Question: How do I enrich Investigations to sufficiently challenge gifted students?
Answer: The difficulty catering to students whose conceptual understanding of math is beyond those of their peers is a frequent concern of most teachers whether they are teaching Investigations or using a textbook. Some solutions isolate "gifted" children by providing them with a more advanced book or with a separate teacher or with work assignments that keep the student busy and away from the rest of the class for long periods of time. A few suggestions will be made at the end for resources the strong students can pursue on their own or with a little help. The bulk of this answer will consider tasks that draw attention to deeper or broader aspects of the Investigations mathematics the class is doing.
It is often tempting to challenge able students by increasing the size of the numbers with which they work. Although there are times when this adds an appropriate increase in difficulty, just using larger numbers doesn't adequately expand the skills of strong students. The kinds of problems described below involve finding, creating, and solving problems with multiple solutions and investigating relationships among numbers. Once students become familiar with some of these ways of creating more challenging problems based on a given problem, they can begin to pose them for themselves.
Be sure to challenge all of your students to adapt tasks to show their best mathematical thinking, to play with numbers, and to look for interesting patterns that occur. Who the "gifted" students in your class are is likely to change as the math content changes. Some students who have great difficulty with number work may excel with geometric tasks. Encourage the gifted students to include other students in their investigations and to do much of their work as an integral part of the Investigations work.
Here are some categories of questions, with a few examples of each, to suggest to students who need more challenging work. Many make good homework assignments over one or several days. Many of the challenges below involve number work, as this is the area that most teachers find most difficult to adjust for students with different levels of conceptual understanding. Many of the Investigations geometry, data, and mathematics of change units have their own challenging work for gifted students, who are expected to work precisely and to report their findings clearly, using equations, accurate graphs, and geometric drawings where appropriate. Some of these problems come directly from Investigations units. Others are variations related to the content and style of Investigations. Most are adaptable for many grade levels.
 Create problems that have solutions with certain characteristics.
 Put in order from least to most.
 How many different ways are there?
 Is your conjecture always true? Can you prove your conjecture?
 Additional Resources
Create problems that have solutions with certain characteristics.
Write about or make a poster showing what you learned in creating these different kinds of problems.
Find a set of 6 digits that allows for many perfect solutions to Close to 100, and another set that provides no solutions closer than 3 away. Adapt this challenge for Close to 20 or Close to 1000.
Make an addition problem of two or more addends with no zeroes that has a sum with some zeroes. Make multiplication, subtraction, and division problems with zeros in the answers but no zeros in the problem numbers. Make a multidigit multiplication problem with several odd digits in the factors (the problem numbers) but only even digits in the product.
Create or describe different probability situations where the chances are exactly 1 in 2 (or other ratios such as 1 in 3, or 2 in 3 or 3 in 4). Make a poster to illustrate situations for one of the probabilities.
Create two or more rectangular buildings of connecting cubes that have the same surface area; put them in order by volume. Or try the easier task of creating buildings with the same volume and putting them in order by surface area.
Create fractions with large numerator and denominator that simplify to single digit numerator or denominator, e.g. 495/825; as an extension to this, make mixed number multiplication problems that have whole number answers or answers with simple fractions, e.g. 2 1/2 x 1 3/5.
Invent problems that look hard but can be done mentally, e.g for younger students: 36 + 40; 10 x 13; 50 x 6; or for older students: 284 x 25; 2001  98; 999,999 + 523; 498^{2}. Display them for other students to try, and discuss mental solutions.
Write Guess My Number Puzzles of the sort in Mathematical Thinking at Grade 5 by providing four clues. Exchange these with classmates and even with another class.
Invent Closest Answer problems for TenMinute Math; put one good answer and three misleading answers for each problem. The teacher can choose some from among the invented problems to present to the class, and ask the student to write them neatly on an overhead or as a handout to be duplicated.
Put in order from least to most.
If there are many possibilities, find the 5 or 6 examples with the most or least. Show and explain how you determined the order.
Find out how many different rectangular arrangements can be made for numbers of chairs between 10 and 30 (number of factor pairs for each number).
Which numbers between 1 and 100 (or between 100 and 200) have the most factors?
Order cereals by price per weight (per 100 gm or per oz.); find the most expensive food substance or the most expensive body lotion by weight and do a presentation about camouflaging packaging.
Order rectangular boxes by volume; less advanced fill the boxes with cubes or with sand to compare size; more advanced figure out order by using only linear measurement. Order irregular shape liquid containers by volume without filling them; check by filling them.
Arrange all of the digits 1, 2, 3, 4, 5 into two numbers (of 1, 2, 3, or 4 digits) to multiply to make the largest product; make the smallest product; make other products and arrange the problems in order from smallest to largest product. Younger children can do the same with addition and subtraction problems to make largest and smallest sum and difference.
Make all the different fractions (including fractions with equal numerator and denominator and with larger numerator than denominator) using only the numbers 2, 3, and 4 as numerators and denominators; put the fractions in order from smallest to largest (2/4, 2/3, 3/4, 2/2 = 3/3 = 4/4, etc.). Do this with a different set of numbers, perhaps 1, 2, and 5; try it with 2, 6, 9. What is always true about the order of the fractions made from 3 numbers? Investigate using 4 different numbers.
Find the greatest and least in real data outside the classroom. What class in the school has the most/fewest students? (Find out by looking at names in the classroom when the class is out at recess.) Look in the newspaper to find out which stock gained or lost the most and which stock gained or lost the greatest percentage. What city had the greatest range in temperature? What person buried in a nearby cemetery lived the longest? What person was born in the earliest year?
How many different ways are there?
These problems are directly from Investigations units or slight variations; the challenge is to find all the solutions and organize them to show you have them all.
How many combinations of popcorn and carrot sticks to have 8 pieces of snack in all? To have 9 pieces of snack in all? To have 10 in all? How can you predict how many combinations there are of two things to make other totals?
How many different rectangular picture frames can be made using only 10 cm, 15 cm, and 20 cm. lengths for sides? How many can be made having four different lengths available for sides? How many if you have only two different lengths?
How many ways to combine 4 legged dogs, 2 legged people, and 6 legged ants to get 20 legs in all? Write a word problem about this that has one of the combinations as a unique answer.
How many ways to make fourths of a regular 16 unit geoboard with vertices only at the nails? Draw all the different fourths on dot paper. How many ways can you find to split a geoboard area into four congruent fourths?
How many ways can you find to make a rectangular box with a volume of 24 cubic inches using linear dimensions with only whole or half inches? To make a rectangular box with a volume of 60 cubic units?
Is your conjecture always true? Can you prove your conjecture?
Explain your idea to someone else or the class or show it on a poster so others understand and make use of it. Make a list of mathematical ideas that are always true; ask your classmates to add to the list. Here are some conjectures to test and prove.
Adding tens and multiples of tens (20, 30, 40, etc.) to a number keeps the same one's digit; adding on hundreds always keeps the same ones and tens digits.
Multiplying two numbers with 4 in the one's place always makes a product with 6 in the one's place. Investigate this with different combinations of one's digits. Investigate the one's digits of square numbers.
Adding 2 odd numbers always makes an even number, but multiplying two odd numbers always makes an odd number.
When 2 people share equally they each get the number in the middle between the numbers they start with (e.g., one starts with 5 and the other starts with 9, and they each get 7).
In all fractions equal to one half, the denominator is twice as big as the numerator; the numerator is half as big as the denominator. What generalizations can you make about other fractions such as 1/3 or 3/4?
A square always has a smaller perimeter than other rectangles of the same area.
Any side of a triangle is always shorter than the lengths of the other two sides added together.
A triangle cannot have more than one right angle. Prove this and then investigate how many right angles shapes with more sides can have (quadrilateral, pentagon, hexagon, etc.). Can you find any sort of pattern?
Additional Resources.
Here are some resources that elementary students can work with independently and introduce to the class.
Web sites:

Elementary Problem of the Week on the Math Forum.

Aunty Math.
 Figure This. Math challenges for families.
Books:
TenMinute Math by Cornelia Tierney and Susan Jo Russell. Dale Seymour Publications 2001. Order number DS21260. 8003213106. This is a compilation of TenMinute Math activities and games from the Investigations curriculum, with some additional suggestions for variation. Note that some games from younger grades are useful in older grades, for example the harder variation of "Fraction Cookie Game" from grade 3 Fair Shares or "Capture Five" from grade 2 Putting Together and Taking Apart. Activities meant for grade 5 can be used in grades 3 and 4. For example, "Nearest Answer", "The Estimation Game", and "The Digits Game".
Mathematics: A Human Endeavor by Harold Jacob. Freeman Press. Long a favorite of some of the Investigations authors, here are notes from a review on a homeschooling web site: "The topics range widely: number tricks and deductive reasoning, graphing the path of billiard balls, number sequences, functions and their graphs, large numbers and logarithms, polygons and symmetry, mathematical curves (including ingenious ways to fold these curves with a piece of paper, and problems exploring the logarithmic spiral of the chambered nautilus shell), probability and chance, and topology. Always there are experiments to do, and intriguing problems that help you learn about the world."
Get It Together (Grades 412) and United We Solve (Grades 510) by Tim Erickson. These are books of Math Problems at varied content and levels of difficulty to be done collaboratively by groups of 4 students who each get one clue to read and share with the group. These can be done by 2 and 3 students as well with some getting more than one clue. They can be done by one student alone, but the value of the format is to get students working together. Spanish edition available. Available together at: http://www.eeps.com/.
Favorite Problems. Dale Seymour. 1982. Order number DS01234. 8003213106. This is a collection of classic problems on black line masters with solutions and extensions on the back. Students can work on these alone or with a partner.
Mental Math in the Middle Grades by Jack Hope, Barbara Reys, and Robert Reys. Dale Seymour 1987. Order number DS01615 8003213106. This is especially for students who may or may not be mathematically gifted, but who have been taught to do written computation with huge numbers and find it boring. This is to move them away from paper and pencil to doing mental arithmetic.
From Marilyn Burns:
 The Book of Think
 The I Hate Mathematics Book
 This Book's About Time
 Math for Smartypants
Software:
Tabletop. The Learning Company 8008678322. Tabletop Jr for grades K4 allows students to work with data to develop understanding of attributes, logical relationships, and plotting points. Tabletop Senior for grades 412 is a general purpose tool for making plots and graphs to organize and analyze data. These tools can be learned by skilled students and introduced to the class to greatly enrich the data work.
Logical Journey of the Zoombinis. The Learning Company 8008678322. A prizewinning software that engages students in figuring out successively more difficult logic problems embedded in a simulation.
Cornelia Tierney, TERC
September 2000
This information was reprinted with permission of CESAME, Northeastern Univ., and the Educational Alliance, Brown University.