Unit 4: Dividing Fractions
Lesson 5: How Many Groups? (Part 2)

In this lesson, students continue to work with division situations involving questions like “how many groups?” or “how many of this in that?” Unlike in the previous lesson, they encounter situations where the quotient is not a whole number, and they must attend to the whole when representing the answer as a fraction (MP6). They represent the situations with multiplication equations (e.g., “? groups of 1/2 make 8” can be expressed as ? x 1/2 = 8) and division equations (8 / 1/2 = ?).

Unit 2: Introducing Ratios
Lesson 8: How Much for One?

This lesson introduces students to the idea of unit price. Students use the word "per" to refer to the cost of one apple, one pound, one bottle, one ounce, etc., as in “$6 per pound” or “$1.50 per avocado.” The phrase “at this rate” is used to indicate that the ratios of price to quantity are equivalent. (For example, “Pizza costs $1.25 per slice. At this rate, how much for 6 slices?”) They find unit prices in different situations, and notice that unit prices are useful in computing prices for other amounts (MP7).

Students choose whether to draw double number lines or other representations to support their reasoning. They continue to use precision in stating the units that go with the numbers in a ratio in both verbal statements and diagrams (MP6).

Note that students are not expected to use or understand the term "unit rate" in this lesson.

Unit 4: Dividing Fractions
Lesson 8: How Much in Each Group? (Part 1)

Previously, students looked at division situations in which the number of groups (or the fraction of a group) was unknown. They interpreted division expressions as a way to find out that number (or fraction) of groups. In this lesson, students encounter situations where the number of groups is known but the size of each group is not. They interpret division expressions as a way to answer “how much in a group?” questions.

Students use the same tools—multiplication and division equations and tape diagrams—and the same structure of equal-sized groups to reason about “how much in a group?” questions (MP7). They also continue to relate their reasoning in quantitative contexts to their reasoning on abstract representations (MP2). Students find both whole-number and non-whole-number quotients, recognizing that, like the number of groups, the amount in one group can also be a whole number or a fraction.

Unit 4: Dividing Fractions
Lesson 9: How Much in Each Group? (Part 2)

This lesson allows students to practice finding the amount in one group, and to interpret, represent, and solve different kinds of division problems with less scaffolding. In one activity, students are not explicitly told whether the division situations involve finding the number of groups or finding the amount in each group. They decide on an interpretation that would enable them to solve a division problem. Students are also required to identify relevant information (from a video, a picture, or written statements) that would help them answer questions.

Because the tasks in this lesson are not scaffolded, students will need to make sense of the problems and persevere to solve them (MP1). As students move back and forth between the contexts and the abstract equations and diagrams that represent them, they reason abstractly and quantitatively (MP2).

Students will use the ten frame card figure out how many more you need to make ten.

Amy Guglielmo, Jacqueline Tourville, and Giselle Potter tell the story of autism advocate Dr. Temple Grandin’s childhood and her quest to experience the sensation of a hug. 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: Students will work to develop an assistive technology solution for people with autism. Alternatively, students may develop assistive technology solutions for students with differing abilities. Students are encouraged to work with peers in a local special education classroom to combine their love of technology and engineering to help positively influence their peers’ lives.

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

This Nrich task is an unusual way to explore number patterns in a well-known context. The activity will reinforce the construction of the hundred square, and increase children's familiarity with the sequences contained within it. Using a common resource, such as a hundred square, is a good way for children to begin to use visualisation, which they may find quite difficult at first. The act of visualising in this problem tests children's understanding of how the number square is created.

I Know It is an engaging, interactive math website for elementary students. Teachers create their own classroom and assign questions to either individual students or the whole class. Teachers can also provide hints to the questions if they want. The student is also provided with feedback if they get a question incorrect.

Unit 9: Putting It All Together
Lesson 2: If Our Class Were the World

This lesson is optional. In this lesson, students look at ratios of different populations in the world and determine what their class would be like if its ratios were equivalent (MP1, MP2). In the process, they again work with percentages that are not whole numbers, using knowledge gained in a previous unit. Moreover, the ratios will be “close” to being equivalent because the exact world population is not known and all populations need to be whole numbers (MP6). The activities in this lesson could take anywhere from one to four days, depending on how much time is available and how far the class takes it. Earlier activities are needed for later ones in this lesson. A variant on this activity involves developing, administering, and analyzing a survey: If the school were our class. Students brainstorm some questions they would like to know about the students in their school. Questions might include:

How many people in the school play an instrument?
How many people in the school eat school lunches?
How many people in the school ride the bus to school?
How many people in the school have a cell phone?
How many people in the school plan to attend a four-year college or university?
How many people in the school were born outside of this state?
How many people in the school have traveled outside of this country?
As with all lessons in this unit, all related standards have been addressed in prior units; this lesson provides an optional opportunity to go more deeply and make connections between domains.

In this Nrich activity students are asked to evaluate data representations based on the book "If the World Were a Village."

The purpose of this task is for students to compare two problems that draw on the same context but represent the two different interpretations of division, namely, the "How many groups?" interpretation and the "How many in each group?" interpretation.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Mrs. Moore’s third grade class wants to go on a field trip to the science museum.

Math Pickle is a free website resource filled with math puzzles and games to engage students in problem-solving. Puzzles are organized by grade level and subject. They are meant to be solved within a 45-60 minute time period.

7th and 8th Graders learn about Volume and Surface Area using the tool Goose Chase.

This video explains some interesting facts about the last digits of prime numbers.

An interactive applet and associated web page that demonstrate the relationship of the interior and exterior angles of a polygon. The applet shows an irregular polygon where one vertex is draggable. As it is dragged the interior and exterior angles at that vertex are displayed, and a formula is continuously updated showing that they are supplementary. The tricky part is when the vertex is dragged inside the polygon making it concave. The applet shows how the relationship still holds provided you get the signs of the angles right. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com.

Unit 4: Dividing Fractions
Lesson 3: Interpreting Division Situations

In an earlier lesson, students were reminded of the connection between multiplication and division. They revisited the idea of division as a way to find a missing factor, which can either be the number of groups, or the size of one group.

In this lesson, students interpret division situations in story problems that involve equal-size groups. They draw diagrams and write division and multiplication equations to make sense of the relationship between known and unknown quantities (MP2).

Unit 7: Rational Numbers
Lesson 10: Interpreting Inequalities

In this final lesson on inequalities, students explore situations in which some of the solutions to inequalities do not make sense in the situation’s context. Students learn to think carefully about a situation’s constraints when coming up with reasonable solutions to an inequality. Students also see that inequalities can represent a comparison of two or more unknown quantities.