Unit 7: Rational Numbers
Lesson 17: Common Multiples
In this lesson, students use contextual situations to learn about common multiples and the least common multiples of two whole numbers. They develop strategies for finding common multiples and least common multiples.
Unit 8: Data Sets and Distribution
Lesson 14: Comparing Mean and Median
In this lesson, students investigate whether the mean or the median is a more appropriate measure of the center of a distribution in a given context. They learn that when the distribution is symmetrical, the mean and median have similar values. When a distribution is not symmetrical, however, the mean is often greatly influenced by values that are far from the majority of the data points (even if there is only one unusual value). In this case, the median may be a better choice.
At this point, students may not yet fully understand that the choice of measures of center is not entirely black and white, or that the choice should always be interpreted in the context of the problem (MP2) and should hinge on what insights we seek or questions we would like to answer. This is acceptable at this stage. In upcoming lessons, they will have more opportunities to include these considerations into their decisions about measures of center.
Unit 7: Rational Numbers
Lesson 3: Comparing Positive and Negative Numbers
Returning to the temperature context, students compare rational numbers representing temperatures and learn to write inequality statements that include negative numbers. Students then consider rational numbers in all forms (fractions, decimals) and learn to compare them by plotting on a number line and considering their relative positions. Students abstract from “hotter” and “colder” to “greater” and “less,” so if a number a is to the right of a number b, we can write the inequality statements a>b and $b\text-100$. Students are briefly introduced to the word sign (i.e., algebraic sign) since it is often used to talk about whether numbers are positive or negative. Students use the structure of the number line to reason about relationships between numbers (MP7).
Unit 2: Introducing Ratios
Lesson 10: Comparing Situations by Examining Ratios
In previous lessons, students learned that if two situations involve equivalent ratios, we can say that the situations are described by the same rate. In this lesson, students compare ratios to see if two situations in familiar contexts involve the same rate. The contexts and questions are:
Two people run different distances in the same amount of time. Do they run at the same speed?
Two people pay different amounts for different numbers of concert tickets. Do they pay the same cost per ticket?
Two recipes for a drink are given. Do they taste the same?
In each case, the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating how-many-per-one and then scaling. The reason for this is to reinforce the concept that equivalent ratios describe the same rate, before formally introducing the notion of unit rate and methods for calculating it. However, students can use any method. Regardless of their chosen approach, students need to be able to explain their reasoning (MP3) in the context of the problem.
Unit 3: Unit Rates and Percentages
Lesson 5: Comparing Speeds and Prices
Previously, students found and used rates per 1 to solve problems in a context. This lesson is still about contexts, but it's more deliberately working toward the general understanding that when two ratios are associated with the same rate per 1, then they are equivalent ratios. Therefore, to determine whether two ratios are equivalent, it is useful to find and compare their associated rates per 1. In this lesson, we also want students to start to notice that dividing one of the quantities in a ratio by the other is an efficient way to find a rate per 1, while attending to the meaning of that number in the context (MP2).
Calculating rates per 1 is also a common way to compare rates in different situations. For example, suppose we find that one car is traveling 30 miles per hour and another car is traveling 40 miles per hour. The different rates tell us not only that the cars are traveling at different speeds, but which one is traveling faster. Similarly, knowing that one grocery store charges $1.50 per item while another charges $1.25 for the same item allows us to select the better deal even when the stores express the costs with rates such as “2 for $3” or “4 for $5.”
An interactive applet and associated web page that demonstrate the concept of complementary angles (angles that add to 90 degrees). The applet shows two angles. You can drag the endpoints of each angle and the other angle changes so that they always add to 90 degrees. They are drawn in such a way that it is visually obvious that together they form a right angle, although they are separate on the page. The angle measure readouts can be turned off for class discussions. 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.
The Starfall Compose & Decompose numbers game is a great option for practicing adding, place value for ones and tens, and number identification in a steady, consistent format. This game is a great option for independent center practice, whole group instruction or small group instruction.
An interactive applet and associated web page that demonstrate the concept of congruent angles. Three angles are shown which always remain congruent as you drag any defining point on any angle. They all change together. This is designed to demonstrate that the angles are considered congruent even if they are in different orientations and the line segments making them up are different lengths. 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.
An interactive applet and associated web page that demonstrate the congruence of polygons. The applet presents nine polygons that are in fact congruent, but don't look it because they are reflected and rotated in various ways. If you click on one, it rotates and flips as needed, then slides over the top of another to show it is congruent. The web page describes how to determine if two polygons are congruent. 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.
An interactive applet and associated web page that demonstrate the concept of congruent triangles. Applets show that triangles a re congruent if the are the same, rotated, or reflected. In each case the user can drag one triangle and see how another triangle changes to remain congruent to it. The web page describes all this and has links to other related pages. 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 2: Introducing Ratios
Lesson 9: Constant Speed
In the previous lesson, students used the context of shopping to explore how equivalent ratios and ratios involving one can be used to find unknown amounts. In this lesson, they revisit these ideas in a new context—constant speed—and through concrete experiences. Students measure the time it takes them to travel a predetermined distance—first by moving slowly, then quickly—and use it to calculate and compare the speed they traveled in meters per second.
Here, double number lines are used to represent the association between distance and time, and to convey the idea of constant speed as a set of equivalent ratios (e.g., 10 meters traveled in 20 seconds at a constant speed means that 0.5 meters is traveled in 1 second, and 5 meters is traveled in 10 seconds). Students come to understand that, like price, speed can be described using the terms per and at this rate.
The idea of a constant speed relating the quantities of distance and time is foundational for the later, more abstract idea of a constant rate, and is important in the development of students’ ability to reason abstractly about quantities (MP2).
Unit 7: Rational Numbers
Lesson 12: Constructing the Coordinate Plane
In this lesson, students explore the idea of scaling axes appropriately to accommodate data where coordinates are rational numbers. Students attend to precision as they plan where to place axes on a grid and how to scale them to represent data clearly (MP6). In an optional activity, students practice working with coordinates in all 4 quadrants as they navigate a maze on a coordinate grid. This lesson gives students the opportunity to develop fluency with plotting coordinates in all 4 quadrants and scaling axes to fit data that is essential for the context-driven work over the next few lessons.
Unit 3: Unit Rates and Percentages
Lesson 4: Converting Units
In grade 4, students began converting units of measurements by multiplying. The work in grade 5 expanded to include conversion by dividing, but was still restricted to units within the same measurement system. In this lesson, students progress to converting units that may be in different systems of measurement, using ratio reasoning and recently-learned strategies such as double number lines, tables, and multiplication or division of unit rates.
Unit 1: Scale Drawings
Lesson 2: Corresponding Parts and Scale Factors
This lesson develops the vocabulary for talking about scaling and scaled copies more precisely (MP6), and identifying the structures in common between two figures (MP7).
Specifically, students learn to use the term corresponding to refer to a pair of points, segments, or angles in two figures that are scaled copies. Students also begin to describe the numerical relationship between the corresponding lengths in two figures using a scale factor. They see that when two figures are scaled copies of one another, the same scale factor relates their corresponding lengths. They practice identifying scale factors.
A look at the angles of scaled copies also begins here. Students use tracing paper to trace and compare angles in an original figure and its copies. They observe that in scaled copies the measures of corresponding angles are equal.
Countdown Fractions in this Nrich game offers a motivating context in which to practice calculating with fractions. There is usually more than one way of hitting the target, which offers an opportunity for rich discussion on the merits of alternative methods.
This Nrich activity, in line with the theme for this month, offers an 'action' to perform on a group of numbers which pupils can continue and explore. Or, you could think of the writing down of the 'description' of a sequence as an action performed on that sequence. It might particularly appeal to those pupils who enjoy number work but who are perhaps not used to succeeding in this area.
It’s easy to take today’s ubiquitous colored crayons for granted, but they were the result of one individual’s innovation. Biebow introduces Edwin Binney—a mustachioed man and head of a carbon black factory—who wished to make color-pigmented wax crayons that reflected the world outside. 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 school/community/world in which you live and identify a problem that could be solved with a new invention.
A document is included in the resources folder that lists the complete standards-alignment for this book activity.
Unit 2: Introducing Ratios
Lesson 7: Creating Double Number Line Diagrams
In this lesson, students create double number line diagrams from scratch. They see that it is important to use parallel lines, equally-spaced tick marks, and descriptive labels. They are also introduced to using the word "per" to refer to how much of one quantity there is for every one unit of the other quantity.
Double number lines are included in the first few activity statements to help students find an equivalent ratio involving one item or one unit. In later activities and lessons, students make their own strategic choice of an appropriate representation to support their reasoning (MP5). Regardless of method, students indicate the units that go with the numbers in a ratio, in both verbal statements and diagrams.
Note that students are not expected to use or understand the term "unit rate" in this lesson.
This Nrich interactive environment can be used with students, perhaps on an interactive whiteboard, for many different purposes. These might include, for example, exploring ideas associated with factors and multiples, or addition and subtraction. It is also an ideal context in which to investigate fractions and ratio, or to look at finding combinations.
This Nrich problem is intentionally difficult as it stands, but this allows us to focus on promoting resilience and perseverance with children.