7.2 Measurement & Data
Key Takeaways
- Measurable attributes include length, weight/mass, capacity/volume, time, money, and temperature; students progress from non-standard units (paper clips, hands) to standard US customary and metric units.
- Conversions within a system use multiplication or division by base ratios: 1 ft = 12 in, 1 yd = 3 ft, 1 m = 100 cm, 1 kg = 1000 g, 1 L = 1000 mL; cross-system conversions are not a P-5 expectation.
- Perimeter (P = 2l + 2w) measures distance around a shape; area (A = l × w for rectangles, A = b·h/2 for triangles) measures the square units covering a surface; volume of a rectangular prism (V = l × w × h or V = B × h) counts unit cubes filling space.
- Angles are measured in degrees with a protractor; angle types are acute (< 90°), right (90°), obtuse (90°–180°), straight (180°), and reflex (180°–360°).
- Data displays include bar graphs, picture graphs, line plots, line graphs, pie charts, and histograms; mean, median, and mode summarize a distribution and students choose the best measure for the data context.
Measurable Attributes
Students begin measuring by comparing objects directly ("this pencil is longer than that one") and then by using non-standard units (paper clips, cubes, hand-widths). Non-standard units build the concept of iteration — laying the same unit end to end without gaps or overlaps.
Standard units make measurements shareable. The two systems in use are US customary and metric.
Measurement Units Reference
| Attribute | US Customary | Metric |
|---|---|---|
| Length | inch (in), foot (ft = 12 in), yard (yd = 3 ft), mile (mi = 5280 ft) | millimeter (mm), centimeter (cm = 10 mm), meter (m = 100 cm), kilometer (km = 1000 m) |
| Weight/Mass | ounce (oz), pound (lb = 16 oz), ton (2000 lb) | gram (g), kilogram (kg = 1000 g) |
| Capacity/Volume | fluid ounce (fl oz), cup (8 fl oz), pint (2 cups), quart (2 pints), gallon (4 quarts) | milliliter (mL), liter (L = 1000 mL) |
| Time | second, minute (60 s), hour (60 min), day (24 h), week (7 days), year (12 months/365 days) | same units used in elementary classrooms |
| Money | penny (1¢), nickel (5¢), dime (10¢), quarter (25¢), half-dollar (50¢), dollar bill (100¢) | — |
| Temperature | degrees Fahrenheit (°F) | degrees Celsius (°C) |
Conversions within a system use multiplication or division by base ratios. To convert 5 feet to inches: 5 × 12 = 60 inches. To convert 3000 grams to kilograms: 3000 ÷ 1000 = 3 kg. Cross-system conversions (e.g., inches to centimeters) are not a P-5 focus.
Perimeter, Area, and Volume
Perimeter is the distance around a figure. For a rectangle: P = 2l + 2w. For a 6 cm × 4 cm rectangle, P = 2(6) + 2(4) = 20 cm.
Area is the number of square units covering a figure. For a rectangle: A = l × w, so a 6 cm × 4 cm rectangle has area 24 cm². For a triangle: A = (b × h) / 2, so a triangle with base 8 in and height 5 in has area (8 × 5) / 2 = 20 in².
Volume is the number of unit cubes filling a 3-D figure. For a rectangular prism: V = l × w × h or equivalently V = B × h where B is the area of the base. A box 5 cm × 3 cm × 2 cm holds 30 unit cubes, so V = 30 cm³.
Young students fill prisms with unit cubes physically before learning the formula. The multiplication principle (count cubes in one layer, multiply by the number of layers) is the conceptual bridge.
Angle Measurement
An angle is formed by two rays sharing an endpoint (the vertex). Angles are measured in degrees with a protractor. A full turn is 360°.
Angle Types
| Type | Measure | Example |
|---|---|---|
| Acute | less than 90° | corner of a sharp pencil |
| Right | exactly 90° | corner of a sheet of paper |
| Obtuse | between 90° and 180° | hands of a clock at 4:00 |
| Straight | exactly 180° | unfolded ruler lying flat |
| Reflex | between 180° and 360° | the larger turn of a door swung most of the way open |
Two angles are complementary if they sum to 90° and supplementary if they sum to 180°.
Representing and Interpreting Data
Students choose a data display that matches the question and data type:
- Bar graph: compares categories with rectangular bars whose heights show values.
- Picture graph: uses symbols to represent counts; each icon stands for a fixed number (e.g., one star = 2 students).
- Line plot: places Xs above a number line to show frequency of measurements; ideal for a small set of fractional measurements like shoe lengths to the nearest 1/4 inch.
- Line graph: shows change over time; the horizontal axis is usually time.
- Pie chart: shows parts of a whole; slices are percentages of 360°.
- Histogram: like a bar graph but bars touch; shows frequency of numeric intervals (bins).
When reading graphs, students answer: What is the mode? What is the range? Which category is largest? What story does the graph tell?
Measures of Central Tendency
- Mean: the arithmetic average; sum of values divided by the count. For data 4, 6, 8, 10, 12: sum = 40, count = 5, mean = 8.
- Median: the middle value when data are ordered. For 4, 6, 8, 10, 12, the median is 8. With an even count, the median is the mean of the two middle values: for 4, 6, 8, 10, median = (6 + 8)/2 = 7.
- Mode: the value that appears most often. For 4, 4, 6, 8, 10, the mode is 4. A data set can have no mode, one mode, or several modes.
The mean is sensitive to outliers; the median is robust. For income data with one very high value, the median often describes a typical family better than the mean. Students in upper elementary begin choosing the best measure based on context.
Worked Example: Converting and Comparing Units
A Georgia fifth-grade class measures the length of the hallway in two ways. Group A reports 12 yards; Group B reports 33 feet. Are the measurements consistent?
Convert yards to feet: 12 yd × 3 ft/y = 36 feet. Group A's measurement (36 ft) is longer than Group B's (33 ft), so the two groups measured different starting or ending points. This example shows why unit choice matters: a smaller unit gives a more precise number, and converting to a common unit is the only fair way to compare.
Choosing the Right Graph
Matching the data display to the question is itself a learning goal:
- Categorical question ("Which snack is most popular?") → bar graph or picture graph.
- Change over time ("How did the plant grow each week?") → line graph.
- Distribution of measurements ("Heights of seedlings to the nearest 1/4 cm") → line plot.
- Parts of a whole ("How is our day spent?") → pie chart.
Students who can justify their choice of display are reasoning statistically, not just producing a picture.
A rectangular garden is 8 feet long and 5 feet wide. What is the perimeter?
Which measurement is equivalent to 3 liters?
The data set 5, 5, 7, 9, 12 has which mode, median, and mean?