# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.05

## The Number Line as a Mathematical Tool

Your feedback is important to us!

After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

## Sample Lesson Plans

### Lesson: Creating fraction rulers and number lines.

*
Objective:
*

Students will place fractions on a number line 0 through 2. Using fourths, fifths, and twentieths will compare, add and subtract fractions.

*
Materials:
*

Each student will be given 3 blank acrylic rulers already cut on 20 inch lengths, a set of non-permanent markers, and a standard inch ruler.

*
Procedure:
*

Students will draw a 20 inch line in the middle of each blank ruler using a different colored marker for each ruler.

1. With the red marker at the 10 in. mark students will write 1 above the line, 2 at the 20 inch mark, 1/2 at the 5 in. mark, and 0 at the beginning. Students will then proceed to label at 1/4 intervals, 1/4 at the 21/2in., 2/4 (1/2) at the 5in. mark, 3/4 at the 71/2 in mark, 4/4 at 10 in., 5/4 at 121/2 in., 6/4 at 15 in., 7/4 at 171/2 in., 8/4 at the 20 in. mark.

2. Using a green marker label the beginning of the line, 0; at the 10 in. mark label 1 above the line; and at the 20 in. label 2. At 2 in. intervals label below the line,1/5, 2/5, 3/5, 4/5, 5/5, 6/5, 7/5, 8/5, 9/5, and 10/5.

3. Using a blue marker, label ruler at 1/2 inch intervals 1/20 through 40/20.

When rulers are complete students should using one ruler at a time add 1/4 sections to 3/4, or 3/4 + 2/4 = 6/4, and subtract 1/4 segments to develop an understanding that the operation of addition and subtraction for fractions is the same as adding whole numbers. The denominators do not change. Students should do the same with the 1/5 ruler and 1/20 ruler. When using their fraction rulers students can add, subtract or compare using their eye or a separate piece of paper as a measure to move right or left of the number line.

*
Problems:
*

1/4 + 3/4 = 4/4 6/4 – 2/4 = 4/4 1/4 + 3/4 + 5/4 = 8/4

1/5 + 1/5 = 2/5 1/5 + 3/5 = 4/5 6/5 – 3/5 = 3/5 2/5 + 3/5 + 1/5 = 6/5

4/20 + 5/20 = 9/20 using three rulers add 1/4 to 1/5 and place on top of 1/20 acrylic ruler students will see that 1/4 +1/5 is the same as 4/20 + 5/20.

Students should make up a series of problems on their own to become more familiar with their new mathematical tool.

*
Closure:
*

If I need 3/4 of a cup of four for one recipe and 3/4 of a cup for another will 1 1/2 cups be enough?

**
**

### Lesson: Solving Word Problems using fraction rulers.

*
Objective:
*

Students will solve word problems using fourths, fifths, improper fractions and mixed numbers.

*
Materials:
*

Fractions rulers.

*
Procedure:
*

Students should be familiar with the properties of their fraction rulers. Point out that 1 represents a whole, 4/4, and 5/5 are the same value as 1. 2/4 is 1/2. The denominator represents the sections to make a whole. On the 1/4 ruler there are 4 sections to make 1 whole and 8 sections are equivalent to 2. On the fifths ruler there are five equivalent lengths to make a whole and 10, 1/5 lengths are equivalent to 2. A good question to ask, “How many 1/5 lengths are equivalent to 1/2?” Students should also continue to add fourths and fifths and place on the twentieths ruler to compare answers. This will develop an understanding of common denominators.

*
Problems:
*

See word problems in Section 1 and 2.

### Lesson: Solving Word Problems using percent rulers.

*
Objective:
*

Students will create a percent ruler to use as a tool to solve word problems. Students will identify equivalent fractions and percents.

*
Materials:
*

Blank 20 inch acrylic rulers and a black non-permanent marker.

*
Procedure:
*

Students will draw a 20 inch line in the center of their ruler. Label 100% at the 10 inch mark, 200% at the 20 inch mark, 50% at 5 in., 25% at 21/2 in., 75% at 71/2in., 125% at 121/2 in., 150% at 15in. On this ruler students will also label, 20% at 2in., 40% at 4in., 60% at 6in., 80% at 8 in., 120% at 12in., 140% at 14in., 160% at 16in., and 180% at 18 in. I would then have the students write the equivalent fractions to the percents using the appropriate color, red for fourths, and green for fifths.

Discuss properties of their new tool in relation to the fraction rulers they have already been using. Students should solve simple addition and subtraction problems.

20% + 20% = (40%) 1/5 + 1/5 = (2/5) Are the values equivalent?

75% - 25% = (50%) 3/4 -1/4 = (2/4) Are the values equivalent?

When adding and subtracting percents it is easier than adding and subtracting fractions with unlike denominators:

20% + 25% = 45% 1/5 + 1/4 = 9/20 The values are the same. 9/20 = 45/10.

Word problems are presented in Section 2 and 3.

*
*

Models for fractions and percent acrylic rulers:

1 2

1/4 2/4 3/4 4/4 5/4 6/4 7/4 8/4

**
**

1 2

1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 10/5

**
**

1 2

… * 5/20 … 10/20 … 15/20 … 20/20 … 25/20 … 30/20 … 35/20 … 40/20

1 2

20% 25% 40% 50% 60% 75% 80% 100% ...* 125% ... 150% ... 175% ... 200%

The 1/4 ruler should be marked in red.

The 1/5 ruler should be marked in green.

The 5/20 ruler should be marked in blue.

The percent ruler should be marked in black.

*When using 20 inch acrylic rulers space will allow including all equivalent fractions and percents.

**
**

Mathematical Standards

This unit addresses the following content standards.

*
Content Standard 1.0
*
Number Concepts, Arithmetic, and Operation Concepts.

Performance Standard 1.2:

Students will describe and compare quantities and compute using fractions and decimals.

Performance Standard 1.4:

Students will interpret percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.

- a. Students will interpret percent as part of 100.
- b. Students will use percents to compare quantities of different or changing sizes.
- c. Students will apply percent concepts to solve problems.

Performance Standard 1.6:

Students will order numbers.

- b. Students will locate numbers on a number line.

*
Content Standard 5.0
*
Problem Solving and Mathematical Reasoning.

Performance Standard 5.2:

Students will formulate and solve a variety of meaningful problems.

*
Content Standard 6.0
*
Mathematical Skills and Tools.

Performance Standard 6.7:

Students will use recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, advice from peers, as appropriate, to achieve solutions.

*
Content Standard 7.0
*
Mathematical Communication.

Performance standard 7.1:

Students will use mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.

**
**