ASDM Final
What is the process of using mathematics to analyze and provide insight into real-world issues, dilemmas, and other phenomena?
Mathematical Modeling
Patterns are found in all areas of mathematics. Which is an example of a repeating pattern?
Patterns with a core that repeats.
What can a teacher do if a student confuses the variable x for a multiplication symbol?
Use other variables, such as n.
A tool called __________, is normally thought of as teaching numeration but can help students to connect place-value and algebraic thinking.
hundred chart
What can strengthen students' understanding of number, for example, multiples?
Repeating patterns.
Which action below will help build students' understanding of the equal sign?
The equal function can be represented concretely by a number balance scale.
The __________ property is central to learning multiplication basic facts and algorithms for the operation.
distrubutive
The term algebraic thinking is used instead of the term algebra because algebraic thinking goes beyond the topics that are typically found in an algebra course. What would be considered a way to describe algebraic thinking?
Noticing patterns and making generalizations.
What is a reason for students to explore and compare linear functions in abstract forms like graphs?
Helps students notice what makes a function linear.
Growing patterns can be represented in multiple ways. Identify the representation below that actually illustrates covariation:
Graph
Which view of the equal sign means, students view the equal sign as a balance and can use relationships between the two sides of the equal sign to reason about equivalence?
Relational-structural view.
What is an early misconception about variables?
A placeholder for one exact number.
What is a way to support functional thinking in elementary school?
Input output activities.
This method of recording can help students think about how two quantities vary from step to step:
Table
Which of the following is not an example of algebraic thinking activity for kindergarten?
Applying properties of addition.
Making sense of properties of the operations is a part of learning about generalizations. Identify the statement below that a student might use to explain the associative property of addition:
when you add three numbers you can add the first two and then add the third or add the second and third and then the first. either way you get the same answer.
These patterns are technically referred to as sequences and they involve a step-to-step progression:
Recursive
Complete this statement, "The use of a balance scale or semi-concrete drawings of a balance help develop a strong understanding of ..."
Abstract concept of equality.
What kind of statements can help build relational understanding and address student misconceptions?
True false statements.
Students need to be familiar and use the language to describe functions of graphs. What are important vocabulary that students need to know and use?
Independent and dependent variables.
Some manipulatives, like fraction circles, can mislead students to believe inaccurately that fractional parts must be the same shape as well as the same size. What can be used to address this misconception?
Color tiles can be used to create rectangles.
Complete this statement, "Comparing two fractions with any representation can be made only if you know the ..."
Fractional parts are parts of the same size whole.
The following visuals/manipulatives support the development of fractions using the length model?
Cuisenaire rods.
Counting precedes whole-number learning of addition and subtraction. What is another term for counting fraction parts?
Iterating.
What is a good visual for connecting the concept of equivalence to the standard algorithm for finding equivalent fractions:
An area model.
Which manipulative is effective for modeling fractions greater than one?
Connecting Cubes.
A __________ is a significantly more sophisticated length model than other models.
number line
What can impede students' conceptual understanding of fractions and fraction equivalence?
Rushing too quick to the algorithm.
Which fraction construct focuses on cardinal size?
Measurement
Understanding that parts of a whole must be partitioned into equal-sized parts across different models is an important step in conceptualizing fractions and provides a foundation for what kind of tasks?
Exploring sharing and equivalence tasks.
What is a consideration when teaching cross multiplication?
Students overuse it when they could use a reasoning strategy.
How do you know ________that ? Identify the statement below that demonstrates a conceptual understanding:
If you have 6 items and you take 4 that would be 4/6. You can make 6 groups into 3 groups and 4 into 2 groups and that would be 2/3.
Fraction understanding is developmental in nature and can begin as early as kindergarten with what type of experiences?
Equal sharing experiences.
Locating a fractional value on a number line can be challenging but is important for students to do. What is a benefit of number line work with fractions?
Useful in helping students see the relative size of a fraction.
Which mental strategy for comparing fractions would be most appropriate for comparing and ?
Same numerators.
Models provide an effective visual for students and help them explore fractions. Identify the statement that is the definition of the length model:
A unit or length involving fractional amounts.
What does a strong understanding of fractional computation rely on?
Fraction Equivalence.
Teaching considerations for fraction concepts include which of the following?
Emphasize that fractions are numbers, making extensive use of number lines.
What is a common misconception with fraction set models?
Knowing the size of the subset rather than the number of equal sets.
What is the definition of the process of partitioning?
Spitting equally
Identify the manipulative used with length models that you can decide what to use as the "whole.":
Cuisenaire rods
Repeated subtraction is also known as what?
Division
Length models are best represented by what manipulative?
Number line
What is the most effective way to build understanding and procedural fluency with fraction operations?
Estimation activities
What model is exceptionally good for illustrating and generalizing fraction multiplication?
Area model
A(n) __________ interpretation is a good method to explore division by a fraction because students can draw illustrations to show the model.
measurement
Which activity does not guide students to understand the algorithm for fraction multiplication?
Subdividing the whole number.
Complete the statement, "Developing the algorithm for adding and subtracting fractions should ...":
Be done side by side with visuals and situations.
It is recommended that division of fractions be taught with a developmental progression that focuses on four types of problems. Which statement below is not part of the progression?
A whole number divided by a mixed number.
Common misconceptions or challenges occur because students tend to overgeneralize what they know about whole number operations. Identify the misconception that is not relative to fraction operations.:
Use of invert and multiply.
A student thinking an answer is wrong because it is less than the factor is a common misconception. How can a teacher respond?
Use visuals and a context to illustrate
Teachers should help students understand why procedures for computations with fractions make sense. Which is an effective teaching practice?
Let estimation and reasoning strategies play a big role in the development of strategies.
What is one of the methods for finding the product of fractional problems when one of the numbers is mixed number?
Compute partial products.
What statement is true about adding and subtracting with unlike denominators?
Is sometimes possible for students, especially if they have a good conceptual understanding of the relationships between certain fractional parts and a visual tool, such as a number line.
What is helpful when subtracting mixed number fractions?
Deal with the whole numbers first and then work with the fractions.
Expectations for competency in today's workforce as well as in daily life mean that changes are warranted in how computation is taught. Identify the true statement below:
Instructional changes include focusing more on reasoning strategies than on computations that are otherwise done on a phone or calculator.
Which statements is true regarding computational estimation?
Offer or accept a range of estimates.
What can a teacher do if a student has difficulty with regrouping, specifically the recording of regrouping in the standard algorithm is reversed:
Have students model using base - ten materials with a place - value mat to carry out the computation.
Which of the following is a true statement about standard algorithms?
In order to use them, students should be required to understand why they work and explain their steps.
The general approach for teaching the subtraction standard algorithm is the same as addition. What statement below would not be a problem when using the standard algorithm for addition?
Exercises with zero.
Cultural differences are evident in algorithms. What will assist teachers in supporting students and responding to families?
Encouraging a variety of algorithms.
All of the following could be examples of invented strategies for obtaining the sum of two-digit numbers EXCEPT:
Using fives, numbers to estimate ( e.g,, to solve 24 +47, think 24 is close to 25 and 47 is closer to 45 so 24 +47 = 25 +45 = 70).
What contributes to students' overall number sense of addition and subtraction?
Strategies that build on decomposing and composing numbers in flexible ways.
Students who have learned this strategy for their "basic facts" can use it effectively with solving problems with multidigit subtraction problems:
Think addition strategy.
Which is one type of reasoning strategies for addition and subtraction?
Jump strategy
What makes it easier to teach the standard algorithm?
The understanding students gain from working with reasoning strategies makes.
Complete the statement, "When creating a classroom environment appropriate for reasoning strategies ...
The teacher should use familiar contexts in story problems to build background and connect to individual students perspectives.
What strategy for computational estimation do you adjust to correct for digits or numbers that were ignored?
Front - End.
There are important things to remember when teaching the standard algorithm. Identify the statement that does not belong:
Requires written record first.
What invented strategy is represented by a student multiplying 58 × 6 by adding 58 + 58 to get 116 and then adding another 116 to get 232 and then adding another 116 to find the product of 348:
Complete number
What are two reasoning strategies for division?
Think multiplication and decomposition.
Representing a product of two factors may depend on the methods that students experienced. What representation of 37 × 5 below would indicate that the student had worked with base-ten?
5 groups of 30 lines and 5 groups of 7 dots.
This model uses a structure that automatically organizes proportionate equal groups and offers a visual demonstration of the commutative and distributive properties:
Area model.
What compensation strategy works when you are multiplying with 5 or 50?
Half - then - double
What invented strategy is just like the standard algorithm except that students always begin with the largest values?
Partitioning
Why would it be beneficial to round up the divisor in a long division problem with two digit divisors?
Underestimate how many can be shared.
Which strategy includes breaks apart the numbers by tens and ones and finds the product of each part (the distributive property). They are then added to find the total product:
Partial products
Number strings are an approach to developing the missing-factor strategy and capitalize on the inverse relationship between multiplication and division. Which equation would be part of a number string for 381 ÷ 72?
5 * 70
Which is an example of the compensation strategy?
27 × 4 is about 30 (27 + 3) × 4 = 120 then subtract out the extra 3 × 4, so 120 - 12 = 108.
One strategy for teaching computational estimation is to ask for information, but not the answer. Which statement below would be an example of asking for information?
it between $400 and $700?
What is the reason why mental calculations estimates are more complex?
They require a deep knowledge of how numbers work.
What statement below describes the number string problem approach for multidigit multiplication?
Encourages the use of known facts and combinations
What is the purpose of using a sidebar chart in multidigit division?
Uses a doubling strategy for considering the reasonableness of an answer.
Identify the statement that represents what might be voiced when using the think multiplication strategy (missing-factor strategy)
What number times seven will be close to three hundred forty-five with less than seven remaining?
What are the two conceptualizations of division?
Partitive and measurement
What is one way an open array is different from the area model?
The dimensions of the open array are not usually drawn to scale and therefore are often not precisely proportional.
What should you do when you notice a student who uses an approach that mimics addition by considering the tens and ones separately?
Go back to base - ten materials.
What is an advisable teaching focus for exploring decimal numbers?
Ten to one multiplicative relationship.
The following decimals are equivalent 0.06 and 0.060. What misconception does a student have who believes that 0.060 is a larger number?
Longer is larger.
What can a teacher do to help students who needs help deciding when zero is important in interpreting the decimal quantity?
Have them model what the values look like, using shading of 10 × 10 grids.
Which question below is an estimation question that would support students estimating a reasonable response for the following task: A farmer fills each jug with 3.7 liters of cider. If you buy 4 jugs, how many liters of cider is that?
Is it more than 12 liters?
Decimal multiplication tends to be poorly understood. What is it that students need to be able to do?
Discover it on their own with models, drawings, and strategies.
Estimation of many percent problems can be done with familiar numbers. Identify the idea that would not support estimation:
Use a calculator to get an exact answer.
Understanding that when decimals are rounded to two places (2.30 and 2.32) there is always another number in between. What is the place in between called?
Destiny
What is a strong visual model used for decimal fractions?
10 × 10 grids.
Understanding where to put the decimal is an issue with multiplication and division of decimals. What method below supports a fuller understanding?
Rewrite decimals in their fractional equivalents.
The main link between fractions, decimals, and percents are:
Physical models
The 10-to-1 relationship extends in two directions. There is never a smallest piece or a largest piece. Complete the statement, "The symmetry is around ..."
The ones place
If students can express fractions and decimals as hundredths, what term can be substituted for the term hundredth?
Percent
For decimal instruction beyond hundredths, what kind of experiences are recommended?
Experiences to help students visualize.