Student pairs are given 10 minutes to create the biggest box possible using one piece of construction paper. Teams use only scissors and tape to each construct a box and determine how much puffed rice it can hold. Then, to meet the challenge, they improve their designs to create bigger boxes. They plot the class data, comparing measured to calculated volumes for each box, seeing the mathematical relationship. They discuss how the concepts of volume and design iteration are important for engineers. Making 3-D shapes also supports the development of spatial visualization skills. This activity and its associated lesson and activity all employ volume and geometry to cultivate seeing patterns and understanding scale models, practices used in engineering design to analyze the effectiveness of proposed design solutions.
MathMemos is a teacher space where adult educators share rich math problems, samples of student work, and practical suggestions for bringing the problems to life in the classroom. All math problems are:
(1) Open-ended, meaning that it might have more than one answer or there may be multiple solution pathways to solve the problem.
(2) The prompt does not clearly direct the students towards a procedural pathway in solving the problem.
(3) Students should be able to struggle productively with the problem for an extended period of time: they are student driven.
The problems include and integrate a wide range of math content and topics such as: Algebra, Geometry, Functions, Number & Quantity, and Statistics & Probability. You can also search for specific topics (fractions, systems of equations, equality) or problem solving strategies (guess & check, charts & tables, visual strategies, manipulatives).
The resources is designed so that teachers can be thoroughly connected to a problem before taking it into the classroom, by solving it themselves and looking at samples of student work that other teachers have posted online. Then, teachers are encouraged to reflect on how the problem played out in their own classroom and post their reflections.
This collection is an ongoing project with new problems being added as teachers submit new write-ups.
A PD Module is available for Professional Developers.
MathMemos was created by Tyler Holzer with a grant from the New York State Education Department Office of Adult Career and Continuing Education Services.
This lesson unit is intended to help teachers assess how well students are able to visualize two-dimensional cross-sections of representations of three-dimensional objects. In particular, the lesson will help you identify and help students who have difficulties recognizing and drawing two-dimensional cross-sections at different points along a plane of a representation of a three-dimensional object.
This question examines the algebraic equations for three different spheres. The intersections of each pair of spheres are then studied, both using the equations and thinking about the geometry of the spheres. For two spheres where one is not contained inside of the other there are three possibilities for how they intersect.
This lesson unit is intended to help teachers assess how well students are able to solve problems involving area and volume, and in particular, to help you identify and assist students who have difficulties with the following: computing perimeters, areas and volumes using formulas; and finding the relationships between perimeters, areas, and volumes of shapes after scaling.
Module 3, Extending to Three Dimensions, builds on students understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids. The student materials consist of the student pages for each lesson in Module 3. The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.
This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a Ňdouble-naped coneÓ with vertex at the center of the sphere and bases equal to the bases of the cylinder.
This lesson unit is intended to help teahcers assess how well students solve problems involving measurement, and in particular, to identify and help students who have the following difficulties; computing measurements using formulas; decomposing compound shapes into simpler ones; using right triangles and their properties to solve real-world problems.
Students build scale models of objects of their choice. In class they measure the original object and pick a scale, deciding either to scale it up or scale it down. Then they create the models at home. Students give two presentations along the way, one after their calculations are done, and another after the models are completed. They learn how engineers use scale models in their designs of structures, products and systems. Two student worksheets as well as rubrics for project and presentation expectations and grading are provided.
Reflective of the modernness of the technology involved, this is a challenging geometric modelling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.
This lesson unit is intended to help teachers assess how well students are able to: choose appropriate mathematics to solve a non-routine problem; generate useful data by systematically controlling variables; and develop experimental and analytical models of a physical situation.
This concept-building module contains a variety of simulations for exploring factors that cause molecules to attract each other. It was developed to help secondary students understand both polar and non-polar covalent bonding. Users can manipulate models to see how the strength of attraction is affected by distance from one molecule to another, by heating the substance, and by mixing polar and non-polar substances. Part II of the activity is devoted to hydrogen bonds, and explores why water is one of the most important molecules for life's existence. This item is part of the Concord Consortium, a nonprofit research and development organization dedicated to transforming education through technology.
Through this lesson and its two associated activities, students are introduced to the use of geometry in engineering design, and conclude by making scale models of objects of their choice. The practice of developing scale models is often used in engineering design to analyze the effectiveness of proposed design solutions. In this lesson, students complete fencing (square) and fire pit (circle) word problems on two worksheets—which involves side and radius dimensions, perimeters, circumferences and areas—guiding them to discover the relationships between the side length of a square and its area, and the radius of a circle and its area. They also think of real-world engineering applications of the geometry concepts.
Modeling Our World with Mathematics Unit 2: Environmental Science Topic 2 - Sustainable Forestry
This classroom task gives students the opportunity to prove a surprising fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.
This lesson unit is intended to help teachers assess how well students are able to: translate between the equations of circles and their geometric features; and sketch a circle from its equation.