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Mathematics Curriculum Framework
Achieving Mathematical Power  January 1996
 "None of us really understands what's going on with these numbers."
David Allen Stockman, Director US OMB, on the U.S. Budget, 1981
Number sense is the cornerstone of mathematics. Sound number sense enables us to interpret and represent the world in which we live. It is exemplified every day, whether we consider notions as complex as the consumer price index, as pivotal as the impact of the Great Depression on United States history, or as personal as a blood pressure reading.
Youngsters' first use of numbers is with whole numbers. Soon, they use fractions when they share snacks with friends and decimals when they buy supplies to make valentines. As preadolescents, they explore negative numbers, perhaps with subzero temperature readings in Celsius. In high school, teenagers expand their understanding of real numbers when they collaborate to build a scale model of their remodeled school for a display at City Hall.
Sound number sense enhances mathematical intuition, which plays a significant role in mathematical problem solving. Mathematical intuition suggests appropriate strategies when confronting a problem situation.
Number sense promotes accuracy in estimation and mental math. While calculators and computers are used to do most of the complex computations in today's world, the ability to estimate is critical for lifelong learners. Students estimate whether there is ample time to get to a movie before it begins. They estimate the cost of items they have gathered from recycling bins at the museum before they approach the cash register. Estimates are used to evaluate the reasonableness of a solution produced by a calculator we all know the results of entering an extra zero or misplacing a decimal point.
Number Sense and Computation
Number sense suggests an ability to compute answers to problems involving quantities. Students figure out answers to problems long before they encounter formal mathematical representations. Younger students can quickly share evenly a dozen baseball cards among three friends before they have studied formal multiplication or division. Older students can find several ways to find the length of the side of a square with an area of five before they ever study irrational numbers.
Students should be taught how to do computational procedures. If they have developed their own accurate and sensible computational methods, then they should be encouraged to use them. A fourth grade student who finds the difference between 187 and 536 by mentally adding on 13 to make 200, adding 300 to make 500, and finally adding the 13 to 36 to get an answer of 349 exhibits a strong, internalized sense of number and a laudable proficiency with computation.
As students develop fluency with numbers and computation, they construct the scaffolding necessary for building an understanding of number systems and number theory. This may begin informally when second grade students investigate and explain what they find whenever they add an odd number to an even number. During elementary and middle school years, students extend their understanding from operations with positive numbers to those with negative numbers, and from rational numbers to irrational numbers. High school students examine in more depth the characteristics of the subsystems of the real number system.
Number sense includes the fuller, more accurate view and appreciation of mathematics that is experienced when students explore discontinuous or finite systems, also called discrete mathematics. Computer methods, compound interest, annuities, and arrangements in nature exemplifying Fibonacci sequences are everyday examples of discrete mathematics. Because many of the topics in discrete mathematics are accessible to students who do not have strong computational skills, it provides these students an accessible entry point into mathematics.

What Is Discrete Math?
Look at all the things
That you can see...
And ask, "Can these
be counted
Individually?"
Ice cubes stack
Count 1, 2, 3.
Each cube sits alone
So discretely!
Water falls down
As a cascade from the top.
It is not so discrete
Unless you count each drop.
Look for patterns
They are everywhere
On ceilings, floors, and tables,
And in numbers that are square!
Place your sharp pencil
On any point at the base.
Can you go over an
alphabet letter
Without having to retrace?
Sequence all the steps
Leading to a final result
This defines an algorithm
With numbers, or without.
When completing these examples
Your solutions may be many.
Can you justify why your answer
Is the best one of any?
 Joan Fitton, Worcester Public Schools
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 construct number meaning by using manipulatives and other physical materials to represent concepts of numbers in the real world.
 demonstrate an understanding of our numeration system by relating counting, grouping, and place value concepts.
 interpret the multiple uses of numbers by taking realworld situations and translating them into numerical statements.
Examples of Student Learning
 Students write an autobiographical number story wherein characters take on the identity of numbers. Number of siblings, grandparents, addresses, sports jerseys, Massachusetts zip codes, telephone area codes, and bus route numbers are encouraged.
 When some classes go on field trips to such places as Old Sturbridge Village or Plimoth Plantation, the classes count themselves off as 1, 2, 1, 2, etc. before boarding the school bus. If you were the teacher, why would you use this counting method?
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 model and discuss a variety of problem situations to help students move from the concrete to the abstract.
 relate the mathematical language and symbolism of operations to problem situations.
 identify a variety of problem structures that can be represented by a single operation.
 know when to use the operations of addition, subtraction, multiplication, and division; and describe their relationships.
Examples of Student Learning
 Groups of students make board games that require players to use operations with whole numbers. Students could devise a game that requires them to add or subtract the numbers on dice. Others may use spinners or drawing cards to derive the numbers to be operated upon. Students name their games and write the rules of play. Groups of students then play each other's games, using manipulatives to help them.
 Students work in pairs with four dominoes of the following descriptions: two are 1 dot and 3 dots; one is 2 dots and 1 dot; and one is 2 dots and 3 dots. They use the guess and check strategy to arrange them in a square so that sides each have the same number of dots.
 Students work in pairs to determine the number of outfits that a can be made for a snowman, using 2 hats and 2 scarves.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 demonstrate an understanding of the basic concepts of fractions, mixed numbers, and decimals.
 use models to relate fractions to decimals, find equivalent fractions, and explore operations on fractions and decimals.
 apply fractions and decimals to problem situations.
Examples of Student Learning
 Students brainstorm as a whole group to write a letter to a sick classmate. Since the class has been investigating the relationship between fractions and decimals, members write about what they have learned this week in their mathematics class. Before the letter is mailed, each student draws a picture or writes a story that tells something about fractions and decimals, and they are included in the mailing.
 Keep a monthly chart displaying students' birthdays. Make a grid divided into the appropriate number of days, for example five students, five blocks. Using two different colors, shade the appropriate number of blocks for boys' birthdays and the appropriate number of blocks for girls' birthdays. Express the findings in fractions. Have students use this method to record other findings of parts of wholes.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 describe the strategies used in exploring estimation.
 determine when an estimate is appropriate.
 apply estimation when working with quantities, measurement, and computation.
Examples of Student Learning
 How many kernels of corn do students estimate are on an ear of corn? How did they arrive at their estimates?
 Each student estimates the number of connecting cubes in a bucket by using a stack of ten cubes as a referent. They compare their estimates to the actual number by stacking the cubes in the bucket into towers of ten and counting them by 10s.
 Students are encouraged to revise their estimates as they stack towers. To extend the activity ask students to estimate the number of objects sold in containers, such as a box of apples.
 Explain why you would estimate the number of plants needed at the base of the town's flagpole, but you would not estimate the number of teaspoons of fertilizer needed to keep the same plants healthy.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 model, explain, and develop proficiency with basic facts and algorithms.
 use calculators in appropriate computational situations.
Examples of Student Learning
 A play store is set up in the math corner. Empty boxes, cartons, cans, and plastic fruit (if available) are tagged with price tags. Play money is provided in a home made cash drawer. Groups of students play store. A pair of students work together as the store owners, and others play the role of shoppers. Roles change, to allow all students the opportunity to compute and make change in the context of grocery shopping.
 Give each pair of students a set of dice and a recording sheet with tenspaced columns numbered 112. Have students take turns rolling the dice and recording the addition fact rolled in the appropriate column.
 The Doorbell Rang by Pat Hutchins (available in English and Spanish editions) is a story of visitors arriving consecutively, and then sharing cookies with each other. It provides opportunities for students to develop and share computation strategies. In a first grade classroom, groups of students each have a dozen paper cookies and share them fairly, pasting an equal number on paper plates then writing a number sentence to describe what they did. In a third grade classroom, students arrange cookies in arrays on baking sheets and consider whether they have enough cookies for their guests. Students use calculators to check their work.
 Students generate multiples of 9, using paper and pencil. They discuss the number patterns that they see. The teacher guides students, to help them see that adding the digits of each multiple results in the sum of 9. Students use calculators to generate larger and larger multiples of 9 and mentally add the digits to see that the pattern of the sum as a multiple of 9 continues. As an extension, students explore multiples of other numbers.
 Students predict shortest and longest routes, using a map. They use calculators to check their predictions.
In a Kindergarten class, Ms. Chapman uses autumnal changes as an interdisciplinary theme. Students talk about the seasons and observe how the change from summer to winter affect plants and animals. They listen to stories about autumn, and draw pictures of what they notice when the seasons change. During a walk on school grounds, each student picks a leaf to bring back to the classroom. The class is seated on a rug, alongside their pile of leaves.
"What can we say about our pile of leaves?" she asks.
Soleap says, "There are lots of them."
"How many do you think there are?" asks Ms. Chapman.
"More than 100," Rachel estimates. Actually, there were 23. Ms. Chapman is trying to build the students' sense of quantity, but realizes that to several students, 100 means "a lot." She is sensitive to Rachel's need to develop the concept of onetoone correspondence.
"What else do you notice about the leaves?" asks Ms. Chapman.
"They are different colors," says Pat.
"Some are yellow, and some are green, and some are red, and some are brown, and some are mixedup," said Floyd.
Ms. Chapman suggests sorting the leaves into piles by color, which involves frequent discussions about into which pile a multicolored leaf fits. Ms. Chapman wants students to realize that identifying attributes and categories can be ambiguousdefining the criteria for a particular group is sometimes arbitrary.
"Look at the pile of green leaves and the pile of yellow leaves. What can you say about them?" Ms. Chapman wants them to compare quantities and use the terms more than and less than.
"Some are big and some are small," says Leah.
"Which leaf do you think is the biggest?" Ms. Chapman asks. Discussing which leaves are big and which are little is also ambiguous since some leaves are longer than others, while other leaves are wider, and still others seem to be larger in terms of area. This experience wherein students informally measure by comparing the lengths or widths of leaves helps lead to an intuitive sense of measurement.
"Can we put all the leaves in order in a long line so the smallest leaf is at the end?" This sequencing gives students the opportunity to see increasing/decreasing relationships. Ms. Chapman observes that while some students try to put the leaves in order by using random techniques, others begin to compare each leaf as it is added to the series. "Let's see how many different ways we can sort the leaves, putting them into groups," suggests Ms. Chapman.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 represent and use equivalent forms of numbers, including integers, fractions, decimals, percents, exponents, and scientific notation.
 apply ratios, proportions, and percents.
 investigate and describe the relationships among fractions, decimals, and percents.
 represent numerical relationships in one and twodimensional graphs.
Examples of Student Learning
 Students use a spreadsheet to work with ratios as they consider recipes for small and large groups.
 As a fundraiser to defray the cost of paint, City Year participants are selling pizzas at one of their citywide beautification days. Sixinch individual cheese pizzas were $2.75 last year and will be the same this year. What is a fair price to ask for their new 12" family cheese pizza? Students justify their answers.
 Students go to the supermarket to record prices of different size containers of identical products. Later their data are used to determine the best buy. As students justify their reasoning, using ratios and comparisons, extraneous factors such as food spoilage and storage are considered.
 How many times greater is your height than the circumference of your head? Students predict, then measure a sample to determine the ratio and check their predictions. What other human body ratios can students think of to predict and check? Students state their hypotheses and test them.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 explain the need for numbers other than whole numbers.
 know and use order relations for whole numbers, fractions, decimals, integers, and rational numbers.
 use operations involving fractions, decimals, integers, and rational numbers.
 demonstrate how basic operations are related to one another.
 create and apply number theory concepts, including prime numbers, factors, and multiples.
Examples of Student Learning
 Groups of students chart temperature and wind chill over the winter months. They find and compare the mean and the median, and consider which would be more useful to meteorologists.
 Students choose a company's stock to follow in the newspaper. They note changes and convert the reported fractions to decimals in order to compute the monetary gains or losses.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 compute with whole numbers, fractions, decimals, integers, and rational numbers.
 develop, analyze, and explain procedures for computing, estimating, and solving proportions.
 select and use an appropriate method for computing from among mental arithmetic, paperandpencil, calculator, and computer methods.
 use computation, estimation, and proportions to solve problems.
 estimate to check the reasonableness of results of computations and problems involving rational numbers.
Examples of Student Learning
 Students work in groups to estimate the amount of water wasted and its cost to each tenant in an apartment complex, when a leaking water faucet goes unrepaired. They report their findings, as well as the strategies they used for collecting and analyzing their data. Each group conducts research to determine the most costeffective method of repair. Students write to the Education Department of the Massachusetts Water Resources Authority, 100 First Avenue, Building 394, Charlestown, MA 02129 for literature discussing water conservation.
Ms. Cho: If we know from our reading that the area of the square for a tern nesting site is 2 square feet, how can we find the length of each side to use as a reference point for fencing a larger area to preserve nesting sites at the Cape Cod National Seashore?
Sandy: Find a number that when multiplied by itself equals 2.
Ms. Cho: Tom suggested 1.5. Use your calculators to see what 1.5 squared is.
Ms. Cho: Did 1.5 x 1.5 equal 2?
Stan: No, it was 2.25. That's too big.
Ms. Cho: What's another estimate for the length of the side?
Leon: I think 1.25 . . . that's between 1 and 1.5.
Judy: Yes, but 2.25 is only a quarter away from 2, and 1 is a whole unit away from 2, so the answer must be closer to 1.5 than to 1.
Felipe: I'm going to try 1.3.
Ms. Cho: OK, let's try both those numbers. (pause) What did you get?
Leon: 1.25 times 1.25 is 1.5625. That's too small.
Felipe: 1.3 times 1.3 is 1.69. 1.3 is too small too.
Joe: 1.4 times 1.4 is 1.96. That is real close. It's going to be just a little more than 1.4. Maybe it's 1.5.
Susan: No! That's what we tried first!
Ms. Cho: Let's see what the rest of you think. In the next few minutes, see if you can get closer to the number 2 by squaring a number. You may choose to work by yourself, with a partner, or with your group of four. Keep a list of the numbers you try and their squares. Then we'll share what we find out.
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 represent problem situations, using discrete structures such as finite graphs, matrices, sequences, and recurrence relations.
 represent and analyze finite graphs, using matrices.
 Students write an autobiographical number story
 Students calculate the number of "byes" necessary for a tournament involving many teams or players.
 Students use graphs and matrices to represent and analyze many types of networks. One group decides to analyze the routes involved in getting from their homes to various places in town. Using a map of their town, they label intersections and note whether streets are one or twoway. They arrange the information in matrices and use graphing calculators to determine the number of routes from one point to another.
 Students use the sequence, table, and list features of their graphing calculators (or spreadsheet programs) to discover and explore the Fibonacci sequence and its relationship to other sequences and ratios also found in many natural and manmade structures. (Mathematics Teacher, Vol. 88, No. 2, February 1995, pp. 101105.)
Grades 910 Learning Standards
Students engage in problem solving, communication, reasoning and connections to:
 compare and contrast the real number system and its subsystems with regard to structural characteristics.
 demonstrate the logic of algebraic procedures and their interrelationship with geometric ideas and concepts.
Examples of Student Learning
 Students determine patterns for quilts by using clock arithmetic (see below)to complete fact tables. They assign a color to each number and color their charts, which become the pattern for their quilt patches. They rotate and reflect their patch to get different pattern combinations for a quilt.
Grades 910 Learning Standards
Students engage in problem solving, communication, reasoning and connections to:
 use estimation strategies to judge the reasonableness of results of computation and problem solving involving real numbers.
 use estimation when making graphs.
Examples of Student Learning
 When Benjamin Franklin died in 1790, his will stated that $5000 be given to Boston to be used for public works. Part was to be used after 100 years and the rest was to be used after 200 years. Estimate the amount of money that could be available now, if the funds were invested at 5% compound annual interest. List your assumptions and justify your answer in terms of your assumptions. Explain the differences or similarities between your estimate and that of one of your classmates.
Groups of students have several large drawings of polygons: quadrilaterals, pentagons, hexagons, septagons and octagons. After estimating the sum of the measures of the interior angles of each of the polygons, they measure each angle and try to come to agreement within the group. Students make charts showing their results. They predict the result for an ngon and develop a formula. Using a graphing calculator, students enter the sequence and determine a formula.
Name of Polygon  Number of Sides  Sum of Angles 
triangle  3  180deg. 
quadrilateral   
pentagon   
hexagon   
septagon   
octagon   
Students then look at the problem in another way, by drawing in all the diagonals that can be made from one vertex of the polygon. This allows them to extend their chart and confirm their predictions.
Name of Polygon  # Sides  Sum of Angles  # Triangles  Sum of Angles as multiple of 180 
triangle  3  180deg.  1  1 x 180 
quadrilateral  4  360deg.  2  2 x 180 
pentagon     
hexagon     
septagon     
octagon     
ngon     
Extend the activity to a study of threedimensional figures and crystals. Or, study the sums of the measures of the exterior angles of polygons.
Students with access to computers and software for learning geometry can use them for these explorations.
Grades 1112 Learning Standards
Students engage in problem solving, communication, reasoning and connections to:
 use inductive and deductive reasoning to investigate problem situations involving networks.
 represent and solve problems by using linear programming.
 invent and analyze algorithms.
 use the algebra of matrices to solve problems involving finite graphs.
 investigate and describe problem situations that arise in connection with computer and calculator validation and the application of algorithms.
 solve enumeration and finite probability problems.
Examples of Student Learning
 Students find the location of Steiner points for three cities or towns in Massachusetts. They locate the points experimentally, using geometry computer programs or coordinate geometry and trigonometry with calculators and/or computers. As an extension, students may find and use geometric constructions to locate points, and use deductive reasoning to validate their findings.
 On a history field trip to the Museum of Fine Arts, some mathematics students visit the Nubian Gallery and look at the burial ornaments in the collection. In their science and mathematics classes, they explore carbon dating and find the approximate ages of the ornaments they have seen.
 Stick, Flip, or Switch? A game show host tells you that behind one of three doors is a fabulous prize. The other two doors lead to a booby prize. You choose a door. After you choose, the host, who knows where the prize is, always opens one of the remaining doors and reveals a booby prize. At this stage of the game, you are allowed to switch the door you selected to the other unopened door if you wish, before the prize is revealed. What is your best strategy, stick to your original selection . . . flip a coin to decide . . . always switch?
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 explain why seemingly different mathematical systems may be essentially the same.
Examples of Student Learning
 Students are guided toward finding the connections among designing electrical circuits (series/ parallel), searching an electronic database (and/or), and using binary operations (multiplication/addition).
Ms. Zangari introduces the "Shortest Network" topic to her class by presenting them with a problem that was generated in 1960 when a national airline company claimed that a major telephone company was over charging them for the private lines between their main terminals in three major cities. The mathematicians at the telephone company labs researched the claims. Based on the theory developed in the 1800s by German mathematician Jacob Steiner, the lab decided that the airline was correct. A fourth point could be located so that the shortest route between the cities is a minimum, and this route could be the basis for the phone line charges. Today, the telephone companies are not really concerned with the location of Steiner points, but in the length of the network that results if Steiner points are used. The cost of the service is based on the total length of the network.
In the first part of the investigation, students use rulers and protractors to determine the shortest network of line segments that will interconnect three given points. Presentations of findings and guided discussion leads the students to recognize that the Steiner point will be located so that the angles formed by the segments connecting the point to each of the original points all have the same measure. The students make templates on overhead transparencies that can later be used to locate the point.
The second part of the investigation has students using their "Y" transparencies to locate actual Steiner points, first for the three cities involved in the airline problem, then for any three cities in the United States, and finally here in Massachusetts.
The third (with some classes, this is first or second) part of the investigation has students using Plexiglas models of the three cities and a soap bubble solution. When the model is dipped into a soap bubble solution, and placed on an overhead projector, the network joining the three points is clear. Students use their "Y" transparencies to validate that a Steiner point has been located.
The investigation continues, with students exploring models with more than three vertices and discovering that there may be more than one Steiner point. Students develop an algorithm to cluster subsets of three points, then form Steiner points in these subsets and measure the networks. Students derive formulas for the number of possible Steiner points for a threepoint connection, a fourpoint connection, a fivepoint connection, and so on. Some students may extend this to write a procedure to create Steiner points for various geometric computer programs.
This investigation can be modified for use at any grade level. At the elementary level, it is used for pattern recognition, measurement, and the interdisciplinary purpose of the science of soap bubbles and geography with map reading. In middle grades, teachers use the investigation for skill development in measurement of segments and angles. The handson approach lends itself to numerous openended discussions. Secondary teachers use all of these approaches as an introduction, then proceed to prove the Steiner point theorem and solve realworld problems through scenarios involving connection of points as applied to analytic geometry and calculus.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 explain the strategies used to explore and develop estimation techniques with whole numbers, fractions, decimals, and percents.
 identify when an estimate is appropriate.
 apply estimation techniques to quantity, measurement, computation, problem solving, workplace, and test situations.
Example of Student Learning
 Small groups of students share strategies for figuring out the approximate meals tax and tip at a restaurant, without using a pencil or calculator. Part of the discussion involves ways to determine an appropriate percentage for the tip. Later, the whole class compiles one list.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 represent and use numbers in a variety of equivalent forms and in order relations in realworld, workrelated, and mathematical problem situations.
 compute with whole numbers, fractions, decimals, and integers, using appropriate algorithms and a variety of techniques, including mental math, paperandpencil, calculator, and computer methods.
 analyze and explain procedures for computation and how operations are related to one another, particularly the reversibility of operations.
 estimate to develop number sense, operation sense, and to check the reasonableness of results.
 analyze and explain methods for solving proportions.
 use computation and proportions to solve problems.
 select and use, from among mental arithmetic, paperand pencil, calculator, and computer application, an appropriate method in problemsolving situations.
Example of Student Learning
 Students go to local stores to determine the prices of different brands of a household item such as toothpaste. They decide which is the best buy for them and explain their reasons. Their results may not necessarily be the least expensive toothpaste. They may bring up other factors such as size, taste, or fluoride. They may prefer a more expensive smaller tube over a cheaper larger tube because the larger tube tends to get messier and more toothpaste gets wasted. Unit pricing is an important piece of this activity, but students should see that in real life, the lowest unit price is one of several factors to consider when making a purchase.
 Students work in groups to compare different cultures' procedures for division. Each member compares and contrasts one procedure with another, and explains why each method works.
One teacher asked a group of General Education Development (GED) mathematics students to tell her how much it would cost to buy four shirts for $7.98 each. She told the students to figure it out any way they wanted, except that they could not use pencil and paper. She noticed that they used their fingers in the air or "wrote" on their desks. Most students were able to multiply and get the right answer. When asked how they got their answer, the students responded that they multiplied $7.98 by 4.
The teacher then asked the class what they would do if they were in a store, and needed to determine the total cost for the four shirts. All agreed that they probably would not have solved the problem the same way. Some thought they would have multiplied 4 x 7, plus 4 x 1, and then subtract 8 cents from that total. Others said they would round $7.98 to $8.00, multiply that by 4, and then subtracted $0.08 from the product.
Finally, the teacher asked why no one solved the problem either of these latter ways when given the problem in the classroom, to which students responded, "This is math class, so we need to work out the problems."
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Last Updated: January 1, 1996
