Unit 5: Arithmetic in Base Ten
Lesson 12: Dividing Decimals by Whole Numbers

This lesson serves two purposes. The first is to show that we can divide a decimal by a whole number the same way we divide two whole numbers. Students first represent a decimal dividend with base-ten diagrams. They see that, just like the units representing powers of 10, those for powers of 0.1 can also be divided into groups. They then divide using another method—partial quotients or long division—and notice that the principle of placing base-ten units into equal-size groups is likewise applicable.

The second is to uncover the idea that the value of a quotient does not change if both the divisor and dividend are multiplied by the same factor. Students begin exploring this idea in problems where the factor is a multiple of 10 (e.g. 8/1 = 80/10). This work prepares students to divide two decimals in the next lesson.

Unit 5: Arithmetic in Base Ten
Lesson 11: Dividing Numbers that Result in Decimals

So far, students have divided whole numbers that result in whole-number quotients. In the next three lessons, they work toward performing division in which the divisor, dividend, and quotient are decimals. In this lesson, they perform division of two whole numbers that result in a terminating decimal. Students divide using all three techniques introduced in this unit: base-ten diagrams, partial quotients, and long division. They apply this skill to calculate the (terminating) decimal expansion of some fractions.

Students analyze, explain, and critique various ways of reasoning about division (MP3).

This Nrich problem gives learners practice in fractions (tenths) as well as in addition and subtraction in a challenging form.

Unit 4: Dividing Fractions
Lesson 10: Dividing by Unit and Non-Unit Fractions

This is the first of two lessons in which students pull together the threads of reasoning from the previous six lessons to develop a general algorithm for dividing fractions. Students start by recalling the idea from grade 5 that dividing by a unit fraction has the same outcome as multiplying by the reciprocal of that unit fraction. They use tape diagrams to verify this.

Next, they use the same diagrams to look at the effects of dividing by non-unit fractions. Through repetition, they notice a pattern in the steps of their reasoning (MP8) and structure in the visual representation of these steps (MP7). Students see that division by a non-unit fraction can be thought of as having two steps: dividing by the unit fraction, and then dividing the result by the numerator of the fraction. In other words, to divide by 2/5 is equivalent to dividing by 1/5, and then again by 2. Because dividing by a unit fraction 1/5 is equivalent to multiplying by 5, we can evaluate division by 2/5 by multiplying by 5 and dividing by 2.

Students gain an understanding of the factors that affect wind turbine operation. Following the steps of the engineering design process, engineering teams use simple materials (cardboard and wooden dowels) to build and test their own turbine blade prototypes with the objective of maximizing electrical power output for a hypothetical situation—helping scientists power their electrical devices while doing research on a remote island. Teams explore how blade size, shape, weight and rotation interact to achieve maximal performance, and relate the power generated to energy consumed on a scale that is relevant to them in daily life. A PowerPoint® presentation, worksheet and post-activity test are provided.

This Nrich problem uses the idea of sequences in a very tangible form. Children will need to recognize odd and even numbers as well as be able to count fluently both backwards and forwards. They will also have opportunities to justify their answers.

Unit 8: Data Sets and Distributions
Lesson 4: Dot Plots

In this lesson, students continue to choose appropriate representation (MP5) to display categorical and numerical data, reason abstractly and quantitatively (MP2) by interpreting the displays in context, and study and comment on features of data distributions they show. Here they begin to use the everyday meaning of the word “typical” to describe a characteristic of a group. They are also introduced to the idea of using center and spread to describe distributions generally. Planted here are seeds for the idea that values near the center of the distribution can be considered “typical” in some sense. These concepts are explored informally at this stage but will be formalized over time, as students gain more experience in describing distributions and more exposure to different kinds of distributions.

The Nrich game as introduced is intended for children who are just beginning to become confident with small numbers. However there are many variations, some suggested below, that make it suitable for older children. As with many of the NRICH games, consolidation of basic number facts is combined with an element of strategic thinking.

This Nrich activity provides a context for doubling multiples of 5. The children will also be noticing and describing patterns.

This team-building task is designed to develop students' team-working skills. If you wish to learn more about these skills, and find other team-builder tasks, take a look at this article.

Unit fractions are the first fractions children meet, and here we discover some very surprising and interesting characteristics of these familiar numbers. Some of these characteristics were known to the ancient Egyptians whilst other conjectures are yet to be proved. Whilst meeting both old and new mathematical ideas, students can improve their fluency in addition and subtraction of fractions and be challenged to generalize and explain their findings.

The Fraction Equality Lab allows students to experiment with fractions by creating equivalent fractions. Different shapes are used as well as a number line. When the game is played students select a level and match equivalent fractions. The higher the level the more challenging it will be.

The topic of music can make a good connection between science and mathematics. The nature of sound and the working of the ear are rich areas of applied mathematics. The ratio emphasis follows from harmonics or overtones and rests on ideas like lowest common multiple.

Unit 6: Expressions and Equations
Lesson 8: Equal and Equivalent

In this lesson students are introduced to the idea of equivalent expressions. Two expressions are equivalent if they have the same value no matter what the value of the variable in them. Students use diagrams where the variable is represented by a generic length to decide if expressions are equivalent, and they show that expressions are not equivalent by giving values of the variable that make them unequal. They identify simple equivalent expressions using familiar facts about operations.

An interactive applet and associated web page that demonstrate the equation of a line in point-slope form. The user can move a slider that controls the slope, and can drag the point that defines the line. The graph changes accordingly and equation for the line is continuously recalculated with every slider and / or point move. The grid, axis pointers and coordinates can be turned on and off. The equation display can be turned off to permit class exercises and then turned back on the verify the answers. The applet can be printed as it appears on the screen to make handouts. The web page has a full description of the concept of the equation of a line in point - slope form, a worked example and has links to other pages relating to coordinate geometry. 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 6: Expressions and Equations
Lesson 15: Equivalent Exponential Expressions

In this lesson, students encounter expressions and equations with variables that also involve exponents. Students first evaluate expressions for given values of their variables. They learn that multiplication can be expressed without a dot or other symbol by placing a number, known as a coefficient, next to a variable or variable expression. In the next activity, students are presented with equations that contain a variable. They engage in MP7 by considering the structure of the equations and apply their understanding of exponents and operations to select a number from a list that, when replaced for the variable, makes the equation true. That number is a solution of the equation.

Unit 3: Unit Rates and Percentages
Lesson 7: Equivalent Ratios Have the Same Unit Rates

The purpose of this lesson is to make it explicit to students that equivalent ratios have the same unit rates. For instance, students can see that the ratios 10:4 15:6, and 20:8 all have unit rates of 2/5 and 5/2. Interpreted in a context, this might mean, for example, that no matter how many ounces of raisins are purchased in bulk and how much is paid, the price per ounce will always match the $0.40 per ounce rate marked on the price label.

This understanding gives new insights as students reason with tables. Up to this point, students have often been reasoning about the relationship from row to row, understanding that the rows contain equivalent ratios and the values in any row can be found by multiplying both quantities in another row by a scale factor. Here students see that they can also reason across columns, because the unit rate is the factor that relates the values in one column to those in the other (MP8). In grade 7, students will call the unit rate the constant of proportionality and write equations of the form y=kx to characterize these relationships.

Later in the lesson, students practice using unit rates and tables of equivalent ratios to find unknown quantities and compare rates in context.

Unit 6: Expressions and Equations
Lesson 14: Evaluating Expressions with Exponents

The focus of this lesson is evaluating expressions that have an exponent and one other operation by carrying out operations in the conventional order. This is accomplished through an example with surface area, where the context provides a clear reason for evaluating the exponential expression before performing the multiplication. Students practice evaluating numeric expressions that include exponents.

Students will be able to differentiate between an even or an odd number