Unit 4: Dividing Fractions
Lesson 17: Fitting Boxes into Boxes

In this three-part culminating activity, students use what they have learned to determine the most economical way to ship jewelry boxes using the United States Postal Service (USPS) flat-rate options.

In Part 1, students make sense of the task, outline what they will need to know and do to answer the question, and map out their plan. In Part 2, they model the problem, calculate the number of jewelry boxes that will fit into each shipping box, and determine the associated costs. Students experiment with different orientations for the jewelry boxes to optimize space and minimize cost. In Part 3, they present, reflect, and discuss. Students explain their strategies and reasoning (MP3) and evaluate the decisions about how to fit all 270 jewelry boxes so they ship at the lowest cost (MP4). As a class, students reflect on how the orientation of the jewelry boxes and the size of the shipping boxes affected the unit cost for shipping each box of jewelry.

Depending on the instructional choices made, this lesson could take one or more class meetings. The time estimates are intentionally left blank because the amount of time needed might vary depending on factors such as:

If students will research the flat-rate options themselves, or be provided with this information.
If each group will explore all size options or only one option.
How much organizational support is given to students.
How student work is ultimately shared with the class (e.g., not at all, informally, or with formal presentations).
Consider defining the scope of work further and setting a time limit for each part of the activity to focus students’ work and optimize class time.

This Nrich activity provides a context for practicing counting on in 1s and 10s (and later 100s). Children will also be invited to find out how far from 50 they end up so they will be practicing finding the difference between two numbers. It encourages children to find more than one way of getting to a solution.

This learning video presents an introduction to the Flaws of Averages using three exciting examples: the ''crossing of the river'' example, the ''cookie'' example, and the ''dance class'' example. Averages are often worthwhile representations of a set of data by a single descriptive number. The objective of this module, however, is to simply point out a few pitfalls that could arise if one is not attentive to details when calculating and interpreting averages. The essential prerequisite knowledge for this video lesson is the ability to calculate an average from a set of numbers. During this video lesson, students will learn about three flaws of averages: (1) The average is not always a good description of the actual situation, (2) The function of the average is not always the same as the average of the function, and (3) The average depends on your perspective. To convey these concepts, the students are presented with the three real world examples mentioned above.

Unit: Area and Surface Area
Lesson 9: Formula for the Area of a Triangle

In this lesson students begin to reason about area of triangles more methodically: by generalizing their observations up to this point and expressing the area of a triangle in terms of its base and height.

Students first learn about bases and heights in a triangle by studying examples and counterexamples. They then identify base-height measurements of triangles, use them to determine area, and look for a pattern in their reasoning to help them write a general formula for finding area (MP8). Students also have a chance to build an informal argument about why the formula works for any triangle (MP3).

This Nrich can be very engaging. It has the ability to introduce pupils to some logical reasoning as well as being solvable by trial and improvement.

In this unit, students build on fraction concepts from previous grades to understand fractions as division. They also use visual models to make estimates, add and subtract fractions and mixed numbers, and check the reasonableness of their answers. Finally, students explore strategies for solving fraction-of problems.

This Nrich problem is useful for those pupils who are coming to terms with spatial representation of fractions where area is concerned rather than just length. Pupils' visualizations vary a great deal and this may prove very difficult for some and yet readily accessible to others.

Unit 4: Dividing Fractions
Lesson 12: Fractional Lengths

This is the first of four lessons in which students use multiplication and division of fractions to solve geometric problems. In this lesson, they encounter problems involving fractional lengths. They use their understanding of the two interpretations of division—“how many groups?” and “how much in each group?”—to solve problems that involve multiplicative comparison (MP7).

In these geometry-themed lessons, students work with a wider range of fractions and mixed numbers, which gives them opportunities to choose their methods and tools for problem solving.

Unit 4: Dividing Fractions
Lesson 14: Fractional Lengths in Triangles and Prisms

In this transitional lesson, students conclude their work with area and begin to explore volume of rectangular prisms. First, they extend their work on area to include triangles, using division to find the length of a base or a height in a triangle when the area is known. Second, they undertake a key activity for extending their understanding of how to find the volume of a prism.

In previous grades, students learned that the volume of a prism with whole-number edge lengths is the product of the edge lengths. Now they consider the volume of a prism with dimensions 1 1/2 inch by 2 inches by 2 1/2 inches. They picture it as being packed with cubes whose edge length is 1/2 inch, making it a prism that is 3 cubes by 4 cubes by 5 cubes, for a total of 60 cubes, because 3 x 4 x 5 = 60. At the same time, they see that each of these 1/2-inch cubes has a volume of 1/8 cubic inches, because we can fit 8 of them into a unit cube. They conclude that the volume of the prism is 60 x 1/8 = 7 1/2 cubic inches.

In the next lesson, by repeating this reasoning and generalizing (MP8), students see that the volume of a rectangular prism with fractional edge lengths can also be found by multiplying its edge lengths directly (e.g., (1 1/2) x 2 x (2 1/2) = 7 1/2).

This Nrich problem provides an opportunity to find equivalent fractions and carry out some simple additions and subtractions of fractions in a context that may challenge and motivate students.

This Nrich activity gives the pupils opportunities to use and develop their visualizing skills in conjunction with the knowledge of fractions. It's quite a contrast to just dealing with fractions numerically.

This simple Nrich game is designed to help children become more familiar with common coins in the UK, and to find fractional quantities of amounts of money. The fractions involved are slightly more challenging than those found in Fraction Card Game. Higher-order thinking is required in order to play strategically.

This Nrich game offers students an opportunity to practice calculating fractions and percentages of amounts.

This Nrich problem gives practice in calculating with fractions in a challenging setting. It also requires the use of factors and multiples. While doing the problem learners will need to express a smaller whole number as a fraction of a larger one and find equivalent fractions. This activity will require some estimating and trial and improvement, combined with working systematically.

This lesson prepares students to apply what they know about the area of parallelograms to reason about the area of triangles.

Highlighting the relationship between triangles and parallelograms is a key goal of this lesson. The activities make use of both the idea of decomposition (of a quadrilateral into triangles) and composition (of two triangles into a quadrilateral). The two-way study is deliberate, designed to help students view and reason about the area of a triangle differently. Students see that a parallelogram can always be decomposed into two identical triangles, and that any two identical triangles can always be composed into a parallelogram (MP7).

Because a lot happens in this lesson and timing might be tight, it is important to both prepare all the materials and consider grouping arrangements in advance.

After the Great Depression struck, Ford especially wanted to support ailing farmers. For two years, Ford and his team researched ways to use farmers’ crops in his Ford Motor Company. They discovered that the soybean was the perfect answer. Soon, Ford’s cars contained many soybean plastic parts, and Ford incorporated soybeans into every part of his life. He ate soybeans, he wore clothes made of soybean fabric, and he wanted to drive soybeans, too. The resource includes a lesson plan/book card, a design challenge, and copy of a design thinking journal that provide guidance on using the book to inspire students' curiosity for design thinking. Maker Challenge: Think about the people in your community and the challenges they face. List three challenges that affect their daily life. Consider something you use every day and brainstorm how it could be repurposed or modified to address this problem.

A document is included in the resources folder that lists the complete standards-alignment for this book activity.

This Nrich problem gives practice in calculating with fractions and encourages different ways of representing and recording.

Several full course OER offerings in ELA, Math, Science, Social Studies, World Languages, Fine Arts

This link takes you to MoodleNet where you can download Middle School level math LMS courses that include Geogebra's digital app version of Illustrative Mathematics. The courses will work in Moodle, Canvas, Schoology or D2L. From the list of courses in the GeoGebra Illustrative Mathematics collection on MoodleNet click on the name of the unit you want. Then choose to either send it to Moodle or download the file so that it can be uploaded to Canvas, Schoology or D2L.

This is a project that follows the general PBL framework that can be used to help students master the concept of intermediate geometry. It was specifically designed to help students review the fundamental theorems of geometry involving lines, segments, angles, and basic shapes; use the properties of similarity and congruence to solve problems for geometric figures; master trigonometric ratios to solve right triangle problems; compare & contrast various geometric transformations and models; learn how to do geometric proofs and construct basic geometric figures; and understand the basic concepts related to the geometry of circles. Note that the project was designed and delivered per the North Carolina Math 2 curriculum and it can be customized to meet your own specific curriculum needs and resources.