Every spring, Springfield Middle School allows groups of students to sign up and maintain garden plots. All garden plots are the same size.
Below is a portion of the school garden with seven plots in it. Each group divides a plot into equal pieces for each student.
Inez, Kewan, Tim, and Waya maintain Plot A. They used string to divide their garden plot into four equal pieces.
1. a. Explain how they used string to equally divide Plot A.
b. Use a fraction to describe what part of the plot each student claims.
Reaching All Learners
Intervention
Some teachers choose to start this unit with page 13 and work through page 18, then do Section A.
13 Models You Can Count On
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__1 4
Section B: The Bar Model 13T
Hints and Comments
Overview
Students start by equally dividing a plot that has the shape of a rectangle. They use a fraction to describe a part.
About the Mathematics
In this problem and the problems on the following page, fractions arise as a result of a fair share situation. For example, a plot divided equally into four parts:
One part is one out of four, or __14 .
A paper strip can be used as a measuring tool to make equal parts. This type of strip is a measuring strip:
Comments About the Solutions
1. a. Be sure that all students understand that halving and then halving again results in parts of __14 of the whole.
By now, students should also know the relationship between 1 out of 4 and __14 .
Solutions and Samples
1. a. They could make a string as long as the garden and fold it double and double again. If they mark the folds, then after unfolding they have 4 equal parts. The string can be used as a measuring strip.
b.__14
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The Bar Model
Marc, Melinda, and Joyce maintain Plot B. They also want to divide their plot into equal pieces using strips of tape.
Use Student Activity Sheet 3 (pages 1 and 2) for problems 2–4.
2. a. Cut out one length of the paper strip. Use the strip to divide Plot B into three equal parts.
b. Label each part of Plot B with a fraction.
The other plots will be divided among groups of 5, 6, 2, and 8 students.
One plot is unclaimed.
3. a. Use the paper strip to divide Plots C–F into the number of equal pieces indicated.
b. Label each part with a fraction. Be prepared to explain how you used the strip to divide the plots.
4. Choose a different number of students to share the last garden plot, Plot G. Divide Plot G accordingly.
In problems 2–4 above, you used a paper strip as a kind of measuring strip to make equal parts. You used fractions to describe each part;
for example:
Tim, Waya, and Inez share three-fourths of Plot A. A fraction relationship to describe this situation is1᎑᎑41᎑᎑41᎑᎑434᎑᎑.
5. Use garden Plots B–G to describe five other fraction relationships.
Measuring strips can be used to find parts of a whole.
If you have three parts out of four, you can express this as the fraction 3᎑᎑
4on afraction bar.
6. Reflect How are a measuring strip and a fraction bar the same?
How are they different?
The Bar Model
B
1᎑᎑
4 1᎑᎑4 1᎑᎑4 1᎑᎑4
Tim Waya Inez Kewan
0 14᎑᎑ 1᎑᎑2 34᎑᎑ 1
Notes
2Students may find it easier to cut the strips if they fold them first.
5Have students struggling with this problem look at Student Activity Sheet 3 for ideas.
Students should not be taught the terms numerator and denominator yet. It can wait until later, unless a student mentions these names.
B
14 Models You Can Count On
Reaching All Learners
Vocabulary Building
Have students add the new vocabulary words measuring strip and fraction bar to the vocabulary section of their notebooks.
Accommodation
Pre-cut the strips from page 1 of Student Activity Sheet 3 for students who may have difficulty cutting. Make extra copies of both pages of Student Activity Sheet 3 for students who make mistakes.
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Section B: The Bar Model 14T
Hints and Comments
Materials
Student Activity Sheet 3, pages 1 and 2 (one each per student); scissors
Overview
Students continue to divide plots equally, and they label each part with a fraction. They describe fraction relationships. Finally, the students are introduced to a new model: the fraction bar.
About the Mathematics
The problems on this page review students’ informal understanding of the concept of fractions. On this page, the fraction bar is introduced in the context of plots; on the next pages, the fraction bar is used in other contexts. Later on in this section, the bar model will become an important problem-solving tool.
Planning
Students can work in small groups on these problems.
Observe how students solve them. If they do well, you may want to check students’ responses to problem 5 and discuss more extensively problems 3 and 6.
Comments About the Solutions
3. b. Discuss students’ explanations in class.
Explanations will vary. Sample explanations:
Lot C: each part is__15; it is not easy to fold tape in five equal parts; students may have made an estimate, or they may have measured.
Lot D: each part is__16; fold the tape in thirds and then in half.
Lot E: each part is__12; fold the tape in half.
Lot F: each part is__18; fold the tape in half 3 times.
Solutions and Samples
2. a. and b.
See solution on copy of page 2 of Student Activity Sheet 3 above.
3. a. and b.
See solution on copy of page 2 of Student Activity Sheet 3 above.
4. Answers may differ; easy numbers are: 10 (based on lot C, halve each part, each___101); 9 based on lot B, each part in 3, fraction __19; 16 based on lot F, take half again; 12, half of lot D.
Note: 7 is of course missing. but it is very hard to fold in 7 equal parts.
5. Answers may differ. Sample student answers:
__1
3ⴙ__13ⴝ__23
__1
3ⴙ__13ⴙ__13ⴝ__33ⴝ1
__1
5ⴙ__15ⴙ__15ⴝ__35
___1
12ⴙ___121 ⴝ___122, or __16
Some students may have compared and used parts from different marked tapes and found more complex relationships, for example:
__1
2ⴙ__14ⴝ__34
6. Answers will vary. Sample answer:
Both a measuring strip and a fraction bar can be used to find parts of a whole and express them as fractions.
They are different because in a measuring strip each part is labeled with the same fraction, for example__15.
You can cut out the parts; they all have__15 in them.
On a fraction bar, the parts are in a way “added,”
so you have __15, __25, __35, and so on. The fractions are not inside in the parts but along a line at the bottom of the bar.
1᎑᎑
3
1᎑᎑
3
1᎑᎑5 1᎑᎑5
1᎑᎑5 1᎑᎑5
1᎑᎑5 1᎑᎑
3
᎑᎑1 6
1᎑᎑
6
᎑᎑1 6
᎑᎑1 6
᎑᎑1 6
1᎑᎑
6
᎑᎑12
᎑᎑12
᎑᎑18 ᎑᎑18 ᎑᎑18 ᎑᎑18 ᎑᎑18 ᎑᎑᎑᎑1188 1᎑᎑8 ᎑᎑18
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Students use a supply of rainwater, stored in tanks, to water the garden plots.
The largest tank in the garden holds 400 liters (L) of water. However, during a dry spell, it usually has less than 400 L of water.
The outside of the tank has a gauge that shows the level of the water in the tank.
You can use a gauge like a fraction bar.
The Bar Model B