## The Cost of a Great Looking Floor

• Lesson
6-8
1

Tile floors are common in many homes and businesses.  They are durable, beautiful, and can add value to the home or business but they can also be costly.  In this lesson, students will create and estimate the cost of a tile floor design using geometric shapes, ratios, proportions, and percents. All cost estimates are based on the purchase of full boxes of tiles so students have to weigh cost against design considerations.  Cost estimates also include labor and taxes for a more realistic estimate of what it costs for a great looking floor.

Students will design either individually or with a partner, a tile floor of a predetermined size using pattern blocks. If you do not have pattern blocks, give each group of students the Pattern Blocks Activity Sheet.

This will also add to the design element of the activity because students can color in the shapes to suit their design scheme. Students may alternatively use the Patch Tool.

Students will need The Cost of a Great Looking Floor Activity Sheet to record their bids.

Otherwise, the parameters of the lessons can be modified to suit your students' needs and goals. Refer to the lists below when designing the lesson for your class.

The following are important design parameters for this lesson:

• Design team consists of one or two designers
• Design must use a minimum of 3 out of the 6 different tile shapes/colors
• All tiles must be fit so that there is no space between shapes and shapes do not overlap.
• Interior tiles must be whole tiles. The virtual cutting of tiles to fit within the floor boundaries is only allowed along the edges, which is true for most floor installations.
• Tiles are sold in full boxes only and may not be shared with another student and/or project. For example, hexagon tiles are sold 4 to a box so even if you only need one you will need to purchase an entire box.
• Labor is calculated according to the tiles purchased, not the tiles used. Since materials are paid for by the customer whether installed or not, any extra tiles are the property of the customer and left on the job site.
• Students must submit design and bid document.

This group of design parameters can be modified to meet the needs of your students and to work within various time constraints:

• Students should come up with a design and estimate the cost for a 6-foot by 8-foot custom tile floor
• How student use edge tiles that are cut is a way that the lesson can be differentiated. For lower level students you may want to consider having them consider the remainder of a cut tile as waste. For higher-level students you may want to have them problem solve how to minimize waste.
• Decide in advance if you want your students to color the design to match the pattern blocks or if you want them to be able to exercise creative freedom with color. Allowing students the option to choose is another way to differentiate and meet the needs of your more creative students while allowing less creative students the option of using the block colors.

As you walk around assisting students, watch for overlapping tiles in the design. If you notice this, ask them if there is another way they could work their design to avoid overlapping tiles. You can share with students the fact that contractors avoid overlapping which then requires cutting the tile to create a flat surface. If you notice blank spaces remind them that their design cannot contain any blank spaces and ask what changes they could make to correct this problem.

### Suggestions for Differentiation

Like ability grouping will allow for modification of design requirements based on ability level. Mixed ability grouping where a student with strength in artistic design is grouped with a student with strength in mathematical calculations would allow both students to succeed in their area of expertise while strengthening their area of weakness.

If you are not going to use partners for this activity you might consider assigning different design criteria to different students based on ability. For example, stronger students might be challenged to use all 6 shapes where students who struggle with the design might be given the option to use only two shapes. You can also modify the size of the floor making it larger for students who work quickly on this type of assignment and smaller for students who work slower.

Assessment Options

1. Collect the The Cost of a Great Looking Floor Activity Sheets and drawings of their floor designs. Assess student understanding by checking for accuracy in following instructions and calculation.
2. Give students the option of designing a second floor as a take-home assignment that could be completed with the help of family members.
3. Consider assigning various groups with different size or shaped rooms. Irregular polygon shapes can be quite challenging.

Extensions

1. Flooring contractors estimate their material at 10% over the required amount. Have student recalculate their projects by adding 10% more of each tile and recalculating. Remind students that the labor costs are calculated from whole boxes, so if only part of a box is used, it's possible that they've already paid 10% extra.
2. When you calculate the cost of your design, would you calculate sales tax on materials and labor or just materials?
[The answer to this question will vary by state and location. If you want this activity to be as realistic as possible consider checking with a local flooring company as to how tax is calculated. You could also assign this task to a student as an extension activity.]
3. Propose to your students that the customer has indicated the cost is too high. Ask what changes they would reduce the cost of their design by 25% if possible. If a 25% reduction is not possible, what percentage is possible?
4. What would happen to the total cost of the floor if labor was 20% higher?
5. Invite a local flooring contractor to come in and talk to the students about how projects such as these would be bid and what costs are associated with a bid.

Questions for Students

1. What can we say about the sides of the various shapes?

[They are all equal except for the sides of the triangle and the longer base of the trapezoid.]

2. If a hexagon tile is too expensive or the wrong color, what other tiles could you use to achieve the same shape?

[You could use 6 triangles or 2 trapezoids or 3 blue rhombi. All of these can be put together to form 1 hexagon.]

3. How many triangles are needed to make a trapezoid?

[3.]

4. How many triangles are needed to make a big rhombus?

[2.]

5. What shape or shapes could be used in place of the small rhombus? How is the small rhombus different from the big rhombus?

[None of the shapes can be substituted for the small rhombus. The rhombi have sides congruent to each other, but the angles in the shapes are different. The angles in the small rhombus are 30° and 150°, while the angles in the big rhombus are 60° and 120°.]

Teacher Reflection

• Did students understand the relationships between shapes before starting this activity? after completing the activity? If not, what would you do differently next time?
• Did the students understand the concept of buying by the box versus by the tile?
• Was students’ level of enthusiasm/involvement high or low? Explain why.
• Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it?
• What were some of the ways that the students illustrated that they were actively engaged in the learning process?
• Did you set clear expectations so that students knew what was expected of them? If not, how can you make them clearer?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
• What worked with classroom behavior management? What didn't work? How would you change what didn’t work?

### Learning Objectives

Students will:

• Manipulate pattern blocks to create a design without gaps between tiles.
• Accurately calculate the cost of their design using predetermined pricing structures.
• Calculate cost using percentages, rounding, and other numeric skills.

### NCTM Standards and Expectations

• Work flexibly with fractions, decimals, and percents to solve problems.
• Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
• Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
• Solve problems involving scale factors, using ratio and proportion.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

• CCSS.Math.Content.6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

• CCSS.Math.Content.7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

• CCSS.Math.Content.7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.

• CCSS.Math.Content.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

### 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.