Question 1

Cluster problems are an approach to developing the missing-factor strategy and capitalize on the inverse relationship between multiplication and division.All of equations below represent clusters that would help solve 381 divided by 72 EXCEPT:

Accepted Answer

Option A (81 x 70) does not directly relate to solving 381 divided by 72 because it involves numbers (81 and 70) that are not closely related to the original division problem. Options B, C, and D involve multiples of 72 or numbers close to 72 (70), which can help in estimating or breaking down the division problem into smaller, more manageable parts.

Question 2

Identify the statement that represents what might be voiced when using the missing-factor strategy.

Accepted Answer

The missing-factor strategy involves determining the factor that is missing from a multiplication problem. In this case, we need to find the number that, when multiplied by 7, will give us a product close to 345 with less than 7 remaining.

Question 3

Division may be easier for students if they are familiar with the concepts.All of the statements below are related to division of whole numbers EXCEPT:

Accepted Answer

Compensating is not directly related to the division of whole numbers. It is more associated with adjusting numbers to make calculations easier, rather than the process of dividing itself. Partitioning, fair sharing, and repeated subtracting are all concepts that directly relate to understanding and performing division.

Question 4

Which of the following is a strategy that is more applicable for multiplying single digits than multidigits?

Accepted Answer

Doubling is a strategy that involves multiplying a number by 2, which can be more applicable for multiplying single digits as it is easier to keep track of the numbers. For example, to multiply 3 by 8, one can use the doubling strategy: 3 doubled is 6, doubled again is 12, and doubled one more time is 24, which is the product of 3 and 8. However, this strategy may not be as efficient when multiplying multidigit numbers.

Question 5

What statement below describes the cluster problem approach for multidigit multiplication?

Accepted Answer

The cluster problem approach for multidigit multiplication encourages the use of known facts and combinations, helping students to break down complex problems into simpler parts that they can solve using their existing knowledge.

Question 6

What compensation strategy works when you are multiplying with 5 or 50?

Accepted Answer

The "half-then-double" strategy is effective when multiplying by 5 or 50 because it simplifies the multiplication. For example, multiplying by 5 can be seen as halving the other number and then doubling the result (since 5 is half of 10). Similarly, for multiplying by 50, you can half the other number and then double it twice (since 50 is half of 100). This method simplifies calculations, especially for large numbers or in mental math.

Question 7

When developing the written record for multiplication computation it is helpful to encourage students to follow these suggestions EXCEPT:

Accepted Answer

The other three options are helpful suggestions for developing the written record for multiplication computation, but marking the subdivisions of the factors is not a necessary step.

Question 8

Identify and discuss methods for supporting students' development of the standard algorithm for division.

Accepted Answer

The answer of Identify and discuss methods for supporting students'...

Question 9

What invented strategy is just like the standard algorithm except that students always begin with the largest values?

Accepted Answer

Partitioning is the invented strategy that is just like the standard algorithm except that students always begin with the largest values.

Question 10

This model uses and a structure that automatically organizes proportionate equal groups and offers a visual demonstration of the commutative and distributive properties.

Accepted Answer

The area model uses a structure that organizes equal groups and visually demonstrates the commutative and distributive properties through the layout of a rectangle.

Question 11

One strategy for teaching computational estimation is to ask for information,but no answer.Which statement below would be an example of NOT gathering information?

Accepted Answer

Option C does not provide any information that can help in the computational estimate. The other options (A, B, and D) provide specific information or a range that can be used to estimate the answer.

Question 12

What is the reason why mental calculations estimates are more complex?

Accepted Answer

Mental calculations estimates are more complex because they require a deep knowledge of how numbers work, including their relationships, properties, and operations. This knowledge allows individuals to break down a complex problem into smaller, manageable parts and apply mental strategies to arrive at an accurate estimate. While knowledge of multiplication and division procedures and partitioning can be helpful, they are not the main reason why mental calculations estimates are more complex.

Question 13

Representing a product of two factors may depend on the methods student experienced.What representation of 37 x 5 below would indicate that the student had worked with base-ten?

Accepted Answer

The answer of Representing a product of two factors may...

Question 14

Which is an example of the compensation strategy?

Accepted Answer

The compensation strategy involves adjusting one number in a problem to make it easier to solve mentally. In this case, adding 3 to 27 to make it 30 makes it easier to multiply by 4. Then, subtracting out the extra 3 × 4 gives the correct answer of 108.

Question 15

Developing the standard algorithm for division teachers should used all of the following guides EXCEPT:

Accepted Answer

The area model is not a standard algorithm for division.

Question 16

An intuitive idea about long division with two digit divisors is to round up the divisor.All of examples below support this idea EXCEPT:

Accepted Answer

Pretending there are fewer sets to share than there really are does not support the idea of rounding up the divisor; it suggests underestimating rather than rounding up.

Question 17

What invented strategy is represented by a student multiplying 58 x 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.

Accepted Answer

The strategy used by the student is the "Complete number" strategy, where the student repeatedly adds the complete number (58) to itself to find the product with 6.

Question 18

A number line can be helpful with teaching this estimation strategy.

Accepted Answer

A number line can be a useful tool for teaching many strategies, including front end estimation and mental computation, but it is particularly helpful with rounding. Using a number line can show students how to round a number to the nearest ten or hundred, for example, by visualizing where the number falls between two marks on the line. This can help students better understand the concept of rounding and how to apply it to estimation.

Question 19

What is the purpose of using a side bar chart in multidigit division?

Accepted Answer

Using a side bar chart in multidigit division helps to use the doubling strategy for determining the reasonableness of an answer. This can be helpful in double-checking the accuracy of the calculation and identifying any potential errors.

Quiz 13: Developing Strategies for Multiplication and Division Computation