Balanced Assessment Test –Seventh Grade 2008
Core Idea
Task
Score
Will it Happen
Probability
This task asks students to describe events as likely or unlikely and calculate numerical
probabilities for simple and compound events. Students need to explain their thinking
and show a sample space for the situation. Successful students understand all the ways
to get a favorable outcome, recognizing that getting a number on one die is different
from getting the same number on the other die.
Algebra and Functions
Odd Numbers
This task asks students to draw and extend geometric patterns. Students need to also
recognize and extend numeric patterns involving odd numbers and square numbers.
Students should recognize the relationship between the number squared and the
number of elements in the pattern. Successful students could also work backward
from a total to describe the elements of the pattern for that result.
Pedro’s Tables
Number Properties
This task asks students to work with number properties including divisibility.
Students need to use properties of numbers, such as factors, multiples, prime numbers,
odd, and even to develop logical reasons for why numbers do or do not match a set of
constraints. Successful students could solve problems with multiple constraints, such
as factors of 12 less than 25, which are multiples of 3, to find solutions.
Winter Hat
Geometry and
Measurement
This task asks students to calculate the dimensions of material needed for a hat. They
need to be able to find circumference of a circle, and area of a rectangle, circle, and
trapezoid in order to find the surface area of a complex shape. Successful students had
strategies for organizing their work to make sure all the pieces in the pattern were
calculated and understood how to use the dimensions of a trapezoid to calculate its
area.
Sale!
Number Operations
This task asks students to reason about sales discounts and percents. Students need to
find a common unit to compare offers and develop a comparison of the different
options. Successful students were able to pick a single measure for comparing all the
options.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
1
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
2
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
3
Will it Happen?
This problem gives you the chance to:
• describe events as likely or unlikely as appropriate
• find the numerical probability of various outcomes of rolling a number cube
What does the future hold?
Select just one of these five words and write it after the following statements.
impossible
unlikely
equally likely
1. a. If today is Monday, tomorrow will be Tuesday.
likely
certain
__________________
b. Today you will meet President Lincoln on the way home from school. __________________
c. When you flip a coin it will land head up.
__________________
2 a. When you roll a number cube with faces numbered 1, 2, 3, 4, 5, 6, what is the numerical
probability of getting the number 4?
__________________
b. When you roll a number cube with faces numbered 1, 2, 3, 4, 5, 6, what is the numerical
probability it will land on an odd number?
Explain how you figured it out.
__________________
________________________________________________________________________________
________________________________________________________________________________
3. The faces of one red number cube and one blue number cube are labeled 1, 3, 5, 7, 9, 11.
The two cubes are rolled and the results are added.
What is the numerical probability of getting a total of 20?
Show how you figured it out.
_______________________
8
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
2
Will it Happen?
Rubric
•
• The core elements of performance required by this task are:
• describe events as likely or unlikely as appropriate
• find the numerical probability of various outcomes of rolling a number cube.
•
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
a.
Gives correct answers:
certain
b.
impossible
c.
equally likely
2
Partial credit
2 correct
(1)
2.a. Gives correct answer 1/6
b.
3.
2
1
Gives correct answer: 3/6 or 1/2
1
Gives correct explanation such as: there are 3 of 6 equally likely
possibilities
1
Gives correct answer 2/36 or 1/18
1
Shows work such as: there are 36 equally likely outcomes.
and 20 = 9 + 11 and 11 + 9
2
Partial credit
Allow partial credit for some correct work.
3
(1)
3
8
Total Points
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
3
Will It Happen?
Work the task. Look at the rubric. What are some of the big ideas about probability that
are being assessed in this task? What do you think might be problematic for students?
In part one, how many of your students chose unlikely for part b?_________________
What might they have been thinking?
Now look at student work for part 2a. How many of your students put:
1/6
2/3
Equally likely
Likely
Unlikely
Other
Why might students choose 2/3? Why might students use words instead of numbers?
What is the logic behind choosing equally likely or unlikely? What are the different
misconceptions for each answer?
Now look at part 2b of the task. How many of your students put:
3/6 or 1/2
1/3 or 2/6
Likely
Equally likely
3
Other
What are the misconceptions shown by these responses?
2/36
4/12
1/12
2/12
1/20
Unlikely
Likely
1/36
11 & 9
How were students thinking about sample space? Were they thinking about 1 die, 2 dice,
the total from adding the two dice or the total possible outcomes?
What experiences have your students had with probability?
How might probability be reinforced in the curriculum when teaching other concepts?
What kind of experiences do students bring to the classroom from previous years? How
should their knowledge of probability deepen at this grade level?
What is your role in helping to prepare students for the high school exit exam?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
4
Looking at Student Work on Will it Happen?
Student A is able to describe probabilities in words and numbers. Student A is able to
use an organized list to define the sample space in part 3. Student A is also able to verify
the total number of outcomes for 2 dice using multiplication.
Student A
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
5
Student B uses a diagram to find the sample space in part 3 and confirms the total
outcomes with multiplication. Notice the careful use of language in describing the
numerical probability in part 2b. Watch how the use of language changes for students
with lower total scores. How does understanding a working definition effect the type of
thinking in part 3?
Student B
Many students were successful in solving parts 1 and 2, but struggled with the thinking in
part 3. Student C is able to figure out the number of total outcomes and see that there are
2 combinations that will add to 20. However, the student confuses the numbers on the
dice with possible outcomes, similar to counting the odd numbers in part 2b. He should
have been counting the number of combinations.
Student C
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
6
Student D also receives credit for parts 1 and 2. The student uses the numbers on the
dice, which sum to 20 to make the probability. Look at the explanation in 2b. How
might the imprecision here have contributed to making errors in 3?
Student D
Student E again uses a weak explanation in part 2b. The student finds a combination to
make 20, but then thinks of the sample space as the total numbers of the 2 dice rather
than the total number of outcomes. How does the concept of sample space change when
going from one object or set to combining options? How do we help students grasp this
significant change? What questions might we ask to make these changes more explicit?
Student E
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
7
Student F has a good concept of finding numerical probabilities for 1 die. However the
student does not think about combining the rolls of two dice in part 3, but treats the
situation as an event with one die, using the logic of finding the odd numbers on one die.
Student F
Student G is only looking at the possibilities and loses the idea of context. The student
finds 4 numbers that will add to 20 and considers that as a possible outcome for rolling 2
dice.
Student G
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
8
Students with lower scores do not know how to find numerical probabilities. Student H
says that rolling a 4 is likely. Here the student is not thinking in numerical terms, getting
a 4 isn’t very likely because there are a lot of other choices; but she is thinking that it is a
possible outcome so it might happen. Do we think about the number sense students need
to choose the appropriate words versus the everyday use of the words? How do we help
students attach quantitative thinking to the probability language? Notice that the student
does not understand that the numbers on each die do not have to match in part 3.
Student H
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
9
Student I is good at reducing fractions. The student is on the borderline of understanding
very simple probability. In part 2b the student can explain how to find the numerical
probability for rolling an odd number. However in 2a the student confuses the roll of the
die with the number of possible outcomes. In part 3 the student considers only 1 die in
defining the sample space. The student here knows to think about outcomes instead of
numbers on the die. How are the three parts using the numerator in slightly different
ways?
Student I
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
10
Student J struggles with the quantitative thinking needed to use the probability words
correctly. In 1c the student is thinking that heads is likely to come out, but doesn’t
consider that it is 1 of 2 possible outcomes so it should be equally likely. In part 2 the
student does not attempt to use numbers to describe the probabilities. In part 3 the
student looks at the numbers needed to make the desired outcomes rather than thinking
about probability.
Student J
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
11
7th Grade
Student Task
Core Idea 2
Probability
Task 1
Will it Happen?
Describe events as likely or unlikely. Find the numerical probability of
various outcomes of rolling a number cube.
Apply and deepen understanding of theoretical and empirical
probability.
• Represent the sample space for simple and compound events in
an organized way.
• Represent probabilities as ratios, proportions, decimals or
percents.
• Determine theoretical and experimental probabilities and use
these to make predictions about events.
Mathematics of this task:
• Understand the quantitative thinking needed to use probability words, such as
impossible, likely, equally likely, certain, etc.
• Understand the difference between numbers on a die and outcomes
• Describe the numerical probability of a situation
• Understand and calculate the sample space for combined probabilities
• Distinguish between the number of ways an outcome can be obtained numerically
and the number of possible outcomes from a situation (9+11= 20 are the numbers
that make the desired outcome 20. The 9 could be on either the red or the blue die,
so there are two possible ways of making the 9 + 11.)
Based on teacher observations, this is what seventh graders know and are able to do:
• Use words to describe probabilities.
• Write a numerical probability for a single event, like rolling a 4 on a die or rolling
an even number.
• Describe how to find the probability for a simple event like rolling an odd
number.
Areas of difficulty for seventh graders:
• Finding sample space for a combined event, like finding the total of 2 die.
• Finding the number of possible outcomes for a combined event.
• Distinguishing between a number on a die and a desired combination or outcome.
Strategies used by successful students:
• More complete definitions for part 2b, using words like total possibilities instead
of the numbers on the die.
• Making an organized list or diagram to define the sample space in part 3
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
12
The maximum score available for this task is 8.
The minimum score needed for a level 3 response, meeting standards, is 4 points.
Most students, 86%, could use words to describe the probability of simple events. More
than half the students, 60%, could use words to describe two of the three events in part
one and could write numerical probabilities for events with one standard die and explain
how they figured it out in part two. 5% of the students could meet all the demands of the
task including finding the sample space for rolling two dice and finding the number of
favorable outcomes. 5% of the students scored no points on this task. All the students in
the sample with this score attempted the task.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
13
Will it Happen?
Points
Understandings
All the students in the sample
0
with this score attempted the
task.
1
Students could describe
probabilities in words for 2 of
the 3 choices in part 1.
2
Students could use words to
describe probabilities in part 1.
4
Students could use words to
describe 2 of the probabilities
in part 1. They could also
describe events with a single
die in numerical terms and
describe how they figured it
out.
Students could use words and
numbers to describe
probabilities in part 1 and 2.
Students realized that the
numbers 9 + 11 equaled the
desired outcome of 20.
6
8
Misunderstandings
14% of the students thought it was unlikely
rather than impossible to run into President
Lincoln. 10% of the students thought that it
was likely or unlikely to get heads when
tossing a coin. Students had trouble attaching
quantitative thinking to their word choices.
Students continued to use word descriptions
in part 2. 11% of the students thought rolling
a 4 was unlikely. 5% thought it was equally
likely to roll a 4 and another 5% thought it
was likely to roll a 4.
About 5% of the students thought that there
was a 4/6 or 2/3 chance of rolling a 4. 8% of
the students thought that it was equally likely
to get an odd number. They did not attempt
to write a numerical probability. 8% thought
it was likely to roll an odd number.
Students struggled with combined
probabilities in part 3. 11% still used words
to describe the probability in part 3.
Students did not know how to define the
sample space for the total number of possible
outcomes. 10% thought the probability was
1/12. 11% thought the probability was 2/12.
They were thinking about 12 numbers on the
two die, rather than the results of one die.
Many students wrote a probability with 6 in
the numerator, because there are 6 numbers
on a die. Some students had difficulty
distinguishing between the numbers on the
die and the outcomes of adding; so two
numbers are needed to make 20: 4/12, 2/6 or
1/3.
Students could use words and
numbers to describe simple
probabilities. Students could
find the sample space for a
combined event and find the
number of favorable outcomes
to write a probability.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
14
Implications for Instruction
Students in fourth grade learn to use words to describe the “likelihood” of an event. Some
students at this grade level are using these words with a more common usage than with a
quantitative sense. For example, students might say that it is likely to roll a 4 on a die,
because it is something that could happen rather than thinking there are more numbers on
a die that aren’t 4 than are 4. Many students do not know how to use numbers to write a
probability. Probability should be taught for mastery in sixth grade and students are held
accountable for this knowledge on the high school exit exam. Because of the changing
games used by students today, some students do not even understand the context of
rolling dice. For example Student H thought both die would generate the same numbers.
Teachers at later grades need to work these concepts into the curriculum to make sure
ideas are fully developed. For example, many students had trouble distinguishing the
numbers on the dice from the sample space when thinking about combined probabilities.
Students also need to know the difference between the actual numbers on the dice and the
numbers, which can be combined to get a favorable outcome. Example: If a nine and an
eleven are needed to make an outcome of 20, that is 2 numbers on the die, but one event
or outcome. Students need to go through several phases of understanding probability,
from the very concrete level of understanding how rolling dice work and how to combine
the results, to a pictorial level being able to make an organized list or diagram to show the
sample space, to an abstract level of using formulas to calculate probabilities. Students,
who had more complete definitions of what the two numbers are that make up a
numerical probability, seemed to do a better job of calculating a combined probability.
The students need to be engaged in rich discussions to bring out some of these subtleties
in writing probabilities that are not necessarily apparent when acquiring the procedural
knowledge. Some students tried to apply the knowledge or procedures for calculating the
probability for an event(s) on one die to finding the probability of combined event.
Ideas for Action Research: Re-engagement
One useful strategy when student work does meet your expectations is to use sample
work to promote deeper thinking about the mathematical issues in the task. In planning
for re-engagement it is important to think about what is the story of the task, what are the
common errors and what are the mathematical ideas that students need to think about
more deeply. Then look through student work to pick key pieces of student work to use
to pose questions for class discussion. Often students will need to have time to rework
part of the task or engage in a pair/share discussion before they are ready to discuss the
issue with the whole class. This reworking of the mathematics with a new eye or new
perspective is the key to this strategy.
To plan a follow-up lesson using this task, pick some interesting pieces of student work
that will help students confront and grapple with some of the major misconceptions.
Make the misconceptions explicit and up for public debate.
During the discussion, it is important for students to notice and point out the errors in
thinking.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
15
I might start the lesson, by trying to develop an agreed definition for writing a numerical
probability to help bring all students to common ground for thinking about the
complexities later on. I might start by posing the question:
In class we have been trying to improve how we explain our thinking. Here is one
response I saw.
“ 3/6 Well I counted the number of odd numbers between 1,2,3,4,5, and 6.”
Is this information clear? What information might make this clearer?
I would hope that students would discuss where the 3 came from, where the 6 came from.
Then I might give students some solutions to compare to help them build an internal
rubric for good explanations. I might pose the question:
Here are some other explanations. What is each person thinking? Where do their
numbers come from? Are there some parts that could be improved? If so, how?
“I know that there are six numbers and only three of them are odd. So, I put the
three over six and reduced, to get 1/2.”
“3/6 First I counted out how many odd numbers there was and then I counted the
number of all numbers after.”
“There are 3 even numbers & 3 odd numbers. 3 out of 6 = 1/2.”
After spending time on digging into this basic understanding of probability, I think the
class would be ready to dig into the more complex issues in part 3. I might start with the
issue of numbers on the die versus outcomes. I might ask:
Jeff wrote 1, 3, 5, 7, 9, 11
9 + 11 = 20, so the probability if 2:6. What do you
think Jeff was thinking? What might he be confused about?
I am hoping the students will also bring up that Jeff is only looking at one die not two.
Then I would look at a sample space 12.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
16
Sarah put 1/12. Only 9 + 11 = 20 There are 2 dice and 6 numbers on each dice.
Fred put:
Gena wrote:
What is each person thinking? Where do their numbers come from? Can you use this
thinking to make a correct solution?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
17
Odd Numbers
This problem gives you the chance to:
• work with shapes to make a number pattern
Kate makes a pattern of squares.
She starts with 1 square, then adds 3 more, then 5 more, and so on.
1 x 1 square
2 x 2 square
3 x 3 square
1. Draw the next shape in Kate’s pattern.
2. How many new squares did you add?
____________
3. What size square did you make? _______________________________
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
20
The numbers of squares make a number pattern.
1=1x1=1
1+3=2x2=4
1+3+5=3x3=9
4. Write the next two lines of the number pattern.
______________________________________________________________________________
______________________________________________________________________________
5. Use the number pattern to total the numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
________________________
Show your work.
6. Write down the number pattern that gives a total of 169.
Explain your work.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
7
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
21
Odd Numbers
Rubric
The core elements of performance required by this task are:
• work with shapes to make a number pattern
section
points points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Draws a correct shape.
1
2.
Gives correct answer: 7
1
3
Gives correct answer: 4 x 4 Accept 16
1
4.
Writes correct lines:
5.
6.
1 + 3 + 5 + 7 = 4 x 4 = 16
1
1 + 3 + 5 + 7 + 9 = 5 x 5 = 25
1
Gives correct answer: 100
and
Shows correct work = 10 x 10
1
1
1
1
2
1
Gives correct answer:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25
and
1
2
Gives correct explanation such as: 169 = 13 so the number pattern contains
the sum of 13 odd numbers>
1
Total Points
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
7
22
Odd Numbers
Work the task. Look at the rubric. What are some of the important elements of the
pattern that you want students to notice?______________________________________
Were your students able to draw the next figure in the pattern?
How did students describe the size of the square? Did they continue the pattern of using
dimensions (4 x 4), did they use the total squares (16), or did they use some other
numbers?
Look at the work in part 4, continuing the number pattern. How many of your students:
• Successfully completed both lines?_____
• Only attempted one line?______
• Only showed the addition and total?_____
• Did not use consecutive odd numbers but got the correct total?_______
When doing pattern work in class, are students asked to describe in detail how the pattern
is growing? Do they talk about the significant attributes of the pattern? Do they talk
about the connections between the numbers in the numerical patterns and the attributes of
the geometric pattern? How often are students asked to make connections between
representations?
Now look at the work in part 5. While most students were able to get the correct total,
many did not have strategies to simplify the process. How many of your students:
• Used a pattern (like thinking about groups of 20)?
• Counted the number of addends (10) and then knew that the total was 10 x 10?
• Chunked out small groups at a time looking for convenient numbers?
• Chunked out small groups at a time in order?
What opportunities do students have to practice mental math and think about strategies or
patterns for simplifying the work? How might mental math help students develop their
number sense and understanding of place value and number properties?
How does a strong number sense and understanding of number properties lay the
foundation for understanding algebra?
Finally look at the work in part 6. How many students could:
• Write the expression of 13 consecutive odd numbers to total 169?
• How many students thought about finding the square root to help them with the
pattern (13 x13)?
• How many students did not give any explanation?
• How many students just added on from the previous answer without thinking
about the attributes of the pattern?
• How many students gave number sentences totaling 169 that did not relate to the
odd number pattern?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
23
Did students use mathematical language, such as consecutive numbers, odd numbers,
and square numbers, when discussing their strategies?
How can we provide students more opportunities to use academic language for a
purpose? How does having academic language to look at different types of numbers
help shape or contribute to your thought processes? How does it contribute to the
types of patterns or attributes that you notice?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
24
Looking at Student Work on Odd Numbers
Student A is able to see and extend the visual pattern. The student recognized the
pattern is made up of two parts: the list of consecutive odd numbers and the square
numbers. Notice that in part 6 the student is able to connect the number of
consecutive addends to the number being squared.
Student A
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
25
Student A, part 2
Not all students with full marks noticed the same level of detail in the patterning.
Student B does not connect the total in part 5 to dimensions of the square, 10 x 10.
The student just continues the pattern of consecutive odd numbers until reaching the
desired total of 169.
Student B
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
26
Student C also needs to keep adding to the 100 from the previous total to find the
numbers that total to 169. Notice that the student needs to look at every element in
the pattern to solve for part 5. The student is not able to form a generalization to
think about how the pattern grows independently of order.
Student C
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
27
Student D is able to continue the pattern in part 4. The student does not look for a
pattern to help with the addition in part 5, nor does the student think of 100 in
connection with square numbers. In part 6, the student does not think about
continuing the pattern of consecutive odd numbers, but just scrambles to find odd
numbers to total 169. Do we continue asking probing questions in the classroom far
enough to find out where student thinking breaks down? While in the early parts, the
student seemed to have a good handle on the mathematics; the student could not
extend that thinking to later parts of the task.
Student D
Student E is able to pick up on some elements or attributes of the pattern, but needs to
use a recursive rule to find the number being squared. This makes the rule more
difficult to apply to later parts of the task. In parts 5 and 6 the student needs to rely on
just arithmetic skills to crunch out the solutions.
Student E
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
28
Being able to identify the attributes of the pattern and how those attributes contribute
to the growth in the pattern is crucial to solving some of the later aspects of the task.
Student F does not connect the addends to odd numbers and ignores the square
numbers part of the pattern as irrelevant. In part 4 the student just tried to reach a
total. The student relies on adding only 2 numbers at a time to cope with the string of
numbers in part 5. The student doesn’t have a pattern to use in solving part 6 and just
adds 69 to the previous string of numbers.
Student F
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
29
Student G does not notice the square numbers in part 4, but only that two numbers are
being multiplied together. The student is able to find a pattern of making 20’s to
simplify the addition in part 5. In part 6 the student adds in even numbers in an
attempt to make 169. Without identifying a pattern, the answers to part 6 become
random.
Student G
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
30
Student H does not see the relationship between the addends and the numbers being
multiplied. In trying to add the string of numbers in part 5, the student uses number
sense to find friendly combinations. However, the student’s process is random so
when he tries to extend the strategy into part 6 he loses sight of the idea of both
consecutive and odd as elements of the pattern. How do we help students develop the
habits of mind of searching for order, defining or describing the parts of the pattern
before solving a problem? What other habits of mind might be useful for this child?
Student H
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
31
Student I focuses on the square numbers in the pattern, but does not notice the
consecutive odd numbers that make up the addends.
Student I
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
32
Student J struggles with addition. In order to cope with limited number fluency, the
student simplifies or overlooks crucial elements of the pattern. What classroom
activities are appropriate at this grade level to help all students continue developing
their number skills?
Student J
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
33
A few students struggled at just the basic level of seeing the visual pattern. Student K
did not see the key attributes of the geometric or numeric patterns in the first 4 parts.
The student is able to get the total in part 5, but loses points for not showing work.
Student K
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
34
Student L does not notice the physical attributes of the pattern. Look at the responses
for parts 1,2, and 3. What aspects of the diagram or prompt might the student be
paying attention to? Where might these answers be coming from? What types of help
do students L and K need prepare them to work on grade level tasks? What is
feasible within the classroom? What other types of help might be available at your
school?
Student L
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
35
7th Grade
Student Task
Core Idea 3
Algebra and
Function
Core Idea 1
Number and
Operation
Task 2
Odd Numbers
Work with shapes to make a number pattern.
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity and
change.
• Represent, analyze, and generalize a variety of functions
including linear and simple exponential relationships.
• Express mathematical relationships using expressions and
equations.
• Use the associative and commutative properties of addition and
multiplication and the distributive property of multiplication
over addition to simplify computations with rational numbers.
Mathematics in this task:
• Ability to identify attributes of a geometric pattern and extend the pattern
• Connect square numbers to the dimensions of a physical square
• Recognize consecutive odd numbers and extend them
• Addition with a series of addends
• Recognize and extend a numeric pattern
• Generalize about the elements of a pattern in order to extend it to later stages
Based on teacher observations, this is what seventh graders knew and were able to do:
• Draw and extend a geometric pattern
• Quantify the dimensions of a square
• Count the number of squares added to the next figure in the pattern
• Add a sequence of numbers totaling 100
Areas of difficulties for seventh graders:
• Make connections between different representations of the same pattern
• Adding a large string of numbers
• Finding a connection between the number of addends and the dimensions of the
square
• Using a pattern when the numbers become too large
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
36
The maximum score available on this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 89%, could extend a geometric pattern, count the added squares, and give
the dimensions of the square. Many students, 81%, could also add a sequence of numbers
to equal 100. 71% of the students could extend and quantify the geometric pattern,
extend one step of the numeric pattern and add numbers to equal 100. 37% of the
students could meet all the demands of the task including extending the pattern of
consecutive odd numbers to total 169. 3% of the students scored no points on this task.
All the students in the sample with this score attempted the task.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
37
Odd Numbers
Points
0
3
4
5
7
Understandings
Misunderstandings
All of the students in the sample Students had difficulty extending the
with this score attempted the
geometric pattern. Some students made 2
task.
x2 squares or made a vertical line of 4
squares. Some students colored in 1 extra
square to the side of the 3 x 3 square. Some
students thought only 3 or 4 new squares
were added.
Students could extend a
Students struggled with adding the
geometric pattern, count the
numbers to 100 in part 5 or did not show
added squares, and give the
any work.
dimensions of the square.
Students could complete part
10% of the students only attempted one
1,2, and 3. Students could add
line of part 4. 10% added numbers that
numbers to total 100.
were not odd or not consecutive odd when
extending the pattern. Some students did
not continue to include the multiplication
of square numbers as part of the pattern in
4.
Students with this score could
Students with this score had difficulty with
usually complete the pattern for continuing the pattern in part 6 because of
one of the lines in part 4.
the size of 169 and the number of elements
needed to make the pattern. Students
looked for friendly numbers to make 169,
such as 25 + 75 + 69 or
10 +10+10+10 . . .+9.
Students could extend a
geometric and numeric pattern.
Recognize the relationship
between elements of the pattern
and the total to extend the
pattern given a total.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
38
Implications for Instruction
Students need to develop certain habits of mind when looking at patterns. They should
routinely ask themselves questions, such as, “What stays the same? What changes? How
are the numbers related to the geometric attributes? What are the important features of the
numeric patterns? Are there any special types of numbers occurring in the pattern? Why
is this outcome happening? How does one thing help me find something else? Can I find
a rule that would help me extend the pattern without going one step at a time?” Students
at this grade level should be reaching for how to make generalizations about solution
strategies.
While it was not necessary to give a written generalization or rule to solve the elements in
this task, students who noticed that the numbers were odd or odd consecutive numbers
did better that students who just noticed that numbers were being added together. Some
students could connect the number being squared with number of elements being added.
Hopefully some students could connect the dimensions of the squares with the square
numbers in the pattern.
At earlier grades students have been extending patterns and using totals to work
backwards to find the place in the pattern. What is new and different for this grade level
is to reach for generalizations and find ways to not draw and count every single item.
Students need to be pushed to let go of the comfort of number crunching and try to find
relationships, which in the long run will simplify their work and also allow them to use
the pattern for any number.
Ideas for Action Research – Exploring Number Theory
While the format of this task is a familiar geometry pattern problem, the problem for
students was not recognizing attributes in the numeric patterns. Many students did not
pick up on the square numbers or the consecutive odd numbers. Try working some
problems relying heavily on interesting number features. Consider trying the following
problems with your class:
Sums of Consecutive Numbers
3+4=7
2 + 3 +4 = 9
4 + 5 + 6 + 7 = 22
For each number from 1 to 35, find all the ways to write it as a sum of two or more
consecutive numbers.
What can you discover about the sums of consecutive numbers? Find as many patterns
as you can. Try to find at least 3.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
39
Without doing any calculations, predict whether each of the following numbers can be
made with 2 consecutive numbers, 3 consecutive numbers, 4 consecutive numbers, and
so on.. Explain how you made your predictions.
a. 45
b. 57
c. 62
d. 75
e. 80
Use your discoveries to come up with shortcuts for writing the same numbers as the sum
of two or more consecutive numbers. Describe you shortcuts and tell how you used them
to write the numbers as sums of consecutive numbers.
From: Fostering Algebraic Thinking, by Mark Driscoll
Other related problems from this source: Something Nu, Stretching Problem, Differences
of Squares
Another related problem is: King Arthur Problem
Once upon a time, there were 19 knights seated around King Arthur’s round table
numbered 1 through 19. The king decided to use this method for choosing the one knight
who would get to marry is beautiful daughter: starting with 1, he eliminated every second
knight in order until only one knight remained. Thus, for 19 knights, he eliminated
numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 5, 9, 13, 17, 3, 11, 19 and 15. The remaining
knight was knight number 7. The king decided to use the same method to determine the
best mathematician in the realm. He called together 100 of the best thinkers and told
them to sit at a circular table with 100 numbered seats. Of course YOU are one of these
brilliant mathematicians. Which seat should you choose to be selected the best
mathematician in the realm? Which seat should you choose if there are 500 seats? 1000
seats?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
40
Pedro’s Tables
This problem gives you the chance to:
• work with number properties including divisibility
• explain your reasoning
Pedro chooses numbers to go in a table.
He can choose any whole number from 1 to 25.
Multiples
of 5
Multiples
of 3
Square
numbers
Even numbers
Factors
6
of 12
Prime
numbers
Pedro says,
I can put 6 in this box.
6 is a factor of 12
and
it’s a multiple of 3.
1. What other numbers could Pedro put in this box? ___________________________________
2. The number 4 can go in two different boxes in the table.
Write 4 in these two boxes.
3. Give a description of numbers that can go in the Even numbers and Multiples of 3 box.
_____________________________________________________________________________
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
41
4. Explain why there are no numbers that can go in the Factors of 12 and Multiples of 5 box.
_____________________________________________________________________________
_____________________________________________________________________________
5. Explain why there is only one number that can go in the middle box on the bottom row.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
7
Grade 7 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
42
Pedro’s Tables
Rubric
The core elements of performance required by this task are:
• work with number properties including divisibility
• explain your reasoning
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answers: 3, 12
(deduct 1 mark if additional numbers listed)
points section
points
2x1
2.
Writes 4 in the correct boxes: Right hand column, first and second rows
1
3
Gives correct answer such as: Multiples of 6
1
4.
Gives correct explanation such as:
‘The factors of 12 are 1, 2, 3, 4, 6 and 12. None of these are multiples of 5.
12 is not divisible by5.
Partial credit
for a partially correct explanation
5.
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
1
2
1
1
7
Total Points
Grade 7 – 2008
1
2
(1)
Gives correct explanation such as:
3 is a prime number and a multiple of 3. All other multiples of 3 have more
than two factors so are not prime numbers.
2
43
Pedro’s Table
Work the task. Look at the rubric. What are the key mathematical concepts being
assessed?_____________________________________________________________
_______________________________________________________________________
_What types of arguments do you think students might make for parts 4 and 5? What
qualities do you want in an explanation?
Look at student work for part 1, finding multiples of 3 that are factors of 12. How many
of your students put:
3,12
Only 12
Only 3
24
9
10
Many extras
Other
What might be the cause for these errors? Were students using incorrect definitions?
Were students only using the vertical clue or only the horizontal clue? Were lack of math
facts part of the problem?
Now look at student work for part 2, placing the fours in the diagram. How many of your
students could:
• Place both 4’s correctly?_____
• Omitted the 4, square numbers and even?_____
• Omitted the 4, square numbers and factors of 12?_______
• Put a 4 in the box for multiples of 3 and even numbers?______
• Put a 4 in the box for multiples of 3 and prime?_____
• Other errors?_______
What might be the cause for these errors? Is it diagram literacy, lack of understanding of
how the two labels connect? Is it incorrect or faulty mathematical definitions? Are
students struggling with the math facts needed to think about the situation? Are students
trying to give answers without doing any work?
Look at work in part 3. How many of your students:
• Recognized that the numbers were multiples of 6?___________
• Gave a description about even multiples of 3 or every other multiple of 3 with
examples?________
• Gave correct examples without any description?________
• Other?_____
How do we help students learn academic language to describe numbers? What
opportunities do they have to use language, like multiples of 6, in conversation?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
44
Now look at student work in part 4. Think about the ways mathematics could quantify
this information. Think about the types of statements used to make a conclusion. How
would you categorize student writing?
• Made statements about multiples of 5:
o End in 5 or 0
o 5,10,15,20
• Listed or described the multiples of 12
o All even
o 1,2,3,4,6,12
o 12,24
• Gave arguments about odd and even
• Numbers are too large or too small
• Restated the prompt
• Confused factors and multiples
• Put 60
• Used incorrect vocabulary, such as primes
• Gave information, but made no concluding statement
• Didn’t relate information to how it makes the case or proves the point
How do we help students develop the logic of making a complete justification? How can
we use student work to provide examples of faulty logic, incomplete logic, and
exemplary arguments?
Look at student work for part 5. Divide them into four groups and give 1 or 2 examples
of each:
Complete explanation:
Some use of definitions but incomplete justification:
Incorrect use of definitions:
Restatement of prompt:
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
45
Looking at Student Work on Pedro’s Tables
Student A notes a pattern to the numbers that fit in the center box, but does not name that
pattern as multiples of 6. How do we help students begin to think about sets of numbers?
What types of sets are appropriate for this grade level? In part 4, the student gives a good
explanation with details of the factors of 12. Notice that in part 5 the student justifies why
further multiples of 3 are not prime.
Student A
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
46
Student A, part 2
Student B shows the thinking behind the statements by listing factors of 12, even
numbers, etc. How do we help students develop this habit of mind? See how the list
strategy offers a complete justification in part 5. Notice that in part 4, the student
confuses the factors of 12 with even numbers. The student also confuses multiples of 5
for factors of 5.
Student B
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
47
Student B, part 2
Student C has problems with the explanation in part 4. even though the student has listed
them to solve part 1. (The scoring for part 1 is incorrect. 6 is not an extra as it is already
given and is correct. Part 4 should not have received points.) Why might the student think
other multiples of 5 are too big to be considered? Notice how the student makes a
complete argument in part 5, stating which number is a prime and explaining why further
multiples of 3 are not prime.
Student C
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
48
Student C, part 2
In part 1 Student D does not appear to understand how the diagram works. What might
the student be thinking? In part 3 the explanation is very weak, just giving the prompt
again. In part 4 the student describes properties of multiplies of 5 but does not show how
this proves that they don’t divide into 12 evenly. However the student does make a
complete justification in part 5.
Student D
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
49
Student D, part 2
Student E confuses factors and multiples in the explanation for part 4. In part 5 the
student makes a statement, but gives no reason for the statement. It is merely a
restatement of the prompt. What is needed to make this a complete justification?
Student E
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
50
Student F gives information about multiples of 5, but makes no assertion and makes no
link to factors of 12. How do we help students think about the components needed to
make a justification? Do you think the argument is complete in part 5? How could you
improve the statement?
Student F
Student G seems to have trouble with diagram literacy. Look at how the table is filled out
at the top of the page. How do you think the student is thinking about the labels? What
definitions is the student confused about? Why do you think that? In part 3 the student
just restates the prompt. Student G has several errors in logic and use of mathematical
language. How many can you find? What is wrong about the argument in part 5? What
needs to be added to this explanation?
Student G
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
51
Student H also struggles with diagram literacy and mathematical definitions. In part 3
the student is only thinking about one of the constraints and does not consider factors of
12. In part 4 the student uses multiples and factors interchangeably and finds a common
multiple. Understanding the intersection of sets is a complex idea. How can we introduce
students to this concept informally? What types of questions might we ask to push at
intersections? Can you identify the student’s working definitions for: multiples, square
numbers, prime numbers, factors?
Student H
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
52
Student H, part 2
Look at the work of Student I. The student has given correct examples that fit both
constraints in the table. The student has also left blank the 2 boxes that have empty sets.
So why does the student struggle with the rest of task? What is problematic for the
student?
Student I
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
53
7th Grade
Student Task
Core Idea 1
Number and
Operation
Task 3
Pedro’s Tables
Work number properties including divisibility. Explain reasoning about
combining constraints to define numbers.
Understand number systems, the meanings of operations, and ways
of representing numbers, relationships, and number systems.
• Describe classes of numbers according to characteristics such as
the nature of their factors.
• Use factors, multiples, prime factorization, and relatively prime
numbers to solve problems.
The mathematics of the task:
• Diagram literacy, understanding each box as the intersection of two sets
• Mathematical vocabulary: multiples, factors, prime numbers, square numbers,
even numbers
• Justification: giving facts or information, making an assertion, connecting
information to various parts of the assertion to close the argument
Based on teacher observations, this is what seventh graders knew and were able to do:
• Understood square numbers, even numbers, multiples of 3
• Could correctly place the two 4’s in the diagram (square even number and square
factor of 12)
• List multiples of 6
Areas of difficulty for seventh graders:
• Recognizing multiples of 6 when looking at a list of examples
• Confusing factors and multiples
• Finding the intersection of two sets (multiples of 3 that are factors of 12)
• Making justifications
• Working with multiple constraints
Strategy used by successful students:
• Making a list of all the numbers for each constraint
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
54
The maximum score available for this task is 7 points.
The minimum score needed for a level 3 response, meeting standards, is 4 points.
Many students, 79%, could explain why there is only 1 prime multiple of 3. More
than half the students, 61%, could also locate the 4’s in the diagram (square even
number and square factor of 12). Some students, 32%, could explain why there is
only one prime multiple of 3, locate the 4’s in the diagram, and either give one
number that is a factor of 12 and multiple of 3 or make some part of the argument that
there are no multiples of 5 that are factors of 12. Only 3% of the students could meet
all the demands of the task, including making a complete argument for why there are
no multiples of 5 that are factors of 12 and recognizing that the intersection of
multiples of 3 and even numbers are multiples of 6. 21% of the students received no
points on this task. 93% of the students with this score attempted the task.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
55
Pedro’s Tables
Points
0
Understandings
93% of the students with this
score attempted the task.
1
Students could explain why the
number 3 is the only prime
multiple of 3.
2
Students could explain why the
number 3 is the only prime
multiple of 3 and locate the 4 in
the appropriate boxes.
4
Students could generally name of
the numbers that fit in the
multiples of 3 and factors of 12
box. They could also answer part
2 and 5.
6
7
Misunderstandings
Students had difficulty explaining why 3
is the only prime multiple of 3. 10%
only restated the prompt. 10% only
discussed why 3 is prime, but did not
mention anything about other multiples
of 3.
Students had difficulty placing the two
4’s in the diagram. 22% did not
recognize that 4 is a square multiple of
12. 10% did not recognize that 4 is an
even square number. 10% thought 4 was
an even multiple of 3. 5% thought 4 was
a prime multiple of 3.
Students had difficulty explaining why
no multiples of 5 are factors of 12. 16%
merely restated the prompt. 5% thought
that 60 fit the constraints. 5% said that 5
doesn’t go into 12, ignoring other
multiples of 5 and other factors of 12.
Many students confused multiples and
factors.
Students struggled with naming both
numbers in part 1, multiples of 3 that are
factors of 12. 29% did not use 12 as a
choice. 19% did not use 3. 16% used a
list of several numbers. Single incorrect
answers 10, 9, 24, and 4.
Students struggled with having both
answers for part 1 or making a complete
argument included for part 4.
Students understood how to work
with multiple constraints, using
terms like multiples, even, prime,
factors. Students could make
complete justifications for why
numbers would or would not fit in
the boxes in the diagram by
making an assertion, giving
reasons connected to both
constraints and connect them to
the assertion.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
56
Implications for Instruction
Students need more work with factors and multiples and using them to solve problems.
While many students seem to know the definitions, they couldn’t use the definitions to
write justifications or make exhaustive lists. Part of the core of middle school math is to
move from thinking about individual numbers to thinking about the number system.
Students should be able to think about categories of numbers or number sets and find
intersections of sets; not in the formal notation of set theory, but in terms of distinct
elements within the set. Students need opportunities to think about how number
properties interact. This is one of the big ideas on the National Assessment Exam
(NAEP).
Students need opportunities to use properties of numbers to solve problems and to use
them to make convincing arguments about whether something is or is not possible. They
need help developing the logic of making and quantifying their arguments. Students
struggled when given multiple constraints, such as the factors of 12 which are also square
numbers. Many students just restated prompts rather than giving supporting evidence and
tying that evidence to their assertions.
Ideas for Action Research – Developing the Logic of Justification
Using student work can be a powerful tool to help students develop the logic of
justification. A good argument should make an assertion, give supporting evidence, then
tie the evidence back to the assertion.
Consider the following plan to work with the class on justification. The teacher might
introduce the idea by saying that in reading their work for Pedro’s table, she thought that
some of the arguments were weak. She would like the class’s help to give her
suggestions on notes to write to students to help improve their ideas.
Here are some student responses that puzzled me. What were these students thinking?
“ Because 12 is an even number and 5 is an odd number.”
“So the factors of 12 and multiples of 5 box should be 60.”
“Because 12 is not a multiple of 5 because no matter how many times it multiples times
itself you can’t get 10 or 5.”
“If you multiply 5 by any number, it will already be higher than 12.”
“Since 12 is an even number all of its factors are even and small. The smallest multiple
of 5 is 5. The other even multiples are too big.”
Did they make any factual errors? Is there something else missing or incorrect in their
thinking?
Hopefully, this part of the discussion will surface students’ definitions for factors and
multiples so that they can be compared and checked for accuracy.
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
57
The teacher can now dig deeper into the ideas with a follow up question, such as:
One answer that confused me said, “Everything you multiply by 5 is either going to end
with 0 or 5.” How does that help me prove the statement?
Someone said, “The factors are 1,2,3,4,6,12.” Does that prove why nothing will fit in the
box? Why or why not?
Then the teacher might press students to start a list of what needs to go into a
justification, hoping to get them to start forming some idea that the evidence needs to be
tied back to the original statement or connected to the other part of the assertion.
Finally the teacher might give students a list of explanations and ask students to write
down what the students did well and how the explanation can be improved.
“If you list out all the factors of 12: 1,2,3,4,6,12, none of them are multiples of 5.”
“Because 5’s multiples always end with 5 or 0 and none of the 12” factors end with 5 or
0.”
“Because 5 doesn’t produce any factors of 12.”
“Because 5 can’t go into 1,2,3,4,6,12.”
“Because the factors of 12(1,12,2,6,3,4) are all even numbers and 5 does not have any
except 1 and 5.”
The teacher might end with a look at part 5.
Now that we have looked at what makes a good mathematical argument. Which of these
do you think is the best argument and why? (Give students a chance to debate different
options.)
Sam: There is only one number that can go there, and that is “3”, because multiples of
three are not prime except for “three” and prime are numbers that can’t go I any
multiples except for “1”. Prime# - 2,3,5,7,11,13,17,19,23 and Multiples of 3- 3.
6,9,122,15,18,21,24.
John: “There is only one number that can go in the multiples of 3 and prime numbers,
which is 3, because if 3 multiplies a numbers larger than 1, it is not prime anymore.”
Sara: “Because 3 can’t go into any prime numbers except 3.”
Joanna: “Because 3 is the only prime number that 3 can produce 3,6,9,12 . . . 3 x 1 =3,
3 x 2 = 6, 3 x 3 = 9, 3 x 4 = 12 . . . .
Trevor: “3 is the only number that can go into that box because 3 is the only prime that’s
a multiple of 3. All the other multiples of 3 can’t be prime because they’re divisible by
3.”
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
58
After the lesson, reflect on student interest in the lesson. Were they engaged? Did they
like trying to solve the puzzle of finding errors and making improvements? Were the
points that they were making relevant or important? Did their points change or deepen
over time? Are there other parts of the mathematics that were not adequately covered?
What additional work might you use to pose further questions?
Grade 7 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
59