Unit: Area and Surface Area
Lesson 19: Designing a Tent

In this culminating lesson, students use what they learned in this unit to design a tent and determine how much fabric is needed for the tent. The task prompts students to model a situation with the mathematics they know, make assumptions, and plan a path to solve a problem (MP4). It also allows students to choose tools strategically (MP5) and to make a logical argument to support their reasoning (MP3).

The lesson has two parts. In the first part, students learn about the task, gather information, and begin designing. The introduction is important to ensure all students understand the context. Then, after answering some preparatory questions in groups and as a class, students work individually to design and draw their tents. They use their knowledge of area and surface area to calculate and justify an estimate of the amount of fabric needed for their design.

The second part involves reflection and discussion of students’ work. Students explain their work to a partner or small group, discuss and compare their designs, and consider the impact of design decisions on the surface areas of their tents.

Depending on instructional choices made, this lesson could take one or more class meetings. The time estimates are intentionally left blank, as the time needed will vary based on instructional decisions made. It may depend on:

whether students use the provided information about tents and sleeping bags or research this information.
whether the Tent Design Planning Sheet is provided or students organize their work with more autonomy.
expectations around drafting, revising, and the final product.
how student work is ultimately shared with the class (not at all, informally, or with formal presentations).
Note: Students will need to bring in a personal collection of 10–50 small objects ahead of time for the first lesson of the next unit. Examples include rocks, seashells, trading cards, or coins.

This video explains how cicadas emerge every 13 or 17 years, which are prime numbers.

Unit 5: Arithmetic in Base Ten
Lesson 10: Using Long Division

This lesson introduces students to long division. Students see that in long division the meaning of each digit is intimately tied to its place value, and that it is an efficient way to find quotients. In the partial quotients method, all numbers and their meaning are fully and explicitly written out. For example, to find 657/3 we write that there are at least 3 groups of 200, record a subtraction of 600, and show a difference of 57. In long division, instead of writing out all the digits, we rely on the position of any digit—of the quotient, of the number being subtracted, or of a difference—to convey its meaning, which simplifies the calculation.

In addition to making sense of long division and using it to calculate quotients, students also analyze some place-value errors commonly made in long division (MP3).

Unit 2: Introducing Ratios
Lesson 17: A Fermi Problem

This unit concludes with an opportunity for students to apply the reasoning developed so far to solve an unfamiliar, Fermi-type problem. Students must take a problem that is not well-posed and make assumptions and approximations to simplify the problem (MP4) so that it can be solved, which requires sense making and perseverance (MP1). To understand what the problem entails, students break down larger questions into more-manageable sub-questions. They need to make assumptions, plan an approach, and reason with the mathematics they know.

Unit 3: Unit Rates and Percentages
Lesson 2: Anchoring Units of Measurement

This lesson is optional. Students have worked with standard units of length since grade 2, and standard units of volume and mass since grade 3. This lesson is designed to anchor students’ perception of standard units of length, volume, weight, and mass with a collection of familiar objects that they can refer to in later lessons in preparation for using ratio reasoning to convert measurement units.

The main task of this lesson is a card-sorting activity in which students match common objects with their closest unit of length, volume, mass, or weight to establish anchor quantities for each unit of measurement. Since this lesson reinforces standards from previous grade levels instead of introducing grade 6 standards, if you believe that your students already have a firm grasp of these units of measurement, you may choose to skip this lesson.

Unit 3: Unit Rates and Percentages
Lesson 6: Interpreting Rates

In previous lessons students have calculated and worked with rates per 1. The purpose of this lesson is to introduce the two unit rates, a/b and b/a, associated with a ratio a:b. Each unit rate tells us how many of one quantity in the ratio there is per unit of the other quantity. An important goal is to give students the opportunity to see that both unit rates describe the same situation, but that one or the other might be preferable for answering a given question about the situation. Another goal is for students to recognize that they can just divide one number in a ratio by another to find a unit rate, rather than using a table or another representation as an intermediate step. The development of such fluency begins in this section and continues over time. In the Cooking Oatmeal activity, students have explicit opportunities to justify their reasoning and critique the reasoning of others (MP3).

Unit 3: Unit Rates and Percentages
Lesson 8: More about Constant Speed

This lesson allows students to practice working with equivalent ratios, tables that represent them, and associated unit rates in the familiar context of speed, time, and distance. Students use unit rates (speed or pace) and ratios (of time and distance) to find unknown quantities (e.g., given distances and times, find a constant speed or pace; and given a speed or pace, solve problems about distance and time).

Unit 3: Unit Rates and Percentages
Lesson 16: Finding the Percentage

While students have found percentages with easy numbers before now, in this lesson they will develop a general structure (MP7) that will work for any numbers.

Unit 4: Dividing Fractions
Lesson 2: Meanings of Division

In this lesson, students revisit the relationship between multiplication and division that they learned in prior grades. Specifically, students recall that we can think of multiplication as expressing the number of equal-size groups, and that we can find a product if we know the number of groups and the size of each group. They interpret division as a way of finding a missing factor, which can either be the number of groups, or the size of one group. They do so in the context of concrete situations and by using diagrams and equations to support their reasoning.

As they represent division situations with diagrams and equations and interpret division equations in context, students reason quantitatively and abstractly (MP2).

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 4: Dividing Fractions
Lesson 6: Using Diagrams to Find the Number of Groups

This is the second lesson in a series of three lessons exploring the “how many groups?” interpretation of division in situations involving fractions.
In the preceding lesson and in this one, the number of groups in each given situation is 1 or greater. In the next lesson, students find the number of groups that is less than 1 (“what fraction of a group?”).

Students have used different diagrams to represent multiplication and division. In this lesson, tape diagrams are spotlighted and used more explicitly. They are more abstract and more flexible than other representations students may have chosen for thinking about division problems that involve fractions. Because they use measurement along the length of the tape, tape diagrams are closer to the number line representation of fractions, and ultimately help students visualize division problems on the number line. (Students are not required to do that in this lesson, however.)

Students continue to make the journey from reasoning with concrete quantities to reasoning with abstract representations of fraction division (MP2).

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 5: Arithmetic in Base Ten
Lesson 2: Using Diagrams to Represent Addition and Subtraction

This lesson is optional. Prior to grade 6, students have added and subtracted decimals to the hundredths using a variety of methods, all of which focus on understanding place value. This lesson reinforces their understanding of place-value relationships in preparation for computing sums and differences of any decimals algorithmically.

In this lesson, students use two methods—base-ten diagrams and vertical calculations—to find the sum and differences of decimals. Central to both methods is an understanding about the meaning of each digit in the numbers and how the different digits are related. Students recall that we only add the values of two digits if they represent the same base-ten units. They also recall that when the value of a base-ten unit is 10 or more we can express it with a different unit that is 10 times higher in value. For example, 10 tens can be expressed as 1 hundred, and 12 hundredths can be expressed as 1 tenth and 2 hundredths. This idea is made explicit both in the diagrams and in vertical calculations.

Unit 5: Arithmetic in Base Ten
Lesson 7: Using Diagrams to Represent Multiplication

Students continue to use area diagrams to find products of decimals, while also beginning to generalize the process. They revisit two methods used to find products in earlier grades: decomposing a rectangle into sub-rectangles and finding the sum of their areas, and using the multiplication algorithm.

Students have previously seen that, in a rectangular area diagram, the side lengths can be decomposed by place value. For instance, in an 18 by 23 rectangle, the 18-unit side can be decomposed into 10 and 8 units (tens and ones), and the 23-unit side can be expressed as 20 and 3 (also tens and ones), creating four sub-rectangles whose areas constitute four partial products. The sum of these partial products is the product of 18 and 23. Students extend the same reasoning to represent and find products such as (1.8) x (2.3). Then, students explore how these partial products correspond to the numbers in the multiplication algorithm.

Students connect multiplication of decimals to that of whole numbers (MP7), look for correspondences between geometric diagrams and arithmetic calculations, and use these connections to calculate products of various decimals.

This Nrich activity can be printed or used interactively. Students will write down and discuss similarities and differences they see.

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

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.

Doing this Nrich problem is an excellent way to work at problem solving with learners. The problem lends itself to small group work, and provides an engaging context for pupils to use the skills of trial and error, and working systematically.

At the basic level, these Nrich challenges offer chances for children to practice number recognition, one-to-one correspondence and counting. However, some will begin to analyze and compare the three versions, explaining their findings and possibly drawing on ideas associated with probability.