Math Methods Test 3 (CH 12, 13, 14, 15)
Which of the following is important to do before students learn the formal algorithms?
Address misconceptions.
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.
Lynne used the partitioning strategy to multiply 27 x 4. Which problem below shows this strategy?
27 x 4, 20 x 4 plus 7 x 4 =
All the following are recommendations for effective fraction computation instruction except:
Carefully introduce procedures
Establishing a culture where students are making their own conjectures develops their skills at justification. Which of the following would foster this culture?
Always, sometimes or never mathematical statements
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 them to explain the meaning of 8 ÷ 2, using cutting ribbon as a context, and then ask them to re-explain to you using 8 ÷ 1/2, still using cutting ribbon as a context.
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.
The way we write fractions with a top and bottom number is a convention. What method focuses on making sense of the parts rather than the symbols?
Begin by using words (i.e., one-fourth)
Which of the following statements about multiplication strategies is true?
Cluster problems use multiplication facts and combinations that students already know in order to figure out more complex computations.
Which of the following strategies would you like students to use when determining which of these fractions is greater 7/8 or 5/7?
Compare how far from 1
All the following are representative of how algebraic thinking is integrated across the curriculum except:
Composing and decomposing shapes
All of the methods below would work to support students' knowledge about what is happening when multiplying a fraction by a whole number except:
Compute with a calculator
Locating a fraction on a number line can be challenging but is very important. Which is a common error that students make in working with the number line?
Count the tick marks that appear without noticing any missing ones
A critical aspect of understanding divisions of fractions is:
the divisor is the unit
The teachers have identified three manipulatives to use when teaching fractional concepts. Each teacher intended to select one manipulative to show each fraction model. Which teacher succeeded in selecting manipulatives for each type?
Denise selected tangrams, color tiles, and number lines.
What is the primary reason to not focus on specific algorithms for comparing two fractions?
Developing number sense about relative size of fractions is less likely
Marek was asked to multiply 34 × 5. He said, "30 × 5 = 150 and 4 × 5 = 20, so I can add them to get 170." Which property did Marek use to solve this multiplication problem?
Distributive property of multiplication over addition
The goal is to rename a fractional amount. What is the concept that requires the use of many contexts and models?
Equivalent fractions
What is a problem with learning only designated (standard) algorithms for fraction operations?
Follow a procedure in a short term, but not retain
Language plays and important role in thinking conceptually about division. Identify the statement below that would not support students thinking about the problem 4 ÷ 583.
Four goes into 5 how many times?
Which of the following analyzes how the pattern is changing with each new element in the pattern?
Geometric growing patterns
All of the following statements are research-based recommendations for teaching and learning about fractions except one. Which one?
Give greater emphasis to specific algorithms for finding common denominators
Research findings support all of the following fraction teaching ideas but one. Which of the following is the unsupported method?
Give students area models that are already partitioned and ask them to record the fractional amount shaded.
A fraction by itself does not describe the size of the whole. A fraction tells us only:
the relationship between part and whole.
Identify which statement below would not be considered a common or limited conception related to fractional parts?
Knowing that answers can be left as fractions rather than writing them as mixed numbers
When asked to solve the division problem 143 ÷ 8, a student thinks, "What number times 8 will be close to 143 with less than 8 remaining?" Which strategy is the student using?
Missing factor
What form of algebraic reasoning is the heart of what it means to do mathematics?
Noticing generalizations and attempting to prove them true
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.
Arithmetic and algebra are closely connected. Identify the reason below that best describes why?
Place value and operations are generalized rules; a focus on algebraic thinking can help students make connections across problems and strengthen understanding.
Which instructional method does not support purposeful teaching of mathematical properties?
Providing opportunities for students to name and match properties to examples
Guiding students to develop a recording scheme for multiplication can be enhanced by the use of what tool?
Recording sheet with base-ten columns
When students use the break apart of decomposition strategy with division, what must they remember?
Remember that you cannot break apart the divisor
What division approach is good for students with learning disabilities that allows them to select facts the already know?
Repeated subtraction
Which of the following can be presented to students that will open opportunities for them to generalize?
Set of related problems
Which model below would not provide a clear illustration of equivalent fractions?
Show an algorithm of multiplying the numerator and denominator by the same number
Identify the example below that represents a relational-structural approach for the problem 8 + 4 = n + 5
Since 4 is one more than 5 on the other side, that means n is one less than 8
Providing students with many contexts and visuals is essential to their building understanding of equivalence. More examples of linear situations are needed to make comparisons more visible. Which of the following would not be best to model on a number line?
Slices of pizza eaten
Computational estimation refers to which of the following?
Substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally
Which of the following options would be misleading for student understanding of fractions?
Tell students that fractions are different from whole numbers, so the procedures are also different.
There are multiple contexts that can guide students understanding of fractions. Which of the following would involve shading a region or a portion of a group of people?
part-whole
All the following are reasons that data and algebra are good topics to integrate except:
There isn't enough time in the year to address everything, so it is more efficient to teach these two together. OR Real data can be gathered and used to see if the data covary, for example in a linear manner, which builds knowledge of both algebraic and statistics.
The benefits of using a rectangular area to represent multiplication of fractions include all the following except which?
They are easy for students to draw.
What statement below best describes functions?
They describe a relationship between two variables and may be linear or not.
When adding fractions with like denominators it is important for students to focus which key idea?
Units are the same
Teaching fractions involves using strategies that may not have been part of a teacher's learning experience. What is a key recommendation to teachers from this chapter?
Use multiple representations, approaches, explanations, and justifications
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 is not representative of the current thinking about arithmetic and algebra in the elementary classroom?
Variables are not appropriate for elementary-age students; a box is a more concrete representation.
Mathematical modeling is appropriate for investigating real challenges. Which of these examples requires some mathematical modeling?
What would be the better deal, buy-one-get-one half off, 25% off, buy-two-get-one-free?
When teaching computational estimation, it is important to:
accept a range of reasonable answers.
Proficiency with division requires understanding:
place value, multiplication, and the properties of the operations
Using contextual problems with fraction division works in providing students with an image of what is being:
shared or partitioned
Mixed numbers:
can be changed into fractions or "improper" fractions and added.
An important concept in working with repeating patterns is for the student to identify the:
core of the pattern
One way to effectively model multiplication with large numbers is to:
create an area model using base-ten materials.
Conceptualizing the symbol for equal as a balance can support students' understanding of:
equality or inequality.
A common misconception with set models is:
focusing on the size of the subset rather than the number of equal sets.
Writing fractions in the simplest terms means to write it so:
fraction numerator and denominator have no common whole number factors.
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.
Research recommends that teachers use one of the following to support students' understanding that fractions are numbers and they expand the number system beyond whole numbers.
number lines
Students need experiences with variables that vary, and pairs of variables that covary, early in the elementary curriculum. It is important to emphasize the:
variable stands for the number of.
Delia was asked to estimate 489 + 37 + 651 + 208. She said, "400 + 600 + 200 = 1200, so it's about 1200, but I need to add about 150 more for 80 + 30 + 50 + 0. So, the sum is about 1350." Which computational estimation strategy did Delia use?
Front-end
Children as early as first grade can explore functional thinking by using:
input-output activities.
