Contributed by:

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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Good Q uestions for Math Teaching

•

Why Ask Them and What to Ask K–

Peter Sullivan

Pat Lilburn

Math Solutions Publications

Sausalito CA

•

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

student learning; and

• free online support, including grade-level lessons, book reviews, inservice

information, and district feedback, all in our quarterly Math Solutions

Online Newsletter.

For information about all of the products and services we have available,

please visit our website at www.mathsolutions.com. You can also contact us to

discuss math professional development needs by calling (800) 868-9092 or by

sending an email to [email protected]ions.com.

We’re always eager for your feedback and interested in learning about your

particular needs. We look forward to hearing from you.

Math Solutions ®

P U B L I C AT I O N S

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

student learning; and

• free online support, including grade-level lessons, book reviews, inservice

information, and district feedback, all in our quarterly Math Solutions

Online Newsletter.

For information about all of the products and services we have available,

please visit our website at www.mathsolutions.com. You can also contact us to

discuss math professional development needs by calling (800) 868-9092 or by

sending an email to [email protected]ions.com.

We’re always eager for your feedback and interested in learning about your

particular needs. We look forward to hearing from you.

Math Solutions ®

P U B L I C AT I O N S

4.

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

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

6.

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■ ❚ ❙

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■ ❚ ❙

8.

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■ ❚ ❙

•

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

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■ ❚ ❙

•

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

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?

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

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■ ❚ ❙

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

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

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■ ❚ ❙

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

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

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

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

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■ ❚ ❙

•

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

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■ ❚ ❙

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

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

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

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

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

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

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

“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

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

■ 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

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

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