2009 Leaders’ Notes Grade 7 Module 5page 1
General Materials and Supplies:Laptop, Projector, Power cord Clay Company Task Handout Scissors
Packing to Perfection Handout Construction paper It’s Good to be Square Handout
Snap Cubes Scotch tape Square Tiles
Slide / Tasks/Activity / Personal Notes
/ (slide 1) Module Five
This module focuses on measurement and geometry.
/ (slide 2) And Now a Word From Our Sponsor
The Partners for Mathematics Learning Project is a Mathematics/Science Partnership (MSP) funded through the North Carolina Department of Public Instruction.
Recall the focus of this professional development is to improve our content and pedagogical knowledge. Equally important is the opportunity to take an introductory look at the Essential Standards and relate mathematical content to those Standards.
- We’ve talked about algebra, proportional reasoning, number, and measurement, problem solving and data
- Look back at the new Standard Course of Study
- What Essential Standards have we addressed?
- What Essential Standards do you still have questions about?
/ (slide 3) It’s Good to Be Square?
The development and understanding of square numbers is an important concept needed for conceptually understanding area of a square, square roots, irrational numbers and factoring.
Having students identify and represent squares geometrically and symbolically will help them remember their perfect squares and understand the meaning of a perfect square.
Ask participants to find the Which Numbers are Square? handout (Module Five, Handout One). Give each pair of participants 50 square color tiles. Encourage participants to build the squares while completing the handout.
/ (slide 4) It’s Good to Be Square!
The numbers on the right side of your table are called square numbers or perfect squares. Why?
- A square can be made from this many square tiles.
- 49, 64, 81
- Multiplied the side number by itself (May want to mention area at this point.)
- At this point, teachers would instruct students on using an exponent of 2. (72 = 7 7), and that this expression is read as “the square of seven or seven squared”
- These are the only numbers between 36 and 81 that result from a product of whole numbers
/ (slide 5) It’s Good to Be Square?
In part two, what pattern do you notice in the number of tiles added each time to create the next size square?
Two consecutive square numbers differ by an odd number. Add consecutive odd numbers to get to the next square number.
1 = 1 1 + 3 + 5 = 9
1 + 3 = 4 1 + 3 + 5 + 7 = 16
Patterns that can be represented both numerically and geometrically help students link number and geometry.
Model that pattern with your square tiles.
Ask a participant to model how each consecutive square increases by consecutive odd numbers, using their square tiles.
How does a simple activity like this help students make a connection between perfect squares and other mathematical concepts?
- Area of a square
- Patterns in numbers
- Sums of Consecutive Odd Numbers
/ (slide 6) Packing to Perfection
Ask participants to think about the following questions.
Is there a relationship between surface area and volume?
Answers will vary. This question will be revisited at the conclusion of the activity.
Can rectangular prisms with different dimensions have the same volume?
Yes a rectangular prism with dimensions of 2 x 1 x 4 has the same volume as a rectangular prism with dimensions of 1 x 8 x 1.
Do rectangular prisms with the same volume have the same surface area?
No, rectangular prisms with the same volume can have different surface areas.
/ (slide 7) Packing to Perfection
Provide each pair of participants with 24 snap cubes. Ask participants to find the Packing to Perfectionhandout (Module Five, Handout Two). Have participants read the scenario and answer any questions that might be asked.
Take 24 snap cubes and imagine that each cube is a fancy chocolate candy. For shipping purposes, these candies need to be packaged in boxes that are rectangular prisms. Knowing the company only sells their candies in groups of 24, what are the possible dimensions for the boxes?
Allow participants ample time to create their packages and calculate the surface areas. (You may or may not allow participants to use calculators. You might want participants to physically count surface area.)
/ (slide 8) Packing to Perfection
Which of your packages requires the least amount of material? The greatest amount of material?
3 x 4 x 2 has the smallest surface area of 52 square units
1 x 24 x 1 has the largest surface area of 98 square units
Why is the amount of material important?
The amount of material needed represents surface area.
What do you notice about the shape of the package with the smallest surface area? How about the package with the greatest surface area?
The 3 x 4 x 2 has the smallest surface area and is the most compact (closest shape to a cube).
The 1 x 24 x 1 package is more spread out, flat, and elongated.
/ (slide 9) Packing to Perfection
Which package would you recommend to the chocolate company? Why?
Answers will vary.
If sold to a store, how would you suggest that the package be displayed on the shelf? Why?
Answers will vary.
/ (slide 10) Packing to Perfection
Revisit the questions asked at the beginning of the activity to see if answers have changed or been revised.
What is the relationship between surface area and volume?
There is no specific relationship between surface and volume. In this activity, volume remains the same, however the surface area may increase or decrease based on compactness of the shape. The closer the package is to a cube, the smaller the surface area.
Can rectangular prisms with different dimensions have the same volume?
Yes. All of the packages have the same volume, but different dimensions.
Do rectangular prisms with the same volume have the same surface area?
No, there are 5 different packages with the same volume, but different surface areas.
/ (slide 11) Packing to Perfection
How does this activity build conceptual understanding of surface area and volume?
- Students build models from 24 cubes (i.e. volume is 24) and count the faces of the cubes to find the area of each face of the package (i.e. surface area).
Perimeter/circumference and area of 2-D figures are related to surface area and volume of 3-D figures
/ (slide 12) Clay Company Task
Ask participants to find the Clay Company Task handout (Module Five, Handout Three). Have a participant read the scenario out loud. Have participants discuss with their partner what their task involves. Have a participant summarize to the group what they have been asked to do. Then have each group discuss what mathematics they think they will need to complete the task.
You have been asked by Magic Modeling to design containers for their modeling clay. You have been assigned the 360 cubic cm clay product. The company would like you to design at least three different containers: a rectangular prism, a triangular prism, and a cylinder.
Provide participants with cardboard, cardstock, construction paper, etc. to create their containers. Require participants to draw and label the nets for each container on a large sheet of paper for display.
Provide participants with ample time to create their containers. It is important for participants to create these packages. Significant learning and understanding occurs while creating the packages.
/ (slide 13) Clay Company Task
How did you determine the dimensions you used for each package?
- Used factors of 360
- Thought about area of the base times the height
- With the cylinder, think about being approximately 3; keep changing length of diameter to get closer to a base area of 36 square units if using a height of 10.
/ (slide 14) Clay Company Task
What mathematical relationships did you think about while designing a package that will hold a specific amount of modeling clay?
- A participant may begin with dimensions of 36 x 10 x 1 and then change to dimensions of 18 x 5 x 4. Trying to make the package more compact, a participant may make a third to dimensions of 9 x 10 x 4. (Factors of 360)
- Begin with a triangular prism that has a base area of 36 square units and a height of 10. (Area of the base times the height of the prism)
- Begin with a cylinder that has circular base with a diameter of 12 units and a height of 10 units. (Area of the base times the height of the cylinder)
What is the relationship between the area of the base and the volume of the container?
- The ratio of volume to the area of the base is equal to the height of the prism. (Area of base times the height of the prism is equal to the volume.)
/ (slide 15) Clay Company Task
How could you modify this activity to meet the needs of struggling learners?
- Require building only one package
- Discuss possible dimensions with students
- Use other polyhedra
Perimeter/circumference and area of 2-D figures are related to surface area and volume of 3-D figures
Formulas are derived from the measures of the attributes and relationships of 2-D and 3-D figures
/ (slide 16) Back to Our Sponsor
In the current curriculum the measurement strand involves indirect measurement of surface area and volume- that is, using formulas to calculate
In the 2009 Standard Course of Study, how do the Essential Standards address measurement?
Allow participants time to look over the Essential Standards once more, focusing on measurement.
/ (slide 17) Reflection
Students need a variety of hands-on and visual experiences to master measurement and geometry concepts.
How might today’s activities change the teaching and learning of measurement and geometry in your classroom?
(slides 18-21) Credits for project and closing slides