Balanced Assessment Test –Sixth Grade 2008
Core Idea
Task
Score
Snail’s Pace
Number Operations
This task asks students to work with distances, time and speeds in inches per minutes
in the context of snails. Students are asked to find equivalent rates. Successful
students were able to find a common unit to make a comparison of the rates for all
four snails.
Probability
Black and White
This task asks students to use probability in the context of picking balls from a bag.
Students needed to think about probabilities and the meaning of fractions for a variety
of contexts. Successful students understood the probability of “not” getting
something.
A Number Pattern
Algebra
This task asks students to describe and extend a number pattern. Successful students
recognized a pattern growing by an increasing amount each time.
Percent Cards
Rational Numbers
This task asks students to relate fractions, decimals, and percents and locate them on a
number line. Successful students could work with decimals in the thousandths place
and locate 3/8 on a number line.
Area and Perimeter
Geometry and
Measurement
This task asks students to calculate area and perimeter for rectangles. Successful
students could keep a given area and create new rectangles with larger and smaller
perimeters.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
1
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
2
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
3
Snail Pace
This problem gives you the chance to:
• work with distances, time and speeds in inches and minutes
These snails move very slowly. Here are their speeds.
Snail A
5 inches in 10 minutes
Snail B
3 inches in 20 minutes
Snail C
1 inch in 15 minutes
Snail D
6 inches in 30 minutes
1. How far can snail D travel in 1 hour?
____________ inches
2. How far can snail C travel in half an hour?
____________ inches
3. How far can snail B travel in 2 hours?
Show how you figured this out.
____________ inches
4. Which snail moves more quickly than the others? ____________
Explain how you figured this out.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
8
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
4
Snail Pace
Rubric
•
• The core elements of performance required by this task are:
• • work with distances, time and speeds in inches and minutes
•
Based on these, credit for specific aspects of performance should be assigned as follows
points
1.
Gives correct answer: 12 inches or 1 foot
1
2.
Gives correct answer: 2 inches
1
section
points
1
1
3.
Gives correct answer: 18 inches or 1 foot 6 inches or 1 1/2 feet
1
Shows correct work such as: 60 divided by 20 = 3
3 x 3 = 9 inches in 1 hour
9 x 2 = 18 inches
4.
2
Gives correct answer: Snail A Accept 5
3
1
Gives correct explanation such as:
In 1 hour Snail A travels 30 inches.
In 1 hour Snail B travels 9 inches.
In 1 hour Snail C travels 4 inches.
In 1 hour Snail D travels 12 inches.
Partial credit
For 1 error
2
(1)
Total Points
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
5
3
8
Snail Pace
Work the task and look at the rubric. What are the big mathematical ideas in this task?
What do we want students to understand about rates and making a mathematical
comparison?
Look at student work for part 3. How many of your students put:
18
9
24
40
6
24
Other
What kinds of errors or misconceptions did you see in their thinking or strategies?
Now look at the work for part 4. how many of your students:
• Could make a comparison in the same unit for all 4 snails?________
• Made a correct comparison between 3 of the 4 snails?____________
• Compared all the snails but used different units?________________
• Only discussed A or a rate for A?_________________
• Tried to make a logical comparison between D and A?___________
• Chose Snail B?________Chose Snail C?_______Chose Snail D?_________
• Made an additive comparison?___________
• Did not mention rates?______________
What were some of the different rates that successful students used?
Were there some interesting strategies used by successful students?
What are the implications for instruction?
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
6
Looking at Student Work on Snail Pace
Student A is able to change the rates to different times in parts 1, 2, and 3. Notice that the
student uses a ratio table strategy for part 3. In part 4 the student changes all the rates to
the same unit by comparing them in inches per hour.
Student A
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
7
Student B also attempts to change everything to the same unit. Interestingly the student
fixes the inches instead of the time for the first 3 comparisons, which is a great strategy.
How might you share this strategy with students to help them learn a new approach to
rates? However for snail D the students makes a comparison with C in minutes instead of
keeping the fixed inches. How could the student change the rate for D to make it
comparable to the others?
Student B
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
8
Student C uses a ratio table to solve for part 3. This is a correct strategy. What is the
error or misunderstanding of Student C in using the table? In part 4 the student has used
a fraction strategy to solve the task. This strategy does make it possible to compare the
rates. Can you explain why? What further work would you have wanted to complete the
explanation for part 4?
Student C
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
9
Student D uses a scale factor to solve for part 3. In part 4 the student attempts a
comparison to eliminate Snail B and C, but doesn’t use comparable units. The student
does use the same rates to compare Snail A and D.
Snail D
Snail E makes an assertion about the rates of Snail A and D, but doesn’t change them to
the same units. Notice that there is no mention of why B and C weren’t even considered.
What do we want as the norms for making a mathematical comparison?
Student E
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
10
Student F only mentions a rate in inches per hour for Snail A, as if that would be a
complete explanation. However on this paper there are rates for 2 other snails in hours.
Can you find them? Would better use of labels have helped this student to think a little
deeper about the final answer?
Student F
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
11
Student G makes an unusual choice of time units. The student is able to get three of the
rates correctly, but is off slightly on the rate for snail C. The student puts 1 1/4 inches per
20 minutes. What should the correct rate be? Notice that in the final part, the student is
unable to make a correct comparison between the rates calculated. What do you think the
student was thinking in part 3? Where did his thinking get derailed?
Student G
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
12
Some students have not realized that proportions are multiplicative. They are instead
trying to apply additive strategies. Look at the work of Student H, who uses additive
thinking in both part 3 and part 4. Student I is also using additive thinking in part 4. What
experiences help students to see why this strategy doesn’t make sense? What kind of
problem can we construct to make this more apparent?
Student H
Student I
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
13
6th Grade
Task 1
Snail Pace
Student Task
Work with distances, time and speeds in inches and minutes. Compare
rates after first converting to either an equivalent distance or equivalent
time period.
Core Idea 1
Understand number systems, the meanings of operations, and ways
Number and of representing numbers, relationships and number systems.
Operation
• Understand and use proportional reasoning to represent
quantitative relationships.
Core Idea 4
Apply appropriate techniques, tools, and formulas to determine
Geometry
measurements.
and
• Solve simple problems involving rates and derived
Measurement
measurements for such attributes as velocity.
The mathematics of this task:
• Converting rates to different times
• Making a comparison between rates by changing to the same units by either
fixing the distance or by fixing the time
• Making a convincing mathematical justification
Based on teacher observations, this is what sixth graders know and were able to do:
• Students know the more inches traveled the faster the snail
• Students could change rates to equivalent rates, e.g. 6 in. in 30 min. = 12 in. in 1
hour
Areas of difficulty for sixth graders:
• Understanding the logic of a mathematical comparison, using an equal distance
with different times or using an equal time for different distances
• Understanding that all items in the list need to be compared
• Making a justification with numerical quantities or rates
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
14
The maximum score available for this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, 90%, could find equivalent rates in parts 1 and 2. Many students, 79%,
could also find the snail with the fastest rate. More than half the students could also find
equivalent rates in part 3 and explain their thinking. 21% of the students could meet all
the demands of the task including comparing all the snails using equivalent units of either
time or distance. Almost 6% of the students scored no points on this task. All the
students in the sample with this score attempted the task.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
15
Snail Pace
Points
Understandings
All the students in the sample
0
2
with this score attempted the
task.
Students could change rates to
different amounts of time in
parts 1 and 2.
3
Students could change rates in
part 1 and 2 and identify the
fastest snail.
5
Students could change the rates
in parts 1, 2, and 3.
Students could change the rates
in parts 1, 2, and 3 and identify
the fastest snail.
6
8
Misunderstandings
Students had difficulty changing the rates
in parts 1 and 2.
Students didn’t recognize which Snail was
the fastest. 10% of the students thought
Snail D was fastest. 9% picked Snail B. 3%
picked Snail c.
Students had difficulty changing to a rate
of 2 hours for Snail B in part 3. 4% gave
the rate for 1 hour. Other common answers
were 24, 6, 21, 8, 40.
Students did not know how to write a
justification. 17% just mentioned the rate
for Snail A. 8% just gave a verbal reason
for choosing A over D with no
quantification. 6% gave explanations with
no rates. 6% made comparisons in different
units.
Students could change rates to
find equivalent rates. Students
could make a mathematical
comparison by either fixing the
time or the distance and
considering all units in the list.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
16
Implications for Instruction
Students at this grade level should be able to make conversions between rates for
different units of time. Students should have a variety of strategies for making
conversions, such as: using division to find a unit rate, finding equivalent fractions, and
using a ratio table.
Students should draw upon their developing understandings around fractions to look for
common multiples or common denominators and working in number theory to find
common multiples and common factors to solve new problems. Many students should
look at the given times and think, “What might be a convenient unit to make all the times
the same.” Students with more experience with rates might think, “Is it more convenient
to change all the rates to a common time or a common distance?” Students need to
frequently be asked questions like, “How does this solution relate to other topics I have
learned? When is one strategy better or more convenient than another?”
Students should think about the idea of comparison as a type of problem, rather than
strictly thinking about what calculations should I use with these numbers. Students
should learn to recognize that when making a choice or comparison the best choice needs
to be compared to all of the other available options. Students need opportunities to
justify their thinking and compare it with the thinking of others in order to develop the
skills at making a complete argument. At this grade level students need to think about
applying a solution strategy to a type of problem, rather than solving each problem fresh.
Ideas for Action Research – Comparing and Contrasting Strategies
Students at this grade level need to start thinking about why different strategies work.
What do these strategies have in common? How are they different? In order to better
manage classroom discussion, teachers need to think about how to present the strategies
to the class and facilitate the mathematics of the solutions rather than just presenting
variety for varieties sake. One helpful tool is to meet with colleagues and try to anticipate
student strategies and discuss these ideas as a learner first.
Consider the following solutions.
• Can you describe which solutions are related?
• How are the strategies different?
• If you were presenting them to the class, what might be the best order to have
students share out in order to bring all the students along? Why would you pick
that particular order?
Some teachers use seating charts during a lesson to record strategies to help them get
ideas out in an order to help everyone follow the discussion. Work below is presented
without labels or in a partial condition, to make thinking about more interesting or reengaging. The idea is for the viewer to really have to think about the mathematics from a
new perspective each time.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
17
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
18
After having students examine and compare the strategies, you might want to give
students back their papers to see if they want to revise their work with a red pen or you
might want to give them a similar problem; such as, 2001 Cans of Kola or 2002
Grandpa’s Knitting, to see how the discussion effected their thinking.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
19
Black and White
This problem gives you the chance to:
• show your understanding of fractions and probability
There are 4 black balls and 7 white balls in a bag.
Jasper picks a ball without looking.
He says,
The probability of getting a black ball is
Jasper is wrong!
4
7
1. What is the probability of getting a black ball?
!
_______________
Explain why Jasper is wrong.
__________________________________________________________________
__________________________________________________________________
2. The fractions
4 7 7 4
, ,
,
are answers to the questions below.
7 4 11 11
Put each fraction in a correct place.
! !
! probability
!
a. What
is the
of getting a white ball?
_______________
b. What is the probability of not getting a black ball?
_______________
c. What is the fraction of black balls in the bag?
_______________
d. What is the number of black balls as a fraction of the number of white balls?
_______________
e. What is the number of white balls as a fraction of the number of black balls?
7
_____________
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
20
Black and White
Rubric
The core elements of performance required by this task are:
• show understanding of fractions and probability
Based on these, credit for specific aspects of performance should be assigned as follows
1.
points section
points
Gives correct answer: 4/11
1
Gives correct explanation such as: The probability of getting a black
ball is the number of black balls divided by the total number of balls.
1
2
2.
Gives correct answer: 7/11
1
Gives correct answer: 7/11
1
Gives correct answer: 4/11
1
Gives correct answer: 4/7
1
Gives correct answer: 7/4
1
5
7
Total Points
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
21
Black and White
Work the task and look at the rubric. What are the big mathematical ideas needed to
solve this task?
Look at student responses to part 1 how many of your students could give reasonable
answers about how to write a probability. Make a list of some of the misconceptions that
you saw.
Now look at work for part 2. Use the chart below to mark student work.
Question
2a
Correct
answer
7/11-
Other
Choice
4/7
Other choice Other choice
7/4
4/11
2b
7/11
4/7
7/4
4/11
2c
4/11
4/7
7/4
7/11
2d
4/7
7/4
7/11
4/11
2e
7/4
4/7
7/11
4/11
Other
Answer
What surprises you about the student errors?
How many students were giving whole number responses? What types of intervention
might you use to help these students?
Why do you think part 2b was so difficult for students? What did they have to
understand in order to be successful?
What further questions would you like to ask students to probe their understanding of
fractions and probability?
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
22
Looking at Student Work on Black and White
Student A is able to explain probability in part 1. Notice the clarification of thinking for
the responses in part 2. How do we help all students to develop the mathematical
language to talk about situations with the thoroughness of Student A?
Student A
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
23
Student B gives an example of an incomplete explanation in part 1. How do we plan
discussions to help students see the difference between complete and incomplete
responses?
Student B
In mathematics we want students to make sense of the situation. One child said that the
probability is one or black because that is the black ball that is at the top. Student C
seems to say that mathematics doesn’t make sense and can’t be relied upon to make
sense. How do we foster sense-making in the classroom?
Student C
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
24
Some students are still not comfortable with non-integer numbers. Look at the work of
student D. What kinds of interventions are needed with students like this? What kind of
help can be provided beyond the scope of the classroom?
Student D
Student Work on Part 2
Question
2a
2b
2c
2d
2e
Correct
answer
7/11- 82%
7/11- 53%
4/11-82%
4/7- 86%
7/4-82%
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
Other
Choice
4/7- 1%
4/7- 9%
4/7-9%
7/4-0%
4/7- 4%
Other
choice
7/4- 4%
7/4-15%
7/4-1%
7/11-7%
7/11- 7%
Other
choice
4/11- 4%
4/11- 13%
7/11-1%
4/11-3%
4/11- 3%
Other
Answer
10%
10%
7%
4%
4%
25
6th Grade
Student Task
Core Idea 2
Probability
Task 2
Black and White
Show your understanding of fractions and probability. Analyze common
misconception and explain how to correct the error.
Demonstrate understanding and use of probability in problem
situations.
• Determine theoretical and experimental probabilities and use
these to make predications about events.
• Represent probabilities as ratios, proportions, decimals, or
percents.
The mathematics of this task:
• Understanding how to express probability as chances of favorable outcomes over
total possibilities
• Understanding fractions as part to whole
• Working with improper fractions
• Looking at an error and being able to determine what is mathematically incorrect
and being able to describe that in academic language
Based on teacher observation this is what sixth graders know and are able to do:
• Express a simple probability numerically and explain how it is written
• Describe situations in terms of fractions and probabilities
Areas of difficulty for sixth graders:
• Interpreting a negative assertion: find the probability of not getting black
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
26
The maximum score available on this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 93%, could find the probability of getting a black ball and explain why
Jasper’s answer was incorrect mathematically. Many students, 83%, could also give the
probability of a drawing a white ball and give one other correct response for either 2c,d,
or e. More than half the student, 62%, could find the correct response for all the
situations except not getting a black ball. Almost 37% of the students could meet all the
demands of the task. 3% of the students scored no points on this task. All of the students
in the sample with this score attempted the task.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
27
Black and White
Points
0
2
4
6
7
Understandings
Misunderstandings
All the students in the sample
with this score attempted the
task.
Students had difficulty writing the
probability for black balls. They also had
difficulty explaining why Jasper made a
mistake, giving nonmathematical reasons
like, “he counted wrong”, only 1 ball is near
the top, or more than 7 marbles.
Students had difficulty writing the
probability for white balls.
Students could give a simple
probability and explain why
someone else made a mistake.
Students could give a simple
probability, explain it, and
solve for 2a and 1 more
situation from 2c,2d,or 2e.
Students could solve for all
the situations except the
negative statement, “not
black”.
Students could work
comfortably with fractions
and probabilities to describe
relationships numerically.
Students could also explain
how to write a probability.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
See chart pg. 25.
28
Implications for Instruction
Students should be able to think about how to set up a probability to describe or quantify
a simple situation. They should also be able to use fractions to make comparisons
between quantities and show part/whole relationships. This is the grade level where
probability should be mastered to prepare for the high school exit exam.
This task is probing only the basic level of understanding for probability. Before leaving
sixth grade students should master compound probabilities with and without replacement.
Because the game culture for students is changing, students do not get enough
opportunities to experiment directly with probabilities by flipping coins, spinning
spinners, tossing dice, picking cards. Students need concrete experiences first to help
them understand probability contexts, before they can interpret and understand the
information generated by technology that can flip a thousand coins in seconds or toss a
thousand dice.
Ideas for Action Research – Exploring Fair Games
There are several interesting units on examining games and using experimental and
theoretical probabilities to test for fairness. One website with several such activities
www.nzmaths.co.nz/statistics/probability/FairGames.aspx prepared in New Zealand. Try
some of this games with students. Probability could be a year-long project integrated into
the content of other mathematical topics or just re-visited every couple of weeks to
deepen the ideas over time.
Best Egg box Three half-dozen egg boxes are numbered as below.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
29
Working in groups of three, each student is allocated an egg box. Two dice are
rolled and their scores multiplied together. A counter is dropped into the hole
with that number. The winner is the student whose egg box gets all the
numbers covered first. Is the game fair? Is one egg box better than the others?
Is one egg box worse than the others? The same approach used with “Marble
Snap Revisited” may be used here also. Start with a class discussion of how
we can determine whether the game is fair. Ensure that students are clear that
they need to determine whether there are equal chances of each egg box
winning. Also discuss the need to use both theoretical and experimental
approaches in order to verify solutions. The same format should be used for
recording their work. Sub-headings of ‘The Problem’, ‘Method’, ‘Solution’ and
‘Verification’ are useful. Students will need to record experimental results
systematically – tally charts are the easiest method. Discuss the desirability to
record the number of times each score is rolled, rather than just the number of
times each egg box wins.
While students are using an experimental approach they can be scaffolded
with questions such as: How will you record the results? Are you recording how
often each number turns up? What do you think of these results? Do they mean
that the game is fail? What would happen if you did the experiment again? How
could you improve the experiment? It may be useful to hold a class discussion
and summarize the experimental results from the whole class in order to
determine the long-run frequency.
Another resource for Fair Games is the Problem of the Month: Fair Games available on
the Noyce website. www.noycefdn.org/math/members/POM/pom.html
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
30
A Number Pattern
This problem gives you the chance to:
• describe and extend a numeric pattern
This is a number pattern. It can go on and on.
Sum of numbers in row
1
1
1
1
1
1
1
2
3
4
5
________
1
3
6
10
2
1
4
10
________
1
________
1
5
1
________
________
1. Which numbers appear just once in the part of the pattern that is shown above?
_____________________
2. In this pattern, each row begins and ends with the number 1.
The other numbers are the sum of the two numbers above it.
For example, 10 = 6 + 4.
Continue the pattern, by writing numbers in the row of empty squares.
3. a. Find the sum of the numbers in each of the rows. The first two have been done for you.
Write your answers on the diagram above.
b. What do you notice about the sequence of numbers in the Totals column?
________________________________________________________________________________
4. Look at the numbers that have been shaded.
What do you notice about the sequence of numbers that have been shaded?
________________________________________________________________________________
8
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
31
A Number Pattern
Rubric
•
• The core elements of performance required by this task are:
• • describe and extend a numeric pattern
•
Based on these, credit for specific aspects of performance should be assigned as follows
points
1.
Gives correct answer: 2 Accept 6 and 20
1
2.
Gives correct answers: 1, 6, 15, 20, 15, 6, 1
2
Partial credit
One error
section
points
1
(1)
2
3.a. Gives correct answer: 1, 2, 4, 8, 16, 32, 64
2
Partial credit
One error
(1)
b. Gives correct answer such as: The numbers double each time.
or Powers of 2
4.
1
3
Gives correct answer such as:
The difference between consecutive numbers (2, 3, 4, 5) increases by one
each time.
Partial credit
For one error or an incomplete statement.
2
(1)
Total Points
Grade 6 – 2008
Copyright © 2008 by Mathematics Assessment Resource Service
All rights reserved.
32
2
8
A Number Pattern
Work the task and look at the rubric. What are the big mathematical ideas in this task?
Now look at student work for part 1. How many of your students thought the answer was
2? ____________ How many just put 6?__________ How many put 1?________ How
many put 10?_________Other answers:____________________ What do you think
caused students difficulty about this part of the task? What might they have been
thinking?
Look at student work for part 2, filling in the bottom row of the table. How many
students put:
• Correct pattern: 1,6,10,15,20,15,10, 6,1?
• Pattern without 1/s on both or either end?
• Pattern of increasing or decreasing numbers: example: 1,2,6,10,15,21,28?
• Patterns that weren’t symmetrical?
What other types of misconceptions did you notice in student errors?
In part 3a students are asked to add the numbers in the rows. Did your students:
• Give the correct numbers?
• Make one calculation error in the final number?
• Make a list of counting numbers: 1,2,3,4,…?
• Come up with a pattern rather than adding numbers?
What don’t students understand? What clues are they not paying attention to?
In part 3b students are asked to describe the pattern of the totals.
• Could your students describe the pattern of doubling or multiplying by 2?
• Did they give a vague answer, like it’s a pattern, they’re in order, or they get
bigger?
• Did they talk about odd and/or even relationships?
• Did they describe a different pattern in the figure, like every row begins and ends
in 1?
• Give a non-mathematical answer, such as the numbers are going down ⇓?
How do we help students develop the ability to look for more simple patterns, like odd
and even? How do we help students express more details about what they’re thinking
rather than stop short of a full description?
In part 4 students are asked to describe the pattern of the shaded numbers.
• How many of your students could describe the pattern? _______Partially describe
the pattern?
• How many described a different pattern in the figure, such as every row begins
and ends with one? To get the bottom number, add the two numbers above?
• How many gave vague answers, such as it’s a sum, a pattern, or its skipping
numbers?
• How many discussed something about odd and even?
• Just noticed that 1 + 3 + 4 = the final number 10?
What types of responses did you like? What qualities made them different?
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
33
Looking at Student Work on A Number Pattern
Student A is able to look at all the numbers in the figure to find the numbers that appear
only once. The student can make the generalization for the numbers in the sum column
and give a description of the progression of numbers in the shaded row and how they are
formed.
Student A
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
34
Student B notices the 6 but doesn’t notice that the 2 also appears only once. The student
gives only a vague description in part 3b and is just difficult to understand in part 4.
While “getting bigger” might be an interesting observation at primary grades, it is not at
the level of observation expected for a sixth graders. How do we help students get
progressively deeper in their thinking and quality of responses as they move through the
grades?
Student B
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
35
Like many students, Student C does not look at the entire figure and misses that “2” is
only used once. In 3b the student makes on observation about the structure of the figure
rather than answering the question about the totals column. In part 4 the student is vague
about which two numbers are added to equal the shaded pattern.
Student C
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
36
Student D gives a similar response to Student C for part 4. In filling out the bottom row,
the student has failed to notice that the numbers in each row are symmetrical or that the
middle or middle 2 numbers are the highest numbers in the row. Do we give students
ample opportunities to describe in detail rich patterns and all the things they notice?
How do we help students start to piece together what are the relevant features in the
structure being described?
Student D
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
37
Student E is able to list most of the numbers in the totals row. The student seems so used
to working with patterns and tables that the student inserts a figure number to the left of
the total. However, the student’s pattern descriptions are very vague. The student may
notice the patterns, but doesn’t have facility in being specific. How do we communicate
to students the qualities we want in a good response? How do we give feedback about not
giving enough details so that students learn to write better mathematically?
Student E
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
38
Look at the work of Student F. How has this student interpreted the instructions “add
the two numbers above”? Notice that the student is looking at discrete digits when trying
to think about the totals column rather than thinking about the number, thirty-two versus
a three and a two. What types of experience does this student need? Also the student sees
3 and 6 and thinks about skipping 3’s. What are some of the errors in this thinking? Why
do you think the student got confused?
Student F
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
39
Student G also has difficulty interpreting “add the two numbers above”. What error has
this student made? The student didn’t understand what was expected in the sums column,
but has worked out the totals in the empty space. The student has actually described the
correct pattern, but not for the numbers appearing in his own sums column.
Student G
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
40
Student H doesn’t look at enough of the diagram to find the pattern for the bottom row.
The student seems to look at the 5’s and 10’s in the row above and then put multiples.
The student has noticed a pattern that the 2nd number in each row is one more than the
previous and that the rows are symmetrical. The student is still struggling with basic
computation, so he doesn’t see the pattern of doubling within the rows. Notice that the
student attempts to total the totals.
Student H
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
41
6th Grade
Student
Task
Task 3
Number Pattern
Describe and extend a number pattern.
Core Idea 3
Algebra and
Functions
Understand relations and functions, analyze mathematical
situations, and use models to solve problems involving quantity and
change.
• Represent, analyze, and generalize a variety of relations and
functions with tables, graphs, and words.
• Describe classes of numbers according to characteristics such as
Core Idea 1
the nature of their factors.
Number and
Operation
Mathematics of this task:
• Identifying a variety of number patterns and being able to describe the patterns
mathematically and completely
• Looking at a pattern with many different structural elements and identifying the
relevant features
• Making connections between structural elements in a pattern
Based on teacher observation, this is what sixth graders know and are able to do:
• Add the totals of the rows
• Notice and describe a doubling pattern
• Continuing the pattern for the first two and last two numbers in the bottom row
Areas of difficulty for sixth graders:
• Finding the pattern for the middle numbers in the bottom row
• Describing the pattern for the shaded numbers
• Looking through the entire pattern to see that the 2 is only used once
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
42
The maximum score available for this task is 8 points.
The minimum score needed for a level 3 response, meeting standards, is 5 points.
Most students, 90%, could notice that the 2 is only used once or do most of the totals
row. Many students 80% could do both. More than half the students, 52%, could fill in
the bottom row, give most of the totals, and describe the doubling pattern in the totals.
Some students, about 40%, could also note that the 2 is only used once. Almost 20% of
the students could meet all the demands of the task, including describing how the shaded
numbers are formed by adding a progression of consecutive numbers, starting with 2.
Almost 10% of the students scored no points on the task. 85% of the students with this
score attempted the task.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
43
A Number Pattern
Points
0
1
2
5
6
8
Understandings
Misunderstandings
85% of the students with this
score attempted the task.
Students had difficulty identifying the
number that was used only once. 21% of
the students thought it was just the 6. 4%
thought it was 1.
Students either knew that “2”
Some students did not notice that each row
was only used once or they can starts and ends with one, 3%. Some
fill in most of the bottom row.
students did not notice the pattern was
symmetrical, 19%. Some students did not
notice the middle number(s) are largest,
20%.
Students noted 2 was used only Students had difficulty with the totals
once and could fill in most of
column. 13% put 1,2,3… 9% just had an
the bottom row.
incorrect answer for the final sum. 5% did
some other pattern.
Students could fill in the bottom Some students still had difficulty with
row, add the totals, and notice
noticing that 2 was only used once.
the doubling pattern.
Students could do all of the task Almost 8% of the students did not attempt
except part 4, the pattern of the this part of the task. 8% might have
shaded numbers.
understood the pattern, but writing was too
unclear. 6% noticed that 1+3+4=10. 4%
gave vague answers, such as its getting
bigger.
Students could work with a
variety of patterns by describing
them or extending them, find
totals, and look at all the clues
in the pattern to see that 2 is
only used once.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
44
Implications for Instruction
Students need to be able to find and extend patterns. At this grade level they should be
exposed to a variety of patterns, beyond looking for odd and even numbers or growing or
decreasing by a set amount. They should recognize doubling patterns or growing by an
increasing number. Students need to be pushed to describe what they see in more detail.
Students often try to reach for some minimum level of explanation. This habit of mind
prevents them looking at numbers closely enough to find other patterns.
Giving students feedback on the quality of their explanations and examples of types of
explanations that are possible helps to raise the bar on student thinking. An important
instructional question to think about at this grade level is how to increase the cognitive
demand from expectations at previous grade levels.
Ideas for Action Research – Using Tasks for Instruction and the
Importance of Feedback
Most of the time MARS tasks are used for assessment purposes. But in following years,
consider using them for instructional purposes. At a recent lesson study, a group of
teachers asked the question about how they could improve the quality of answers given
by students. They didn’t feel that students challenged themselves to think deeply enough.
Teachers were also concerned about how to give feedback to students when their class
sizes were so large. They didn’t feel they had the time to write notes to every student, yet
they knew from articles, such as “Inside the Black Box” by Black and Wilam, that
specific feedback is one of the foremost factors in furthering student learning.
Consider how this task might be used as a whole class learning activity to work on this
issue.
You might start by giving pairs of students just the diagram. Ask them to find and
describe as many patterns in the diagram as they can. Have the pairs glue their diagram in
the middle of a large piece of poster paper and then write out their patterns, using colored
markers to help highlight what they are describing.
Students might then share out in groups of 3 to 5 pairs, with students asking each other
clarifying questions.
After everyone has had a chance to explore the patterns and make sense of the context,
think about asking a re-engagement question to push their thinking. For example:
Margie says, “I think this might be like other pattern problems. I bet the teacher will
want to know how to predict future numbers. What patterns will help us know what
comes in the next rows?”
See if this stimulates students to find new or different patterns. Now try another push.
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
45
Ford says, “I also bet the teacher will ask how we know if our predictions are correct.”
Kristi adds, “I think that if we add the total for each row it might make a pattern that can
help us.”
Do you think Kristi is correct? Why or why not?
Finally push students to evaluate responses and give their own feedback. It is important
for them to develop their own internal logic about what makes a detailed explanation.
Give them the rest of the task then pose a question, such as:
I noticed some patterns from other classes. Look at part 3b. What do you think each
student is thinking about?
The number is getting bigger and bigger.
Each row has a “1” at the end and the beginning.
Each number is the product of the number above it times 2.
Except for the 2 ones they are all different.
What do you like about their patterns? How might their explanations be improved?
Now look at part 4. Which responses do you like the best and why?
They are the sum of the numbers above it.
That the first three are skipping by 3’s.
I noticed that each number except 1 is the sum of the number plus the numbers in
numerical order. So, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = . . .
The difference between each number gets bigger by one like 1 and 3 are a
difference of 2, 3 and 6 are a difference of 3, . . .
They are added numbers.
How might you improve these explanations?
How does this lesson help all students follow the mathematics of the task? How does this
lesson help push students to think about the qualities of a good explanation?
Grade 6 – 2008
Copyright © 2008 by Noyce Foundation
All rights reserved.
46