ELEMENTARY
FALS
FORMATIVE
ASSESSMENT LESSONS
Formative Assessment Lesson
Alpha
Counting Dots in Various Arrangements
Mathematical goals
This lesson unit is intended to help you assess how well students are able to count objects up to 20 no matter how they
are arranged and also how well they are able to represent their counts with written numerals. It will help you to identify
students who have the following difficulties:
Not being able to “mark” where they start/stop counting
Arranging objects in order to be counted
One-to-one correspondence
Common Core State Standards
This lesson involves mathematical content in the standards from across the grade, with emphasis on:
Know number names and the count sequence.
Count to tell the number of objects.
(K.CC.5) Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a
circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
(K.CC.3)Write numbers from 0-20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
This lesson involves a range of mathematical practices, with emphasis on:
2. Reason abstractly and quantitatively.
7. Look at and make use of structure.
Introduction
This lesson is structured in the following way:
Before the lesson, students work individually on an assessment task that is designed to reveal their current
understandings and difficulties. You then review their work and create questions for students to answer in
order to improve their solutions.
Students work in small groups on collaborative discussion tasks, to match cards with dots in regular patterns,
structured 10 frame patterns, unstructured arrangements, and numerals. As they do this, they interpret the
cards’ meanings and begin to link them together. Throughout their work, students justify and explain their
decisions to their peers.
Students return to their original assessment tasks, and try to improve their own responses.
Materials required
Each individual student will need:
Two copies of the assessment task Counting Dots.
A sheet of sticky dots.
1
Formative Assessment Lesson
Alpha
A mini-whiteboard, marker, and eraser.
Each small group of students will need the following resources:
Card sets: A, B, C, and D. All cards should be cut up before the lesson and it would be helpful if each set were a
different color.
There is also a projector resource to help with the whole class discussion.
Time needed
Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes
for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact
timings will depend on the needs of the class.
Before the Lesson
Assessment task: Counting Dots (15 minutes)
Have students do this task individually in class a day or more before the formative assessment lesson. This will give you
an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to
target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or
in small groups (they should still work individually.
Give each student a copy of the assessment task Counting Dots.
Count the dots in each space and write the numeral that tells how many dots on the line.
Put stickers in the space to match the numeral.
It is important that the students are allowed to answer the questions without your assistance, as far as possible.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will
engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should
expect to answer questions such as these confidently. This is their goal.
Assessing students’ responses
Collect students’ responses to the task. Make some notes about what their work reveals about their current levels of
understanding, and their different problem solving approaches.
We suggest that you do not score student’s work. The research shows that this will be counterproductive, as it will
encourage students to compare their scores, and will distract their attention from what they can do to improve their
mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some
suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of
this unit.
2
Formative Assessment Lesson
Alpha
We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You
may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that
will be of help to the majority of students. These can be written on the board at the end of the lesson.
The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help
students to find a variety of ways that work in different situations and make sense to them.
Common Issues:
Student writes numbers that are more than the number
of dots.
Student writes numbers that are less than the number of
dots.
Suggested questions and prompts:
Did you count each dot only once?
Is there a way to know if you have already counted
a dot?
How do you know where you started counting and
where you stopped?
Think about how you could remember where you
started.
How can you be sure that you counted every dot?
Suggested lesson outline
Collaborative Activity: matching Card Sets A, B, C, and D (30 min.)
Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the
task.
Give each group Card Sets A(numerals) and B (structured 10-frames).
Introduce the lesson carefully:
I want you to work as a team. Take turns placing a numeral card with a dot card that has the same number of
dots. Each time you do this, explain your thinking clearly to your partner. If your partner disagrees with your
match then challenge him or her to explain why. It is important that you both understand why each card is
matched with another one. There is a lot of work to do today and you may not all finish. The important thing is
to learn something new, so take your time.
Your tasks during the small group work are to make a note of student approaches to the task, and to support student
problem solving.
Make a note of student approaches to the task
You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice
any common mistakes. For example, students may consistently loose count or forget where they started and re-count
dots.
3
Formative Assessment Lesson
Alpha
Support student problem solving
Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help
students clarify their thinking. Encourage students to use each other as a resource for learning.
If one student has placed a particular card to make a match, challenge their partner to provide an explanation.
If you find students have difficulty articulating their decisions, then you may want to use the questions from the
Common Issues table to support your questioning.
If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and
organize a whole class discussion. The projector resources may be useful when doing this.
Placing Card Set C (regular patterns)
As students finish with matching card sets A and B, hand out card set C. These provide students with a different way of
interpreting the numbers.
Do not collect card set B. an important part of this task is for students to make connections between different
representations of numbers.
As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. The
following patterns may be noticed:
Placing Card Set D (unstructured)
As students finish matching card set C, hand out card set D. Continue to monitor work and listen to discussion.
Taking two class periods to complete all activities
If you have to divide the lesson into two class periods, you may want to have a way for students to save the work they
have done with matching the card sets. You may give each group a chart paper or poster board and have them tape the
cards down with their matches. You may chose to have them do this even if you are not dividing up the class period just
to use as a visual during the class discussion.
Sharing Work (10 minutes)
When students get as far as they can with matching the card sets, ask one student from each group to visit another
group’s work. Students remaining at their desk should explain their reasoning for the matched cards on their own desk.
If you are staying at your desk, be ready to explain the reasons for your group’s matches.
If you are visiting another group, check to see which matches are different from your own. (they might need
some kind of representation of their own matches to take with them—perhaps a cell phone picture?)
If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. When you
return to your own desk, you need to consider, as a group, whether to make any changes to your work.
Students may now want to make changes.
4
Formative Assessment Lesson
Alpha
Extension activities
Ask students who finish quickly to make another set of cards that shows each number in a different way.
Plenary whole-class discussion (10 minutes)
Give each student a mini-whiteboard, pen, and eraser.
Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending
what has been learned to new examples, and then examining some of the conclusions listed above.
Show students the examples of number configurations on the slide resources and have them write the matching
numeral on their white board. Then, showing them the slide with all three representations, discuss which was easiest to
count and why. Have students explain their process for counting the dots on other slides, always asking, “Is there
another way?”
Have students draw a configuration on their whiteboards for numerals on the slides and discuss their approaches and
why they chose them. You might even group students by the configuration they drew (scattered, regular, 10-frame).
Improving individual solutions to the assessment task (10 minutes)
Return to the students their original assessment, Counting Dots, as well as a second blank copy of the task (and more
sticky dots if needed).
Look at your original responses and think about what you have learned during this lesson.
Using what you have learned, try to improve your work.
If you have not added questions to individual pieces of work then write your list of questions on the board. Students
should select from this list only the questions appropriate to their own work. (with kindergarten, you may only focus on
the questions students really seemed to need and state these out loud and ask them again as you move around the
room or work with a small group at a time.)
5
Formative Assessment Lesson
Alpha
Counting Dots
Count the dots in each picture.
Write the numeral that tells how many dots in the box beside the picture.
6
Formative Assessment Lesson
Alpha
Count out sticker dots that match the number in the box. Place them in the box beside of the number.
7
9
8
5
7
Formative Assessment Lesson
Alpha
Card Set A: Numeral Cards
8
7
5
4
9
8
Formative Assessment Lesson
Alpha
Card Set B: Structured Ten Frames
9
Formative Assessment Lesson
Alpha
Card Set C: Regular Patterns
10
Formative Assessment Lesson
Alpha
Card Set D: Unstructured
11
Math Formative Assessment Lesson
Alpha Version
Pieces of the Hundred Chart Grade 1
Mathematical goals
This lesson unit is intended to help you assess how well students are able to count to count from any number and
recognize patterns on the 100 chart that will help them to count on or back. It will also help you to assess how well they
write numbers between 1 and 100. It will help you to identify students who have the following difficulties:
Recognizing patterns on the hundred chart.
Counting on or back from a given number
Common Core State Standards
This lesson involves mathematical content in the standards from across the grade, with emphasis on:
Extend Counting Sequence
Understand Place Value
(1.NBT.1) Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a
number of objects with a written numeral.
(1.NBT.2) Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the
following as special cases:
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0
ones).
This lesson involves a range of mathematical practices, with emphasis on:
2. Reason abstractly and quantitatively.
7. Look at and make use of structure.
Introduction
This lesson is structured in the following way:
Before the lesson, students work individually on an assessment task that is designed to reveal their current
understandings and difficulties. You then review their work and create questions for students to answer in
order to improve their solutions.
Students work in small groups on collaborative discussion tasks, to place cards with numerals onto a section
taken from a hundred chart. Throughout their work, students justify and explain their decisions to their peers.
Students return to their original assessment tasks, and try to improve their own responses.
Materials required
Each individual student will need:
Two copies of the assessment task Torn Hundred Chart.
A mini-whiteboard, marker, and eraser.
Each small group of students will need the following resources:
Two place cards with squares, some of which are marked with numerals in the pattern of the hundreds chart.
Two sets of colored cards that will fill in the blanks on the two charts along with a few numbers that will not fit
either chart.
Hundred Charts on hand for groups that need more support
1
Math Formative Assessment Lesson
Alpha Version
Time needed
Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes
for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact
timings will depend on the needs of the class.
Before the Lesson
Assessment task: Torn Hundred Chart (15 minutes)
Have students do this task individually in class a day or more before the formative assessment lesson. This will give you
an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to
target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or
in small groups (they should still work individually.)
Give each student a copy of the assessment task Torn Hundred Chart
Fill in the missing numbers in the white spaces of the torn hundred charts.
It is important that the students are allowed to answer the questions without your assistance, as far as possible.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will
engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should
expect to answer questions such as these confidently. This is their goal.
Assessing students’ responses
Collect students’ responses to the task. Make some notes about what their work reveals about their current levels of
understanding, and their different problem solving approaches.
We suggest that you do not score student’s work. The research shows that this will be counterproductive, as it will
encourage students to compare their scores, and will distract their attention from what they can do to improve their
mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some
suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of
this unit.
We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You
may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that
will be of help to the majority of students. These can be written on the board at the end of the lesson.
The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help
students to find a variety of ways that work in different situations and make sense to them.
2
Math Formative Assessment Lesson
Common Issues:
Students are placing cards in incorrect spot.
When going down to the next line, students write the
number that comes next.
33
34
35
36
37
38
39
40
Alpha Version
Suggested questions and prompts:
Why did you put that card here?
Think about our hundreds chart, how is each row
organized?
--You may even want to carry a hundred chart with
you to allow students a resource when questioning.
What would go into the spaces between the
number that is already there and the one that you
wrote?
What are some of the patterns you notice in a
hundred chart?
Why did you put that card here?
----You may even want to carry a hundred chart
with you to allow students a resource when
questioning.
Suggested lesson outline
Collaborative Activity: Placing Card set onto torn hundreds charts
Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the
task.
Give each group Set A with two place cards and a set of numeral cards.
Introduce the lesson carefully:
I want you to work as a team. Take turns placing a numeral card onto the piece of the hundred chart. Each time
you do this, explain your thinking clearly to your partner. If your partner disagrees with your placement then
challenge him or her to explain why. It is important that you both understand why each card is placed where it
is. There is a lot of work to do today and you may not all finish. The important thing is to learn something new,
so take your time.
Your tasks during the small group work are to make a note of student approaches to the task, and to support student
problem solving.
Make a note of student approaches to the task
You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice
any common mistakes. For example, students may consistently try to place numbers in counting order without noticing
that some of the numbers will not show on their section of the chart.
Support student problem solving
Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help
students clarify their thinking. Encourage students to use each other as a resource for learning.
If one student has placed a particular card on the chart, challenge their partner to provide an explanation.
If you find students have difficulty articulating their decisions, then you may want to use the questions from the
Common Issues table to support your questioning.
If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and
organize a whole class discussion. The projector resources may be useful when doing this.
3
Math Formative Assessment Lesson
Alpha Version
Place Card Set B
As students finish with matching Place card set A, hand out card set B. These provide students with a bit less structure
in placing the numerals.
Do not collect Place card set A. An important part of this task is for students to make connections between different
arrangements of numerals.
As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. The
following patterns may be noticed:
Students observe that the numbers in vertical lines have the same digit in the ones place and one higher in the
tens place.
Students observe that numbers in horizontal lines have the same digit in the tens place and one higher in the
ones place.
Students observe that diagonally the digits in each place either rise or fall depending on the direction.
Place Card Set C
When students move to Place Card Set C have them fill out there puzzles by writing in their responses. When they have
completed this, give them Card Set C to see if their responses match.
Place Card Set D
When students move to Place Card Set D have them fill out there puzzles by writing in their responses. When they have
completed this, give them Card Set D to see if their responses match.
Taking two class periods to complete all activities
If you have to divide the lesson into two class periods, you may want to have a way for students to save the work they
have done with the Place card sets. You may have each group tape the cards down with on their place cards. You may
chose to have them do this even if you are not dividing up the class period just to use as a visual during the class
discussion.
Sharing Work (10 minutes)
When students get as far as they can with placing the card sets, ask one student from each group to visit another
group’s work. Students remaining at their desk should explain their reasoning for the placement of the cards on their
own desk.
If you are staying at your desk, be ready to explain the reasons for your group’s placements.
If you are visiting another group, check to see which placements are different from your own. (They might need
some kind of representation of their own placement to take with them—perhaps a cell phone picture?)
If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. When you
return to your own desk, you need to consider, as a group, whether to make any changes to your work.
Students may now want to make changes.
Extension activities
Give students who finish quickly a mini set of place cards and cards to make their own puzzle.
Plenary whole-class discussion (10 minutes)
Give each student a mini-whiteboard, marker, and eraser.
4
Math Formative Assessment Lesson
Alpha Version
Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending
what has been learned to new examples, and then examining some of the conclusions listed above.
Show students the examples of place cards on the slide resources and have them write the missing numeral on their
white board. Discuss why they think that number goes there. Have students explain their process for deciding which
number goes in the blank and have them describe patterns that they see.
Improving individual solutions to the assessment task (10 minutes)
Return to the students their original assessment, Torn Hundreds Chart, as well as a second blank copy of the task.
Look at your original responses and think about what you have learned during this lesson.
Using what you have learned, try to improve your work.
If you have not added questions to individual pieces of work then write your list of questions on the board. Students
should select from this list only the questions appropriate to their own work. (With first graders, you may only focus on
the questions students really seemed to need and state these out loud and ask them again as you move around the
room or work with a small group at a time.)
5
Math Formative Assessment Lesson
Alpha Version
Torn Hundreds Chart
These are chunks taken from a hundred chart. Fill in the empty boxes with the correct numbers.
1
4
13
32
6
27
6
Math Formative Assessment Lesson
Alpha Version
Place Card Set A
1
13
22
34
7
Math Formative Assessment Lesson
Alpha Version
Place Card Set A
7
16
28
39
8
Math Formative Assessment Lesson
Alpha Version
Place Card Set B
17
38
46
9
Math Formative Assessment Lesson
Alpha Version
Place Card Set B
42
64
71
10
Math Formative Assessment Lesson
Alpha Version
Place Card Set C
57
79
11
Math Formative Assessment Lesson
Alpha Version
Place Card Set C
43
12
Math Formative Assessment Lesson
Alpha Version
Place Card Set D
39
57
Place Card Set D
13
Math Formative Assessment Lesson
Alpha Version
36
Card Set A
14
Math Formative Assessment Lesson
1
Alpha Version
2
3
4
11 12 13 14
21 22 23 24
31 32 33 34
15
Math Formative Assessment Lesson
Alpha Version
Card Set A
6
7
8
9
16 17 18 19
26 27 28 29
36 37 38 39
16
Math Formative Assessment Lesson
Alpha Version
Card Set B
15 16 17 18
25 26 27 28
35 36 37 38
45 46 47 48
17
Math Formative Assessment Lesson
Alpha Version
Card Set B
41 42 43 44
51 52 53 54
61 62 63 64
71 72 73 74
18
Math Formative Assessment Lesson
Alpha Version
Card Set C
57 58 59 60
67 68 69 70
77 78 79 80
87 88 89 90
19
Math Formative Assessment Lesson
Alpha Version
Card Set C
22 23 24 25
32 33 34 35
42 43 44 45
52 53 54 55
20
Math Formative Assessment Lesson
Alpha Version
Card Set D
36 37 38 39
46 47 56 58
59 69
21
Math Formative Assessment Lesson
Alpha Version
Card Set D
6
13 14 15
16 23 33 34
35 36
22
Math Formative Assessment Lesson
Alpha Version
Extension
23
Math Formative Assessment Lesson
Alpha Version
Extension
24
Math Formative Assessment Lesson
Alpha Version
Extension
25
Mathematics Formative Assessment Lesson Alpha Version What is the Value of the Place?
Mathematical goals This lesson unit is intended to help you assess how well students are able to recognize the value of the place in a number. It will help you to identify students who have the following difficulties: • Recognizing place value of ones, tens, hundreds. • Use correct mathematical vocabulary to explain place value to a partner. • Make connections of place value to previously taught lessons. • Common Core State Standards This lesson involves mathematical content in the standards from across the grades, with emphasis on: 2.NBT: Understand Place Value (2.NBT.1AB) Understand that the three digits of a three‐digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following special cases: a. 100 can be thought of as a bundle of ten tens—called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). (2.NBT.3) Read and write numbers to 1000 using base‐ten numerals, number names, and expanded form. This lesson involves a range of mathematical practices, with emphasis on: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. Introduction This lesson is structured in the following way: • Before the lesson, students work individually on an assessment task that is designed to reveal their current understandings and difficulties. You then review their work and create questions for students to answer in order to improve their solutions. • Students work in small groups on collaborative discussion tasks to represent 3‐digit numbers in multiple ways. Throughout their work, students justify and explain their decisions to their peers. • Students return to their original assessment tasks, and try to improve their own responses. Materials required Each individual student will need: • Two copies of the assessment task What’s the Value? • A mini‐whiteboard, marker, and eraser. Each small group of students will need the following resources: • A set of arrow cards. • A set of numeral cards. There is also a projector resource to help with the whole class discussion. 1 Mathematics Formative Assessment Lesson Alpha Version Time needed Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes for a follow‐up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact timings will depend on the needs of the class. Before the Lesson Assessment task: What’s the Value (15 minutes) Have students do this task individually in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow‐up lesson. Depending on your class you can have them do it all at once or in small groups (they should still work individually.) Give each student a copy of the assessment task What’s the Value. Circle all the statements that are true about the number, and explain your thinking. It is important that the students are allowed to answer the questions without your assistance, as far as possible. Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal. Assessing students’ responses Collect students’ responses to the task. Make some notes about what their work reveals about their current levels of understanding, and their different problem solving approaches. We suggest that you do not score student’s work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the end of the lesson. The solution to all these difficulties is not to teach one particular way of counting‐one to one matching‐
but to help students to find a variety of ways that work in different situations and make sense to them. 2 Mathematics Formative Assessment Lesson Common Issues: Students group cards based on hundred place only. Students are selecting the wrong arrow cards. For example, if the number is 243, the students just may get out the 2…4…and 3 rather than 200…40…3 Drawings are not represented correctly in set C. Alpha Version Suggested questions and prompts: • Why do you think those are a match? •
I see that number is 362 and there are 3 hundred flats represented, but what about the rest of the blocks, did you count all those too? •
How are you tracking that you have counted all the blocks? •
Is there a way we can make sure we have counted everything? •
Did you line your arrows up? •
What does it mean when we line our arrows up and the numbers cover each other up? •
Look at you 2, what is the value of the two in you number and what is the value on your arrow card? •
How are you tracking that you have counted all the blocks? •
Is there a way we can make sure we have counted everything? Suggested lesson outline Collaborative Activity: Match Base Ten Representations and Build Numbers with Arrow Cards Organize the class into groups of two or three students. With larger groups, some students may not fully engage in the task. Give each group Set A of numeral cards, arrow cards, and base ten cards Introduce the lesson carefully: I want you to work as a team. Match each of the base ten cards to the correct numeral card. Take turns building the numeral using the arrow cards. Each time you do this, explain your thinking clearly to your partner. If your partner disagrees with your placement then challenge him or her to explain why. It is important that you both understand why each number is built the 3 Mathematics Formative Assessment Lesson Alpha Version way it is. There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so take your time. Your tasks during the small group work are to make a note of student approaches to the task, and to support student problem solving. Make a note of student approaches to the task You can then use this information to focus a whole‐class discussion towards the end of the lesson. In particular, notice any common mistakes. Partners should be engaged in checking their partner, asking for clarification, and taking turns. When calling on students make sure you allow the struggling groups to share first. Support student problem solving Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help students clarify their thinking. Encourage students to use each other as a resource for learning. If one student has built a number in a particular way, challenge their partner to provide an explanation. If you find students have difficulty articulating their decisions, then you may want to use the questions from the Common Issues table to support your questioning. If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and organize a whole class discussion. The projector resources may be useful when doing this. Place Card Set B As students finish with matching Place card set A, hand out card set B. These are developed to be more difficult. There are three representations of each number using base ten cards which manipulate the tens and ones. Do not collect Place card set A. An important part of this task is for students to make connections between different ways of building the numerals. As you monitor the work, listen to the discussion and help students to look for patterns and generalizations. Place Card Set C As students finish with Card Set B, hand out Card Set C. Students will now be challenged to draw two different representations of each number. Do not take up the previous sets of cards. Students may use these for guidance in making their decisions. Sharing Work (10 minutes) When students get as far as they can with the card sets, ask one student from each group to visit another group’s work. Students remaining at their desk should explain their reasoning for the placement of the cards on their own desk. 4 Mathematics Formative Assessment Lesson Alpha Version If you are staying at your desk, be ready to explain the reasons for your group’s work. If you are visiting another group, check to see which matches are different from your own. (They might need some kind of representation of their own placement to take with them—perhaps a cell phone picture?) If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. When you return to your own desk, you need to consider, as a group, whether to make any changes to your work. Students may now want to make changes. Extension activities Extension 1: Challenge those students who complete card set C to go back and add a third representation of the number. Extension 2: Provide the students with Card Set D and let them explore representations in the thousands. Plenary whole‐class discussion (10 minutes) Give each student a mini‐whiteboard, marker, and eraser. Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending what has been learned to new examples, and then examining some of the conclusions listed above. Show students the examples of number arrows on the slide resources and have them write the numeral they represent on their white board. Discuss why they think that number is built. Have students explain their process for deciding which number goes in which place. Improving individual solutions to the assessment task (10 minutes) Return to the students their original assessment, What’s the Value?, as well as a second blank copy of the task. Look at your original responses and think about what you have learned during this lesson. Using what you have learned, try to improve your work. If you have not added questions to individual pieces of work then write your list of questions on the board. Students should select from this list only the questions appropriate to their own work. (With second graders, you may only focus on the questions students really seemed to need and state these out loud and ask them again as you move around the room or work with a small group at a time.) Resources: Arrow Cards http://www.kentuckymathematics.org/intervention/doc/2008/Arrow%20Card%20Template%200‐
thous.pdf (Adapted From: Uncovering Student Thinking in Mathematics Grades K‐5: 25 Formative Assessment Probes for the Elementary Classroom, Tobey and Minton, Corwin Press, 2011) 5 Mathematics Formative Assessment Lesson Alpha Version What’s the Value? 243
Circle all the true statements about this number, and explain
your thinking.
A) 2 tens and 43 ones
B) 243 ones
C) 2 hundreds 403 ones
D) 24 tens and 3 ones
E) 2 hundreds and 43 ones
F) 1 hundred and 143 ones
6 Mathematics Formative Assessment Lesson Alpha Version Card Set A
347 136 254 477 Card Set B 502 337 482 357 7 Mathematics Formative Assessment Lesson Alpha Version Card Set A
8 Mathematics Formative Assessment Lesson Alpha Version 9 Mathematics Formative Assessment Lesson Alpha Version 10 Mathematics Formative Assessment Lesson Alpha Version 11 Mathematics Formative Assessment Lesson Alpha Version 12 Mathematics Formative Assessment Lesson Alpha Version 13 Mathematics Formative Assessment Lesson Alpha Version Card Set B
14 Mathematics Formative Assessment Lesson Alpha Version 15 Mathematics Formative Assessment Lesson Alpha Version 16 Mathematics Formative Assessment Lesson Alpha Version 17 Mathematics Formative Assessment Lesson Alpha Version 18 Mathematics Formative Assessment Lesson Alpha Version 19 Mathematics Formative Assessment Lesson Alpha Version Card Set C 547
547
20 Mathematics Formative Assessment Lesson Alpha Version 338
338
21 Mathematics Formative Assessment Lesson Alpha Version 630
630
22 Mathematics Formative Assessment Lesson Alpha Version 421
421
23 Mathematics Formative Assessment Lesson Alpha Version Extension 1 Students will return to card set C and will be asked to draw each number a third way. Extension 2 4390
7310
2902
5912
6934
9022
24 Math Formative Assessment Lesson
Alpha Version
What Subtraction Strategy Did You Use? Grade 3
Mathematical goals
This lesson unit is intended to help you assess how well students are able to apply and understand a variety of different
strategies to subtraction. It will help you to identify students who have the following difficulties:
 Misunderstandings regarding methods of subtracting
 lack of conceptual understanding of the properties of numbers
 misconception that there is only one correct algorithm for each operation
 misconception that there is no need to understand other methods of subtracting
Common Core State Standards
This lesson involves mathematical content in the standards from across the grades, with emphasis on:
Use place value understanding and properties of operations to perform multi-digit arithmetic.
(3.NBT.2) Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of
operations, and/or the relationship between addition and subtraction.
This lesson involves a range of mathematical practices, with emphasis on:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others
Introduction
This lesson is structured in the following way:
 Before the lesson, students work individually on an assessment task that is designed to reveal their current
understandings and difficulties. You then review their work and create questions for students to answer in
order to improve their solutions.
 Students work in small groups on collaborative discussion tasks, match student work with student explanation
and solve subtraction problems using various strategies. Throughout their work, students justify and explain
their decisions to their peers.
 Students return to their original assessment tasks, and try to improve their own responses.
Materials required
Each individual student will need:
 Two copies of the assessment task What Subtraction Strategy Did You Use?
Each small group of students will need the following resources:
 Card Set A
 Have Activity B ready, but do not pass out
 Have Activity C ready, but do not pass out
1
Math Formative Assessment Lesson
Alpha Version
Time needed
Approximately 15 minutes before the lesson (for the individual assessment task), one 40 minute lesson, and 15 minutes
for a follow-up lesson (for students to revisit individual assessment task). Timings given are only approximate. Exact
timings will depend on the needs of the class.
Before the Lesson
Assessment task: What Subtraction Strategy Did You Use? (15 minutes)
Students should have been exposed to lessons that promote each of the strategies used in this activity. (think addition,
compensation, and split/jump, standard algorithm).
Have students do this task individually in class a day or more before the formative assessment lesson. This will give you
an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to
target your help more effectively in the follow-up lesson. Depending on your class you can have them do it all at once or
in small groups (they should still work individually.)
Give each student a copy of the assessment task What Subtraction Strategy Did You Use?
Circle the method that most closely matches how you solved the problem. Explain whether each method makes
sense mathematically.
It is important that the students are allowed to answer the questions without your assistance, as far as possible.
Students should not worry too much if they cannot understand or do everything, because in the next lesson they will
engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should
expect to answer questions such as these confidently. This is their goal.
Assessing students’ responses
Collect students’ responses to the task. Make some notes about what their work reveals about their current levels of
understanding, and their different problem solving approaches.
We suggest that you do not score student’s work. The research shows that this will be counterproductive, as it will
encourage students to compare their scores, and will distract their attention from what they can do to improve their
mathematics.
Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some
suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of
this unit.
We suggest that you write a list of your own questions, based on your students’ work, using the ideas that follow. You
may choose to write questions on each student’s work. If you do not have time to do this, select a few questions that
will be of help to the majority of students. These can be written on the board at the end of the lesson.
The solution to all these difficulties is not to teach one particular way of counting-one to one matching-but to help
students to find a variety of ways that work in different situations and make sense to them.
2
Math Formative Assessment Lesson
Common Issues:
Students always subtract the small number from the
larger number rather than regrouping.
46 students subtracted 8-6.
-28
22
Students do not demonstrate place value understanding.
They struggle with breaking two-digit numbers into tens
and ones.
Alpha Version
Suggested questions and prompts:
 Which numbers did you subtract?
 Does it matter which order numbers are in when
you subtract?




Students do not think about decomposing numbers into
tens and ones for easier adding and subtracting.


What do the digits in 37 stand for?
What is their value?
How does the position of a number affect its
value?
How would the value of the 3 and 7 change if we
swapped their places?
Can you write 37 in expanded form?
How could you make this number easier to work
with?
Suggested lesson outline
Collaborative Activity: Subtraction Card Sort and Use of Multiple Strategies
Organize the class into groups of three or four students. With larger groups, some students may not fully engage in the
task. You may want to consider heterogeneous grouping kids who anchor to different strategies.
Give each group a copy of the problem or show the problem on the board or slide.
Introduce the lesson carefully:
I want you to work as a team. You will have to work together and match students’ work with the matching
explanation. Explain your thinking clearly to your partner. If your partner disagrees with your thinking then
challenge him or her to explain why. It is important that you both understand how each problem was solved.
There is a lot of work to do today and you may not all finish. The important thing is to learn something new, so
take your time.
Your tasks during the small group work are to make a note of student approaches to the task, and to support student
problem solving.
Make a note of student approaches to the task
You can then use this information to focus a whole-class discussion towards the end of the lesson. In particular, notice
any common mistakes. For example, students may not consider or understand the thought process behind the
algorithm of subtraction. They may try to “borrow” without knowing the foundation math behind the algorithm.
Therefore they do not recognize when they conclude with an incorrect answer. Students may become confused with
the problems that show multiple ways to subtract if they have not been exposed to decomposing numbers.
Support student problem solving
3
Math Formative Assessment Lesson
Alpha Version
Try not to make suggestions that move students toward a particular approach to the task. Instead, ask questions to help
students clarify their thinking. Encourage students to use each other as a resource for learning.
When a student creates a match, challenge their partner to provide an explanation.
If you find students have difficulty articulating their decisions, then you may want to use the questions from the
Common Issues table to support your questioning.
If the whole class is struggling on the same issue, then you may want to write a couple of questions on the board and
organize a whole class discussion. The projector resources may be useful when doing this.
As you monitor the work, listen to the discussion and help students to look for patterns and generalizations.
Activity B
As students finish with Card Set A and are able to explain their reasoning give them Card Set B. Card Set B will consist of
a subtraction problem for kids to solve using 4 different strategies (standard algorithm, think addition, compensation,
and split/jump).
Activity C
As students finish with Activity B and are able to explain their reasoning give them Activity C. Activity C will consist of a
subtraction problem for kids to solve using 4 different strategies (standard algorithm, think addition, compensation, and
split/jump).
Sharing Work (10 minutes)
When students get as far as they can with working the problems, ask one student from each group to visit another
group’s work. Students remaining at their desk should explain their reasoning for the way they worked the problem at
their own desk.
If you are staying at your desk, be ready to explain the reasons for your group’s work.
If you are visiting another group, check to see which answers or explanations are different from your own. (They
might need a copy of their own groups work to take with them.
If there are differences, ask for an explanation. If you still don’t agree, explain your own thinking. When you
return to your own desk, you need to consider, as a group, whether to make any changes to your work.
Students may now want to make changes.
Extension activities
Students who finish quickly may work another problem or may try to find a different strategy or strategies for working
the original problem.
Plenary whole-class discussion (10 minutes)
Conclude the lesson by discussing and generalizing what has been learned. The generalization involves first extending
what has been learned to new examples, and then examining some of the conclusions listed above.
Choose students to share how their group used the different strategies to work the problem. Discuss which strategy
they liked best and why.
Improving individual solutions to the assessment task (10 minutes)
Return to the students their original assessment, Which Subtraction Strategy Did You Use? as well as a second blank
copy of the task.
4
Math Formative Assessment Lesson
Alpha Version
Look at your original responses and think about what you have learned during this lesson.
Using what you have learned, try to improve your work.
If you have not added questions to individual pieces of work then write your list of questions on the board. Students
should select from this list only the questions appropriate to their own work. (With first graders, you may only focus on
the questions students really seemed to need and state these out loud and ask them again as you move around the
room or work with a small group at a time.)
5
Math Formative Assessment Lesson
Alpha Version
Which Subtraction Strategy Did You Use?
Katrina, Jenny, Debbie, and Seth each subtracted 28 from 65.
1. Circle the method that most closely matches how you solved the problem.
2. Explain whether each method makes sense mathematically.
A) Katrina’s Method
65 +5 60
+30
-28 +2 30
______
37
B) Jenny’s Method
65-28
65-20=45
45-8=37
C) Debbie’s Method
65 – 28
+2 +2
67 – 30
=37
D) Seth’s Method
5 1
65
-28
_____
37
_____
(Adapted From: Uncovering Student Thinking in Mathematics Grades K-5: 25 Formative Assessment Probes for the Elementary Classroom, Tobey
and Minton, Corwin Press, 2011)
6
Math Formative Assessment Lesson
Alpha Version
Card Set A
A) Jessica’s Method B) Charlie’s Method
56-39
56
56-30=26
-39
______
26-10=16
17
16+1=17
+6
50
+10
+1
40
C) Renee’s Method
56 – 39
+1 +1
57 – 40 = 17
D) Teresa’s Method
4 1
56
-39
_____
17
_____
7
Math Formative Assessment Lesson
Alpha Version
8
Math Formative Assessment Lesson
Alpha Version
Card Set A
9
Math Formative Assessment Lesson
Alpha Version
10
Math Formative Assessment Lesson
Alpha Version
Activity B
42-28
11
Math Formative Assessment Lesson
Alpha Version
Activity C
81-56
12
Formative Assessment Lesson Materials
Alpha Version
A Snail in the Well
Mathematical goals
This lesson unit is intended to help you assess how well students are able to use addition and
subtraction in a problem solving situation. In particular, this lesson aims to identify and help
students who have difficulties with:
• Choosing an appropriate, systematic way to collect and organize data.
• Examining the data and looking for patterns
• Describing and explaining findings clearly and effectively.
Common Core State Standards
This lesson involves a range of mathematical practices from the standards, with emphasis on:
1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
8. Look for and make use of repeated reasoning.
This lesson asks students to select and apply mathematical content from across the grades,
including the content standards:
K-OA: Understand addition as putting together and adding to, and understand subtraction as
taking apart and taking from.
1-OA: Represent and solve problems involving addition and subtraction.
2-OA: Represent and solve problems involving addition and subtraction.
3-OA: Solve problems involving the four operations, and identify and explain patterns in
arithmetic.
Introduction
This lesson unit is structured in the following way:
• Before the lesson, students attempt the task individually. You then review their work and
formulate questions for students to answer in order for them to improve their work.
• At the start of the lesson, students work individually to answer your questions.
• Next, they work collaboratively, in small groups, to produce a better collective solution than
those they produced individually. Throughout their work, they justify and explain their decisions
to peers.
• In the same small groups, students critique examples of other students’ work.
• In a whole-class discussion, students explain and compare the alternative approaches they have
seen and used.
• Finally, students work alone again to improve their individual solutions.
Materials required
•
Each individual student will need two copies of the worksheet A Snail in the Well.
• Each small group of students will need d a copy of Sample Responses to Discuss and whichever
samples of student work chosen.
Time needed
Approximately fifteen minutes before the lesson, a one-hour lesson, and ten minutes in a followup lesson. All times are approximate. Exact timings will depend on the needs of the class.
Before the lesson
Assessment task:
Have the students do this task in class a day or more before the formative assessment lesson. This
will give you an opportunity to assess the work and to find out the kinds of difficulties students
have with it. Then you will be able to target your help more effectively in the follow-up lesson.
Give each student a copy of A Snail in the Well. Introduce the task briefly and help the class to
understand the problem and its context.
Spend fifteen minutes on your own, answering
this question. Show your work.
Don’t worry if you are not sure if your solution
is correct. There will be a lesson on this
material [tomorrow] that will help you improve
your work.
Your goal is to be able to answer this question
with confidence by the end of that lesson.
It is important that students answer the question without assistance, as far as possible. Students
who sit together often produce similar answers, and then, when they come to compare their work,
they have little to discuss. For this reason, we suggest that when students do the task individually,
you ask them to move to different seats. Then at the beginning of the formative assessment
lesson, allow them to return to their usual places. Experience has shown that this produces more
profitable discussions.
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their
current levels of understanding and their different problem solving approaches. The purpose of
this is to forewarn you of the issues that will arise during the lesson, so that you may prepare
carefully.
We suggest that you do not score students’ work. The research shows that this is
counterproductive, as it encourages students to compare scores, and distracts their attention from
how they may improve their mathematics.
Instead, help students to make further progress by asking questions that focus attention on aspects
of their work. Some suggestions for these are given on the next page. These have been drawn
from common difficulties observed in trials of this unit.
We suggest that you write your own lists of questions, based on your own students’ work, using
the ideas below. You may choose to write questions on each student’s work. If you do not have
time to do this, select a few questions that will be of help to the majority of students. These can be
written on the board at the beginning of the lesson.
Common issues - Suggested questions and prompts:
Common Issues
Student forgets to consider what the snail does
each day and each night.
Suggested questions and prompts
•
•
•
Student work is unsystematic.
•
•
•
Student assumes that the initial pattern continues
indefinitely and over-generalizes.
•
•
Student writes answer without explanation.
•
•
Student correctly identifies when the snail gets out
of the well.
•
•
How could you simplify this into an easier
task?
What sort of diagram might be helpful?
How can you show the path the snail
follows until he gets out of the well?
What pattern do you notice?
What is the same and what is different
about the how the snail moves during the
day and at night?
How can you organize your work?
What assumptions can you make about
how far the snail travels each day?
Does the snail always fall back?
How could you explain/show how you
reached your conclusions so that someone
in another class understands?
How can you use numbers, words, or
diagrams to describe the path of the snail?
Think of another way of solving the
problem. Is this method better or worse
than your original one? Explain your
answer.
Can you make a new problem with a
different size well and/or a snail that travels
different amounts each day and night?
Suggested lesson outline
Improve individual solutions to the assessment task (10 minutes)
Return your students’ work on the A Snail in the Well problem. Ask students to re-read both the A
Snail in the Well problem and their solutions. If you have not added questions to students’ work,
write a short list of your most common questions on the board. Students can then select a few
questions appropriate to their own work and begin answering them.
Recall what we were working on previously. What was the task?
Draw students’ attention to the questions you have written.
I have read your solutions and I have some questions about your work.
I would like you to work on your own to answer my questions for ten minutes.
Collaborative activity 1:
Organize the students into small groups of two or three. In trials, teachers found keeping groups
small helped more students play an active role.
Students should now work together to produce a joint solution.
Put your solutions aside until later in the lesson. I want you to work in groups now.
Your task is to work together to produce a solution that is better than your individual
solutions.
You have two tasks during small-group work, to note different student approaches to the task, and
to support student problem solving.
Note different student approaches to the task
Notice how students work on finding the path of the snail. Notice what strategies they use and
how they organize their data. Note the representations they use, including incorrect versions, for
use in whole-class discussion. You can use this information to focus the whole-class plenary
discussion towards the end of the lesson.
Support student problem solving
Try not to make suggestions that move students towards a particular approach to this task.
Instead, ask questions to help students clarify their thinking. If several students in the class are
struggling with the same issue, you could write a relevant question on the board. You might also
ask a student who has performed well on one part of the task to help a student struggling with that
part of the task.
The following questions and prompts would be helpful:
What information have you been given?
What do you need to find out?
What changes in the diagram? What stays the same?
How will you write down your pattern?
Why do you think your conjecture might be true?
Teachers who have used this activity noted that the students sometimes lose sight of both color
patterns and how the patterns are related. If this issue arises in your class, help the student to
focus his or her attention on both the white and black bead patterns and the connections between
the patterns.
How are the white beads changing?
How are the black beads changing?
What effect does the change of the white bead have on the change in the black beads?
You may find that some students do not work systematically when organizing their data.
What can you do to organize your data to show both patterns?
If students have found function rules or equations, focus their attention on improving
explanations, or exploring alternative methods.
How can you be sure your explanation works in all cases?
Ask another group if your argument makes sense.
Show me that this equation works.
Some stronger explanations are shown in the Sample Responses to Discuss.
Make a note of student approaches to the task
Give each small group of students a copy of the Sample Responses to Discuss. Choose the
samples of student work that match your students’ level of understanding. Display the following
questions on the board or OHP using the provided sheet: Analyzing Student responses to discuss.
Describe the problem solving approach the student used.
You might, for example:
• Describe the way the student has organized the data.
• Describe what the student did to calculate the day the snail gets out of the well.
Explain what the student could do to make his or her solution correct or clearer if they
calculated correctly.
This analysis task will give students an opportunity to evaluate a variety of alternative approaches
to the task, without providing a complete solution strategy.
During small-group work, support student thinking as before. Also, check to see which of the
explanations students find more difficult to understand. Identify one or two of these approaches to
discuss in the plenary discussion. Note similarities and differences between the sample
approaches and those the students took in small-group work.
Plenary whole-class discussion comparing different approaches (20 minutes)
Organize a whole-class discussion to consider different approaches to the task. The intention is
for you to focus on getting students to understand the methods of working out the answers, rather
than just numerical solutions. Focus your discussion on parts of the two small-group tasks
students found difficult.
Let’s stop and talk about different approaches.
Ask the students to compare the different solution methods.
Read through your original responses and think about what you have learned this lesson.
Using what you have learned, try to improve your work.
Which approach did you like best? Why?
Which approach did you find it most difficult to understand?
Sami, your group used that method. Can you explain that for us?
Improving individual solutions to the assessment task (10 minutes)
If you are running out of time, you could schedule this activity for the next lesson or for
homework.
Make sure students have their original individual work on the A Snail in the Well task on hand.
Give them a fresh, blank copy of the A Snail in the Well task sheet.
If a student is satisfied with his or her solution, ask the student to try a different approach to the
problem and to compare the approach already used.
Solution
The snail will reach the top of the well on the 8th day. He does gain one foot per day,
so that at the end of the 7th day he is 7 feet from the bottom of the well. When he
travels up 3 ft on the 7th day he does not slide back down because he reaches the top
at 10ft and is out of the well.
Analysis of Student Responses to Discuss
Will’s Method
Will noticed that up 3 then down 2 always results in a gain of 1 ft. He overgeneralizes this fact and concludes in will take 10 days for the snail to get out of the
well.
Timothy’s strategy of noticing a gain of 1 foot per day is correct but he did not
systematically organize the data to show the complete path of the snail.
Will’s Solution
Whitney’s Method
Whitney made a list of each day and the amount the snail gained. She overgeneralizes this fact and concludes it will take 10 days for the snail to get out of the
well. Whitney’s strategy of noticing a gain of 1 foot per day is correct but she did not
show the complete path of the snail.
Whitney’s Solution
Chuck’s Method
Chuck correctly shows subtraction and addition calculations organized by day so
that he arrives at the correct solution to the problem. He could enhance his solution
by labeling each day and night clearly and/or showing an additional representation
(picture, words, chart, etc.) to validate the solution.
Chuck’s Solution
Tim’s Method
Tim draws a diagram and attempts to show the path of the snail including how the
snail moves up the well during each day and slides back each night. However, his
diagram is not clear and consistent so he cannot accurately tell how many total feet
the snail has traveled.
Tim’s diagram could be enhanced with additional labels showing feet, day, night, etc.
Tim’s Solution
Denise’s Method
Denise clearly and accurately shows the path of the snail, labeling feet, day and
night. Her distinction in bold versus slashed lines makes the distinction between
night and day easy to see. To enhance her solution, Denise could verify her solution
with another representation (chart, words, etc.) to show how they are related.
Denise’s Solution
Bill’s Method
Bill made a chart to organize the data in this problem. He makes columns to identify
the overall day in the well, the distance the snail is from the top of the well each day
and each night and the total feet the snail is from the top at the end of the day. On
the 8th day Bill shows there are 0 feet left for the snail to travel. His work is
organized and labeled so that he correctly arrives at an accurate solution.
Bill’s labels may be a bit confusing for someone else reading his work. He could
enhance his solution by clearly labeling the columns so it easy to tell the difference
between day and d, and N and total. He could also further verify the solution with
alternative representation.
Bill’s Solution
A Snail in the Well
Student Materials
Alpha Version
A Snail in the Well
Student Materials
Alpha Version
Sample Responses to Discuss
Here is some work on A Snail in the Well from students in another class.
For each piece of work:
1. Write the name of the student whose solution you are analyzing.
2. Describe the problem solving approach the student used.
For example, you might:
• Describe the way the student has organized the data.
• Describe what the student did to calculate the day the snail reaches the top of
the well.
3. Explain what the student needs to do to complete or correct his or her solution.
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
A Snail in the Well
Student Materials
Alpha Version
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Will’s Solution
Whitney’s Solution
Chuck’s Solution
Tim’s Solution
Denise’s Solution
Bill’s Solution
Formative Assessment Lesson Materials
Alpha Version
Multiplication, Division, and Interpreting
Remainders
Mathematical goals
This lesson unit is intended to help you assess how well students are able to use a variety of strategies to
multiply and divide. In particular, this unit aims to identify and help students who have difficulties with:
• The traditional multiplication algorithm.
• The traditional division algorithm.
• Interpreting remainders.
Common Core State Standards
This lesson involves a range of mathematical practices from the standards, with emphasis on:
2. Reason abstractly and quantitatively.
7. Look for and make use of structure.
8. Look for and make use of repeated reasoning.
This lesson asks students to select and apply mathematical content from across the grades, including the
content standards:
3-OA: Represent and solve problems involving multiplication and division.
3-OA: Understand properties of multiplication and the relationship between multiplication and
division.
4-OA: Use the four operations with whole numbers to solve problems.
5-OA: Operations and Algebraic Thinking
3-NBT: Use place value understanding and properties of operations to perform multi-digit
arithmetic.
4-NBT: Use place value understanding and properties of operations to perform multi-digit
arithmetic.
5-NBT: Perform operations with multi-digit whole numbers.
Introduction
This lesson unit is structured in the following way:
• Before the lesson, students work individually on an assessment task that is designed to reveal their
current understanding and difficulties. You then review their work, and formulate questions for students
to answer, to help them improve their solutions.
• During the lesson, students work in pairs and threes to match the word problem, model, and multiple
strategies of the same multiplication or division problem.
• In a whole-class discussion, explain their answers.
• Finally, students return to their original assessment task, and try to improve their own responses.
Materials required
Each individual student will need two copies of the worksheet Multiplication, Division, and Interpreting
Remainders.
• Each small group of students will need a packet of Card Set A and Card Set B copied in color.
• The card sets should be cut up before the lesson.
Formative Assessment Lesson Materials
Time needed
Alpha Version
Approximately fifteen minutes for the assessment task, a one-hour lesson, and 15 minutes for the students
to review their work for changes. All timings are approximate. Exact timings will depend on the needs of
the class.
Before the lesson
Assessment task:
Have the students do this task in class a day or more before the formative assessment lesson. This will
give you an opportunity to assess the work and to find out the
kinds of difficulties students have with it. Then you will be able
to target your help more effectively in the follow-up lesson.
Give each student a copy of Multiplication, Division, and
Interpreting Remainders. Introduce the task briefly and help the
class to understand the problem and its context.
Spend fifteen minutes on your own, answering these
questions.
Don’t worry if you can’t figure it out. There will be a
lesson on this material [tomorrow] that will help you
improve your work. Your goal is to be able to answer
these questions with confidence by the end of that
lesson.
It is important that students answer the question without
assistance, as far as possible.
If students are struggling to get started, ask them questions that
help them understand what is required, but do not do the task for
them.
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their current
levels of understanding and their different problem solving approaches. The purpose of this is to forewarn
you of the issues that will arise during the lesson, so that you may prepare carefully.
We suggest that you do not score students’ work. The research shows that this is counterproductive, as it
encourages students to compare scores, and distracts their attention from how they may improve their
mathematics.
Instead, help students to make further progress by asking questions that focus attention on aspects of their
work. Some suggestions for these are given on the next page. These have been drawn from common
difficulties anticipated.
We suggest that you write your own lists of questions, based on your own students’ work, using the ideas
below. You may choose to write questions on each student’s work. If you do not have time to do this,
select a few questions that will be of help to the majority of students. These can be written on the board at
the beginning of the lesson.
Formative Assessment Lesson Materials
Alpha Version
Common issues: Suggested questions and prompts:
Common Issues
Student doesn’t match the cards correctly
because he or she doesn’t have a conceptual
understanding of multiplication.
Student doesn’t understand Distributive
Property.
Student doesn’t understand the area model
for multiplication.
Students don’t know what to do with the
remainder.
Suggested questions and prompts
• If you are multiplying 27x4, what
does the 2 represent? the 7?
• What would happen if you multiplied
20 x 4 and 7 x 4? Could you take
those answers and do a calculation to
get the answer to 27x4?
• How can these number(s) we are
multiplying be broken apart?
• What could you do with those
numbers to solve this problem?
• In the problem 27x14 let’s look at the
number 27. How many 10s are in 27?
How many ones? How could you
model 27? Now let’s look at 14? How
many tens? ones? How could you
model 14?
• Is there a way to take those two
models and fit them on a rectangle to
discover 27x14 without doing any
calculations?
• What does this remainder represent?
Suggested lesson outline
If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, then we
suggest you end the lesson after the first collaborative activity and continue in a second lesson.
Collaborative activity 1: matching Card Sets Models 1A, 1B, 1C, 1D, 1E, and 1F (30 minutes)
Organize the class into groups of two or three students. With larger groups, some students may not fully
engage in the task.
Give each group Card Sets 1A, 1B, and 1C.
Formative Assessment Lesson Materials
Alpha Version
Introduce the lesson carefully:
I want you to work as a team. Take it in turns to match
a Model card with either a Lattice card or a
Distributive Property card.
Each time you do this, explain your thinking clearly and
carefully. If your partner disagrees with the placement
of a card, then challenge him/her. It is important that
you both understand the math for all the placements.
There is a lot of work to do today, and it doesn't matter
if you don't all finish. The important thing is to learn
something new, so take your time.
Your tasks during the small group work are to make a note of student approaches to the task, and to
support student problem solving
You can then use this information to focus a whole-class discussion towards the end of the lesson. In
particular, notice any common mistakes. For example, students may know Lattice multiplication and the
Distributive Property but may not understand the model.
Make a note of student approaches to the task
Try not to make suggestions that move students towards a particular approach to this task. Instead, ask
questions to help students clarify their thinking. Encourage students to use each other as a resource for
learning.
Students will correct their own errors once the Partial Product and Traditional Algorithm cards are added.
For students struggling to get started:
There is more than one way to tackle this task. Can you think what
one of them might be? [Working out the
answer in either the Distributive Property
problem or the Lattice multiplication model
and matching that card to the model or
finding the answer from the model and
matching it to either the Lattice or
Distribution Property card.]
How can you calculate products with the
Distributive Property? with the Lattice
model?
This Distributive Property card shows (20 +
2) x (10+5)? What would the original multiplication problem be for this model? Does that
multiplication problem match any of the other cards on the table?
Formative Assessment Lesson Materials
Alpha Version
If one student has placed a particular card, challenge their partner to provide an explanation.
Maria placed this Lattice card with this Model. Martin, why has Maria placed it here?
If you find students have difficulty articulating their decisions, then you may want to use the questions
from the Common Issues table to support your questioning.
If the whole class is struggling on the same issue, then you may want to write a couple of questions on the
board and organize a whole class discussion.
Placing Card Set 1D and 1E: Partial Products and Traditional Algorithm
As students finish placing the Model, Distributive Property, and Lattice cards, hand out Card Set 1D and
1E: Partial Products and Traditional Algorithm. These provide students with a different way of
interpreting the situation.
Do not collect the card sets they have been using. An important part of this task is for students to make
connections between different representations of multiplication problems.
As you monitor the work, listen to the discussion and help students to look for patterns and
generalizations.
Extension activity
Hand out Card Set 1F: Word Problems to students who finish. These are word problems that will match
each set of representations.
Wrap up the lesson when students are completely satisfied with their own work by handing out the poster
template Representations of Multiplication. Students should use it to record the position of their cards.
The poster template allows students to record their finished work. It should not replace the cards during
the main activities of this lesson as students can more easily make changes when working with the cards,
and they encourage collaboration.
Sharing work (10 minutes)
When students get as far as they can with matching cards, ask one student from each group to visit
another group's work. Students remaining at their desk should explain their reasoning for the matched
cards on their own desk.
If you are staying at your desk, be ready to explain the reasons for your group's matches.
If you are visiting another group, write your card placements on a piece of paper. Go to another
group's desk and check to see which matches are different from your own.
If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
Formative Assessment Lesson Materials
Alpha Version
When you return to your own desk, you need to consider, as a group, whether to make any
changes to your work.
Students may now want to make changes.
Taking two lessons to complete second card sort
The second card sort, which deals with division, will be completed after the Representation of
Multiplication card sort. Repeat the process used in the previous card sort when using the Division Card
Sort.
Hand out Card Sets 2A and 2B. Once students have matched these sets correctly, give them sets 2C and
2D.
Use Card Set 2E for extensions.
Improve individual solutions to the assessment task (10 minutes)
Return to the students their original assessment, Multiplication, Division, and Interpreting Remainders, as
well as a second blank copy of the task.
Look at your original responses and think about what you have learned this lesson.
Using what you have learned, try to improve your work.
If you have not added questions to individual pieces of work then write your list of questions on the
board.
Students should select from this list only the questions appropriate to their own work.
If you find you are running out of time, then you could set this task in the next lesson, or for homework.
Solutions
Assessment Task: Multiplication, Division, & Interpreting Remainders
For questions 1 and 2 student responses will vary depending on how they solve multiplication
problems. Be sure that the responses have correct interpretations of each model.
Question 3
a) $50.00 ÷ 8 = $6 R 2 (or $6.25)
b) 25 ÷ 7 = 3 R 4
c) $60.00 ÷ $8.00 = 7 R 4
Report remainder as a decimal because it is money.
Round the remainder up because the 4 people will
need a bench to sit on.
Ignore the remainder because you don’t have
enough left over to buy a whole pizza.
These materials were adapted from Everyday Mathematics, Uncovering Student Misconceptions in
Mathematics, and the National Library of Virtual Manipulatives.
When teaching these multiplication and division strategies, Teaching Student Centered Mathematics
by Van de Walle will be a useful resource.
Student Materials
Alpha Version
CARD SET 1 – REPRESENTATIONS OF MULTIPLICATION
Model
Card Set 1A
Lattice
Card Set 1B
Distributive Property
Card Set 1C
Partial Products
Card Set 1D
Traditional
Card Set 1E
Problem
Card Set 1F
15
X 22
10
20
100
200
15
X 22
30
300
Each pack of
baseball cards has
15 cards. How
many cards are in
22 packs?
12
X 12
4
20
20
100
12
X 12
24
120
How many eggs
are in 12 dozen?
11
X 23
3
30
20
200
11
X 23
33
220
The boy scout
troop eats about
23 grapes each on
their campout.
How many total
grapes did the
troop of 11 boys
eat?
26
X 17
42
140
60
200
26
X 17
182
260
14
X 19
36
90
40
100
14
X 19
126
140
13
X 28
24
80
60
200
13
X 28
104
260
The deck Scott is
building needs 26
boards and each
board needs 17
nails. How many
nails does Scott
need to buy?
An opossum
sleeps an average
of 19 hours per
day. How many
hours does it
sleep in a 2-week
time period?
Cam bought 13
different colored
folders and each
had 28 dots. How
many total dots
are on her
folders?
21
X 25
5
100
20
400
21
X 25
105
420
15
X 15
25
50
50
100
15
X 15
75
150
Bags of reese’s
cups have 21
individually
wrapped peanut
butter cups. How
many pieces are
in 25 bags?
The zoo has 15
monkeys who eat
15 bananas each
day. How many
bananas does the
need each day for
the monkeys?
CARD SET 2 - DIVISON
Model
Groupings
Remainder
CARD SET 2A
CARD SET 2B
CARD SET 2C
2 7/18
Interpreting
Remainders
CARD SET 2D
Write the
remainder as a
fraction.
?
7
?
Round up the
remainder.
Problem
CARD SET 2E
The 18 cheerleaders each
want a piece of pink ribbon
to wear for the breast
cancer march. There is 43
inces of ribbon. How much
ribbon should each girl
get?
You are organizing a trolley
ride for 95 total students,
teachers & parents. If each
trolley can seat 15 people,
how many trolleys do you
need?
5
There is no
remainder.
Mr. Jones bought 95 new
pencils to give his class of
19 students. How many
pencils will each student
get?
3.67
Write the
remainder as a
decimal.
The soccer team bought
their coach a $55.00
sweatshirt. The 15 players
split the bill evenly. How
much did each pay?
6
Ignore the
remainder.
Compact discs are on sale
for $13.00 including tax.
How many can you buy
with $84.00?
?
?
?
14
There is no
remainder.
There are 84 girls in a
basketball league and 6
girls on each team. How
many teams are there?
7
Ignore the
remainder.
Sally scooped out 87 pieces
of candy to buy at the
store. She wants to divide
the candy evenly among 12
people. How many pieces
of candy will each person
get?
?
?
Formative Assessment Lesson
Alpha Version
Arrays, Number Puzzles, and Factor Trees
Mathematical goals
This lesson unit is intended to help you assess how well students are able to use “clues” about numbers
including: factors, multiples, prime, composite, square, even, odd, etc. In particular, this unit aims to
identify and help students who have difficulties with:
• Understanding the difference between primes and composites, or factors and multiples.
Common Core State Standards
This lesson involves a range of mathematical practices from the standards, with emphasis on:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
6. Attend to precision.
7. Look for and make use of structure.
This lesson asks students to select and apply mathematical content from across the grades, including the
content standards:
4-OA: Use the four operations with whole numbers to solve problems.
4-OA: Gain familiarity with factors and multiples.
5-NBT: Understand the place value system.
6-NS: Compute fluently with multi-digit numbers and find common factors and multiples.
Introduction
This lesson unit is structured in the following way:
• Before the lesson, students work individually on an assessment task that is designed to reveal their
current understanding and difficulties. You then review their work, and formulate questions for students
to answer, to help them improve their solutions.
• During the lesson, students work in pairs and threes to match the array cards, number clues and factor
trees.
• In a whole-class discussion, explain their answers.
• Finally, students return to their original assessment task, and try to improve their own responses.
Materials required
Each individual student will need two copies of the worksheet Arrays, Number Puzzles, and Factor Trees.
• Each small group of students will need a packet of Card Set 1: Arrays A-J, Number Puzzles 1-8, &
Factor Trees a-g; and Card Set 2: Arrays K-T, Number Puzzles 9-16, & Factor Trees h-n copied and cut
up before the lesson.
Formative Assessment Lesson
Alpha Version
Time needed
Approximately fifteen minutes for the assessment task, a one-hour lesson, and 15 minutes for the students
to review their work for changes. All timings are approximate. Exact timings will depend on the needs of
the class.
Before the lesson
Assessment task:
Have the students do this task in class a day or more before the formative assessment lesson. This will
give you an opportunity to assess the work and to find out the kinds of difficulties students have with it.
Then you will be able to target your help more effectively in the follow-up lesson.
Give each student a copy of Arrays, Number Puzzles & Factor Trees. Introduce the task briefly and help
the class to understand the problem and its context.
Spend fifteen minutes on your own,
answering these questions.
Don’t worry if you can’t figure it out.
There will be a lesson on this material
[tomorrow] that will help you improve
your work.
Your goal is to be able to answer these
questions with confidence by the end of
that lesson.
It is important that students answer the question without assistance, as far as possible.
If students are struggling to get started, ask them questions that help them understand what is required,
but do not do the task for them.
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their current
levels of understanding and their different problem solving approaches. The purpose of this is to forewarn
you of the issues that will arise during the lesson, so that you may prepare carefully.
We suggest that you do not score students’ work. The research shows that this is counterproductive, as it
encourages students to compare scores, and distracts their attention from how they may improve their
mathematics.
Formative Assessment Lesson
Alpha Version
Instead, help students to make further progress by asking questions that focus attention on aspects of their
work. Some suggestions for these are given on below.. These have been drawn from common difficulties
anticipated.
We suggest that you write your own lists of questions, based on your own students’ work, using the ideas
below. You may choose to write questions on each student’s work. If you do not have time to do this,
select a few questions that will be of help to the majority of students. These can be written on the board at
the beginning of the lesson.
Common issues - Suggested questions and prompts:
Common Issues
Student doesn’t find all factor pairs for a
given number.
Suggested questions and prompts
• How can you make sure you haven’t
left out any factor pairs for a number?
• How can you use rectangular arrays
to decide if you have all the factor
pairs? How can you use a factor tree
to decide if you have all the factor
pairs? How could you make a chart to
decide if you have all the factor pairs?
Conceptual understanding of prime numbers.
• How many other rectangular arrays
Student doesn’t realize that a prime number
can you find for each number? What
will have only one factor pair, only one
do you notice about the number of
rectangular array, and it will not have a factor
rectangular arrays possible for each
tree if it is prime. Student should also
number? Which number(s) do not
recognize the only EVEN prime number is 2.
have factor tree? Why not?
• Why are more of the numbers even
than odd?
One is neither prime nor composite. Students
• Why don’t any of the factor trees
do not understand that you do not use one as
include the number 1 as a factor?
a factor in factor trees.
Not understanding that if there are an odd
• Which rectangular arrays are actually
number of factors for a number, the number
squares? How many total factors do
is a square number.
each of your numbers have? How can
you tell if a number will have a square
as one of its arrays?
Suggested lesson outline
If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, then we
suggest you end the lesson after the first collaborative activity and continue in a second lesson.
Collaborative activity 1- matching Card Set 1: Arrays A-J, Number Puzzles 1-8, & Factor Trees a-g
(30 minutes)
Formative Assessment Lesson
Alpha Version
Organize the class into groups of two or three students. With larger groups, some students may not fully
engage in the task.
Give each group Card Set 1: Arrays A-J, Number Puzzles 1-8, and Factor Trees a-g.
Introduce the lesson carefully:
I want you to work as a team. Take it in
turns to match am Array with either a
Number Puzzle or a Factor Tree.
Each time you do this, explain your
thinking clearly and carefully. If your
partner disagrees with the placement
of a card, then challenge him/her. It is
important that you both understand the
math for all the placements.
There is a lot of work to do today, and
it doesn't matter if you don't all finish.
The important thing is to learn
something new, so take your time.
Your tasks during the small group work are to make a note of student approaches to the task, and to
support student problem solving
You can then use this information to focus a whole-class discussion towards the end of the lesson. In
particular, notice any common mistakes. For example, students may know how to find the number
represented by each array or factor tree, but have difficulty finding the answer for each number puzzle.
Formative Assessment Lesson
Formative Assessment Lesson
Alpha Version
Alpha Version
Make a note of student approaches to the task
Try not to make suggestions that move students towards a particular approach to this task. Instead, ask
questions to help students clarify their thinking. Encourage students to use each other as a resource for
learning.
Students will correct their own errors as they find matches for all the cards.
For students struggling to get started:
There is more than one way to tackle this task.
Can you think what
one of them might be? [Finding the number
represented by the rectangular arrays is the
most efficient way to find the numbers that will
match the number puzzle and factor trees.]
How can you narrow down which numbers
might be the solution to each number puzzle?[It
may be better to give students who were unable to
complete much of the pre-assessment task, only the
arrays & factor trees to begin with, then give them
the set of number puzzles once they have the list of
possible numbers].
The factor trees are missing numbers, how can
you figure out which numbers go in the blanks?
The use of a hundred’s board may be beneficial
to some students to “mark off” numbers that do
not fit the criteria for the number puzzles. All
of the solutions are for numbers of 100 or less.
If one student has placed a particular card, challenge their partner to provide an explanation.
Maria placed this Array card with this Number Puzzle. Martin, why has Maria placed it here?
If you find students have difficulty articulating their decisions, then you may want to use the questions
from the Common Issues table to support your questioning.
If the whole class is struggling on the same issue, then you may want to write a couple of questions on the
board and organize a whole class discussion.
Collaborative activity 2 - matching Card Set 2:
Arrays K-T, Number Puzzles 9-16, & Factor Trees h-n (30 minutes)
Formative Assessment Lesson
Formative Assessment Lesson
Alpha Version
Alpha Version
Card Set 2 can be used in the exact same way as card set one. Set 2 does include clues using the
exponents which are not included in set 1. You may have students work on set one the first day and set
two a second day, or have all groups of students use card set one and only have card 2 as an extension
activity for who finish the first card set quickly. If you have all students complete both sets, an
Extension Activity on either day would be to have students write their own number puzzles and make
corresponding arrays and factor trees to match the puzzles they have made. Groups who finish
making their own number puzzles can switch creations with other groups to test out the new puzzles.
Wrap up the lesson when students are completely satisfied with their own work by them glue or tape their
cards together in solution sets including arrays, puzzles & factor trees that match. They should have eight
matching solution sets for Card Set 1 and another eight for Card Set 2.
Sharing work (10 minutes)
When students get as far as they can with matching cards, ask one student from each group to visit
another group's work. Students remaining at their desk should explain their reasoning for the matched
cards on their own desk.
If you are staying at your desk, be ready to explain the reasons for your group's matches.
If you are visiting another group, write your card placements on a piece of paper. Go to another
group's desk and check to see which matches are different from your own.
If there are differences, ask for an explanation. If you still don't agree, explain your own thinking.
When you return to your own desk, you need to consider, as a group, whether to make any
changes to your work.
Students may now want to make changes.
Taking two lessons to complete second card sort
The second card set, will be completed after Card Set 1 has been taped or glued on a large sheet of paper.
Have students make up their own number puzzles as an extension n either day as needed for an extension.
Improve individual solutions to the assessment task (10 minutes)
Return to the students their original assessment, Arrays, Number Puzzles & Factor Trees, as well as a
second blank copy of the task.
Look at your original responses and think about what you have learned this lesson.
Using what you have learned, try to improve your work.
If you have not added questions to individual pieces of work then write your list of questions on the
board. Students should select from this list only the questions appropriate to their own work.
If you find you are running out of time, then you could set this task in the next lesson, or for homework.
Formative Assessment Lesson
Alpha Version
Solutions
Assessment Task: Arrays, Number Puzzles & Factor Trees
For questions 1 and 2 student responses will vary depending on how they solve multiplication
problems. Be sure that the responses have correct interpretations of each model.
Question 1:
# of
shifts
# of
hours
1
24
2
12
3
8
4
6
6
4
8
3
12
2
24
1
Question 2: The dimensions of the sides of the rectangles should be the factors for 32. Students
should drawn rectangles with the following dimensions:
1x32, 2x16, 4x8 (or the reverse 32x1, 16x2, 8x4)
Question 3: Numbers with exactly three factors would be square numbers. Square numbers
between 12 and 40 are: 16, 25, & 36. Only 25 has only three factors.
Question 4: only one because prime numbers only have two factors – one and itself.
For example 7 is prime and has only factors 1 and 7 so the only array would be 1x7 (or 7x1).
Question 5: Rosa is correct. Strings of factors at the bottom of a factor tree do not include 1.
Question 6: All prime numbers are odd, except for the number 2 so ODDS have more prime
numbers.
These materials were adapted from Connected Mathematics: Prime Time, and Investigations in
Number Data & Space: Mathematical Thinking.
Student Materials
Arrays, Number Puzzles & Factor Trees
Alpha Version
1.) A restaurant is open 24 hours a day. The manager wants to divide the day into workshifts of equal
length. Show the different ways this can be done. The shifts should not overlap, and all shifts should be
a whole number of hours long.
2.) Sammie’s Game Store wants to rent a space of 32 square units. Find all the possible ways Sammie
can arrange the squares. How are the rectangles you found related to the factors of 32?
3.) Lewis has chosen a mystery number. His number is larger than 12 and smaller than 40 and it has
exactly three factors. What could his number be?
4.) How many rectangles can you build with a prime number of square tiles?
5.) Rosa claims the longest string of factors at the bottom of a factor tree for 30 is 2x3x5. Lon claims
there is a longer string at the bottom of the factor tree: 1x2x1x3x1x5. Who is correct? Why?
6.) Which group of numbers – evens or odds- has more prime numbers? Why?
Set 1: Array Cards A – J
A
C
B
D
F
G
E
H
I
J
Set 1: Number Puzzles 1 – 8
Puzzle 1
1-This number of tiles will make a rectangle
3 tiles wide.
2-This number of tiles will make a rectangle
4 tiles wide.
3- This number is greater than 20.
4- This number is less than 30.
Puzzle 2
1-This number is a multiple of 5 and less
than 50.
2-This number is not an odd number.
3-This number has exactly 8 factors.
Puzzle 3
1-This number is a factor of 60.
2-This number has 2-digits
3-The sum of the digits is less than 5.
4-One factor of this number is 4.
Puzzle 4
1-This number has 2 digits
2-This number is a multiple of 9.
3-This number is even.
4-This number is a square number.
Puzzle 5
1-This number is a multiple of 3 and a
multiple of 5.
2-Add the digits in this number and the sum
is odd.
3-Multiply the digits and the product is
even.
4-This number is odd.
Puzzle 6
1-This number divided by 5, has a remainder
of 4.
2-This number has 2 digits which are both
even.
3-The sum of the digits of this number is 10.
Puzzle 7
1-This number is not an even number, and
less than 50.
2-The difference of the digits in this number
is 3.
3-This number is not a multiple of 3, 5, or 7.
4- This number is less than 50.
5-The sum of the digits in this number is 11.
Puzzle 8
1-The difference between the digits in this
number is greater than 3.
2-This number is a multiple of 7, and is not
odd.
3-Both digits are even, and the first digit is
larger than the second digit.
4-The sum of the digits in this number is
great than 10.
Set 1: Factor Trees a - g
a)
What number is represented by this factor tree?
x
14
x
2
x
7
x
3
x
x
2
x
3
x
16
x
2
x
x
2
x
4
x
x
2
x
x
3
f)
What number is represented by this factor tree?
2
g)
What number is represented by this factor tree?
2
3
5
2
x
x
3
6
3
x
2
d)
What number is represented by this factor tree?
x
4
x
3
e)
What number is represented by this factor tree?
2
2
9
x
2
2
x
4
c)
What number is represented by this factor tree?
2
b)
What number is represented by this factor tree?
x
x
5
ANSWER KEY SET 1
Array Card
E
C
F
G, J
B
H, I
A
D
Number Puzzle
1
2
3
4
5
6
7
8
Factor Tree
b
f
e
c
d
g
-a
Number
24
30
12
36
45
64
47
84
Set 2: Array Cards K – T
L
K
N
M
P
O
Q
R
S
T
Set 2: Number Puzzles 9 – 16
Puzzle 9
1-This number is a multiple of 2 and 7.
2-This number is less than 100 and larger
than 50.
3-This number is the product of 3 different
primes.
Puzzle 10
1-This number has two digits, and has 4 as a
factor.
2-The sum of the digits in this number is 10.
3-The difference between the digits is 6.
Puzzle 11
1-This number is less than 20.
2-The sum of the digits in this number is
greater than 7.
3-This number is prime.
Puzzle 12
1-This number is a factor of 103.
2-This number is less than 45 and greater
than 12.
3-This number is a multiple of 4.
Puzzle 13
1-This number is a multiple of 5, but does
not end in 5.
2-This number is bigger than the 7th square
number but smaller than the 9th square
number.
3-Two of the three numbers in the prime
factorization of this number are the same.
Puzzle 14
1-The sum of the two digits in this number is
odd.
2-The only prime number in this number’s
prime factorization is 2.
3-This number of tiles will make a square.
Puzzle 15
1-This number is a composite number.
2-This number is a factor of 103, and 102.
3-This number has exactly 9 factors.
4-The sum of the digits in this number is
neither prime nor composite.
Puzzle 16
1-This number is a multiple of 3 and a
multiple of 5.
2-The sum of the digits is even.
3-This number is the largest number less
than 100 that fits all the other clues.
4-When you multiply the digits of this
number you get an odd number for the
product.
Set 2: Factor Trees h – n
h)
What number is represented by this factor tree?
x
x
5
2
j)
What number is represented by this factor tree?
x
x
x
5
x
3
x
2
5
2
x
2
x
x
5
m)
What number is represented by this factor tree?
x
2
x
k)
What number is represented by this factor tree?
14
5
l)
What number is represented by this factor tree?
7
x
10
5
2
5
i)
What number is represented by this factor tree?
2
5
x
2
n)
What number is represented by this factor tree?
2
x
8
x
4
x
2
ANSWER KEY SET 2
Array Card
R
D, T
K
N, P
S
L
O
M
Number Puzzle
9
10
11
12
13
14
15
16
Factor Tree
k
l
-m
h
n
i
j
Number
70
28
17
20
50
16
100
75
Formative Assessment Lesson Materials
Alpha Version
Beads Under the Cloud
Mathematical goals
This lesson unit is intended to help you assess how well students are able to identify patterns
(both linear and exponential) in a realistic context: the number of beads of different colors that are
hidden behind the cloud. In particular, this unit aims to identify and help students who have
difficulties with:
• Choosing an appropriate, systematic way to collect and organize data.
• Examining the data and looking for patterns
• Describing and explaining findings clearly and effectively.
Common Core State Standards
This lesson involves a range of mathematical practices from the standards, with emphasis on:
3. Construct viable arguments and critique the reasoning of others.
7. Look for and make use of structure.
8. Look for and make use of repeated reasoning.
This lesson asks students to select and apply mathematical content from across the grades,
including the content standards:
4-OA: Generate and analyze patterns.
5-OA: Analyze patterns and relationships
6-EE: Represent and analyze quantitative relationships between dependent and
independent variables.
8-F: Use functions to model relationships between quantities.
F-LE: Linear, Quadratic, and Exponential Models★
Introduction
This lesson unit is structured in the following way:
• Before the lesson, students attempt the task individually. You then review their work and
formulate questions for students to answer in order for them to improve their work.
• At the start of the lesson, students work individually to answer your questions.
• Next, they work collaboratively, in small groups, to produce a better collective solution than
those they produced individually. Throughout their work, they justify and explain their decisions
to peers.
• In the same small groups, students critique examples of other students’ work.
• In a whole-class discussion, students explain and compare the alternative approaches they have
seen and used.
• Finally, students work alone again to improve their individual solutions.
Materials required
•
Each individual student will need two copies of the worksheet Beads under the Cloud.
• Each small group of students will need d a copy of Sample Responses to Discuss and whichever
samples of student work chosen.
Pg.1
Time needed
Approximately fifteen minutes before the lesson, a one-hour lesson, and ten minutes in a followup lesson (or for homework). All timings are approximate. Exact timings will depend on the
needs of the class.
Before the lesson
Assessment task:
Have the students do this task in class or for homework a day or more before the formative
assessment lesson. This will give you an opportunity to assess the work and to find out the kinds
of difficulties students have with it. Then you will be able to
target your help more effectively in the follow-up lesson.
Give each student a copy of Beads under a Cloud. Introduce the
task briefly and help the class to understand the problem and its
context.
Spend fifteen minutes on your own, answering these
questions.
Show your work.
Don’t worry if you can’t figure it out. There will be a
lesson on this material [tomorrow] that will help you
improve your work. Your goal is to be able to answer
this question with confidence by the end of that
lesson.
It is important that students answer the question without
assistance, as far as possible. Students who sit together often produce similar answers, and then,
when they come to compare their work, they have little to discuss. For this reason, we suggest
that when students do the task individually, you ask them to move to different seats. Then at the
beginning of the formative assessment lesson, allow them to return to their usual places.
Experience has shown that this produces more profitable discussions.
Assessing students’ responses
Collect students’ responses to the task. Make some notes on what their work reveals about their
current levels of understanding and their different problem solving approaches. The purpose of
this is to forewarn you of the issues that will arise during the lesson, so that you may prepare
carefully.
We suggest that you do not score students’ work. The research shows that this is
counterproductive, as it encourages students to compare scores, and distracts their attention from
how they may improve their mathematics.
Instead, help students to make further progress by asking questions that focus attention on aspects
of their work. Some suggestions for these are given on the next page. These have been drawn
from common difficulties observed in trials of this unit.
We suggest that you write your own lists of questions, based on your own students’ work, using
the ideas below. You may choose to write questions on each student’s work. If you do not have
time to do this, select a few questions that will be of help to the majority of students. These can be
written on the board at the beginning of the lesson.
Pg.2
Common issues - Suggested questions and prompts:
Common Issues
Student forgets to look at one of the sets of beads
(only looking at white or black, and not both).
Suggested questions and prompts
•
•
How could you simplify this into an easier
task?
What sort of diagram might be helpful?
Student work is unsystematic.
Student sees the patterns as two separate entities
and does not see the relationship between the
white and black patterns or how they alternate on
the string.
•
•
•
How do the black beads grow?
What patterns do you notice?
What is the same and what is different
about the patterns of the black & white
beads?
Student assumes the picture of the cloud is to scale
and that not very many beads can fit under the
cloud.
•
What assumptions can you make about the
size of the cloud?
Are all math diagrams always drawn to
scale?
Student writes answers without explanation.
•
•
•
How could you explain/show how you
reached your conclusions so that someone
in another class understands?
How can you use words and/or variables to
describe the patterns?
Student does not generalize.
•
Can you describe a visual pattern in the
black beads and the white beads? How
could I find out the number of white beads
that follow the set of ten black beads? Or
the set of 20 black beads?
Student correctly identifies the pattern for both
the black and the white beads.
•
Think of another way of solving the
problem. Is this method better or worse
than your original one? Explain your
answer. Can you extend your solution to
include exponents?
Pg.3
Suggested lesson outline
Improve individual solutions to the assessment task (10 minutes)
Return your students’ work on the Beads under a Cloud problem. Ask students to re-read both the
Beads under the Cloud problem and their solutions. If you have not added questions to students’
work, write a short list of your most common questions on the board. Students can then select a
few questions appropriate to their own work and begin answering them.
Recall what we were working on previously. What was the task?
Draw students’ attention to the questions you have written.
I have read your solutions and I have some questions about your work.
I would like you to work on your own to answer my questions for ten minutes.
Collaborative activity 1:
Organize the students into small groups of two or three. In trials, teachers found keeping groups
small helped more students play an active role.
Students should now work together to produce a joint solution.
Put your solutions aside until later in the lesson. I want you to work in groups now.
Your task is to work together to produce a solution that is better than your individual
solutions.
You have two tasks during small-group work, to note different student approaches to the task, and
to support student problem solving.
Note different student approaches to the task
Notice how students work on finding the patterns for the white and black beads. Notice what
strategies they use and how they organize their data. Notice also whether and when students
introduce algebra. If they do use algebra, note the different formulations of the functions they
produce, including incorrect versions, for use in whole-class discussion. You can use this
information to focus the whole-class plenary discussion towards the end of the lesson.
Support student problem solving
Try not to make suggestions that move students towards a particular approach to this task.
Instead, ask questions to help students clarify their thinking. If several students in the class are
struggling with the same issue, you could write a relevant question on the board. You might also
ask a student who has performed well on one part of the task to help a student struggling with that
part of the task.
The following questions and prompts would be helpful:
What information have you been given?
What do you need to find out?
What changes in the diagram? What stays the same?
How will you write down your pattern?
Why do you think your conjecture might be true?
Pg.4
Teachers who have used this activity noted that the students sometimes lose sight of both color
patterns and how the patterns are related. If this issue arises in your class, help the student to
focus his or her attention on both the white and black bead patterns and the connections between
the patterns.
How are the white beads changing?
How are the black beads changing?
What effect does the change of the white bead have on the change in the black beads?
You may find that some students do not work systematically when organizing their data.
What can you do to organize your data to show both patterns?
If students have found function rules or equations, focus their attention on improving
explanations, or exploring alternative methods.
How can you be sure your explanation works in all cases?
Ask another group if your argument makes sense.
Show me that this equation works.
Some stronger explanations are shown in the Sample Responses to Discuss.
Make a note of student approaches to the task
Give each small group of students a copy of the Sample Responses to Discuss. Choose the
samples of student work that match your students’ level of understanding. Display the following
questions on the board or OHP using the provided sheet: Analyzing Student responses to discuss.
Describe the problem solving approach the student used.
You might, for example:
• Describe the way the student has organized the data.
• Describe what the student did to calculate number of beads under the cloud.
Explain what the student could do to make his or her solution correct or clearer if they
calculated correctly.
This analysis task will give students an opportunity to evaluate a variety of alternative approaches
to the task, without providing a complete solution strategy.
During small-group work, support student thinking as before. Also, check to see which of the
explanations students find more difficult to understand. Identify one or two of these approaches to
discuss in the plenary discussion. Note similarities and differences between the sample
approaches and those the students took in small-group work.
Plenary whole-class discussion comparing different approaches (20 minutes)
Organize a whole-class discussion to consider different approaches to the task. The intention is
for you to focus on getting students to understand the methods of working out the answers, rather
than either numerical or algebraic solutions. Focus your discussion on parts of the two smallgroup tasks students found difficult.
Pg.5
Let’s stop and talk about different approaches.
Ask the students to compare the different solution methods.
Read through your original responses and think about what you have learned this lesson.
Using what you have learned, try to improve your work.
Which approach did you like best? Why?
Which approach did you find it most difficult to understand?
Sami, your group used that method. Can you explain that for us?
Improving individual solutions to the assessment task (10 minutes)
If you are running out of time, you could schedule this activity for the next lesson or for
homework.
Make sure students have their original individual work on the Beads under the Cloud task on
hand. Give them a fresh, blank copy of the Beads under the Cloud task sheet.
If a student is satisfied with his or her solution, ask the student to try a different approach to the
problem and to compare the approach already used.
This Formative Assessment Lesson was created around a task taken from Mathematics
for Elementary Teachers (A Contemporary Approach).
Solution
Pg.6
Analysis of Student Responses to Discuss
Timothy’s Method
Timothy made a list of the number of black and white beads. He found the correct
pattern of both sets of beads. However, because he did not systematically organize
the data, Timothy omits the 64 white beads.
Timothy’s strategy of only counting the beads under the cloud is correct and he
knew to subtract the beads from the pattern that were showing.
Timothy could use a table to better organize his data. By doing this, he could see
that the 16 black beads correspond to the 4 white beads, the 32 black beads
correspond with the 5 white beads, etc.
Timothy’s Solution
Hannah’s Method
Hannah followed the pattern correctly because she uses the numbers 16, 5, 32, 6,
and64 in her calculations even though she did not show the data in an organized list.
She miscalculated 64-5 and used 58 instead of 59 in her final sum of beads under
the cloud.
Hannah’s Solution
Curtis’ Method
Curtis correctly identified the pattern of the black beads. He knew to subtract the 2
beads from 32 and 5 beads from 8,192.
Curtis did not generalize the pattern of the white beads correctly. He multiplied the
two previous numbers to generate the next number of beads. If he were asked to
explain how this generalization could be applied to grow from 2 white beads to 4
white beads, he might realize that this rule does not apply to all numbers in the
pattern.
Curtis’ Solution
Tony’s Method
Tony generated both patterns correctly and
attempted to organize the data in a systematic
way. However, he does not see the
correspondence between the black and white
beads. Even though he knew to add the 5 and
6 black beads that he represented by the
question mark in his table, he doesn’t add the
32 and 64 white beads that correspond to
these numbers.
Tony could continue his pattern of the white
beads to possibly see that there would be
more white beads to add in to the total of
beads under the cloud.
Tony’s Solution
Claire’s Method
Claire’s Solution
Claire correctly generated both
patterns. She has labeled the
sequence of beads with a term
number.
Claire uses the correct strategy
for determining how many beads
are under the cloud even though
she does not show how she
arrived at the 14 and 60.
Claire’s total number of beads
under the cloud was incorrect
because she subtracted 4 from 64
instead of 5.
Victoria’s Method
Victoria drew a picture to demonstrate the pattern of the black and white beads
hidden by the cloud.
Victoria does not have a numerical answer about how many beads are hidden but
does have 5 black beads, 32 white beads, 6 black beads, and 64 white beads drawn.
She has 59 of the 64 white beads under the cloud. Victoria has not drawn the
correct number of white beads under the cloud (14) but does have 14 white beads
drawn.
Victoria’s Solution
Sam’s Method
Sam correctly calculated the number of beads hidden under the cloud. He listed the
pattern and showed his calculations. He attempts generalizing a rule to repeat the
pattern but does not identify the relationship between the corresponding terms.
Sam’s Solution
Rick’s Method
Rick uses algebra to determine the patterns of the beads. He generated the rule to
determine the next numbers in the pattern. Rick recognized the linear relationship
of the black beads and the exponential relationship of the white beads. He also
graphed the data.
Rick subtracted the correct number of white beads showing in the 4th and 6th set of
beads.
Rick’s Solution
Beads under the Cloud
Student Materials
Alpha Version
Beads under the Cloud
Student Materials
Alpha Version
Sample Responses to Discuss
Here is some work on Beads under the Cloud from students in another class.
For each piece of work:
1. Write the name of the student whose solution you are analyzing.
2. Describe the problem solving approach the student used.
For example, you might:
• Describe the way the student has organized the data.
• Describe what the student did to calculate a number of beads under the cloud.
3. Explain what the student needs to do to complete or correct his or her solution.
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Beads under the Cloud
Student Materials
Alpha Version
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
_____________’s_ Solution
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Timothy’s Solution
Hannah’s Solution
Curtis’ Solution
Tony’s Solution
Claire’s Solution
Victoria’s Solution
Sam’s Solution
Rick’s Solution