Prepare a sandwich bag of 100 candies for each group. Choose candies with 4–6 different colors. Depending on your objectives, you can divide the candies in specific ratios (e.g., 50/25/25 or 50/20/15/15) or leave the number of each color to chance. Try to find candies that are slightly less than 1 cm in diameter so that 100 candies fit well along the edge of a meter stick. If candies are larger than 1 cm, students will have to offset each candy piece slightly to fit 100 pieces along the meter. As this may cause confusion, it is best to find candies that are an appropriate size.
Also prepare for each group a paper strip cut slightly longer than 100 cm. Provide one mark on the paper strip 2–3 cm from one end for students to label as 0. Adding machine tape works well for the strips.
Lastly, trace a circle with a diameter of 12.5 in or 32 cm onto a poster board for each group and mark the center of the circle.
A Meter of Candy (Set/Linear Model)
Using the A Meter of Candy Overhead. Ask:
- What does it mean to have 100% of something covered up? 50%?
- Can you picture 10%?
- Why do we say that 100% is the whole thing? Is that the same as 100/100?
- What is 50% of 20 objects? Would that be the same as 50/100 of 20 objects?
- What is 50% of 100 objects? Would that be the same as 50/100 of 100 objects?
You may have students work in small groups to explore these and similar questions, and then share their ideas with the entire class.
Organize students into groups of 2 or 3. Don't tell them how many candies are in each bag. Let them estimate and discuss individual estimates.
Next, have groups place their 100 candies randomly along a meter stick, one candy per hundredth. Ask, "Can you easily tell what percentage of the candies are red? green? Why or why not? What would help you to determine the percentages?"
Guide students to understand that grouping the candy pieces by color along the meter stick does not change the percentage of each candy color, but it does provide a clearer visual representation of the percentage of each color.
For the moment, ask students to return their candies to the sandwich bag; the candies will be used again later in the lesson.
Have each group make a linear representation of their collection of 100 candies. First, they should label 0 on their strip, at the mark you made previously. Then, have students lay their paper along a meter stick, lining up the 0 on the paper strip with the 0 cm mark on the stick. Ideally, they should place a pencil mark at each centimeter from 0–100. However, the paper meters become too messy if every centimeter is labeled with a numeral. Introduce decimeter as you have students count and label by 10s from 0–100 cm.
Next, ask students to make piles of their candy pieces by color. Ask students:
- How easily can you estimate the percentage of each color?
[Not very; large groups need actual counting.]
- How can the meter stick help you?
[It shows hundredths.]
Reinforce the connection between hundredths (written as fractions and decimals) and percentages. Have students count and record their candy data (colors/numbers) on the A Meter of Candy Activity Sheet.
Finally, have students place the candies by color along their paper meter strip and color the paper according to the colors of their candy pieces. Students can complete Questions 1–3 on the activity sheet.
A Grid of Candy (Rectangular Area Models)
Begin by having students share their colored paper meters. Post the meters around the classroom. Emphasize that the meter is a linear model showing percentages. You can verify understanding by having students do a round robin between paper meters and share percentage values of colors verbally using the terms hundredth and percent. Have one member of each group remain by his/her paper strip while other students visit and ask questions. Rotate the students from each group so everyone has a chance to present to classmates (and you can listen in).
Suggest to students that percentages can be shown on a grid. Ask students:
- How many squares should be in the grid? [100.]
- Is the number of squares important? [Yes.]
- What shape should the grid be? [It can vary.]
- Does the grid shape matter? [No.]
- Will the percentage stay the same? [Yes.]
Use the Grids Activity Sheet, which has grids of 10×10, 4×25, and 5×20. All the grids use the same unit size. You may want to enlarge the activity sheet so students have room to place their candies on the grids prior to coloring. The members of the small groups can do the same grids or different ones. Depending on students' understanding, have them lay out their candies prior to coloring or just color according to their data sheet.
Have students think about and discuss the best ways to group the colors. Let them discuss and decide choices. Students should then color the grids according to the percentages of their candy colors. Once the grids are posted, students can discuss similarities and differences. If a student randomly colors individual squares, it will be apparent that counting is required to determine the percentages of color. After the grid work, students can complete Questions 4–6 of the A Meter of Candy Activity Sheet.
A Circle of Candy (Region Model)
Review with students the grids they created, and compare the linear and area representations. Spend time discussing the different rectangular shapes of the area models. Have students brainstorm other figures that could show percentage. Lead the discussion towards the idea of a pie graph, which is a circular model that can show percentages.
To begin creating their pie graphs, have students connect the ends of their linear meter to form a circle. Students match the 0 cm mark with the 100 cm mark and tape the circle closed. Have students lay their meter strip around the circumference of their poster-board circle. They should mark where each color begins and ends. Then, have students connect these marks to the center of the circle to create each piece of the pie. The pie pieces become area representations of the percentages of each color of candy. Students should color and then label each sector of the pie graph with decimals, fractions, and percentages.
To help students contemplate all three models (linear, area, and the pie graph), direct them to individually write one true mathematical statement about each model. This can be done in journals or on cardstock (for posting later). Have them review their statements with peers for clarification. Then, as a class, share their statements aloud. This is a great time to highlight statements that are similar even though they are about different types of models because this shows the interconnectedness of the representations. You can also challenge students to count how many unique statements are made throughout the sharing.