What’s Your Problem?

Ways to Modify Questions

Given limited time Focus on three categories Not the only ones Prompt other methods

Three Ways to Modify Questions

Un-Doing Error Analysis Snap Shot

Examples of the Three Types

A Typical Textbook Item

Solve for x:

x2 9x 36 0

What might it look like ….

as an Un-Doing problem?

Solve for x:

x2 9x 36 0

Un-Doing Example

Write and graph 2 quadratic functions that have zeros x = -12, x = 3

Un-Doing Example

Write and graph a quadratic function that has

a. one x-intercept.b. two x-intercepts.c. no x-intercepts

What might it look like ….

as a snap shot problem?

Solve for x:

x2 9x 36 0

Snap Shot Example

Yesterday in class we solved some equations graphically. What was the equation we were solving below? What was the solution?

Snap Shot Example

Chelsea dropped her homework in a puddle. Help her reconstruct the blurry areas. What question was she answering?

What might it look like ….

as an error analysis problem?

Solve for x:

x2 9x 36 0

Error Analysis Example

Ralph:Factor

Eleanor:Solve:

Sanna: Solve

x2 9x 36 0

(x 12)(x 3) 0

x 12, 3

x2 9x 36

x2 9x 36

(x 12)(x 3)

x2 9x 36 0

x2 9x 36 0x2 9x 36

x2 9x 36

x 3, 6

Error Analysis Example

Diane: Graph

x2 9x 36 0

(x 12)(x 3) 0

x 12, 3

yx2 9x 36

TAKS Item (9th grade 2004)

The area of a rectangle is 3x 2 + 14x + 8, and the width is x + 4. Which expression best describes the rectangle’s length?

3x 22x 4

2x 23x 2

a.

b.

c.

d.

TAKS Item (10th grade 2004)

What are the roots of the function graphed below?

a. (-1, -9) and (0, -8)

b. (0, -4) and (2, 0)

c. (-4, 0) and (2, 0)

d. (0, 2) and (0, -4)

TAKS Item (10th grade 2004)

Which ordered pair represents one of the roots of the function ?

a. (-5/2, 0)

b. (-4, 0)

c. (-5, 0)

d. (-20, 0)

f x 2x2 3x 20

Earlier with Roger

Earlier we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

Earlier with Roger

1. Match the graph with the appropriate table.

2. What are the roots, solutions shown?

Earlier with Roger3. When will Roger and the baseball be at the same height?

4. Equations have ____ or ____, whereas functions have

______ or _______.

Earlier with Roger

5. What is Roger’s new starting height?

Earlier with RogerYesterday we looked at a series of graphs and tables thatmodeled Roger throwing a baseball upward from adownward moving elevator. Based on that work, Rauland Marco answered the following question. Who isright and why?

1. Given the table and graph representing Roger throwinga baseball upward from a downward moving elevator, whatare the roots, solutions shown?

RaulRoots and solutions are the same as zeros.Y1 is zero at x = 0 and in between x = 3and x = 4, so the roots are x = 0 and x isabout 3.5

MarcoA roo t is the solution t o an equa tion sow e are l ooking fo r w here Y1 = Y2 . Inthe table, both Y1 and Y2 have t he sameva lues a t x = 0 and x = 4. On the graph,Y1 and Y2 in ter sect at the se sa me xva lues x = 0 and x = 4.

Earlier with Roger

Bo was absent yesterday. When you started telling him about the Roger-throwing-the-ball-elevator activity, he said, “What’s the big deal? Roots, zeros, solutions, x-intercepts – they are all the same thing.”How do you respond to Bo? A complete answer includes graphs, tables, equations and discussion.

Part 2: As A Class Everyone should have one problem from

the set Discuss the problems in your group. Decide where the items would best fit. Post your problem Gallery walk - do you agree? Choose one to discuss as a group

Discussion

Un-Doing Error Analysis Snap Shot

Advantages and Disadvantages

Grading Conceptual understanding Memorization

Write your own

Choose a TEKS statement Write a typical question to assess it. Write it as an Un-Doing question Write it as an Error Analysis question Write it as a Snap Shot question

Snap Shot Problems

What are two ideas, processes, or representations that students mix up? Juxtapose them and ask which is which.

What part of a large activity can you grab to assess if students got the gist of the large activity?

Un-Doing Problems

Can you start with the answer? Can you start in the middle? Can you change one constraint? Can you start with a different

representation? Ask students to create or invent the

beginning of a problem.

Error Analysis

What are the typical errors that students make?

Pose an incorrect solution Ask students to explain what went wrong. Sometimes show the incorrect process,

sometimes just show the incorrect answer

The Assessment Principle

Assessment should become a routine part of the ongoing classroom activity rather than an interruption.

NCTM’s Principles and Standards for School Mathematics (2000)

Another Example

The following slides begin with a different stem problem based on geometry and scale factors.

A Typical Textbook Item

The length of a rectangle is 8 cm and the width is 6 cm. Find the perimeter of the new rectangle created by when the original width is dilated by a scale factor of 4.

What might it look like ….

as an Un-Doing problem?

The length of a rectangle is 8 cm and the width is 6 cm. Find the perimeter of the new rectangle created by when the original width is dilated by a scale factor of 4.

Un-Doing Example

If the volume is increased by a factor of 8, what is the change in the length of the side of a cube?

Un-Doing Example

Below is Craig’s work. What might have been the question?

16

3

34

8+6=144x3=12

What might it look like ….

as a snap shot problem?

The length of a rectangle is 8 cm and the width is 6 cm. Find the perimeter of the new rectangle created by when the original width is dilated by a scale factor of 4.

Snap Shot Example

Yesterday in class we explored figures formed by dilations. Abby dropped her paper in a puddle. Help her fill in the missing titles and values.

2

2

3

3

2 4 9664216

2

What might it look like ….

as an error analysis problem?

The length of a rectangle is 8 cm and the width is 6 cm. Find the perimeter of the new rectangle created by when the original width is dilated by a scale factor of 4.

Error Analysis Example

The length of rectangle OLDR was enlarged by a scale factor of 3 to create rectangle NEWS. If OLDR has a width of 3 cm and perimeter of 16 cm, what is the area of NEWS?Sandi wrote the following. What do you say to her?

35

L

DR

O

9

15E

WS

N

So, A= 9x15 = 90+45

= 135 cm2

TAKS Item (9th grade 2004)

a. 1/3

b. 1/2

c. 2/7

d. 5

What scale factor was used to transform ∆MNP to ∆RST?

TAKS Item (11th grade 2004)

a. The length is 2 times the original length.

b. The length is 4 times the original length.

c. The length is 6 times the original length.

d. The length is 8 times the original length.

If the surface area of a cube is increased by a factor of 4, what is the change in the length of the sides of the cube?

TAKS Item (11th grade 2004)

A rectangle has a length of 4 feet and a perimeter of 14 feet. What is the perimeter of a similar rectangle with a width of 9 feet?

a. 36 ft.

b. 42 ft.

c. 108 ft.

d. 126 ft.