Question 1

Show, using 324 ÷ 28, how to work from the scaffolding algorithm to the standard algorithm.

Accepted Answer

Similar to question 1. Students' work should show an awareness of the scaffolding algorithm format to an abbreviated form to the usual US form.

Question 2

You decide to introduce your fourth-graders to the long-division algorithm with one-digit divisors, using base-ten materials and 96 ÷ 3. You also want to use a story problem that they would find interesting as the basis for their work.
A) From the practical standpoint of acting out the calculation, which way of thinking about division-repeated subtraction or sharing equally-should you use in your story problem for 96 ÷ 3?
B) Write such a story problem (involving 96 ÷ 3).
C) Show how you would act out your story problem, with drawings of the base-ten materials.
D) Write a second story problem for 96 ÷ 3, involving a different way of thinking about division from the way of thinking in your story problem in part
E) Answer parts A-D if you were to use 960 ÷ 320 instead of 96 ÷ 3.

Accepted Answer

A) Sharing equally is more practical for 96 ÷ 3 if you start with nine longs and three small cubes because acting out that calculation with repeated subtraction of 3s would be unwieldy.
B) Various. The 96 should be put into three (equal-sized) amounts or groups.
C) With nine longs and six small cubes, "deal" them out to three locations equally. You would start with the longs to illustrate the usual algorithm.
D) Various. This time the situation should call for how many groups of size 3 are in, or make, a group of 96. Notice that the units for the 3 and the 96 should be the same.
E) For 960 ÷ 320, repeated subtraction is much more practical, and so on.

Question 3

Write a word problem that would require solving 540 ÷ 4.

Accepted Answer

Example: A grandmother had four grandchildren. She had $540 to give as Christmas gifts to the children, who all received the same amount. How much did each grandchild receive?

Question 4

A student places multi-base blocks on the table as follows:   
Write which calculation the student might be doing. A. $ 226 + 49$
B. $226 - 49$
C. $226 - 134$
D. $226 - 118$

Accepted Answer

The answer of A student places multi-base blocks on the...

Question 5

Name two positive and two negative aspects of learning nonstandard algorithms.

Accepted Answer

The answer of Name two positive and two negative aspects...

Question 6

Show 3335 ÷ 23 with a scaffolding algorithm and then by the standard algorithm. Show how each number in the standard algorithm is associated with each number in the scaffolding algorithm.

Accepted Answer

The answer of Show 3335 ÷ 23 with a scaffolding...

Question 7

The Russian peasant method for multiplying uses two basic processes: doubling and _____.

Accepted Answer

The answer of The Russian peasant method for multiplying uses...

Question 8

Use the scaffolding method to compute 5883 ÷ 17.

Accepted Answer

The answer of Use the scaffolding method to compute 5883...

Question 9

Draw how one would act out 200 - 62 (comparison view) to support the usual right-left algorithm, with base-ten materials. (Make clear what represents 1.) Make a separate drawing for each step (add steps if you need them).

Accepted Answer

The answer of Draw how one would act out 200...

Question 10

What would the NEXT line be in a Russian peasant calculation of 23 × 621?
You do not have to do the complete algorithm.
23 × 621

Accepted Answer

A)  11 × 1242 
B)  46 × 310 
C)  23 × 21 
D)  11 × 1242 or possibly 46 × 310 
E)  46 × 310 or possibly 23 × 21 
A)  11 × 1242 
B)  46 × 310 
C)  23 × 21 
D)  11 × 1242 or possibly 46 × 310 
E)  46 × 310 or possibly 23 × 21

Question 11

Consider this arithmetic problem:

A) Write a story problem where the answer would be 6.
B) Write a story problem where the answer would be 7.
C) Write a story problem where the answer would be 1.
D) Write a story problem where the answer would be 6$1 / 4$

Accepted Answer

A) Possible: Jake was buying school supp

Question 12

Make a drawing of base-ten materials that shows the initial setup for 3 × 130.2. Make clear what = 1. Do not take time to draw all the later steps of the calculation with the base ten materials.

Accepted Answer

The answer of Make a drawing of base-ten materials that...

Question 13

Do 32 × 467 using the method of writing all partial products.

Accepted Answer

The answer of Do 32 × 467 using the method...

Question 14

A student's work on a base-ten subtraction problem is shown below. How much does the little 1 above the 5 represent?

Accepted Answer

The answer of A student's work on a base-ten subtraction...

Question 15

Draw how one would act out 200 - 62 (take-away view) to support the usual right-left algorithm, with base-ten materials. (Make clear what represents 1.) Make a separate drawing for each step (add steps if you need them).

Accepted Answer

The answer of Draw how one would act out 200...

Question 16

Use a nonstandard algorithm to find 240five + 314five, but showing all partial sums.

Accepted Answer

This work is all done in base

Question 17

Below is a worked-out calculation of 313 × 42, using the lattice method for multiplication. Explain why the method does give the correct number in the tens place (the circled 4). (Note: Some current textbooks use this algorithm to teach multiplication of whole numbers.)

Accepted Answer

The answer of Below is a worked-out calculation of 313...

Exam 4: Some Conventional Ways of Computing

Show, using 324 ÷ 28, how to work from the scaffolding algorithm to the standard algorithm.

Write a word problem that would require solving 540 ÷ 4.

A student places multi-base blocks on the table as follows: Write which calculation the student might be doing. A. 226+49 226 + 49226+49 B. 226−49226 - 49226−49 C. 226−134226 - 134226−134 D. 226−118226 - 118226−118

Name two positive and two negative aspects of learning nonstandard algorithms.

Show 3335 ÷ 23 with a scaffolding algorithm and then by the standard algorithm. Show how each number in the standard algorithm is associated with each number in the scaffolding algorithm.

The Russian peasant method for multiplying uses two basic processes: doubling and _____.

Use the scaffolding method to compute 5883 ÷ 17.

Draw how one would act out 200 - 62 (comparison view) to support the usual right-left algorithm, with base-ten materials. (Make clear what represents 1.) Make a separate drawing for each step (add steps if you need them).

What would the NEXT line be in a Russian peasant calculation of 23 × 621? You do not have to do the complete algorithm. 23 × 621

Consider this arithmetic problem: A) Write a story problem where the answer would be 6. B) Write a story problem where the answer would be 7. C) Write a story problem where the answer would be 1. D) Write a story problem where the answer would be 6 1/41 / 41/4

Make a drawing of base-ten materials that shows the initial setup for 3 × 130.2. Make clear what = 1. Do not take time to draw all the later steps of the calculation with the base ten materials.

Do 32 × 467 using the method of writing all partial products.

A student's work on a base-ten subtraction problem is shown below. How much does the little 1 above the 5 represent?

Draw how one would act out 200 - 62 (take-away view) to support the usual right-left algorithm, with base-ten materials. (Make clear what represents 1.) Make a separate drawing for each step (add steps if you need them).

Use a nonstandard algorithm to find 240five + 314five, but showing all partial sums.

Below is a worked-out calculation of 313 × 42, using the lattice method for multiplication. Explain why the method does give the correct number in the tens place (the circled 4). (Note: Some current textbooks use this algorithm to teach multiplication of whole numbers.)

Reasoning About Quantities

Numeration Systems

Understanding Whole Number Operations

Using Numbers in Sensible Ways

Meanings for Fractions

Computing With Fractions

Multiplicative Comparisons and Multiplicative Reasoning

Ratios, Rates, Proportions, and Percents

Integers and Other Number Systems

Number Theory

What Is Algebra

A Quantitative Approach to Algebra and Graphing

Understanding Change: Relationships Among Time, Distance, and Rate

Further Topics in Algebra and Change

Polygons

Polyhedra

Symmetry

Tessellations

Similarity

Curves, Constructions, and Curved Surfaces

Transformation Geometry

Measurement Basics

Area, Surface Area, and Volume

Counting Units Fast: Measurement Formulas

Special Topics in Measurement

Quantifying Uncertainty

Determining More Complicated Probabilities

Introduction to Statistics and Sampling

Representing and Interpreting Data With One Variable

Dealing With Multiple Data Sets or With Multiple Variables

Variability in Samples

Special Topics in Probability

Filters

A student places multi-base blocks on the table as follows: Write which calculation the student might be doing. A. $226 + 49$ B. $226 - 49$ C. $226 - 134$ D. $226 - 118$

Consider this arithmetic problem: A) Write a story problem where the answer would be 6. B) Write a story problem where the answer would be 7. C) Write a story problem where the answer would be 1. D) Write a story problem where the answer would be 6 $1 / 4$

Use a nonstandard algorithm to find 240_five + 314_five, but showing all partial sums.