To see how these performance descriptions
compare with the expectations for middle school and high
school, click here.
The examples that follow the performance
descriptions for each standard are examples of the work
students might do to demonstrate their achievement. The
examples also indicate the nature and complexity of activities
that are appropriate to expect of students at the elementary
level. Depending on the nature of the task, the work might
be done in class, for homework, or over an extended period.
The crossreferences that follow the examples
highlight examples for which the same activity, and possibly
even the same piece of work, may enable students to demonstrate
their achievement in relation to more than one standard.
In some cases, the crossreferences highlight examples of
activities through which students might demonstrate their
achievement in relation to standards for more than one subject
matter.
The crossreferences after the examples that begin “E,”
“S,” and “A”
refer to the performance standards for English Language
Arts, Science, and Applied Learning respectively. See, for
example, the crossreferences after the examples of activities
for .



Arithmetic
and Number Concepts 
The student demonstrates understanding of a mathematical
concept by using it to solve problems, representing it in
multiple ways (through numbers, graphs, symbols, diagrams,
or words, as appropriate), and explaining it to someone
else. All three ways of demonstrating understanding—use,
represent, and explain—are required to meet this standard.
The student produces evidence that demonstrates understanding
of arithmetic and number concepts; that is, the student:
a
Adds, subtracts, multiplies, and divides whole numbers,
with and without calculators; that is: 
• 
adds, i.e., joins things together, increases; 
• 
subtracts, i.e., takes away, compares,
finds the difference; 
• 
multiplies, i.e., uses repeated addition,
counts by multiples, combines things that come in groups,
makes arrays, uses area models, computes simple scales,
uses simple rates; 
• 
divides, i.e., puts things
into groups, shares equally; calculates simple rates; 
• 
analyzes problem situations and contexts
in order to figure out when to add, subtract, multiply,
or divide; 
• 
solves arithmetic problems by relating
addition, subtraction, multiplication, and division
to one another; 
• 
computes answers mentally, e.g., 27 +
45, 30 x 4; 
• 
uses simple concepts of negative numbers,
e.g., on a number line, in counting, in temperature,
“owing.” 
b Demonstrates understanding of the base ten
place value system and uses this knowledge to solve
arithmetic tasks; that is: 
• 
counts 1, 10, 100, or 1,000 more than
or less than, e.g., 1 less than 10,000, 10 more than
380, 1,000 more than 23,000, 100 less than 9,000; 
• 
uses knowledge about ones, tens, hundreds,
and thousands to figure out answers to multiplication
and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000,
4,000 ÷ 4. 
c
Estimates, approximates, rounds off, uses landmark
numbers, or uses exact numbers, as appropriate, in
calculations.
d
Describes and compares quantities by using concrete
and real world models of simple fractions; that is: 
• 
finds simple parts of wholes; 
• 
recognizes simple fractions as instructions
to divide, e.g., 1/4
of something is the same as dividing something by 4; 
• 
recognizes the place of fractions on number
lines, e.g., in measurement; 
• 
uses drawings, diagrams, or models to
show what the numerator and denominator mean, including
when adding like fractions, e.g., 1/8
+ 5/8, or
when showing that 3/4
is more than 3/8; 
• 
uses beginning proportional reasoning
and simple ratios, e.g., “about half of the people.” 
e
Describes and compares quantities by using simple decimals;
that is: 
• 
adds, subtracts, multiplies, and divides
money amounts; 
• 
recognizes relationships among simple
fractions, decimals, and percents, i.e., that 1/2
is the same as 0.5, and 1/2
is the same as 50%, with concrete materials, diagrams,
and in real world situations, e.g., when discovering
the chance of a coin landing on heads or tails. 
f
Describes and compares quantities by using whole numbers
up to 10,000; that is: 
• 
connects ideas of quantities to the real
world, e.g., how many people fit in the school’s
cafeteria; how far away is a kilometer; 
• 
finds, identifies, and sorts numbers by
their properties, e.g., odd, even, multiple, square. 

Examples
of activities through which students might demonstrate
conceptual understanding of arithmetic and number include: 

Use examples and drawings to show a third
grader who is having trouble understanding multiplication
why 3 x 6 = 6 x 3. 1a 

Use base ten blocks and numerals or other
models and representations to solve 43 x 38, and show
how the numbers get taken apart in the process of solving
such a problem, e.g., 43 can become 40 + 3, or 38 can
become 40  2. 1a, 1b 

Figure out how many Valentine’s Day
cards would be exchanged in total, if there are 30 students
in the class and if everyone gave everyone else one
card. 1a, 1b, 1c 

Draw and explain many different ways to
make 263, using tens, hundreds and ones. 1a,
1b, 1c 

Organize a budget for a project. 1a,
1b, 1c, 1d, 1e, 1f, A1a, A1b, A1c 

Solve the following problem: “You
have a book to read that is 100 pages long. You must
read it in five days. How many pages should you read
each day?” 1a, 1b, 1c,
1f 

Determine the possible number of objects
in a group given this information: when put in groups
of three, none is left over; when put in groups of two,
one is left over; when put in groups of five, one is
left over; there are fewer than 40 objects altogether.
1a, 1b, 1c, 1f 

Make reasonable estimates and then calculate
accurately the number of beans in a cup, seeds in a
pumpkin, ceiling tiles in a room, and wheels on 37 tricycles.
1a, 1b, 1c, 1f 

Show and explain to a classmate the different
coin combinations that make 75¢, excluding pennies.
1a, 1b, 1c, 1f 

Find one possible answer for addends
that equal the sum of 8,829, when the digits 1, 2, 3,
4, 5, 6, 7, and 8 are used only once in the solution.
1a, 1b, 1c, 5a, 5b, 5c 

Analyze seasonal variations in temperature,
including negative values.
1a, 1c, 1f, S3c, A1a 

Put together fraction pieces to make
a whole in different ways, e.g., 1/8
+ 1/8 +
3/4. 1c,
1d 

Draw diagrams to explain how three pizzas
can be shared equally by four people. 1c,
1d 

Make approximate counts of different kinds
of animals in different zones of a small tide pool area.
1c, 1f, S6b 

Draw and label diagrams to show the fractional
value of each piece of a set of tangram pieces, if all
seven pieces (i.e., the whole set) are equal to one
whole. 1d 

Pretend you see a highway sign that reads,
“Exit A, 1 mile. Exit B, 2¼ miles,”
and shortly thereafter you see a sign that reads, “Exit
A, ¼ mile.” Figure out how far away exit
B must now be. (Balanced Assessment) 1d,
1f 

Make up stories that go with number sentences,
e.g., 5 x 9 = 45; 27 x 6 = ?. 1f




Geometry
and Measurements Concepts 
The student demonstrates understanding
of a mathematical concept by using it to solve problems,
representing it in multiple ways (through numbers, graphs,
symbols, diagrams, or words, as appropriate), and explaining
it to someone else. All three ways of demonstrating understanding—use,
represent, and explain—are required to meet this standard.
The student produces evidence
that demonstrates understanding of geometry and measurement
concepts; that is, the student: 

a
Gives and responds to directions about location, e.g.,
by using words such as “in front of,” “right,”
and “above.” 

b
Visualizes and represents two dimensional views of simple
rectangular three dimensional shapes, e.g., by showing
the front view and side view of a building made of cubes. 

c
Uses simple two dimensional coordinate systems to find
locations on a map, and represent points and simple
figures. 

d
Uses many types of figures (angles, triangles, squares,
rectangles, rhombi, parallelograms, quadrilaterals,
polygons, prisms, pyramids, cubes, circles, and spheres)
and identifies the figures by their properties, e.g.,
symmetry, number of faces, two or three dimensionality,
no right angles. 

e
Solves problems by showing relationships between and
among figures, e.g., using congruence and similarity,
and using transformations including flips, slides, and
rotations. 

f
Extends and creates geometric patterns using
concrete and pictorial models. 

g
Uses basic ways of estimating and measuring the
size of figures and objects in the real world, including
length, width, perimeter, and area. 

h
Uses models to reason about the relationship
between the perimeter and area of rectangles in simple
situations. 

i
Selects and uses units, both formal and informal
as appropriate, for estimating and measuring quantities
such as weight, length, area, volume, and time. 

j
Carries out simple unit conversions, such as
between cm and m, and between hours and minutes. 

k
Uses scales in maps, and uses, measures, and
creates scales for rectangular scale drawings based
on work with concrete models and graph paper.

Examples of activities
through which students might demonstrate conceptual
understanding of geometry and measurement include: 

Locate a point on a map using its distance away from
another given point, the map’s scale, and compass
point directions, e.g., N, NE, E, SE, etc. 2a,
2c, 2k 

Given views of three faces of a three dimensional
figure made of stacked cubes, represent the views of
the other faces and figure out the total number of cubes
used to construct the figure. 2b 

Explain and show why a square is not a cube. 2d 

Describe a shape from among the following,
using as few attributes as possible to distinguish it
from the others: right triangle, equilateral triangle,
trapezoid, and parallelogram. 2d 

Design and label a quilt square which includes two
lines of symmetry, four congruent shapes, and two similar
shapes; then make a final version at twice the original
size, i.e., a “two to one” scale, and add
color while maintaining symmetry. 2d,
2e, 2f, 2k 

Draw
at each of three consecutive 90° rotations in a
clockwise direction. 2e 

Find all the shapes that can be made with five squares
if the sides touch completely. 2e 

Figure out the approximate area and perimeter of the
bottom of a shoe. 2g 

Solve the following problem, using concrete materials
as appropriate: “Given a diagram of a fencedin,
rectangular garden plot with dimensions three meters
by eight meters, find its area and perimeter; design
a second garden plot using less fencing, but providing
greater area; design a third plot using more fencing,
but providing less area. Of all possible rectangular
designs using the original amount of fencing, which
provides the greatest area?” 2g,
2h, 2i 

Put five objects, such as books, rocks, or pumpkins,
in rank order by weight, first by estimating and then
by measuring exactly. 2i 

Figure out exactly how much time would you gain for
the completion of a project by waking up at 6:15 rather
than 7:00 for five school days. 2i,
2j 



Function
and Algebra Concepts 
The student demonstrates understanding
of a mathematical concept by using it to solve problems,
representing it in multiple ways (through numbers, graphs,
symbols, diagrams, or words, as appropriate), and explaining
it to someone else. All three ways of demonstrating understanding—use,
represent, and explain—are required to meet this standard.
The student produces evidence
that demonstrates understanding of function and algebra
concepts; that is, the student: 

a
Uses linear patterns to solve problems; that is: 

• 
shows how one quantity determines another
in a linear (“repeating”) pattern, i.e., describes,
extends, and recognizes the linear pattern by its rule,
such as, the total number of legs on a given number
of horses can be calculated by counting by fours; 

• 
shows how one quantity determines another
quantity in a functional relationship based on a linear
pattern, e.g., for the “number of people and total
number of eyes,” figure out how many eyes 100 people
have all together. 

b
Builds iterations of simple nonlinear patterns, including
multiplicative and squaring patterns (e.g., “growing”
patterns) with concrete materials, and recognizes that
these patterns are not linear. 

c
Uses the understanding that an equality relationship
between two quantities remains the same as long as the
same change is made to both quantities. 

d
Uses letters, boxes, or other symbols to stand for any
number, measured quantity, or object in simple situations
with concrete materials, i.e., demonstrates understanding
and use of a beginning concept of a variable. 
Examples of activities
through which students might demonstrate conceptual
understanding of functions and algebra include: 

Find, make, and describe linear patterns
on the 99chart, e.g., 4, 14, 24, 34. 3a 

Given the situation described in the Christmas carol,
“The Twelve Days of Christmas,” determine
how many gifts would have been given in total; describe
any pattern you notice that helps in solving the problem.
3a, 3b 

Show how the letters “aab, aab,...,” can
represent the pattern “metal, metal, plastic,...,”
“leaf, leaf, rock,...,” or many other patterns.
3a, 3d 

Solve the following problem: When building a staircase
out of cubes, one step uses a total of one cube, two
steps a total of three cubes, three steps a total of
six cubes, etc. Find how many cubes are used for six
steps, nine steps, n steps. Describe the pattern. 3b 

Observe and record, in a twocolumn table, multiplicative
patterns with concrete materials, e.g., how many regions
are produced by increasing the numbers of folds of paper;
and recognize that this type of pattern does not proceed
in a linear way, i.e., not 2, 4, 6, 8, etc., but 2,
4, 8, 16, etc. 3b 

Build the fourth, fifth, and sixth iterations in the
following sequence:
;
use any given number to make a square number; and recognize
that the number of little squares in the pattern does
not proceed in a linear fashion. 3b 

Given x + y = 10 (or, say,+
__=10), figure out all the whole numbers that will make
the equation true, i.e., use the variables to show all
the ways to make ten by adding two whole numbers together.
3c, 3d 

Plot points on a coordinate graph according to the
convention that (x,y) refers to the intersection of
a given vertical line and a given horizontal line. 3b 



Statistics
and Probability Concepts 
The student demonstrates understanding of a mathematical concept
by using it to solve problems, representing it in multiple ways
(through numbers, graphs, symbols, diagrams, or words), and explaining
it to someone else. All three ways of demonstrating understanding—use,
represent, and explain—are required to meet this standard.
The student produces evidence that demonstrates
understanding of statistics and probability concepts in the
following areas; that is, the student: 
a 
Collects and organizes data to answer a question
or test a hypothesis by comparing sets of data. 
b 
Displays data in line plots, graphs, tables,
and charts. 
c 
Makes statements and draws simple conclusions
based on data; that is: 
• 
reads data in line plots, graphs, tables, and
charts; 
• 
compares data in order to make true statements,
e.g., “seven plants grew at least 5 cm”; 
• 
identifies and uses the mode necessary for making
true statements, e.g., “more people chose red”; 
• 
makes true statements based on a simple concept
of average (median and mean), for a small sample size and
where the situation is made evident with concrete materials
or clear representations; 
• 
interprets data to determine the reasonableness
of statements about the data, e.g., “twice as often,”
“three times faster”; 
• 
uses data, including statements about the data,
to make a simple concluding statement about a situation, e.g.,
“This kind of plant grows better near sunlight because
the seven plants that were near the window grew at least 5
cm.” 
d 
Gathers data about an entire group or by sampling
group members to understand the concept of sample, i.e., that
a large sample leads to more reliable information, e.g., when
flipping coins. 
e 
Predicts results, analyzes data, and finds
out why some results are more likely, less likely, or equally
likely. 
f 
Finds all possible combinations and arrangements
within certain constraints involving a limited number of variables. 
Examples of activities
through which students might demonstrate conceptual understanding
of statistics and probability include: 

Generate survey questions, such as: How many
people are in your family? What is your hair color? What do
you do at recess? Predict the survey results, carry out the
survey, graph the data, and write true statements about the
data. 4a, 4b, 4c, 4d 

Figure out a need, e.g., the librarian’s
purchase of new library books; and collect data in order to
make a recommendation. 4a, 4b, 4c,
4d 

Make conjectures, after sampling the particular
situation, about how many raisins are in a given box taken
from a set of boxes or what colors of tiles are in a bag.
4a, 4b, 4c, 4d 

Investigate the temperature over the entire
school year. 4a, 4b, 4c, 4d, S3c,
A1a 

Investigate the possible and likely or unlikely
outcomes when rolling two number cubes and recording the sums.
4d, 4e, 4f 

Design spinners with regions red, blue and green,
according to the following instructions: Spinner #1: red will
certainly win; Spinner #2: red cannot win; Spinner #3: red
is likely to win; Spinner #4: red, blue and green are equally
likely to win; Spinner #5: red or blue will probably win.
4e 

Find all of the different combinations possible
using three ice cream flavors, two sauces, and one topping;
then discover which would allow the most new combinations—an
additional ice cream flavor or a new topping. (Balanced Assessment)
4f 



Problem
Solving and Reasoning 
The student demonstrates logical reasoning throughout
work in mathematics, i.e., concepts and skills, problem
solving, and projects; demonstrates problem solving by using
mathematical concepts and skills to solve nonroutine problems
that do not lay out specific and detailed steps to follow;
and solves problems that make demands on all three aspects
of the solution process—formulation, implementation,
and conclusion.
Formulation
a
Given the basic statement of a problem situation, the
student: 
• 
makes the important decisions about the
approach, materials, and strategies to use, i.e., does
not merely fill in a given chart, use a prespecified
manipulative, or go through a predetermined set of steps; 
• 
uses previously learned strategies, skills,
knowledge, and concepts to make decisions; 
• 
uses strategies, such as using manipulatives
or drawing sketches, to model problems. 
Implementation
b The student
makes the basic choices involved in planning and carrying
out a solution; that is, the student: 
• 
makes up and uses a variety of strategies
and approaches to solving problems and uses or learns
approaches that other people use, as appropriate; 
• 
makes connections among concepts in order
to solve problems; 
• 
solves problems in ways that
make sense and explains why these ways make sense, e.g.,
defends the reasoning, explains the solution. 
Conclusion
c The student moves
beyond a particular problem by making connections,
extensions, and/or generalizations; for example, the
student:

• 
explains a pattern that can
be used in similar situations; 
• 
explains how the problem is
similar to other problems he or she has solved; 
• 
explains how the mathematics
used in the problem is like other concepts in mathematics; 
• 
explains how the problem solution
can be applied to other school subjects and in real
world situations; 
• 
makes the solution into a general
rule that applies to other circumstances. 
Examples of activities through which
students might demonstrate facility with problem solving
and reasoning include: 

Suppose you are given a string that is
sixteen inches long. If you cut or fold it in any two
places, will it always make a triangle? 5a,
5b, 5c 

Figure out how many handshakes there will
be altogether if five people in a room shake each other’s
hand just once. 5a, 5b,
5c 

Prove or disprove a classmate’s claim
that 3/4
and 5/6
are really the same because both have one piece missing.
5a, 5b, 5c 

Given an unopened bag of chocolate chip
cookies, with cookies arranged in rows, show and explain
whether the manufacturer’s claim of “More
than 1,000 chips in every bag!” is reasonable.
5a, 5b, 5c 

Figure out the value of each tangram piece
if the small tangram triangle is worth one cent and
the value is proportional to the area. 5a,
5b, 5c 

For the sum of 8,829, find one possible
answer for addends when the digits 1, 2, 3, 4, 5, 6,
7, and 8 are used only once in the solution. 5a,
5b, 5c 



Mathematical
Skills and Tools 
The student demonstrates fluency with basic and important
skills by using these skills accurately and automatically, and
demonstrates practical competence and persistence with other skills
by using them effectively to accomplish a task, perhaps referring
to notes, books, or other students, perhaps working to reconstruct
a method; that is, the student:
a
Adds, subtracts, multiplies, and divides whole numbers correctly;
that is: 
• 
knows single digit addition, subtraction, multiplication,
and division facts; 
• 
adds and subtracts numbers with several digits; 
• 
multiplies and divides numbers with one or
two digits; 
• 
multiplies and divides three digit numbers
by one digit numbers. 
b
Estimates numerically and spatially. 
c
Measures length, area, perimeter, circumference, diameter,
height, weight, and volume accurately in both the customary
and metric systems. 
d
Computes time (in hours and minutes) and money (in dollars
and cents). 
e
Refers to geometric shapes and terms correctly with concrete
objects or drawings, including triangle, square, rectangle,
side, edge, face, cube, point, line, perimeter, area, and
circle; and refers with assistance to rhombus, parallelogram,
quadrilateral, polygon, polyhedron, angle, vertex, volume,
diameter, circumference, sphere, prism, and pyramid. 
f
Uses +, , x, ÷, /, ,
$, ¢, %, and . (decimal point) correctly in number sentences
and expressions. 
g
Reads, creates, and represents data on line plots, charts,
tables, diagrams, bar graphs, simple circle graphs, and coordinate
graphs. 
h Uses recall, mental computations,
pencil and paper, measuring devices, mathematics texts, manipulatives,
calculators, computers, and advice from peers, as appropriate,
to achieve solutions; that is, uses measuring devices, graded
appropriately for given situations, such as rulers (customary
to the 1/8
inch; metric to the millimeter), graph paper (customary to
the inch or halfinch; metric to the centimeter), measuring
cups (customary to the ounce; metric to the milliliter), and
scales (customary to the pound or ounce; metric to the kilogram
or gram). 

Examples of activities
through which students might demonstrate facility with mathematical
skills and tools include: 

Know that 6 x 7 = 42. 6a 

Use an efficient mental approach to estimate
the total cost of items costing $2.94, $1.28 and $0.74. 6a,
6b, 6d, 6h 

Mentally add two digit numbers correctly during
problem solving. 6a, 6h 

Decide whether to use a calculator, paper and
pencil or mental arithmetic to figure out 6 x 6,000. 6a,
6h 

Figure out rectangular areas correctly when
designing a floor plan for a “dream house.” 6b,
6c 

Measure the circumference of a pumpkin accurately
by using a piece of yarn, and then laying it next to a ruler.
6c, 6e 

Use a calculator to check the arithmetic in
a project. 

Use a table to record functions such as how
many chairs fit at how many tables. 6g 

Use a Venn diagram to record students who wore
a sweater to school and students who walked to school. 6g 

Make a bar graph or simple circle graph to show
how many students like different kinds of vegetables. 6g,
6f 


Mathematical
Communication 
The student uses the language of mathematics, its symbols,
notation, graphs, and expressions, to communicate through reading,
writing, speaking, and listening, and communicates about mathematics
by describing mathematical ideas and concepts and explaining
reasoning and results; that is, the student:
a
Uses appropriate mathematical terms, vocabulary, and language,
based on prior conceptual work. 
b
Shows mathematical ideas in a variety of ways, including
words, numbers, symbols, pictures, charts, graphs, tables,
diagrams, and models. 
c
Explains solutions to problems clearly and logically, and
supports solutions with evidence, in both oral and written
work. 
d
Considers purpose and audience when communicating about
mathematics. 
e Comprehends mathematics
from reading assignments and from other sources. 
Examples of
activities through which students might demonstrate facility
with mathematical communication include: 

Use words, numbers, or diagrams to explain
how to take numbers apart in order to solve problems using
mental math, e.g., 25 x 6. One way is “20 x 6 is 120,
and 5 x 6 is 30; 120 + 30 = 150.” Or, “25 x 4
is 100, and 25 x 2 is 50, and so the answer is 150 because
100 + 50 = 150.” 7a, 7b, 7c 

Explain why 34 + 17 3417
to a first grader or to a visitor from outer space.
7a, 7b, 7c, 7d 

Show 1/2 + 3/4 in pictures and diagrams so
a younger student could understand the sum. 7a,
7b, 7c, 7d 

Use clear, correct mathematical language
to describe a shape composed of several geometric solids
so that it could be reproduced exactly by a person who cannot
see the shape. 7a, 7b, 7c, 7d 

Represent survey data about student school
lunch preferences in graphical, written, and numerical form
in order to make a clear and effective recommendation to
kitchen staff about menu changes. 7a,
7b, 7c, 7d 

Give an oral presentation of a preliminary
investigation of classification of shapes, in order to get
peer feedback; then revise the classification scheme to
make it clearer. 7a, 7b, 7c, 7d,
E3c 

Prepare a report, including graphs, charts,
and diagrams, on the optimal number and location of recycling
containers, based on data from the classroom and the entire
school. 7a, 7b, 7c, 7d, 7e, E2a,
S4b, S6d, S7a, A1b 


Putting
Mathematics to Work 
The student conducts at least one large scale project each year,
beginning in fourth grade, drawn from the following kinds and,
over the course of elementary school, conducts projects drawn
from at least two of the kinds.
A single project may draw on more
than one kind. 
a Data study, in which the
student: 
• 
develops a question and a hypothesis in a situation
where data could help make a decision or recommendation; 
• 
decides on a group or groups to be sampled and
makes predictions of the results, with specific percents,
fractions, or numbers; 
• 
collects, represents, and displays data in
order to help make the decision or recommendation; compares
the results with the predictions; 
• 
writes a report that includes recommendations
supported by diagrams, charts, and graphs, and acknowledges
assistance received from parents, peers, and teachers. 
b
Science study, in which the student: 
• 
decides on a specific science question to study
and identifies the mathematics that will be used, e.g., measurement; 
• 
develops a prediction (a hypothesis) and develops
procedures to test the hypothesis; 
• 
collects and records data, represents and displays
data, and compares results with predictions; 
• 
writes a report that compares the results with
the hypothesis; supports the results with diagrams, charts,
and graphs; acknowledges assistance received from parents,
peers, and teachers. 
c
Design of a physical structure, in which the student: 
• 
decides on a structure to design, the size
and budget constraints, and the scale of design; 
• 
makes a first draft of the design, and revises
and improves the design in response to input from peers and
teachers; 
• 
makes a final draft and report of the design,
drawn and written so that another person could make the structure;
acknowledges assistance received from parents, peers, and
teachers. 
d
Management and planning, in which the student: 
• 
decides on what to manage or plan, and the criteria
to be used to see if the plan worked; 
• 
identifies unexpected events that could disrupt
the plan and further plans for such contingencies; 
• 
identifies resources needed, e.g., materials,
money, time, space, and other people; 
• 
writes a detailed plan and revises and improves
the plan in response to feedback from peers and teachers; 
• 
carries out the plan (optional); 
• 
writes a report on the plan that
includes resources, budget, and schedule, and acknowledges
assistance received from parents, peers, and teachers. 

e
Pure mathematics investigation, in which the student: 
• 
decides on the area of mathematics to investigate,
e.g., numbers, shapes, patterns; 
• 
describes a question or concept to investigate; 
• 
decides on representations that will be used, e.g.,
numbers, symbols, diagrams, shapes, or physical models; 
• 
carries out the investigation; 
• 
writes a report that includes any generalizations
drawn from the investigation, and acknowledges assistance received
from parents, peers, and teachers. 
Examples of projects include: 

Develop questions and a hypothesis for a study of students’
diets; collect, organize, display, and analyze the data; and make
recommendations to the school community based on the data. 8a,
S2a, S4c, A1b 

Compare the growth of a set of plants under a variety of conditions,
e.g., amount of water, fertilizer, duration and exposure to sunlight.
8b, S2a 

Make a design for a tree house that accounts for physical and
financial constraints. 8c, A1a 

Plan a class camping trip, including making a schedule, researching
costs and facilities, developing a budget. 8d,
S2a, S2c, S4b, S4c, A1c 

Plan and conduct a probability study that compares the results
from three different spinners, e.g., . 8e 
