SED 506: Project for Teaching Math Through Engineering/Construction Concepts
Applying Fractions, Decimals, Percents and Ratios to Practical Situations in Beginning Pre-Algebra
Using geometric constructions to learn math vocabulary
Course taught: Beginning Pre-Algebra, Seventh grade
Brown Middle School, Hillsboro, Oregon
Textbook used: Holt Middle School Math. Course 2. 2004.
Context: Students in Beginning Pre-Algebra could be described as non-confident in their math abilities. This lack of confidence could be attributed to a refusal and or reluctance to finish practice. Some of the students have a low level math ability (either perceived or real). Many of the students scored either at the “pass” level or slightly below on the Oregon statewide math assessment (formerly TESA). They are reluctant to learn or practice their math skills. They sometimes mention that neither parent is “good” at math. They frequently question the need to learn specific math concepts. Many are not planning to seek higher education. Hands-on learning opportunities may increase the level of student enthusiasm and participation by more completely addressing various learning styles.
Connie C. Saul
Spring 2007
Oregon Seventh (7) and Eighth Grade (8) Math Standards covered in lessons outlined below:
Calculations and Estimations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
NUMBERS
Model and compare rational numbers with an emphasis on integers (7)
Use rates, ratios, and percents to solve problems (7)
Apply proportions to solve problems (8)
Apply equivalent forms of rational numbers (including percents) to solve problems (8)
COMPUTATION AND ESTIMATION
Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, and part-whole relationships) (7)
ALGEBRAIC RELATIONSHIPS
Understand patterns, relations, and functions
PATTERNS AND FUNCTIONS
Represent, analyze, and determine rules for finding patterns involving integers with tables, graphs, words, and when possible, symbolic rules (7)
Measurement
Understand measurable attributes of objects and the units, systems, and processes of measurement
UNITS AND TOOLS
Select the most appropriate unit to measure surface area and volume(7)
Convert from a measurement expressed in one unit within a system to another using a different unit within the same system to measure surface and volume (7)
Determine an appropriate scale for representing an object in a scale drawing (8)
Carry out unit conversions between the metric and U.S. customary systems of measurement given conversion ratios (e.g., 1 in = 2.54 cm) (8)
Apply appropriate techniques, tools, and formulas to determine measurements
DIRECT AND INDIRECT MEASUREMENT
Solve problems involving rates and derived measurements for such attributes as speed, velocity, and density (8)
Determine actual distances from scale drawings, blueprints, and maps and solve problems involving scale factors (8)
GEOMETRY
Analyze characteristics and properties of two- and three dimensional geometric shapes and develop mathematical arguments about geometric relationships
PROPERTIES AND RELATIONSHIPS
Use proportional reasoning, drawings, models, or technology to demonstrate congruence and similarity of polygons with an emphasis on quadrilaterals (7)
Use proportional reasoning, drawings, models or technology to demonstrate similarity and congruence of polygons with an emphasis on triangles (8)
Use visualization, spatial reasoning, and geometric modeling to solve problems
MODELING
Model, sketch, and label prisms, pyramids, cylinders, and quadrilaterals with specified side lengths or angle measures(7)
Draw to scale two-dimensional representations of rectangular prisms and triangles with specified side lengths or angle measures (8)
Construct and read drawings and models made to scale (8)
Apply transformations and use symmetry to analyze mathematical situations
TRANSFORMATIONS AND SYMMETRY
Classify transformations based on whether they produce congruent or similar non-congruent figures (e.g., compare pairs of shapes (where the image has been transformed, identify the type of translation and use angles, diagonals, and lines of symmetry to determine congruence) (8)
Identify and sketch the figure that is the result of a given rotation, translation, reflection, or dilation or a combination of two transformations (8)
Determine the effects of a transformation on linear and area measurements of the original figure (8)
Description of Instructional Plans and Activities
Lesson Plans
Proportional Reasoning
Lesson 1: RATIOS AND RATES
Learning objective: Students identify, write, and compare ratios and rates.
Warm-up page 259: Are you ready? 6-36, multiples of 6.
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Ratio: A comparison of two quantities by division. There are three ways to write a ratio: fraction form (1/2), using the word “to” (1 to 2), and using a colon (1:2).
Rate: A comparison of two quantities with different units. Example: 110 miles in 2 hours.
Unit Rate: A unit rate is a comparison of two quantities, where the second quantity is one. Unit rate means per ONE unit. Example: 55 miles per (one) hour.
Hand out 10 random die-cut shapes in varying colors. Have students write 10 ratios (in all three forms) with the die-cut pieces.
Practice problems: Page 262: 1-13 all
Lesson 2: IDENTIFYING AND WRITING PROPORTIONS
Learning objective: Students find equivalent ratios and identify proportions.
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Proportion: An equation that states that two ratios are equivalent
Equivalent Ratio: Fractions/Ratios that name the same comparison
The textbook shows two methods that can be used to find if two numbers are proportional:
Simplifying the fractions
Comparing by using a common denominator (the “heart” method)
Practice problems: Page 266: 3-39, multiples of 3
Lesson 3: SOLVING PROPORTIONS
Learning Objective: Students solve proportions by using cross products
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Cross Products: When multiplying a numerator from one ratio and denominator from another ratio
Using the “hook” method to solve cross products
Practice Problems: Page 270: 2-34 Evens
Lesson 4: HANDS-ON MAKING SIMILAR FIGURES USING GRID PAPER
Textbook Page 278
Lesson 5: SIMILAR FIGURES AND PROPORTIONS
Learning Objective: Students use ratios to determine if two figures are similar
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Similar Figures: Figures with the same shape, but not necessarily the same size
Corresponding Sides: Matching sides of two or more polygons
Corresponding Angles: Matching angles of two or more polygons or a pair of angles formed by a transversal and two parallel lines.
Practice Problems: Page 282: 1-13 All
Lesson 6: USING SIMILAR FIGURES
Learning Objective: Students use similar figures to find unknown lengths.
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Indirect Measurement: A way to use similar figures and proportions to find a measure
Practice problems: Page 286-287: 1-10 All
HANDOUT—
Using Similar Figures Quilt Project
Choose a design from the templates provided.
Using graph paper, transfer the small design template to a 6” x 6” square. (5 points)
Using either proportional reasoning or measurement transfer the 6 x 6 to a 12” x 12” square.
Cut the individual pieces of the 12” x 12” square out. (5 points)
Retrace the individual pieces to butcher paper and add ½” on all sides of the pieces.
(By doing this, it will make your quilt pattern pieces.) (5 points)
Select cotton fabrics of your choice.
Iron fabric (as demonstrated)
Iron the butcher paper pieces (shiny side down) to the cotton fabric. (5 points)
Cut out the ironed pattern pieces.
You can choose to take the pieces home to assemble (5 points extra credit), or turn them in to be sewn.
Lesson 7: SCALE DRAWINGS AND SCALE MODELS
Learning Objective: Students understand ratios and proportions in scale drawings and use ratios and proportions with scale.
VOCABULARY—Introduce and discuss. Students will copy the following definitions into notebooks:
Scale Model: A proportional model of a three-dimensional object
Scale factor: The ratio used to enlarge or reduce similar figures
Scale: The ratio between two sets of measurements
Scale drawing: A drawing that is proportional to the original
Practice problems: Page 290: 1-14 All
HANDOUT—
My Dream Home
The time is now to start planning the home of your dreams! You will be using the skills that you have learned in this chapter to build a floor plan of a home that you one day dream to own.
The project:
Complete on graph paper (provided).
Use a pencil and ruler, draw a floor plan (using the scale that you select) of your dream house. (7 points)
Outline the final picture with pen. (3 points)
Determine the scale of your house; make sure that it is identified on the floor plan. (For instance, 1 square on the grid paper = ______feet). (10 points)
List the room dimensions. Calculate the square footage (area) of each room. (10 points)
Calculate the total square footage (area) of the house and the outside perimeter dimensions. (10 points)
Your house must be at least 1,000 square feet. There is no upper limit on the size of the house.
Other Lessons Using Hands-On Engineering Applications….
HANDOUT—
Construct a math manipulative to show fraction, decimal and percent equivalents.
We will be focusing on the equivalents to 10%, 12.5 %, 16.6 %, 20%, 25%, 33.3%, 50%, 66.6%, 75% and 100%.
Materials: colored paper, cut in 1” x 11” strips (a standard paper would produce 8.5 of them)
scissors, pen, ruler
We will determine the lengths of the papers as a class. DO NOT cut the strips of paper until we have decided the correct length of each paper!!
A 10 inch strip of paper will be equal to 1/1, 1.0, and 100%
Q. How do you make a paper that is 10 inches long? (Cut 1 inch off of it!) It might be easier for you to cut 1 inch off of all of the strips to make them all 10 inches.
Cut one inch off of one of the strips. Label the paper with the numbers 1/1, 1.0, 100%.
Q. What is the next easiest percent to remember? (Usually 50%). Cut that length. (5 inches). Label it ½, .50, 50%.
Q. What is the next easiest percent to remember? (Usually 25%-- half of 50%) Cut that length (2.5 inches). What is the fraction equivalent to 25%? (If it is half of 50%, then you can get there by increasing the denominator by twice as much). Label this strip with ¼, .25, 25%.
Q. What is half of 25? (12.5) Cut a strip to a length of 1.25 inches or 1 ¼ inches. This will represent 1/8, .125, 12.5%. Label it.
Q. How would you make a strip to represent 75%? (Multiply 25% by three). Therefore, you can multiply 2.5 by 3 to get the length of the next strip. Label it with ¾, .75, 75%.
Q. What length would you use to represent 10%? (Remember that 100%=10 inches) (One inch). Label it with 1/10, .10. 10%.
Q. What about 20%? (2 inches). Label it with 1/5. (Two times 1/10=2/10=1/5) .20, 20%.
The strips for 1/6, 1/3, and 2/3 seem a bit trickier. Try to figure out what the lengths should be.
Discuss this with your table partner.
(After about 10 minutes of productive struggle…some students discover that 1/6 is between 1/8 and 1/5 and that 1/3 is the same as 2/6 and that 2/3 is the same as 4/6. They need help in figuring out how long they should cut the strips. 1/6 is equal to .166 when multiplied by 10—the size of the strip should be about 1-10/16 inches. The size of the “1/3” strip should be 3-5/16 inches. The size of the “2/3” strip should be 6-10/16 inches.)
Please use these to help you memorize the common equivalents!! They will be collected before the chapter test.
An original example of your own that emphasizes science and engineering concepts
A study of gear ratios utilizing bicycles as the vehicleJ
Reflection and Ideas for Future Content
I think that connecting math concepts to real-life examples has great merit. I would like to pursue this approach more in upcoming years. I think that by providing the students with a variety of lesson types, it would appeal to more learners. In fact, students that might be categorized as “emerging” on a statewide exam might prove to be either “proficient” or “strong” when given an untraditional assessment. This concept became apparent when the students were working on the activities.
Due to the upcoming building expansion at our school, I think that it would be a good idea to have a guest speaker that could talk about how architectural plans/blueprints are made, what the construction timeline is, what the new building will look like and other math related concepts that tie in to the school expansion. I would also like to incorporate a construction type school improvement project or service-learning project into my lessons.
Additionally, Chapter 7 lends itself well to stimulating interest in engineering because it highlights the career of a bridge designer. It also includes lessons on geometric constructions. I have included some student samples of these lessons. In the future, I also plan to further develop lessons that involve measurement and precision.
I would like to cooperate with the counseling department to sponsor a “career” day to invite speakers in to talk to students about their jobs. I think that by talking to the students about how math and science are incorporated into their jobs, it may help make the students’ math and science education more meaningful. And perhaps, interest students in seeking a career related to math and science.
Overall, this study was a positive experience me.
Chapter 7 focuses on Plane figures. The chapter opens with a comparison of bridge types and a paragraph about careers as a bridge designer.
Learning objective: Students learn to identify and describe geometric figures
Vocabulary: point, line segment, ray, line, plane, congruent
Learning objective: Students learn to identify angles and parts of angles
Vocabulary: angle, vertex, right angle, acute angle, obtuse angle, straight angle, complementary angle, supplementary angle
Learning objective: Students learn to identify parallel, perpendicular, and skew lines, and angles formed by a transversal.
Vocabulary: perpendicular lines, parallel lines, skew lines, vertical angles, transversal
Learning objective: Students learn to identify parts of a circle and to find central angle measures