Students consider what happens if you start with two bacteria on a kitchen counter and the number of bacteria doubles every hour. They make a table and graph their results, noting that the graph is not linear. |
Students arrange bowling pins in the shape of equilateral triangles of various sizes. They make a table showing the number n of rows in each triangle and the number b of bowling pins in each triangle. The numbers in the second column - 1, 3, 6, 10, ... - are called the triangular numbers. They find a rule expressing this relationship p = n(n + 1) /2, by putting two triangles of the same size side by side, counting the total number of bowling pins, and dividing by 2. |
Students investigate how many stools with three legs and how many chairs with four legs can be made using 48 legs. They may use objects or draw pictures to make models of the solutions. They look for patterns in the numbers and display their results in a table, as ordered pairs graphed on the rectangular coordinate plane, as a rule like 3s + 4c = 48, and as an equation like s = 16 - 4c/3, which gives the number of stools as a function of the number of chairs. They describe the pattern and how they found it in writing. |
Parallelograms. For each parallelogram, they record the length of the base (b), the height of the parallelogram (h), and the area of the parallelogram (A), found by counting squares. The students look for a relationship among the numbers in the three columns of their table, express this relationship as a verbal rule, and then write the rule in symbolic form. |
Students analyze a given series of terms and fill in the missing terms. Patterns include various arithmetic (repeating patterns) and geometric (growing patterns) sequences and other number and picture patterns. Students develop an awareness of the assumptions they are making. For example, given the sequence 0, 10, 20, 30, 40, 50, one might expect 60 to be next; but not on a football field, where the numbers now decrease! |
Students supply missing fractions between any two given numbers on a number line. They might label each of eight intervals between 1 and 2, or they might label the next 16 intervals from 23 1/2 to 24. They extend this to decimals, labeling each missing number in increments of .1 or .01. For example, students might label each of five intervals between 59.34 and 59.35. |
Students compare different pay scales, deciding which is a better deal. For example, is it better to be paid a salary of $250 per week or to be paid $6 per hour? They create a table comparing the pay for different numbers of hours worked and decide at what point the hourly rate becomes a better deal. |
Students decide how many different double-dip ice cream cones can be made from two flavors, three flavors, and so on up to Baskin and Robbins' 31 flavors. They arrange the information in a table. They discuss whether one flavor on top and another on the bottom is a different arrangement from the other way around, and how that would change their results. |
Students investigate how increasing the temperature measured in degrees Celsius affects the temperature measured in degrees Fahrenheit and vice versa. They collect data using water, ice, and a burner. They use their data to develop a formula relating Celsius to Fahrenheit, summarize the formula in a sentence, and graph the values they have generated. |
Students investigate how the temperature affects the number of chirps a cricket makes in a minute. |
Students investigate the effect of changing the radius or diameter of a circle upon its circumference by measuring the radius (or diameter) and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and describe the relationship they have found in a paragraph. Then they develop a symbolic expression that describes that relationship. |
Students work on problems like this one from the New Jersey Department of Education's Mathematics Instructional Guide (p. 7-69): Two of the opposite sides of a square are increased by 20% and the other two sides are decreased by 10%. What is the percent of change in the area of the original square to the area of the newly formed rectangle? Explain the process you used to solve the problem. |
Groups of students pretend that they work for construction companies bidding on a federal project to build a monument. The monument is to be built from marble cubes, with each cube being one cubic foot. The monument is to have a "triangular" shape, with one cube on top, then two cubes in the row below, then three cubes, four cubes, and so on. The monument is to be 100 feet high. The students make a chart and look for a pattern to help them predict how many cubes they will need to buy so that they can include the cost of the cubes in their bid. |
Students use probes and graphing calculators or computers to collect data involving two variables for several different science experiments (such as measuring the time and distance that a toy car rolls down an inclined plane, or the temperature of a beaker of water when ice cubes are added). They look at the data that has been collected in tabular form and as a graph on a coordinate grid.
They classify the graphs as straight or curved lines and as increasing (direct variation), decreasing (inverse variation), or mixed. For those graphs that are straight lines, the students try to match the graph by entering and graphing a suitable equation. |
| Using a temperature probe and a graphing calculator or computer, students measure the temperature of boiling water in a cup as it cools. They make a table showing the temperature at five-minute intervals for an hour. Then they graph the results and make observations about the shape of the graph, such as the temperature went down the most in the first few minutes or it cooled more slowly after more time had passed, or it's not a linear relationship. The students also predict what the graph would look like if they continued to collect data for another twelve hours. |
