Using Questions to Promote Algebraic Thinking: (being able to think
about functions and how they work and to think about the impact that a
system’s structure has on calculations)
Doing/undoing – reversibility
EX: Solving equations like 9x2 – 16 = 0. Vs What equation has
solutions 4/3 and –4/3?
How does this process work in reverse?
Building rules to Represent Functions
Recognizing patterns and organize data to represent situations in
which input is related to output by well-defined functional rules.
How are things changing in this situation?
Abstracting from Computation.
Thinking about computations independently of particular numbers
that are used.
What are my operation shortcut options to get from here to there?
Role of Classroom questions:
Consistent modeling of algebraic thinking – make explicit what students
have left implicit in their thinking.
Give well timed pointers to students that help them shift or expand their
thinking or that help them pay attention to what is important.
Making it a habit to ask a variety of questions aimed at helping students
organize their thinking and respond to algebraic prompts.
Types:
Managing: Intended to help set students on task, get their work organized.
Clarifying: Request info from student when it’s not clear about what the
student means or intends – helps understand the questions
Orienting: get students started, keep them thinking about the problem,
motivate to the correct answer.
Prompting mathematical reflection: explain their thinking, extend their
thinking about the mathematics
Eliciting Algebraic Thinking:
Intended to ask the student to undo, to build rules for describing functional
relationships; to abstract from computations they have made to ask about the
meaning of the work they are doing; to ask about what statements are
“always” true, about nth terms, and about finding patterns and looking for
what changes; to work forward and backward, and to ask students to justify
generalizations
Describe the different ways learners approach the following two problems:
1.
Debbie bought 36 apples at 25 cents each. She ate 2 of them, and
she sold the rest for 32 cents each. What was her profit?
2.
Debbie bought some apples at 25 cents each. She ate 2 of them,
and she sold the rest for 32 cents each. If her profit was $1.88, how
many apples did she buy?
The first, a typical pre-algebra word problem, can be done with a
straightforward string of computational actions: subtract 2 from 36, multiply
the answer by 32, from that number subtract the product of 25 and 36.
Students often think of each of these computation steps as its own
independent action and do not think about the string of steps as a whole
entity that takes inputs and yields output.
By contrast, the second, typical of algebra, students are given output and
then asked to find input. This kind of challenge demands a different way of
thinking about and dealing with the calculation used: to map back from
output to input, students must think in terms of functions that yield output
from input and they must find a way to package calculations together to
form the appropriate function. In this case the function takes given input,
subtracts 2 from it, multiplies the result by 32, and then subtracts from this
result the result of multiplying the input by 25. Knowing the output is 188
allows one to reason back to the input. In essence, this is a significant shift
because it demands that students start thinking of a computation-based
algorithm not as a string of independent actions, but as a process, an entity
unto itself.
Golden Apples
A prince picked a basketful of golden apples in the enchanted orchard. On
his way home, he was stopped by a troll who guarded the orchard. The troll
demanded payment of one-half of the apples plus two more. The prince gave
him the apples and set off again. A little further on, he was stopped by a
second troll guard. This troll demanded payment of one-half of the apples
the prince now had plus two more. The prince paid him and set off again.
Just before leaving the enchanted orchard, a third troll stopped him and
demanded one-half of his remaining apples plus two more. The prince paid
him and sadly went home. He had only two golden apples left. How many
apples had he picked?
Before reading on, work through the problem. Pay attention to how you are
thinking about it – where you start, planned steps, and so on.
Guess & Check – how can you extend this to algebraic thinking?
Is any starting number as good as any other?
Some starting numbers lead to answers that are not whole numbers. Which
are these.
What information about the numbers in the problem can help in choosing a
starting number that leads to a whole number answer? In fact, what are
starting numbers that lead to whole numbers? What does that tell you?
What would make sense to try? Using 60 as a starting number would leave 4
at the end. We want an output of 2. Do we try a number larger than 60 or
smaller? Why?
Is there information here that lets me predict what’s going to happen?
How are things changing?
When I do the same thing with different numbers, what still holds true?
What changes?
Can I write down a mechanical rule that will do this job once and for all?
How can I describe the steps without using specific inputs?
Extensions:
Suppose there were 6 apples left, how would the problem change?
Sneaking Up the Line
Eric the Sheep is at the end of a line of 50 sheep waiting to be shorn. But
being an impatient sort of a sheep, Eric sneaks up the line two places every
time the shearer takes a sheep from the front to be shorn. So, for example,
while the first sheep is being shorn, Eric moves ahead so that there are two
sheep behind him in the line. If at some point it is possible for Eric to move
only one place, he does that instead of moving ahead two places. How many
sheep get shorn before Eric?
1. Predict a solution “in your head”
2. Use manipulatives or anything else to solve these problems:
Number in front of Eric
Number shorn before Eric
6
11
3. What pattern did you notice that could help you solve the original
problem in which Eric is at the end of the line of 50 sheep.
4. Explore the result with other numbers and create a table.
5. Describe how you could work out the answer for any number of
sheep.
6. Work out the solutions for the boxes left blank.
Number in front of Eric
Number shorn before Eric
37
296
1000
7695
13
21
7. Eric gets more (and more) impatient! Explore how your rule changes
if Eric sneaks past 3 at a time, 4 at a time, or even 10 at a time. As
soon as someone tells you how many sheep are in front of Eric and
how many he sneaks past at a time, describe how you could work out
the answer.
8. Extension: What if the farmer hires another shearer? Suppose there is
still one line, but the first and second sheep get shorn at the same time,
and then Eric sneaks ahead. What does this do to your rule?
DAILY PLAN
date
course
A.
Student Objective(s)
B.
Opening Activity/Purpose
hour
(To involve all students. Purpose might be to find readiness level for
developmental activities.)
C.
Developmental Activities
(Cues, reinforcement, practice, feedback, involvement.)
D.
Closing Activity
(To involve all students. Purpose is to identify students’ success with the daily
objectives.)
E.
Assignment/Purpose
F.
Activity for the last five minutes/Purpose