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 8: Data Sets and Distributions
Lesson 12: Using Mean and MAD to Make Comparisons

In this lesson, students continue to develop their understanding of the mean and MAD as measures of center and spread as well as interpret these values in context. They practice computing the mean and the MAD for distributions; compare distributions with the same MAD but different means; and interpret the mean and MAD in the context of the data (MP2).

Unit 7: Rational Numbers
Lesson 5: Using Negative Numbers to Make Sense of Contexts

In this lesson, students are introduced to conventions for using signed numbers to represent money spent and received, as well as inventory gained and lost. While money contexts can be represented without signed numbers, there are many situations that are more efficiently modeled by signed numbers. For example, if a person has $50 in the bank and writes a $20 check, we can represent the balance as 50 - 20. If they had written an $80 check, we can still write the balance as 50-80, as long as we have adopted the convention that negative numbers represent what the person owes the bank (and assuming the bank allows overdrafts). Since students do not operate on signed numbers in this grade, this lesson is simply an introduction to the convention of using signed numbers to represent a change in money or a change in inventory, an important convention in modeling financial situations with mathematics (MP4). In a later lesson, students will be introduced to the idea of an account balance. In grade 7, students will study addition and subtraction of signed numbers and apply those concepts in accounting situations.

Unit 5: Arithmetic in Base Ten
Lesson 14: Using Operations on Decimals to Solve Problems

In this lesson, students apply their knowledge of operations on decimals to two sporting contexts. They analyze the distance between hurdles in a 110-meter hurdle race. In this situation, students use the given context to determine which arithmetic operations are relevant and use them to solve the problems. Additionally, they draw or use a diagram to help them make sense of the measurements, as well as to communicate their reasoning about the measurements (MP3). The numbers used in the problems reflect measurements that can be accurately measured on site, so the decimals can all be calculated by hand.

Unit 4: Dividing Fractions
Lesson 11: Using an Algorithm to Divide Fractions

In the previous lesson, students began to develop a general algorithm for dividing a fraction by a fraction. They complete that process in this lesson. Students calculate quotients using the steps they observed previously (i.e., to divide bya/b , we can multiply by a and divide by b), and compare them to quotients found by reasoning with a tape diagram. Through repeated reasoning, they notice that the two methods produce the same quotient and that the steps can be summed up as an algorithm: to divide by a/b, we multiply by b/a (MP8). As students use the algorithm to divide different numbers (whole numbers and fractions), they begin to see its flexibility and efficiency.

Unit 5: Arithmetic in Base Ten
Lesson 9: Using the Partial Quotients Method

Prior to grade 6, students reasoned about division of whole numbers and decimals to the hundredths in different ways. In this first lesson on division, they revisit two methods for finding quotients of whole numbers without remainder: using base-ten diagrams and using partial quotients. Reviewing these strategies reinforces students’ understanding of the underlying principles of base-ten division—which are based on the structure of place value, the properties of operations, and the relationship between multiplication and division—and paves the way for understanding the long division algorithm. Here, partial quotients are presented as vertical calculations, which also foreshadows long division.

In a previous unit, students revisited the two meanings of division—as finding the number of equal-size groups and finding the size of each group. Division is likewise interpreted in both ways here (MP2). When using base-ten diagrams or dividing by a small whole-number divisor, it is often natural to think about finding the size of each group. When using partial quotients, it may be more intuitive to think of division as finding the number of groups (e.g., 432 / 16 can be viewed as “how many 16s are in 432?”).

Visnos is an interactive math website that provides many different math tools to allow teachers to show math visually and allow students to explore with the tools.Visnos can be used with grades one-five as there are activities for telling time, fractions, multiplication tables, subtraction facts, angles, measurement using a protractor.

Unit 4: Dividing Fractions
Lesson 15: Volume of Prisms

In this lesson, students complete their understanding of why the method of multiplying the edge lengths works for finding the volume of a prism with fractional edge lengths, just as it did for prisms with whole-number edge lengths. They use this understanding to find the volume of rectangular prisms given the edge lengths, and to find unknown edge lengths given the volume and other edge lengths.

Problems about rectangles and triangles in the previous two lessons involved three quantities: length, width, and area; or base, height, and area. Problems in this lesson involve four quantities: length, width, height, and volume. So finding an unknown quantity might involve an extra step, for example, multiplying two known lengths first and then dividing the volume by this product, or dividing the volume twice, once by each known length.

In tackling problems with increasing complexity and less scaffolding, students must make sense of problems and persevere in solving them (MP1).

Unit 3: Unit Rates and Percentages
Lesson 10: What Are Percentages?

This lesson is the first of two that introduce students to percentages as a rate per 100 (MP6) and the ways they are used to describe different types of situations.

Percentages are commonly used in two ways:

To describe a part of a whole. For example, “Jada drank 25% of the bottle of water.” In this case, the percentage expressing the amount consumed is not bigger than 100% because it refers to a part of a whole.

To describe the size of one quantity as a percentage of another quantity. For example, “Jada drank 300% as much water as Diego did.” In this case, there is no restriction on the size of the percentage, because the percentage is describing a multiplicative comparison between two quantities.

In the first usage there is a single quantity and we are describing a part of it; in the second usage we are comparing two quantities. Students may have prior exposure to percentages, but are likely to have only encountered the first usage and might not be able to make sense of percentages above 100% or those used in comparative contexts. This lesson exposes students to both applications of percentages.

Money is the main context for exploring percentages in this lesson and the warm up asks students to convert between dollars and cents providing an opportunity for the teacher to assess students’ current abilities.

For the first several lessons exploring percentages, double number lines are the primary representation presented to students. This choice is intended to strongly communicate that we are working with percent rates, and that students can and should use all of the reasoning they have developed to deal with equivalent ratios and rates when dealing with rates per 100. That said, if students prefer to reason using tables or by multiplying or dividing by unit rates, they should not be discouraged from doing so.

This Nrich activity offers free exploration which will help learners develop a deep understanding of halves and halving. The task gives a context in which to discuss the importance of the part-whole relationship of fractions so that children realise halves can be different sizes, depending on the whole.

Unit 4: Dividing Fractions
Lesson 7: What Fraction of a Group?

In the previous three lessons, students explored the “how many groups?” interpretation of division. Their explorations included situations where the number of groups was a whole number or a mixed number. In this lesson, they extend the work to include cases where the number of groups is a fraction less than 1, that is, situations in which the total amount is smaller than the size of 1 group. In such situations, the question becomes “what fraction of a group?”.

Students notice that they can use the same reasoning strategies as in situations with a whole number of groups, because the structure
(number of groups) x (size of groups) = (total amount)
is the same as before (MP7). They write multiplication equations of this form and for the corresponding division equations.

Throughout the lesson, students practice attending to details (in diagrams, descriptions, or equations) about how the given quantities relate to the size of 1 group.

In this Nrich activity students look for connections in an array combining colors and shapes.

Unit 1: Scale Drawings
Lesson 1: What are Scaled Copies?

This lesson introduces students to the idea of a scaled copy of a picture or a figure. Students learn to distinguish scaled copies from those that are
not—first informally, and later, with increasing precision. They may start by saying that scaled copies have the same shape as the original figure, or that they do not appear to be distorted in any way, though they may have a different size. Next, they notice that the lengths of segments in a scaled copy vary from the lengths in the original figure in a uniform way. For instance, if a segment in a scaled copy is half the length of its counterpart in the original, then all other segments in the copy are also half the length of their original counterparts. Students work toward articulating the characteristics of scaled copies quantitatively (e.g., “all the segments are twice as long,” “all the lengths have shrunk by one third,” or “all the segments are one-fourth the size of the segments in the original”), articulating the relationships carefully (MP6) along the way.

The lesson is designed to be accessible to all students regardless of prior knowledge, and to encourage students to make sense of problems and persevere in solving them (MP1) from the very beginning of the course.

Unit: Area and Surface Area
Lesson 12: What is Surface Area?

This lesson introduces students to the concept of surface area. They use what they learned about area of rectangles to find the surface area of prisms with rectangular faces.

Students begin exploring surface area in concrete terms, by estimating and then calculating the number of square sticky notes it would take to cover a filing cabinet. Because students are not given specific techniques ahead of time, they need to make sense of the problem and persevere in solving it (MP1). The first activity is meant to be open and exploratory. In the second activity, they then learn that the surface area (in square units) is the number of unit squares it takes to cover all the surfaces of a three-dimensional figure without gaps or overlaps (MP6).

Later in the lesson, students use cubes to build rectangular prisms and then determine their surface areas.

At its most basic this Nrich task is an exercise in reading and recording information from a table. It also offers opportunities for children to do some elementary reasoning as they compare results with each other and work out why they differ.

Math can be solved through story problems. Most stories are divided into three acts. Learn about the 3 Act Math program.

The Nrich interactivity offers an ideal context in which to observe the "messy" randomness of results after a small number of experiments, and the predictability of results after a large number of trials. The problem also offers a good starting point for considering different probability distributions and their features, which could be followed up with the tasks Which List is Which and Data Matching.

Lonnie Johnson tried to create a new cooling system for refrigerators and air conditioners, but instead created the mechanics for one of the top twenty toys of all time, the Super Soaker. From childhood to adulthood, Lonnie had a love for rockets, robots, inventions, and a mind for creativity. He was driven toward innovation through his persistence and passion for problem solving, tinkering, and building. These traits served him well as we went on to work for NASA as an engineer. 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 use materials on hand to invent and design a new toy or game.

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

Emma Lilian Todd was a self-taught engineer who tackled one of the greatest challenges of the early 1900s: designing an airplane. As an adult, typing up patents at the U.S. Patent Office, Lilian built inventions in her mind, including many designs for flying machines. However, they all seemed too impractical. Lilian knew she could design one that worked. She took inspiration from both nature and her many failures, driving herself to perfect the design that would eventually successfully fly. 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: Design a new mode of transportation (air, sea, or ground) or select a current mode of transportation and improve it then use household items to create a prototype of your new or updated invention.

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