Mathematical Thinking at Grade 3 (Introduction)
In this introductory unit, students solve problems in each of the three areas of the curriculum (number, data, and geometry), using basic mathematical tools and materials. They begin to use critical mathematics processes that they will continue to use all year, such as working in small groups and communicating their thinking through writing, drawing, and talking. Students develop an orientation to thinking mathematically through several investigations. They explore the characteristics of 100, using interlocking cubes and 100 charts. They investigate doubles and halves in numerical and geometric contexts. They collect and represent data about themselves.
Things That Come in Groups (Multiplication and Division)
Students become familiar with the multiplication tables up to the 12's, identify patterns in the multiplication tables, and develop their own strategies for doing multiplication and division problems. Students make lists of things that come in groups of 2 to 12 and create and solve multiplication problems and riddles. They explore multiples by skip counting orally, by using a calculator, by highlighting multiples on a 100 chart, and by using cubes to make arrays. Students discuss the different actions suggested by division and multiplication story problems, and write and solve each others' story problems. They solve problems with larger numbers, for example, problems about saving money and problems about the number of legs on combinations of different creatures.
Flips, Turns, and Area (2-D Geometry)
In this unit, which can be done with or without computers, students explore shape, area, and geometric motions (slides, flips, and turns) through tetrominoes--arrangements of four squares with full sides touching. Students find all possible tetrominoes, and use interlocking cubes, paper cutouts, and a specially designed computer program as they try to cover rectangles with their tetrominoes. On geoboards or dot paper, they use squares and triangles of unit and half-unit sizes to create many different (non congruent) shapes with an area of four square units. They explain how they found the area of a shape that is 5, 6, or 7 square units.
From Paces to Feet (Measuring and Data)
Students explore measurement and simple statistics by discussing why we need to measure, using different measuring tools and systems, and interpreting data they collect by measuring. Students use baby steps, giant steps, and paces to measure distances. They compare results and find the "middle-sized pace." Students use standard measurement tools--inch rulers and yardsticks, centimeter rulers and meter sticks--to measure the size of their feet and the size of adult feet and to measure other familiar objects. They locate "benchmarks" on their body to use to estimate lengths without a ruler. The unit culminates with students doing one or two final projects. In one project, Do Our Chairs Fit Us?, students use measurements to analyze whether the classroom furniture fits them and make recommendation about the optimal distribution of chairs for the class. In the other project, Balobbyland, students use one-centimeter graph paper to construct different spaces, such as bedroom floor plans, for Balobbies--creatures who are 5-8 centimeters tall.
Landmarks in the Hundreds (The Number System)
Students build their understanding of the base ten number system by using landmarks in that system--numbers that are familiar landing places, that make for simple calculations, and to which other numbers can be related. Students find factors of different numbers, including all of the factors of 100, by grouping cubes, skip counting orally and on a 100 chart, and dividing a dollar evenly among different numbers of people. They solve problems involving multiplication and division, including problems involving money, and they create and solve their own problems. Students also construct a poster-size chart of 1000 centimeter squares, locate large numbers on the charts, and use the charts to calculate the "distance" (differences) between numbers such as 550 and 950.
Up and Down the Number Line (Changes)
Students investigate ideas about addition and subtraction, numbers below zero, net change, the opposite effects of addition and subtraction, and the many ways to use addition and subtraction to reach a given answer. Students make a diagram of a fantasy skyscraper with an elevator, then use the diagram to model an elevator traveling between floors as they find strategies for making many sets of changes that result in the same net change. Students represent different elevator trips in a graph, compare graphs, and extend the graphs to see what they would look like if the same changes were made repeatedly. They create and play board games that are played on horizontal tracks with the negative spaces to the left and positive to the right.
Combining and Comparing (Addition and Subtraction)
Students develop their own addition and subtraction strategies and use estimation and multiple strategies to double-check their work. Students compare their own data, such as the number of children in their family or the ages of their pets, with world record data. They use pan balances to compare the weight of cut-up pieces of fruits and vegetables before and after one or two days exposure to air. They solve problems with multiple addends in which a total measurement or total amount of money is given and they must figure out a set of numbers to make that sum, such as, "What three items can I buy for $2.50?" Students also solve problems involving comparisons on the calendar, for example, figuring out the number of days until a particular event and determining how much longer children in other countries spend in school than they do.
Turtle Paths (2-D Geometry)
Students explore problems involving paths, lengths of paths, perimeter, and turns. Students create paths on the floor, on paper mazes, and on the computer using Geo-Logo. They write commands, such as forward 5, left turn 90, forward 2, to provide directions for other students or for the computer turtle to move along these paths. They use the Turtle Turner (protractor) to estimate and measure turtle turns in multiples of 30°. Students define triangles and draw them with the Geo-Logo turtle. Students solve problems involving lengths and turns, and draw rectangles (and other paths) having a total path length of 200 turtle steps. The unit ends with each student writing procedures to design a face on the computer.
Fair Shares (Fractions)
Students use fractions and mixed numbers as they solve sharing problems and build wholes from fractional parts. Students find ways to share rectangular "brownies" and hexagonal pattern block "cookies" among different numbers of people, play games based on the sharing model, and connect these sharing problems to division. They decide how to share things (such as balloons) that cannot be cut up into exact shares, and how to share dollars by converting to cents. They explore how to do sharing problems with a calculator, thus seeing their fraction answers as decimals and develop a class chart of "fraction facts," such as, 3/4 = 1/2 + 1/4 or 1/6 + 1/3 =
Exploring Solids and Boxes (3-D Geometry)
Students sort, build, and describe different kinds of polygons and common geometric solids. Using a set of geometric solids, students sort them in several ways, and identify a mystery solid by playing What's My Shape? Using straws or dowels, they build polygons and polyhedra by analyzing pictures and verbal descriptions of the shapes. Students make boxes that will hold a certain number of cubes by designing patterns on squared paper, cutting them out, folding, and taping them. They predict the number of cubes that other box patterns will hold. Students build a model of a city--a set of open-box buildings made from patterns that they draw on graph paper--and determine the total amount of room in the buildings.
Return to 1st Edition Units.