Exam 3: Understanding Whole Number Operations

A visitor to a first-grade classroom sees a teacher ask a student to solve this problem: Jaime gets $5 a week for keeping the yard in good shape. He is saving his money for the county fair. After four weeks, how much has he saved? A) The visitor thinks to himself, "This is a multiplication problem, and first-graders have not yet been taught multiplication, so they can't answer this problem." But after a few minutes, the first-grader says that the answer is 20. The student explains how she did this problem, and, much to the visitor's surprise, she did not do any formal multiplication. What did she most likely do to find the answer? B) This visitor also sees the students work another problem: 8 miles of highway are being paved. If the workers pave 2 miles a day, how long will it take them to pave all 8 miles? The visitor thinks, "This is a division problem and first-graders have not yet learned to divide." But then one student says that it would take four days. How do you suppose she explained this answer without using division?

To determine how much older than you your father is, you need to make an additive comparison of his and your ages.

I am a number with 21 tens, 14 ones, and 11 tenths. What number am I?

Make up a story problem about a bake sale so that the problem could be solved by: A) 3/4 × 12. (Notice the order.) B) 12 × 3/4 .

Tell why the following calculations are INCORRECT. A) Sum: 310 225 980 1895 B) 280 ÷ 70 = 40 C) 480 ÷ 0.4 = 120

The product of a number n by any other number m different from 0 is always greater than n.

Is this child's thinking correct? If it is, complete the second calculation using the child's method. If the thinking is not correct, explain why not. (given) (child does) 675 677 -198 -200 477 Second calculation (or explanation if not correct): 453 -295

Amy finds 32 × 54 as follows: 54 is 4 more than 50, so find 32 × 50 and add 4 back to get 1728. A) Which is TRUE of Amy's mathematical steps? i. Amy's steps are probably mathematically correct, although the phrasing is not perfect (should be "add four 32s back"). ii. Amy's steps are mathematically flawed. iii. It cannot be determined whether Amy's steps are mathematically correct or flawed. B) Which is TRUE of Amy's understanding of multiplication? i. Amy doesn't appear to understand multiplication. ii. Amy may or may not understand multiplication. iii. Amy shows good understanding of multiplication.

Of the expressions 0 ÷ 6, 6 ÷ 0, and 0 ÷ 0, which is/are undefined? Explain why any undefined choice(s) is/are undefined. If the expression is defined, tell what it equals.

Use a nonstandard algorithm to calculate 128 × 67.

A student wrote the following answer to the problem 95 - 34: 95 - 34 = 95 - 35 = 60 60 - 1 = 59 So the answer is 59 Analyze this student's thinking.

Write a missing-addend word problem using $35.95 and $19.50.

A bill for school supplies was $87.35. Josh paid with two $50 bills. Rikki, at the cash register (one that did not tell the change to be given to the buyer), counted Josh's change: "40, 50, $1, and $10 makes $100." How much change did Josh receive? In what currency? Is that what he should have received?

Following is only the start of a child's work (in base ten). What seems lacking in this child's understanding? (Assume, perhaps unfairly, that the child is not thinking of negative numbers.) 402 \ldots7 506 \ldots3

In each scenario, which way of thinking about subtraction is involved? A) Story problem: Basketball score: Aztecs 82, Opponents 69. By how many points did the Aztecs win? B) Thinking/drawing strategy (sometimes used with children having trouble with their basic subtraction facts): For 15 - 7, think of going "up the hill," going to 10 along the way…

Find 21 + 49 using an empty number line.

Following is an example of a child's work. Study the work and then judge the student's understanding. Hiro was asked to divide 4240 by 6. His work is shown below. Hiro's work: A) Is Hiro's work correct or incorrect? B) If the work is incorrect, please explain how.

Give the rest of the "family of facts" for k - 3 = p.

In which word problem (A or B) is there an action involved that may make it easier for young children to solve?

Mitchell decides to get his car painted and to buy new hubcaps. He selects five colors he likes and three styles of hubcaps. Then he decides to paint the roof a different color than the body. He decides to let his wife make the final decision. How many choices does she have?