ask her to re-explain to you using 8 ÷ 1/2 , still using cutting ribbon as a context. This is a common misconception that students carry from their experiences with whole numbers. You cannot tell students how to fix a misconception: They must have an opportunity to compare whole numbers to fractions and see how they are related, as in the cutting ribbon situation. This one prompt may not be enough, but it is a good start to helping the student see why division by a fraction might result in a larger number (when the divisor is less than 1).

The natural progression for teaching students to understand and read analog clocks includes starting with which of the following steps?

Begin with a one-handed clock that can be read with reasonable accuracy. One of the best approaches to teaching an analog clock is to start with a one- handed clock. Then students can focus on the movement of the little hand as they watch the movement during the hour and even predict when it is half past the hour (just with one hand).

What is the cognitive skill that helps students recognize and group shapes according to their attributes and properties?

Classification When students use skills in classification, they are grouping by a common attribute and sorting the shapes into categories.

Which of the following strategies would you like students to decide to use when determining which of these fractions is greater than the other? 7/8 and 5/6

Compare how far from 1 The best choice is a reasoning strategy

The teachers in the following answer choices have identified three manipulatives to use when teaching fraction concepts. Each teacher intended to select one manipulative for each type of model. Which teacher succeeded in selecting one manipulative of each type?

Denise selected tangrams, color tiles, and number lines.

Which of the following transformations is a non-rigid transformation?

Dilation Rigid transformations are changes in the position or shape of the object moved. A dilation is a non-rigid transformation, as it makes a figure larger or smaller depending on the scale factor.

Which of the following words is not one of the words learned as a position description in kindergarten?

Direction Young learners describe the relative positions of objects using terms such as above, below, beside, in front of, behind, next to, between, near, far, over, under, left, and right. The word direction is not used to describe the relative position of an object.

Of the following statements, which is the most central to effectively teaching ratios and proportions?

Engage students in a variety of strategies for solving proportions, including ratio tables, tape diagrams and graphs.

Base-ten models, the rational number wheel with 100 markings around the edge, and a 10- by-10 grid are all models for linking which three concepts together?

Fractions, decimals, and percents They are all based on 100

Which of the following teaching ideas for teaching partitioning is not consistent with the research findings?

Give students area models that are already partitioned and ask them to record the fractional amount shaded. Although this may be only a small part of teaching about partitioning, students will not learn what partitioning means if shapes have already been pre-partitioned and shaded. This approach is overdone in textbooks and worksheets and teachers must plan to include the many other ideas discussed in this section.

Which of the following is not a suggested strategy for estimating?

Guessing An estimation is not a guess. Instead, it is a reasoned way in which to come close to an actual measure of an attribute. There are several strategies for estimation. The other answers are all examples of these strategies.

How can one obtain an accurate measure of the volume of a rectangular prism when given a set of the same-sized cubes?

Layer the cubes on the bottom of the box to fit the dimensions and then see how many layers are needed.

Which of the following is not a common misconception or limited conception related to fractional parts?

Leaving answers as fractions rather than writing them as mixed numbers

Which of the following is not a way to illustrate equivalent fractions?

Multiply the numerator and denominator by the same number. "Multiply the numerator and denominator by the same number" describes a procedure that results in fractions that are equivalent, but it does not illustrate the idea of equivalence. The other three answers each give one way to illustrate equivalence for each conceptual model (area, length, and set).

Which of the following is a nonstandard unit of measure?

Paper clip

Which of the following best describes the relationship between iterating and partitioning?

Partitioning is finding the parts of a whole, whereas iterating is counting the fractional parts.

To compare the weights of two objects, which of the following is the best approach?

Place the two objects in the two pans of a balance

Which of the following reasons best describes why arithmetic and algebra should be closely connected?

Place value and operations are generalized rules; a focus on algebraic thinking can help students make connections across problems and strengthen understanding.

Which of the following is not important when teaching properties?

Providing opportunities for students to name and match properties to examples Unfortunately, traditional instruction around the properties has been a matching activity. Students may be able to name a property, but if they cannot put it to use, then it is, in fact, useless to know it. Conversely, students may know they can reverse numbers in addition, but not know the name of the property. These students can still become fluent in learning the operations.

Which of the following statements is true?

Ratios can be interpreted as composed units or as rates. There are two ways to interpret ratios: as composed units (6 cans of corn for $4) or as rates ($1.50 per can of corn).

All of the following are important considerations for teaching proportions except which?

Teach key words that can support students in effectively setting up proportions correctly.

Which of the following options is not a recommendation for supporting student understanding of fractions?

Tell students that fractions are different from whole numbers, so the procedures are also different The key word here is "tell." You cannot tell students to understand something. The "design situations to address student misconceptions and help them make distinctions between whole numbers and fractions" option addresses the issue of whole numbers and fractions and is an effective way to help students understand fractions.

Which of the following is shown through research to be a common error or misconception when students are comparing or ordering decimals?

The decimal that is the shortest is the largest. Students may think that shorter is larger, as they believe any number of tenths is larger than any number of hundredths. This is not accurate (e.g., 95 hundredths is larger than 7 tenths). See the listing of common errors and misconceptions to confirm the well-researched choices.

Which of the following is true about the van Hiele levels?

They are a progression of ways in which students understand geometric ideas The van Hiele model is a five-level hierarchy of ways of understanding spatial ideas. Each level describes the thinking processes learners use in geometric contexts.

The benefits of using a rectangular area to represent multiplication of fractions include all of the following except which?

They are easy for students to draw.

According to your textbook, which of the following trios of real-world situations represent common uses for estimating percentages?

Tips, taxes, and discounts Percents are based on a proportion of an amount. Tips, taxes, and discounts are all related to everyday uses of proportions of a given amount on a proportion of 100. The other choices include things that relate to percentages, but are not as common experiences.

One of the main goals of the visualization strand is to be able to identify and draw which of the following?

Two-dimensional images of three-dimensional shapes One of the main goals of the visualization strand is to be able to identify and draw two-dimensional images of three-dimensional figures, and then build three-dimensional figures from these two-dimensional images.

Algebraic thinking includes several characteristics. Which of the following statements is not a part of algebraic thinking?

Using manipulatives to reason about situations

Which of the following statements is not true?

Variables are not appropriate for elementary-age students; a box is a more concrete representation. Variables can be introduced at any age, as long as the connection is made to the meaning of the variable. For example, students might be thinking about the ways that 10 horses may be in the paddock or in the field and write it as p + f = 10, stating that p is the number of horses in the paddock and f is the number of horses in the field. Or, they may have selected the variables a + b = 10, illustrating that any letter can be chosen to represent an unknown number.

Mixed numbers:

can be changed into fractions or "improper" fractions and added.

When students explore how shapes fit together to form larger shapes, it is called:

composing shapes

A good teaching option for developing a full understanding of computation with decimals is to focus on:

concrete models, drawings, place value knowledge, and estimation.

Functions

describe a relationship between two variables and may be linear or not.

The role of the decimal point in a number is to:

designate the units position The decimal point indicates the location of the units position. This enables people to figure the unit that is the whole.

Measurement _______________________ is the process of using mental and visual information to measure or make comparisons without the use of measuring instruments.

estimation Estimation is a way in which to make comparisons and reasonable approximations of a given measurement.

The most important factor in moving students up the van Hiele levels is:

geometric experiences that teachers provide to the students. Advancement through the levels requires geometric experiences. Teachers need to help students explore, talk about, and interact with content at the next level while increasing experiences at their current level.

A good task to use with students to assess their understanding of the use of a standard ruler is:

measuring with a "broken ruler." The broken ruler will cause some students to say that it is impossible to measure when beginning and end units are cut off, because there is no starting point. If students fully understand the ruler, they will know they only need the units (not the numbers) and can count the matching units from any point on a ruler.

Mathematical models are useful in both real life and mathematics because:

models such as equations, graphs, and tables can be used to analyze empirical situations, to understand them better, and to make predictions.

Students should explore the area of a triangle after they have a conceptual understanding of the area of a:

parallelogram. When students have the knowledge of how to find the area of a parallelogram, which can always be transformed into a rectangle with the same base and height, they can use two identical copies of a triangle. (See the activity Area of a Triangle to construct a parallelogram related to the two copies of their triangle.)

When a shape can be folded on a line—so that the two halves match—that fold line is also a line of:

reflection.

When you are measuring an object using a tool and choosing the attribute to be measured, then you must:

select a tool with the same attribute to measure with.

The four major content goals in geometry for all grade levels are

shapes and properties, transformation, location, and visualization.

One of the basic ideas of length measurement is that when the unit is longer, the measure is:

smaller. When a unit of length is longer, such as a yard compared to an inch, the measure is smaller. For example, when measuring an item and getting the answer of 1 yard, that would be a smaller measurement (1 unit) than the same item measured with inches, which would be 36 inches (a larger measurement—36 units).

The value of a collection of coins is best learned by having students:

sort the coins starting with the highest value and skip counting.

An area model that demonstrates how figures can have the same area composed of different shapes is:

tangrams Tangrams have seven different pieces, all of which can be represented with a small triangular unit. By creating figures with all of the seven different pieces, the figures will each have the same area regardless of the use of different shapes.

Coordinate grids are often used in geometry to explore:

transformations. Students examine transformations on a coordinate plane in the middle grades. Through this process, they can support their thinking about similarity and congruence

Angles are measured by:

using a smaller angle to fill or cover the spread of the rays.