## A Meter of Candy

• Lesson
3-5
3

In this series of 3 hands-on activities, students develop and reinforce their understanding of hundredths as fractions, decimals, and percentages. Students explore using candy pieces as they physically make and connect a set/linear model to area models.

### Preparation

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.

Assessment Options

1. Ask students to write a question based on the information contained in a pie graph (or any other model) of your choice. For example:
• If there are 20 red and 10 green candies, what is the percent difference?
• What color, written as a fraction, can be simplified to 1/4?

Have students write their questions on a sticky note (including the answer), sign their name to the back, and post the note on the graph that their question is about. The depth of students' understanding is often shown through the level of their questions.

2. Students can write journal entries on topics such as the following:
• If you showed one of your grids to a friend, how would they know each square represents 1%?
• If you explained this percentage activity to a friend, which model would you show and why? Give your reasons based on the mathematics you've done.

Extensions

1. Compare the percentages of candy colors between groups. Notice similarities, differences, or trends. For practice with addition and subtraction of decimals, ask students to complete some tasks. For example, pose these problems: If there are 32 red and 11 green candies, what is the percent difference? [0.32 – 0.11 = 0.21] Choose two pie graphs and find the sum of their orange candies. Tasks can also challenge higher-level students. Adding like colors from different groups may lead to sums greater than 1, leading students to understand the concept of a whole in fractions.
2. Have students collection pie graphs clipped from newspapers, photocopied from encyclopedias, and printed from the Internet. Have students examine the graphs, interpret them, and share their results.
3. Use technology to create pie graphs. Use either the Circle Grapher Tool or the graphing function in Excel. In the latter software, students may also try different graphical models to represent 100% of their candy.
Circle Grapher

Questions for Students

1. What was your thinking when you grouped your candy pieces along the meter strip? Did you have a strategy? Does the order of your groups make a difference mathematically?
[Answers will vary. Some students may have grouped their colors beginning with the largest number or percentage of candies. Others may have used the smallest number or percentage of candies. Others may have grouped randomly. Students should understand that the order of the colors doesn't change the portions of the colors.]
2. How does your grid show the percentages of your colors? If you showed one of your grids to a friend, how would they know each square represents 1%?
[Percentages are based out of 100, so 23% equals 23 colored squares. In this early stage of understanding percentage, students should recognize that it's important to have 100 squares on a grid.]
3. If you wanted to explain the ideas in this lesson to a friend, which model would you show them? Why would you choose that model?

[Answers will vary according to the random colors students received, as well as how each student views each model. If they have numbers that represent, or are near, a benchmark fraction, one type of model or another might make the benchmarks quickly recognizable. The important thing is to have students justify their choice mathematically.]

Teacher Reflection

• Was the use of candy a support or distraction during this lesson? Is there a better material to use? Did using candy help motivate reluctant students?
• Was there a model of percentages that appeared to make more sense to students? Why do you think that is?
• Might student learning be better supported if you began with the area (grid) model to interpret percentages?
• How well does this activity stretch thinking for capable students?

### Learning Objectives

Students will:

• Reinforce understanding of the connections between fractions, decimals, and percentages.
• Connect the set and linear model to area models (rectangular and circular).

### NCTM Standards and Expectations

• Use models, benchmarks, and equivalent forms to judge the size of fractions.
• Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.

### Common Core State Standards – Mathematics

Grade 3, Num & Ops Fractions

• CCSS.Math.Content.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Grade 3, Geometry

• CCSS.Math.Content.3.G.A.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 5, Num & Ops Fractions

• CCSS.Math.Content.5.NF.B.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Grade 4, Num & Ops Fractions

• CCSS.Math.Content.4.NF.C.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

Grade 4, Num & Ops Fractions

• CCSS.Math.Content.4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.