Unit 7: Rational Numbers
Lesson 13: Interpreting Points on a Coordinate Plane

This lesson pays particular attention to choices about what axes represent and the scale used on each axis. Graphs need to present information clearly and legibly to be useful for visualizing relationships between quantities. Students learn to make these choices purposefully when plotting points and to consider the decisions that have been made when reading and interpreting the coordinates of points from a graph. They interpret and label axes appropriately to clearly communicate their correspondence with the quantities in a problem. They reason abstractly and quantitatively as they interpret vertical distance in a coordinate plane in context (MP2).

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 8: Data Sets and Distributions
Lesson 9: Interpreting the Mean as Fair Share

In this lesson, students find and interpret the mean of a distribution (MP2) as the amount each member of the group would get if everything is distributed equally. This is sometimes called the “leveling out” or the “fair share” interpretation of the mean. For a quantity that cannot actually be redistributed, like the weights of the dogs in a group, this interpretation translates into a thought experiment.

Suppose all of the dogs in a group had different weights and their combined weight was 200 pounds. The mean would be the weight of the dogs if all the dogs were replaced with the same number of identical dogs and the total weight was still 200 pounds.

Here students do not yet make an explicit connection between the mean and the idea of “typical,” or between the mean and the center of a distribution. These connections will be made in upcoming lessons.

This Nrich problem challenges children to calculate with fractions and provides a good context in which to encourage learners to be curious about different methods of approach.

Unit 2: Introducing Ratios
Lesson 6: Introducing Double Number Line Diagrams

This lesson introduces the double number line diagram, a useful, efficient, and sophisticated tool for reasoning about equivalent ratios.

The lines in a double number line diagram are similar to the number lines students have seen in earlier grades in that:

Numbers correspond to distances on the line (so that the distance between, say, 0 and 12 is three times the distance between 0 and 4);
We can choose what scale to use (e.g., whether each interval represents 1 unit, 2 units, 5 units, etc.);
The lines can be extended as needed.
In a double number line diagram we use two parallel number lines—one line for each quantity in the ratio—and choose a scale on each line so equivalent ratios line up vertically.

For example, if the ratio of number of eggs to cups of milk in a recipe is 4 to 1, we can draw a number line for the number of eggs and one for the cups of milk. On the number lines, the quantity of 4 for the number of eggs and the 1 for cups of milk would line up vertically, as would 8 eggs and 2 cups of milk, and so on.

Because they represent quantities with length on a number line rather than with counts of objects, double number lines are both more abstract and more general than discrete diagrams. Later in this unit, students will learn an even more abstract representation of equivalent ratios—the table of values. Connecting the concrete to the abstract helps students connect quantitative reasoning to abstract reasoning (MP2). Though some activities are designed to hone students’ facility with particular representations, students should continue to have autonomy in choosing representations to solve problems (MP5), as long as they can explain their meaning (MP3).

Unit 2: Introducing Ratios
Lesson 1: Introducing Ratios and Ratio Language

In this lesson, students use collections of objects to make sense of and use ratio language. Students see that there are several different ways to describe a situation using ratio language. For example, if we have 12 squares and 4 circles, we can say the ratio of squares to circles is 12:4 and the ratio of circles to squares is 4 to 12. We may also see a structure that prompts us to regroup them and say that there are 6 squares for every 2 circles, or 3 squares for every one circle (MP7).

Expressing associations of quantities in a context—as students will be doing in this lesson—requires students to use ratio language with care (MP6). Making groups of physical objects that correspond with “for every” language is a concrete way for students to make sense of the problem (MP1).

It is important that in this first lesson students have physical objects they can move around. Later, they will draw diagrams that reflect the same structures and learn to reason with and interpret abstract representations like double number line diagrams and tables. Working with objects that can be physically rearranged in the beginning of the unit can help students make sense of increasingly abstract representations they will encounter as the unit progresses. Students will continue to develop ratio language throughout the unit and will learn about equivalent ratios in a future lesson.

Introduction to the Modeling and Analysis of Complex Systems introduces students to mathematical/computational modeling and analysis developed in the emerging interdisciplinary field of Complex Systems Science. Complex systems are systems made of a large number of microscopic components interacting with each other in nontrivial ways. Many real-world systems can be understood as complex systems, where critically important information resides in the relationships between the parts and not necessarily within the parts themselves. This textbook offers an accessible yet technically-oriented introduction to the modeling and analysis of complex systems. The topics covered include: fundamentals of modeling, basics of dynamical systems, discrete-time models, continuous-time models, bifurcations, chaos, cellular automata, continuous field models, static networks, dynamic networks, and agent-based models. Most of these topics are discussed in two chapters, one focusing on computational modeling and the other on mathematical analysis. This unique approach provides a comprehensive view of related concepts and techniques, and allows readers and instructors to flexibly choose relevant materials based on their objectives and needs. Python sample codes are provided for each modeling example.

This Nrich problem uses the context of sports training to offer opportunities for learners to explore division and/or multiplication. Pupils will be required to consider the relationships between multiplication, division and fractions, which will help reveal their level of understanding.

This is the first Nrich problem in a set of three linked activities. Egyptian Fractions and The Greedy Algorithm follow on.

It's often difficult to find interesting contexts to consolidate addition and subtraction of fractions. This problem offers that, whilst also requiring students to develop and analyze different strategies and explain their findings.

Kids Math has free interactive learning activities, math games, facts, printable worksheets, quizzes, videos, and other fun resources that will keep students engaged while learning.

Teach K-2 students the importance of labeling their answers when solving a number story.

This Nrich activity encourages students to notice and wonder and to engage in mathematical discussions. Students then use graphs to represent the situations.

This Nrich problem encourages children to work together to develop a method for finding a solution which will always work. It will also help to reinforce understanding of odd and even numbers.

Meg set out to climb up and investigate the rain forest tree canopies — and to be the first scientist to do so. But she encountered challenge after challenge. Male teachers would not let her into their classrooms, the high canopy was difficult to get to, and worst of all, people were logging and clearing the forests. Meg never gave up or gave in. She studied, invented, and persevered, not only creating a future for herself as a scientist, but making sure that the rainforests had a future as well. 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: Your community has many different areas to explore - it might be a park, a grocery store, a forest, or an alley. For some people, it might be difficult to explore these areas because they may have differing abilities. Select one area in your community, and come up with a plan to build a way for it to be more accessible to everyone.

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

Students will be able to learn about tangrams and basic shapes through a read-aloud of Grandfather Tang’s Story, which describes various animals being made with Tangrams, understand how they make shapes by reading the book Tricky Tangrams, and then try making tangrams themselves using a game on digipuzzle.net.

This is an opportunity for pupils to set up their own models using a spreadsheet and investigate what happens when they change variables such as interest, inflation rates and how much is spent each year. Investigating how a small change in interest rates can affect the total income over different time periods can be enlightening. Investigating the impact of spending large amounts of money up-front can also provide valuable insights.

In this Nrich activity, students use manipulatives to investigate triangles. They are asked to construct possible triangles and to search for an impossible combination.

An interactive applet that allows the user to graphically explore the properties of a linear functions. Specifically, it is designed to foster an intuitive understanding of the effects of changing the two coefficients in the function y=ax+b. The applet shows a large graph of a quadratic (ax + b) and has two slider controls, one each for the coefficients a and b. As the sliders are moved, the graph is redrawn in real time illustrating the effects of these variations. 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.