Good Questions for Math Teaching

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Sharp Tutor Mon, Jan 10, 2022 09:41 PM UTC

Good Questions for Math Teaching looks in more detail at a particular type of open question that we call a “good” question. Our goals of education are for our students to think, to learn, to analyze, to criticize, and to be able to solve unfamiliar problems, and it follows that good questions should be part of the instructional repertoire of all teachers of mathematics. In this book, we describe the features of good questions, show how to create good questions, give some practical ideas for using them in your classroom, and provide many good questions that you can use in your mathematics program.

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2. Good Q uestions for Math Teaching
•
Why Ask Them and What to Ask K–
Peter Sullivan
Pat Lilburn
Math Solutions Publications
Sausalito CA

3. A Message from Marilyn Burns
We at Math Solutions Professional Development believe that teaching math
well calls for increasing our understanding of the math we teach, seeking
deeper insights into how children learn mathematics, and refining our
lessons to best promote students’ learning.
Math Solutions Publications shares classroom-tested lessons and teaching
expertise from our faculty of Math Solutions Inservice instructors as well as
from other respected math educators. Our publications are part of the
nationwide effort we’ve made since 1984 that now includes
• more than five hundred face-to-face inservice programs each year for
teachers and administrators in districts across the country;
• annually publishing professional development books, now totaling more
than fifty titles and spanning the teaching of all math topics in kinder-
garten through grade 8;
• four series of videotapes for teachers, plus a videotape for parents, that
show math lessons taught in actual classrooms;
• on-site visits to schools to help refine teaching strategies and assess
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please visit our website at www.mathsolutions.com. You can also contact us to
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P U B L I C AT I O N S

5. C ontents
vii Acknowledgments
1 PART ONE: THE IMPORTANCE OF QUESTIONING
3 What Are Good Questions?
7 How to Create Good Questions
11 Using Good Questions in Your Classroom
17 PART TWO: GOOD QUESTIONS TO USE IN MATH LESSONS
19 Number
20 Money
25 Fractions
29 Decimals
33 Place Value
36 Counting and Ordering
40 Operations
47 Measurement
48 Weight
51 Volume and Capacity
56 Area
61 Time
66 Length and Perimeter
73 Space
74 Location and Position
77 Two-Dimensional Shapes
84 Three-Dimensional Shapes
91 Chance and Data
92 Chance
97 Data

7. A cknowledgments
The idea of open-ended or good questions developed over several years during
ongoing discussions between Peter Sullivan and David Clarke.
Peter and David conducted a number of research studies and classroom
trials of open-ended questions. Many of David’s initial ideas are used in various
places throughout this book. His creativity, energy, and interest in exploring
good questions contributed significantly to the idea of using open-ended activ-
ities in the teaching of mathematics, and for this we thank him.
The idea went through a number of phases before it reached its final form.
Pam Rawson’s contribution to the early planning stages, which ultimately led to
the development of this resource, is greatly appreciated.
Finally, we thank Sheryl and Mike without whose continued support
there would be no book.
PETER SULLIVAN
PAT LILBURN
viii■ ❚ ❙

9. PART O NE
•
The Importance of Questioning
During the course of a normal school day teachers ask many questions. In fact,
something like 60 percent of the things said by teachers are questions and most
of these are not planned.
One way of categorizing questions is to describe them as either open or
closed. Closed questions are those that simply require an answer or a response to
be given from memory, such as a description of a situation or object or the
reproduction of a skill. Open questions are those that require a student to think
more deeply and to give a response that involves more than recalling a fact or
reproducing a skill.
Teachers are usually skilled at asking open questions in content areas such
as language arts or social studies. For example, teachers often ask children to
interpret situations or justify opinions. However, in mathematics lessons closed
questions are much more common.
Questions that encourage students to do more than recall known facts
have the potential to stimulate thinking and reasoning. To emphasize problem
solving, application, and the development of a variety of thinking skills it is vital
that we pay more attention to improving our questioning in mathematics les-
sons. Teachers should use questions that develop their students’ higher levels of
thinking.
Good Questions for Math Teaching looks in more detail at a particular type
of open question that we call a “good” question. Our goals of education are for
our students to think, to learn, to analyze, to criticize, and to be able to solve
unfamiliar problems, and it follows that good questions should be part of the
instructional repertoire of all teachers of mathematics.
1i■ ❚ ❙

10. In this book we describe the features of good questions, show how to
create good questions, give some practical ideas for using them in your class-
room, and provide many good questions that you can use in your mathe-
matics program.
2i■ ❚ ❙ Good Questions for Math Teaching

11. 1 What Are Good Questions?
•
Let us have a closer look at what makes a good question. There are three main
features of good questions.
■ They require more than remembering a fact or reproducing a skill.
■ Students can learn by answering the questions, and the teacher learns
about each student from the attempt.
■ There may be several acceptable answers.
This section explains these features in more detail.
More Than Remembering
A particular grade 6 student, Jane, had just finished a unit on measurement
where she had been asked to calculate area and perimeter from diagrams of rec-
tangles with the dimensions marked. She was able to complete these correctly,
and the teacher assumed from this that Jane understood the concepts of area and
perimeter. However, when she was asked the following good question she
claimed that she could not do it because there was not enough information
given. I want to make a garden in the shape of a rectangle. I have 30 meters of fence
for my garden. What might be the area of the garden?
To find an answer to this Jane needed to think about the constraints that
a perimeter of 30 meters places on the lengths of the sides of the rectangle, as
well as thinking about the area. She needed to use higher order reasoning skills
since she had to consider the relationship of area and perimeter to find possible
whole number answers that could range from 14 x 1 (14m2) to 7 x 8 (56m2).
This certainly required her to do more than remember a fact or reproduce a skill.
It required comprehension of the task, application of the concepts and appro-
3i■ ❚ ❙

12. priate skills, and analysis and some synthesis of the two major concepts
involved.
Through further probing, this question allowed the teacher to see that
Jane had little appreciation of perimeter as the distance around a region, and no
concept of area as covering. She had learned to answer routine exercises without
fully understanding the concepts.
Another example of closed questions commonly found in textbooks is
from the topic of averages. A typical question looks something like What is the
average of 6, 7, 5, 8, and 4? This mainly requires students to recall a technique.
That is, add the numbers and divide by how many there are—in this case five.
However, if this question was rephrased in the form of a good question it would
look something like The average of five numbers is 6. What might the numbers be?
or After five games, the goalie had averaged blocking six goals per game. What might
be the number of goals he blocked in each game?
These questions require a different level of thinking and a different type of
understanding of the topic of averages to be able to give an answer. Students
need to comprehend and analyze the task. They must have a clear idea of the
concept of average and either use the principle that the scores are evenly placed
about the average or that the total of the scores is 30 (that is, 5 x 6) as the basis
of their response. It most definitely requires more than remembering.
Students Learn By Answering the Question
and Teachers Learn from the Students’ Attempts
Good questions are particularly suitable for this because they have the potential
to make children more aware of what they do know and what they do not know.
That is, students can become aware of where their understanding is incomplete.
The earlier question about area and perimeter showed that by thinking about
area and perimeter together the student is made aware of the fact that the area
can change even though the perimeter is fixed. The very act of trying to com-
plete the question can help children gain a better understanding of the concepts
involved. The manner in which some children went about answering the fol-
lowing question illustrates this point.
John and Maria each measured the length of the basketball court.
John said that it was 25 yardsticks long, and Maria said that it was
24 yardsticks long. How could this happen?
Some fifth- and sixth-grade students were asked to discuss this question in
groups. They suggested a variety of plausible explanations and were then asked
4i■ ❚ ❙ Good Questions for Math Teaching

13. to suggest what they need to think about when measuring length. Their list
included the need to:
■ agree on levels of accuracy
■ agree on where to start and finish, and the importance of starting at
the zero on the yardstick
■ avoid overlap at the ends of the yardsticks
■ avoid spaces between the yardsticks
■ measure the shortest distance in a straight line.
By answering the question the students established for themselves these
essential aspects of measurement, and thus learned by doing the task.
As we have discussed, the way students respond to good questions can also
show the teacher if they understand the concept and can give a clear indication
of where further work is needed. If Jane’s teacher had not presented her with the
good question she would have thought Jane totally understood the concepts of
area and perimeter. In the above example, the teacher could see that the children
did understand how to use an instrument to measure accurately. Thus we can
see that good questions are useful as assessment tools, too.
Several Acceptable Answers
Many of the questions teachers ask, especially during mathematics lessons, have
only one correct answer. Such questions are perfectly acceptable, but there are
many other questions that have more than one possible answer and teachers
should make a point of asking these, too. Each of the good questions that we
have already looked at has several possible answers. Because of this, these ques-
tions foster higher level thinking because they encourage students to develop
their problem-solving expertise at the same time as they are acquiring mathe-
matical skills.
There are different levels of sophistication at which individual students
might respond. It is characteristic of such good questions that each student can
make a valid response that reflects the extent of their understanding. Since cor-
rect answers can be given at a number of levels, such tasks are particularly appro-
priate for mixed ability classes. Students who respond quickly at a superficial
level can be asked to look for alternative or more general solutions. Other stu-
dents will recognize these alternatives and search for a general solution.
If we think back to the earlier question on the area of the garden, there is
a range of acceptable whole number answers (14 x 1, 13 x 2, 12 x 3 . . . 8 x 7).
Students could be asked to find the largest or smallest garden possible. They
5i■ ❚ ❙ What Are Good Questions?

14. could be asked to describe all possible rectangles. Other students will be inter-
ested in exploring answers other than those that involve only whole numbers,
for example, 12.5m x 2.5m. It is the openness of the task that provides this rich-
ness. The existence of several acceptable answers stimulates the higher level
thinking and the problem solving.
In this section, we have looked more closely at the three features that categorize
good questions. We have seen that the quality of learning is related both to the
tasks given to students and to the quality of questions the teacher asks. Students
can learn mathematics better if they work on questions or tasks that require
more than recall of information, and from which they can learn by the act of
answering the question, and that allow for a range of possible answers.
Good questions possess these features and therefore should be regarded as
an important teaching tool for teachers to develop. The next section shows two
ways to construct your own good questions.
6i■ ❚ ❙ Good Questions for Math Teaching

15. How to Create
2 Good Questions
•
Good questions can be used as the basis for an entire lesson either as a lesson that
stands alone or as part of a unit of work. It is possible to make up your own good
questions for any topic and any grade level. The important thing is to plan the ques-
tions in advance, as creating them is not something that can be done on your feet.
When you first start using good questions you might find helpful the collec-
tion of questions in Part Two, “Good Questions to Use in Math Lessons.” After
awhile you will want to create good questions for yourself. Detailed on pages 7
and 8 are two methods that can be used to construct good questions. The one
you use is a matter of personal preference.
Method : Working Backward
This is a three-step process.
Step 1: Identify a topic.
Step 2: Think of a closed question and write down the answer.
Step 3: Make up a question that includes (or addresses) the answer.
For example:
Step 1: The topic for tomorrow is averages.
7i■ ❚ ❙

16. Step 2: The closed question might be The children in the Smith family are
aged 3, 8, 9, 10, and 15. What is their average age? The answer is
9.
Step 3: The good question could be There are five children in a family.
Their average age is 9. How old might the children be?
STEP STEP STEP
Identify a topic Think of an answer Make up a question that includes the
answer
rounding 11.7 My coach said that I ran 100 yards in about
12 seconds. What might the numbers on the
stopwatch have been?
counting 4 chairs I counted something in our room. There were
exactly 4. What might I have counted?
area 6cm2 How many triangles can you draw each with
an area of 6cm2?
fractions 3 Two numbers are multiplied to give 3 . What
might the numbers be?
money 35 cents I bought some things at a supermarket and
got 35 cents change. What did I buy and how
much did each item cost?
graphing x What could this be the graph of?
x x x
x x x
x x x x x
x x x x x
1 2 3 4 5
Some more examples of how this works are shown in the following table.
Method : Adapting a Standard Question
This is also a three-step process.
Step 1: Identify a topic.
Step 2: Think of a standard question.
Step 3: Adapt it to make a good question.
8i■ ❚ ❙ Good Questions for Math Teaching

17. For example:
Step 1: The topic for tomorrow is measuring length using nonstandard
units.
Step 2: A typical exercise might be What is the length of your table meas-
ured in handspans?
Step 3: The good question could be Can you find an object that is three
handspans long?
Some more examples of how this works are shown in the following table.
STEP STEP STEP
Identify a topic Think of an standard Adapt it to make a good question
question
space What is a square? How many things can you write about this
square?
addition 337 + 456 = On a train trip I was working out some dis-
tances. I spilt some soft drink on my paper
and some numbers disappeared. My paper
looked like
3?7
+??6
7 9?
What might the missing numbers be?
subtraction 731 – 256 = Arrange the digits so that the difference is
between 100 and 200.
time What is the time What is your favorite time of day?
shown on this clock? Show it on a clock.
The more experience you have with good questions the more you will
want to use them, and the easier it will become for you to make up your own.
Refer to either or both of these methods until you feel confident.
9i■ ❚ ❙ How to Create Good Questions

18.

19. Using Good Questions
3 in Your Classroom
•
Today’s mathematics classrooms should be dynamic places where children are
involved and engaged in their own learning. This can be achieved through activ-
ities that promote higher level thinking, cooperative problem solving, and com-
munication.
We have seen that good questions support these activities and are readily
available for teachers to use. The first part of this section describes generally how
to use a good question as the basis of a mathematics lesson. It sets out the impor-
tant steps of the lesson, explains the roles of the teacher and students, and
advises how to overcome problems that could arise at each stage. The second
part of this section takes you through each of the steps with a specific good ques-
tion.
Before the start of a lesson it is necessary to choose or create a good ques-
tion. This should be aimed at the appropriate level for the children in your class.
At first you might find the question you choose is too easy or too difficult, but
keep practicing because you will soon get the hang of it. Once you have chosen
the question then the following steps should help you to use it with your class.
Step : Pose the Good Question
It is a good idea to have the question written on the blackboard and as you ask
11i■ ❚ ❙

20. the question refer to the words on the board. It is very important to make sure
that all children know what the question is; do not assume they know it because
it is on the board. You could even ask some students to repeat the question in
their own words.
Allow some time for children to ask you about the meaning of the task.
Explain the task to them if necessary but do not give any directions or sugges-
tions on how to do it. This is for the children to work out for themselves.
Step : Students Work on the Good Question
When first using good questions in your classroom it is better to let the children
work in pairs or small groups. This allows them to communicate their ideas to oth-
ers. This communication is an important part of learning. Working together can
also assist those children who may have difficulty starting. If these children have
to wait for the teacher then organizational and attitudinal problems can arise.
If, once children start working, there are too many who cannot make
progress without teacher assistance then it might be necessary to stop and
have a whole class discussion to overcome the general concerns. If the con-
cerns of each group, or individuals within each group, are all different then
this is a sign that the question you have posed is too difficult for the class.
If this happens either make the question easier or suggest that the students
represent the problem in some way, such as by using materials or drawing a
diagram. A variety of concrete materials should always be available for chil-
dren to select from. You could also decide to abandon the question alto-
gether as unsuitable at this stage. If this happens do not worry, as it takes
time and practice to choose appropriate good questions. However, you will
find good questions to be worth the effort and perseverance. Ideally, you
should plan in advance how to help children who may not be able to start
on the question.
Once the groups are working, your task is to monitor their progress. If a
group stops after giving one response, ask them to look for other possible
answers. If they have found all possible answers ask them to describe all their
answers. In this way they can experience the meaning of a general solution. You
could also ask a related question to extend them. For example, a related ques-
tion for the task The stopwatch shows tenths of seconds. My coach said that I ran
100 yards in about 12 seconds. What might the numbers on the stopwatch have
been? could be What if the stopwatch showed hundredths?
It is not vital that you wait until all groups have finished the task before
initiating a discussion. They will all have answered the question to a degree. It
12i■ ❚ ❙ Good Questions for Math Teaching

21. is better to stop while students are still engaged with the question and interested
in the task. This way they do not become distracted or need to be given addi-
tional work of a different type. You could give a five minute warning before you
stop groups so they have time to tie up the loose ends.
Step : Whole Class Discussion
This is an important phase. Ask the pairs or groups in turn to suggest responses
and to explain their thinking. As each does this write their responses on the
blackboard or, if this is not appropriate, display their model or diagram, mak-
ing sure to give each group equal status. If a response is not suitable be sup-
portive, but try to find out the cause of the error. As we saw earlier, good ques-
tions can often make it easier for teachers to pinpoint exactly where their stu-
dents are experiencing difficulty. Also, as students are explaining what they have
done they often see the error for themselves anyway.
Step : Teacher Summary
Usually, if the task is at an appropriate level, some of the students will make the
main teaching points for you during the class review. Nevertheless, just because
one or more students give a response does not mean that they all understand.
Thus it is necessary to summarize the discussion for everyone, emphasizing and
explaining key points. Wherever possible do this using models and teaching
aids. Because different people learn in different ways we need to use as wide a
range of methods and materials as possible to model a situation. Also, make sure
you relate the answers back to the task children have been working on so that
the discussion remains meaningful. It is also helpful to pose more questions
using a similar format so that the students can apply what they have learned to
new situations.
An Example of a Good Question
Now let us have a look at how these steps would apply to the following good
question.
Two-fifths ( ) of the students in a school borrow books from the
library each day. How many students might there be in the school
and how many of them borrow books each day?
13i■ ❚ ❙ Using Good Questions in Your Classroom

22. Step : Pose the Good Question
Have the question written on the blackboard and as you ask the question refer
to the words on the board. Ask some students to read the question out loud and
ask others to tell you what it means in their own words. Let children ask you
any questions they may have. Explain the task to them if necessary but do not
give any directions or suggestions on how to do the task. This is for the children
to work out for themselves.
Step : Students Work on the Good Question
Organize the children to work in pairs or groups. Once they start working check
that they are able to continue without teacher assistance. If necessary stop them
and have a whole class discussion to overcome any general concerns. If most of
the groups are finding it difficult you could make the question easier by chang-
ing the fraction to a unit fraction such as , , or , or suggest that the groups use
counters to represent the school children. If only one or two groups are finding
it difficult let them start on an easier related fraction such as , and when they
understand this extend it to .
Monitor the progress of the groups. If a group stops after giving one
response, ask them to look for other possible answers. If they have found a few
answers you could ask them to think of a way to describe all their answers. For
example, they could look for a pattern or a rule. You could also give a related task
to extend them such as Find the pattern if of the students borrow books each day.
When all groups have at least one response to the question give them a five
minute warning and after this time stop all students. Do not be concerned that
groups are at different stages.
Step : Whole Class Discussion
Ask the groups in turn to present their responses to the class. Some groups may
want to use the counters to show their responses. Remember that students can
respond at a variety of levels. For example, some possible responses are:
■ It could be anything.
■ One hundred students, forty of whom borrow books each day.
■ The number of students in the school is a multiple of 5, such as 5,
10, 15, 20, and so on, and the number borrowing books would then
be 2, 4, 6, 8, and so on.
These three responses differ not only in the level of mathematical under-
standing but also in the quality of thinking that is demonstrated by the answers.
14i■ ❚ ❙ Good Questions for Math Teaching

23. Try to take a positive approach to each group’s response. For example, if the first
response is given you could agree with the group and then ask them if they can
give a specific answer. The group who gave the third response could be asked to
demonstrate it using counters if they have not already done so.
Step : Teacher Summary
The main points from the activity are the pattern that emerges (2:5, 4:10, 6:15,
and so on), and the use of fractions as operators (for example, of 10). Even if
these points have been discussed it is important to go over them again. It would
also be helpful to ask children to suggest how they would calculate of certain
amounts and let them demonstrate using materials. You could also look at what
happens to the answer when the amount is not a multiple of 5. As you are sum-
marizing do not lose sight of the original question. Refer to it when necessary to
make a point.
A similar task that you could pose is In a survey I found that of the people
liked Michael Jordan. How many people did I ask, and how many liked Michael
Thus we can see that using good questions in your classroom requires a
different lesson format from a lesson in which the teacher demonstrates a tech-
nique or skill and follows up with student practice. It places different demands
on a teacher, too. As well as being receptive to all students’ responses, the teacher
must acknowledge the validity of the various responses while making clear any
limitations, drawing out contradictions or misconceptions, and building class
discussion from partial answers. We have seen how good questions provide the
environment for better learning; it is up to the teacher to ensure that the oppor-
tunities for learning become realities.
15i■ ❚ ❙ Using Good Questions in Your Classroom

24.

25. PART T WO
•
Good Questions
to Use in Math Lessons
This section contains many good questions for you to select from and use in
your classroom.
Questions are presented for sixteen mathematics topics in the areas of
number, measurement, geometry, and chance and data. The questions for each
topic are organized into three grade levels:
Grades K–2
Grades 3–4
Grades 5–6
For the topic of decimals, there are questions only for grades 3–4 and 5–6.
At the beginning of each level is a list of experiences that children should
encounter for the particular topic. Not all children will be ready for these expe-
riences at the same time. It is quite possible that some children in grades 3 and
4 might be working on some of the experiences listed for K–2 while other chil-
dren in grades 3 and 4 are working on some of the experiences listed for grades
5 and 6. They should not be treated as a progression of experiences but rather
as a range of possible experiences.
Many of the questions in these levels can be adapted to meet the needs of
the students in your classroom by making them easier or more difficult.
As you are reading through the good questions that follow, you will find
some instances where they have been written as investigations rather than ques-
tions. This has been done where we felt they were better written as investiga-
tions. Use them in exactly the same way as the questions.
17i■ ❚ ❙

26. Below each question there are teacher notes. Sometimes these are to make
you aware of some important teaching points for the particular question. They
may also help you ascertain if children have understood the concept being pre-
sented. At other times they will be useful in helping you assess children so you
can plan to overcome any difficulties. It is a good idea to make notes as you
observe children working to use in future planning.
A list of materials that you might need is provided at the beginning of each
level. You will not need all of these materials unless you complete every question
listed for the topic at that level. Check that you have suitable materials before you
present a question to your class. It is important that children have a variety of
concrete materials to select from when they are working on mathematical tasks.
18i■ ❚ ❙ Good Questions for Math Teaching

27. 4 Number
The six topics included in this strand are:
■ money
■ fractions
■ decimals
■ place value
■ counting and ordering
■ operations
There are links in these number topics with the other areas of the mathematics
curriculum and with each other. It is neither possible nor useful to try to treat
them separately. The questions in each topic do, however, have their main teach-
ing point within that topic.
While answering these questions children will develop a feeling for the
way numbers work. They will develop number sense not only for whole num-
bers but also for where fractions and decimals fit into the number system. They
will understand the importance of estimation and mental calculation skills and
use calculators to enable them to understand key ideas without having to do
complicated calculations before they are ready to do so.
Do not forget to adapt questions where necessary by making numbers or
amounts smaller or larger.
19i■ ❚ ❙

28. Money (Grades K–)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ recognize different coins
■ describe, sort, and classify coins
■ exchange money for goods in play situations and give appropriate
change
■ order money amounts
■ use coins to represent written money amounts and use numbers to
record the value of a group of coins
■ use estimation and a calculator for money calculations
MATERIALS
■ coins and play bills
■ goods marked with varying prices below $1.00 as part of the class
store (Ensure that there are combinations of items that add to
$1.00.)
Good Questions and Teacher Notes
How many different ways can you make cents?
In my pocket I have cents What coins might I have?
In questions #1 and #2, children should realize that there are many different
ways to make a money amount. See if they use only multiples of one coin, for
example, four nickels, as well as combinations of different coins, for example,
10 cents + 5 cents + 5 cents.
Are children confident when counting in 5s, 10s, 20s?
I bought something and got cents change How much did it cost and how
much money did I give to pay for it?
Children’s responses might be:
■ costs 5 cents and gives 10 cents
■ costs 15 cents and gives 20 cents
■ costs 95 cents and gives $1.00
■ costs $1.95 and gives $2.00
Can children see the folly of giving 15 cents for an item costing 10 cents
to receive 5 cents change?
20i■ ❚ ❙ Good Questions for Math Teaching

29. I spent exactly  at our class shop What might I have bought?
Check how children add amounts to $1.00. Note if they calculate multiples of
5, 10, or 20 to make $1.00, for example, do they know five items at 10 cents
each is 50 cents or do they add each one separately?
I am a coin with a building on me What might I be?
The main focus here is to look more closely at the attributes of coins.
I have two coins in one hand and one in the other hand The coins in each
hand are worth the same amount What could the coins be?
Note if children develop a system when recording. How easily do they calculate
amounts?
The answer to a calculation is cents What is the question? Refer to this list
to help you
CAFETERIA PRICE LIST
Peanut butter sandwich $1.10 Salad $1.55
Ham & salad roll $1.40 Bag of chips $1.65
Fruit salad $1.15 Piece of fruit $1.20
Cookie $1.15
Can children write more than one question?
I had one of each of the coins in our currency on my table I sorted them into
two groups What might the groups have been?
It is interesting to note what categories children use. Ask them to tell you their
categories; don’t assume you know their reasoning.
The price tag on a toy car is  What coins would I use to pay for this?
Note if children develop a system when recording. How easily do they calculate
amounts?
I have exactly  in bills in my pocket What bills might I have?
Are children aware of available bills? Check if they can count in 5s, 10s, 20s, 50s.
21i■ ❚ ❙ Number

30. Someone was asked to remember the cost of ﬁve items They knew the most
expensive was  and the least expensive was cents What might the
other three be?
The focus here is on ordering of money amounts. Note if the children can
record different amounts correctly.
Money (Grades –)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ round to the nearest dollar to estimate or check total cost
■ record money amounts
■ pay with appropriate amounts when the exact amount is not
available
■ order money amounts
■ use an appropriate method (mental, written, calculator) to solve
problems involving money
MATERIALS
■ coins and play bills
■ supermarket advertisements from newspapers
■ calculators
Good Questions and Teacher Notes
I bought an item at a shop and got cents change What did I buy and how
much did it cost?
Children need to see the folly of including such things as buying an item cost-
ing 5 cents and giving 40 cents to get 35 cents change. Note if children look for
a pattern when recording answers.
I gave change of  using quarters dimes and nickels What might the
change have looked like?
Note if children record systematically and accurately. Check how easily they
make $1.00.
How could I spend exactly  at the supermarket? (Use a supermarket
advertisement and a calculator to help )
Check if children use estimation skills to help them; for example, they might
22i■ ❚ ❙ Good Questions for Math Teaching

31. round off some amounts to assist their estimation. Note how they use the
calculator.
In my pocket I have  What bills might I have?
This allows you to see how familiar children are with the various bills and if they
use a system when recording.
I spent  on six tickets to the theater How many adults and children are
there and how much are the tickets?
Are the answers realistic? Can children multiply amounts, for example, 4 x $10
or 4 x $5? Note if they figure mentally or use paper and pencil to compute.
When I was in a music shop I saw that a CD cost about  and a tape about
 What might have been the price tag on the CD and the tape?
This question focuses on rounding off. Are children aware that they can round
up and down?
A number sentence uses three of the following amounts or numbers: 
 cents  cents What might the number sentence be?
The main focus here is to see if children use a variety of processes, for example,
6 x .50 = $3, $1.50 ÷ 2 = .75, $3.75 – .75 = $3.00.
My friends and I shared an amount of money equally between us We each
got  How much money was there and how many friends might I have?
It is interesting to see how children do this—mentally, with paper and pencil, or
with coins. When they check their answer do they include themselves or only
the friends?
I bought something and paid for it with three coins What might it have been
and how much did it cost?
Look for a range of responses that are realistic.
I went to get  out of the bank What are the different ways I can ask for
this amount of bills?
Note how children multiply and divide by 2, 5, 10, and 20.
23i■ ❚ ❙ Number

32. Money (Grades –)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ use mental calculation and estimation
■ use +, –, x, and ÷ for written computation of money
■ select an appropriate operation to solve problems involving money
MATERIALS
■ play money, notes, and coins
■ new and used car section of a newspaper
■ calculators
Good Questions and Teacher Notes
Scientiﬁc calculators cost  and basic calculators cost  How much
might it cost for a class set of some basic and some scientiﬁc calculators?
Note how children decide how many of each calculator to purchase. Do they
record their answers systematically? Do they choose appropriate operations to
work out the price? How easily do they handle these operations?
I have  and want to buy two cars What could I buy?
Note if children can justify their answers and if they can provide a range of
answers.
If one of the bills currently in use was to be changed to a coin which one
would you choose? Why?
Children should be able to justify their choice in a reasonable manner. You
could extend this by looking at bills and coins in use in other countries.
You are spending ﬁve nights away You have won  for accommodations
Where could you stay?
Top class hotel $300 per night
4 star hotel $225 " "
3 star hotel $100 " "
2 star hotel $60 " "
Backpackers $25 " "
24i■ ❚ ❙ Good Questions for Math Teaching

33. Note what methods children use to work this out, that is, do they readily mul-
tiply amounts when needed or do they always add amounts? They can stay at
different places.
Design a rounding policy for a supermarket
The way children approach this will tell you if they understand rounding. It is
interesting to note whose side they are on—owner or customer?
Fractions (Grades K–)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ use informal fraction language for objects and collections
■ compare fractional parts of objects and collections
MATERIALS
■ fraction materials such as rods, counters, or shapes
■ lengths of string, tape, or yarn
Good Questions and Teacher Notes
_
You see a sign in a shop window that reads OFF SALE What does this mean
to you?
Listen carefully to children’s responses as these will indicate the depth of their
understanding. Also, give them some prices, ask them how much the sale price
would be, and ask them to explain their reasoning.
Half of the people in a family are males What might a drawing of the family
look like?
Do children understand that there must be the same number of people in each
group? Make sure they see a range of drawings done by class members.
Draw a shape Show how to cut the shape into two halves
Do children show equal parts?
I was listening to the radio and I heard the announcer say “half ” What might
she have been referring to?
25i■ ❚ ❙ Number

34. The answers will indicate depth of understanding. Two possible responses are
“Half-past 3” and “Half-time.”
We want to paint the top half of the room How could we ﬁnd out where the
half way mark is?
Allow children to do this how they want but have some string, tape, and so on,
available for their use. The main focus here is to look at how practical the chil-
dren’s strategies are.
_
What do you know and what can you ﬁnd out about ? Record it on paper or
show it with materials
Note if children understand as part of a whole and as part of a collection. Can
they use a range of materials to represent it?
Fractions (Grades –)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ represent simple fractional parts of objects and collections
■ order and compare fractions with the same denominators
■ write common fractions
■ record simple equivalence, for example, = =
■ add and subtract tenths and fractions with like denominators
MATERIALS
■ concrete materials, for example, counters, shapes, or rods
■ drawing paper for designs, such as origami or other square paper
■ circles cut into quarters to represent “pizzas”
Good Questions and Teacher Notes
_ _
How many different designs can you make that are red and yellow?
Note if the designs are simple or complex. Ask children to explain how they
know is red and is yellow.
One third of a class orders lunches from the cafeteria each day How many
students might be in the class and how many of them order lunches each day?
Let children use counters to represent the students if they wish. Can they find
more than one answer? Do they base their answers on multiples of 3?
26i■ ❚ ❙ Good Questions for Math Teaching

35. My aunt said that when she was half her age she could touch her toes How
old might she be now and how old was she when she could touch her toes?
Check that suggested answers are realistic.
I picked up a handful of M&M’s One third of them were red What might a
drawing of the M&M’s look like?
This requires children to show a fractional part of a collection. Do children pro-
vide a range of answers and does anyone develop a system to do so? Do children
understand why amounts that are not multiples of 3 do not work?
I had some pizzas that I cut into quarters How many pizzas might I have had
and how many quarters might I have after cutting them?
Can children identify a relationship between wholes and quarters?
_
How many different ways can you show ?
Note if children understand as part of a whole or part of a length and as part
of a collection. Do they use a range of materials to represent it?
I folded an origami square to show a fraction How did I fold it and what
might the fraction have been?
Look for equal parts and a range of answers.
_ _ _ _ _
A friend of mine put these fractions into two groups: What might
the two groups be?
Ask children to give reasons for their groups as these could highlight some mis-
conceptions. One possible grouping is to put and in one group because they
are unit fractions; another grouping is to put and in one group because they
are greater than .
Fractions (Grades –)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ use equivalence to compare and order fractions
■ locate fractions on a number line
■ rename fractions in different forms, for example, as percentages or
decimals
27i■ ❚ ❙ Number

36. ■ mentally add and subtract common equivalent fractions
■ add and subtract fractions with related denominators
■ understand the relationship between division and fractions
MATERIALS
■ fraction materials, for example, counters, shapes, rods, or kits
Good Questions and Teacher Notes
_
Two fractions add up to What might those two fractions be?
Do children only use known fraction combinations such as + or do they use
subtraction to find other possibilities, for example, – = , so + = ? Do they
use equivalence, for example, = , so + =
Some numbers add up to I know that at least one of them has a fraction
part in it but none uses decimals What might the numbers be?
As above, note the methods children use to find answers. These will tell you a
lot about their understanding of fractions.
_
What three fractions might I add together and get an answer of ?
Again look at the methods used. Do children guess and then work it out to
check? If so, how do they then adjust the fractions? Do they know which frac-
tions are smaller than ?
_
The answer is What might the question be?
Encourage children to use other processes than just addition.
_
What two fractions might I subtract to get an answer of ?
Do children use equivalence (“I know = , so – = ”) or some other method
to do this?
_
? x ? ? What might the missing numbers be?
The missing numbers do not have to be the same. Can children describe all of
the answers? Can they prove they have all of the answers? (There are nine pos-
sibilities.)
28i■ ❚ ❙ Good Questions for Math Teaching

37. A rectangle has a perimeter of two units What might the area be?
Yes, this is a fraction question! The perimeter must be a combination of frac-
tions, for example, + + + , + + + , and so on. The area will vary
depending on the length of the sides. You may want to remind the students that
a square is a rectangle.
_
Write some different stories about T
The purpose of this is for children to understand the difference between 3 ÷
and of 3. Their stories will indicate this. Stories like “I had $3.00 and I gave
half to my friend” are not appropriate.
______ _ ______
Teacher: “Which is bigger or ?” Student: “ is bigger because has been
added to the top and the bottom ” Is this reasoning correct? Are there any
examples where adding to the top and the bottom makes the fraction bigger?
This question highlights a misconception that some children may have.
_? _
? What might the missing fraction be?
Provide concrete materials for this question. Do not assume that because some
children write, for example, that their understanding is correct. Check why they
write this. Some children may think any number smaller than the 3 or the 4 will
make a smaller fraction and will not consider fractions such as , , and so on,
to be smaller. This question checks the same misconception as question 9.
Decimals (Grades –)
EXPERIENCES AT THIS LEVEL WILL HELP CHILDREN TO:
■ read decimals on a calculator screen
■ write decimal numerals
■ round off to nearest whole number
■ record and order numbers with two decimal places
■ add and subtract numbers involving tenths
■ understand how decimals fit in the number system
MATERIALS
■ calculators
■ materials to model decimals, for example, base 10 materials, Popsicle
sticks, interlocking cubes, or metric rulers
29i■ ❚ ❙ Number

38. Good Questions and Teacher Notes
A decimal number has been rounded off to What might the number be?
This will establish if children understand the concept of rounding. Do they give
one number or do they know it can include 5.5 but must be smaller than 6.5?
I am thinking of some decimal numbers between and What might they
be? Give at least answers
My big sister says that the yard record at her school is between and
seconds What might the record be?
In questions #2 and #3, note which children give answers using only hundredths
or only tenths and which children give a combination of these. Are there any
children who include thousandths? Does anyone give the complete range of
tenths and hundredths?
Using only these keys on your calculator ( ) what numbers can you
make the calculator show?
Children should use a range of processes. Note how comfortable they are with
the functioning of the calculator.
Represent with materials in at least ﬁve different ways
Provide plenty of concrete materials for this question. Some you might find use-
ful are base 10 blocks, Popsicle sticks, metric rulers, interlocking cubes, and play
money.
I added three decimal numbers together to make exactly What might the
three numbers be?
Look for a variety of answers. Do children add two of them and then subtract
from 4 to find the third?
If I use a ﬂat to represent one whole a long to represent tenths and a unit to
represent hundredths what numbers can I represent using exactly ten pieces?
Children need to work with base 10 blocks to do this question. You could
extend it by asking them to show the biggest or smallest possible number using
ten pieces.
In this number sentence what might the missing digits be? ? ?
Can children give the entire range (2.1 to 9.9, including numbers like 5.0)?
30i■ ❚ ❙ Good Questions for Math Teaching

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