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When teachers have a deep conceptual understanding of mathematics, it can help their students develop strong mathematical skills and knowledge.…

To begin the exploration of what topics in number and operations look like in a classroom at your grade level,…

Scroll through the pictures below to see our salad-bar plates and what the larva is eating at that time. See…

Scroll through the pictures below to see our ovipositrons. See the Mother Knows Best activity instructions for questions to consider.

Scroll through the days below to see our Brassicas and butterflies develop through their life cycles. Then, at the bottom of…

Click on one of the three growing systems in either of the pictures below to see a close-up of one…

Below are pictures of our Selective Herbivory systems in its initial state and at one and two days after introducing…

Watch the 30-minute video “Writing in Science.” Apply what you have learned in “Writing Across the Curriculum” as you watch…

Watch the 30-minute video “Writing Across the Curriculum.” If you prefer to watch the video in segments, you can stop…

Watch the 30-minute video “Modeling Revision.” Apply what you learned in “Learning to Revise” as you watch the extended video…

Watch the 30-minute video “Learning To Revise.” If you prefer to watch the video in segments, you can stop at…

Watch the 30-minute video “Peer Conferences.” Apply what you have learned in “Conversations Among Writing Peers” as you watch the…

Watch the 30-minute video “Conversations Among Writing Peers.” If you prefer to watch the video in segments, you can stop…

Watch the 30-minute video “Teacher-Student Conferences.” Apply what you have learned in “Conversations With Student Writers” as you watch the…

Watch the 30-minute video “Conversations With Student Writers.” If you prefer to watch the video in segments, you can stop…

Watch the 30-minute video “Teaching a Specific Writing Strategy.” Apply what you have learned in “Teaching the Writing Craft” as…

Watch the 30-minute video “Teaching the Writing Craft.” If you prefer to watch the video in segments, you can stop…

Watch the 30-minute video “Reading/Writing Connections.” Apply what you learned in “Reading Like a Writer” as you watch the extended…

Watch the 30-minute video “Reading Like a Writer.” If you prefer to watch the video in segments, you can stop…

Watch the 30-minute video “Fostering Choice and Independence.” Apply what you have learned in “Reasons for Writing” as you watch…

Watch the 30-minute video “Reasons for Writing.” If you prefer to watch the video in segments, you can stop at…

In This Part: How Square Can You Be? In Parts A and B, you confirmed that there is a strong positive…

Scroll through the pictures below to see a larger picture of one of our petri dishes. See the Now You…

Scroll through the pictures below to see enlargements of the aquatic habitat of our TerrAqua system. See the What Is…

Click on the links below to see pictures of the components we put into our TerrAqua system. See the Is It…

Watch the 30-minute video “Teacher as Writer.” Apply what you have learned in “Building a Community of Writers” as you…

Watch the 30-minute video “Building a Community of Writers.” If you prefer to watch the video in segments, you can…

Choose Activities In this section, you will apply what you have learned to your own teaching. The three activities are…

Extend Your Knowledge In this section, you will expand your understanding of connecting school and home literacy. You will compare…

Lecture In this section, you will watch Dr. Paratore’s lecture on connecting school and home literacy practices. Print Dr. Paratore’s…

Video Summary The video for this session examines the research and related practices for building positive home-school connections. Dr. Paratore…

Extend Your Knowledge In this section, you will expand your understanding of assessment practices that inform your instruction. You will…

Lecture In this section, you will watch Dr. Paratore’s lecture on using assessment to guide instruction. Print Dr. Paratore’s Assessment…

Video Summary The video for this session examines assessment practices that measure student performance and progress, and inform instructional decisions.…

Differentiate Instruction in a Class In this section, you will explore an activity to better understand how to group students…

Lecture In the session, you will watch Dr. Paratore’s lecture on differentiating instruction for early literacy. Print Dr. Paratore’s Differentiating…

Video Summary In the video for this session, Dr. Jeanne R. Paratore presents research-based principles that support the use of…

Extend Your Knowledge In this section, you will expand your understanding of writing instruction by comparing the ideas from the…

Video Summary The video for this session examines how to teach writing as a process. Dr. Jeanne R. Paratore begins…

Reading for Meaning In this section, you will explore an activity to better understand what factors children use to comprehend…

Lecture In this section, you will watch Dr. Paratore’s lecture on comprehension instruction in two parts. Print Dr. Paratore’s Comprehension…

Video Summary The video for this session examines comprehension and response to literature in grades K-2. In her lecture, Dr.…

Consider Other Points of View In this section, you will expand your understanding of word study by comparing the ideas…

Lecture In this section, you will watch Dr. Paratore’s lecture on Word Study and Fluency. Print Dr. Paratore’s Word Study…

Video Summary The video for this session examines word study and explicit fluency instruction in the primary grades. Dr. Jeanne…

Choose Activities In this section, you will apply what you have learned to your own teaching. The four activities are…

Examine Cueing Strategies In this section you will explore an activity to better understand reading as an English Language Learner.…

Lecture In this section, you will watch Dr. Gort’s lecture on reading instruction for English Language Learners in grades K-2.…

Video Summary In the video for this session, Dr. Mileidis Gort presents research-based principles that support English Language Learners’ literacy…

Extend Your Knowledge In this section, you will expand your understanding of a literate classroom community by comparing the ideas…

Lecture In this section, you will watch Dr. Paratore’s lecture on creating a literate community in early literacy. Print Dr.…

Video Summary In this video session, Dr. Jeanne R. Paratore presents three research-based principles for creating an effective literacy environment…

Each session of The Learning Classroom generally concentrates on a single learning theory and describes how it might be applied in a…

Lecture In this section, you will watch Dr. Paratore’s lecture on teaching writing as a process. Print Dr. Paratore’s Writing…

Classroom Videos Investigating Word Meaning Teacher: Angie Zapata Grade 3, Mathews School, Austin, TX Using a passage from Charlotte’s Web by E.…

We all agree that learning to read is a critical step in a child’s education. But teaching children to read…

All lessons are appropriate for K–12 teachers of any foreign language.

#10009 The focus of this photograph is a group of young children sitting tightly together on the floor in a dimly…

#2003 The color photograph is of a field of cotton. The photograph is divided to depict two different crops, which are…

#4015 The color photograph portrays a landfill area with colorful furniture and debris piled in the foreground. Framing the foreground scene…

#2573 A woman sits in profile on a stationary bicycle in the foreground of the image. Her profile orientation and placement…

#8047 The black and white photograph depicts a Mexican migrant family in front of a makeshift shelter in California. The photograph…

#5020 The black and white photograph depicts a group of young people sitting on the floor in the entry hall of…

#3520 The color photograph depicts four people standing at the edge of a grassy cliff. Large pieces of earth have broken…

#3038 In the black and white photograph, three young boys stand holding stacks of newspapers at their sides. It is nighttime,…

#6023 The digital-colorized image depicts a crater on the planet Mars and includes some of the surrounding area.

Woman holding a small child and a box of Wrigley’s Spearmint gum, in front of a building on Broadway in New…

Photograph shows men in morning coats and top hats standing at the side of Niagara Falls.

The Center for English Learning and Achievement (CELA) has made these two great resources available for teachers. Authored by Dr.…

These four case studies are interactivities based on actual scientific research projects carried out by leading teams in the field.…

This series of problems leads you through the creation of a histogram and its corresponding tables for the data in…

Introduction Purpose Now that you have explored the envisionment-building process and revisited the joy of reading literature, we invite you…

A Response-Based Approach to Reading Literature Judith A. Langer National Research Center on Literature Teaching and Learning University at Albany…

Explore two poems using strategies from the workshops. This activity offers you an opportunity to read two poems and reflect…

The following drafts allow you to evaluate authentic samples of student pieces written in three different genres. Each time you…

A Closer Look is a helpful video to view first, as it provides the basis for further study. This video features:…

Catherine Rohrbaugh teaches students how to take notes and analyze text across different content areas to foster independent learning. Teacher: Catherine…

Rubrics are an important tool for all teachers. They carefully explain the criteria that will be used to judge a…

When you assess or comment on student papers, what are the things that you look at? Do you concentrate on…

No one knows the anguish and joy of writing better than a writer. That’s why numbers of educational organizations believe…

Interviewer: What is your position and who do you work for? STUART: I am the Director of the Center for…

Interviewer: How did you first get involved in the ecosystem in science? BOB: When I was seven years old, and…

Interviewer: Tell me a bit about your position here. PENNY: My name is Penny Chisholm and I’m a professor in…

Interviewer: What is the history of The Global Monitoring Division (GMD)? PIETER: I’ll start further back a little bit. Mankind’s use…

Interviewer: I’d like to begin by asking you how you first got interested in weather phenomenon. Can you think back…

Interviewer: What regulates the average temperature of the Earth over billions of years? PAUL: This only really became a question in…

Interviewer: What makes Mars different from Earth? ANDY: If you look at a picture of Mars from one of the NASA…

Interviewer: Can you tell us who you are and what you do? WENDY: My name is Wendy Graham. I’m a professor…

Problem H1 Read the Grades 6-8 standard on data analysis and probability that was developed by the National Council of Teachers of…

As this course comes to a close and you reflect on ways to bring your new understanding of data analysis,…

The NCTM (2000) data analysis and probability standards state that students should “develop and evaluate inferences and predictions that are…

The National Council of Teachers of Mathematics (NCTM, 2000) identifies data analysis and probability as a strand in its Principles…

A data investigation should begin with a question about a real-world phenomenon that can be answered by collecting data. After…

This is the final session of the Data Analysis, Statistics, and Probability course! In this session, we will examine how statistical…

Problem H1 Read the Grades 3-5 standard on data analysis and probability that was developed by the National Council of Teachers of…

As this course comes to a close and you reflect on ways to bring your new understandings of data analysis,…

The National Council of Teachers of Mathematics (NCTM, 2000) identifies data analysis and probability as a strand in its Principles and…

Problem H1 Read the Grades Pre-K-2 standard on data analysis and probability that was developed by the National Council of Teachers of…

Problem H1 Many people wonder whether students need to know algebra in order to be successful in today’s world. What…

Problem H1 In the Achilles and the tortoise problems in Part C, Achilles runs at a constant rate of 9…

In mathematics, technology can be used to learn new concepts, apply known mathematics, avoid tedious computation, and gather information. It…

Linear functions have come up in many situations so far in this session. In this section, we’ll consolidate some of…

A rate describes how much one variable changes with respect to another. Rates are often used to describe relationships between…

Thinking About Slope Slope is an important concept in mathematics, and in Part B we’ll explore how it is used…

Finding the Pattern A function expresses a relationship between variables. For example, consider the number of toothpicks needed to make…

These homework problems will take you through the process of statistical estimation using numerical data. You will investigate the quality…

In This Part: Sample Size 20 All of our estimates thus far have been based on a sample size of 10…

In the Grades 6-8 curriculum, students are frequently asked to think about patterns, but often their “pattern sniffing” skills end…

At the 6-8 grade level, there are many problems that prepare students for their later work in algebra. There is…

Many students find that a context or situation helps them think about algebraic ideas. In the next problem, we’ll look…

The National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000) identifies algebra as a strand for…

To begin the exploration of what algebraic thinking looks like in a classroom at your grade level, watch a video…

In the grades 3-5 curriculum, students are frequently asked to think about patterns, but often their “pattern sniffing” skills end…

At the 3-5 grade level, there are many problems that prepare students for their later work in algebra. Experiences with…

In This Part: Using a Stem and Leaf Plot In Part B, you obtained several different estimates for the total number…

In the K-2 curriculum, students are frequently asked to think about patterns, but often their “pattern sniffing” skills end with…

At the K-2 grade level, there are many problems that prepare students for their later work in algebra. Experiences with…

In This Part: Fair Sampling You may have noticed that your estimates for the total penguin population vary quite a bit…

In This Part: Counting Penguins Statisticians often use a random sample to estimate characteristics of a population when the population is…

How might you solve equations in this new system? Note 7 Let’s start with the equation 3x = 8. Looking…

Problem H1 Use the table of sums for the sum of two dice to determine the more likely winner of…

Mathematicians have always been interested in solving equations. Over the past 150 years they have studied techniques for solving equations,…

One way to compare algorithms is to play the “Guess My Rule” game. Here’s how it goes: • Someone makes…

In This Part: Developing the Mathematical Probability Let’s return to the statistics question we considered in Part A: Can a person develop…

Students begin to move from arithmetic to algebra (in the structural sense) when they start thinking about properties of operations…

We said previously that algebra has become very much concerned with operations. So far, the only operations we’ve used are…

We’ve seen many different kinds of functions in the past few sessions. We’ve examined characteristics of functions, looked at their…

Splitting a Prize Imagine that the teachers in your school decide to play the lottery together. If they win, the…

Tides The following situation describes a new kind of function: Note 2 The ocean has high tides and low tides.…

You’ve already seen examples of exponential growth. When you have an exponential function where the constant multiple between outputs is…

Quadratic Functions and Differences You know that for linear functions, the difference between successive outputs is a constant. For exponential…

Square Numbers Now that we have a basic understanding of exponential functions, let’s move on to another kind of relationship.…

A Salary Situation Suppose you are given two different options for salary at a temporary job. Note 5 Plan A:…

Problems: Changing + to * In Session 5, we learned that linear functions have constant first differences. That is, every…

In the puzzles that follow, each shape stands for a number value of “weight.” To solve these puzzles, make the…

Here are 3 balanced scales. Use your intuition about balance to determine, for each case, how many blocks have to…

In This Part: Making a Tree Diagram A tree diagram is a helpful tool for determining theoretical or mathematical probabilities. Let’s…

Problem H1 Invent a number game like the one from Problem D1. Draw a network that lets you figure out…

Functions and Non-Functions So far you have been thinking about functions as algorithms or machines. They take an input —…

Problem D1 Draw a picture of this algorithm and use your picture to answer the following questions: If you…

About Function Machines When thinking about inputs and outputs, you are thinking about algorithms as functions: You input a number…

False Position The solutions to mathematical problems often involve writing and solving equations. We generally think of solving equations by…

Equality We have been using the equal sign throughout previous sessions. In this part we are going to look at…

In This Part: Predicting Outcomes You can use statistics to determine whether it’s possible for a player to develop skill in…

An algorithm is a recipe or a description of a mechanical set of steps for performing some task. For example,…

Problem H1 Sandy made some iced tea from a mix, using 12 tablespoons of mix and 20 cups of water.…

There are many situations, both in and outside of mathematics, where the process of doing and undoing helps you organize…

In Parts B and C we examined the proportionality of mixtures and scale models. In this activity, we will take…

In This Part: What Is Probability? Problem A1 Make a list of the topics and ideas that come to mind when…

Playing with Quadperson This activity compares the effects of absolute and relative comparisons on a picture. Think about what would…

Comparing Mixtures The scientists at the research lab for Whodunit Jeans are trying to decide on just the right shade…

Problem A1 Suppose there are two classes in a school, one with 20 students and one with 25. If the…

Is there an association between the length of your forearm (elbow to wrist) and the length of your feet? The…

Problem H1 Look at this table: Input Output 1 6 2 10 3 14 4 18 5 22 6 26…

Introduction to Qualitative Graphs Thinking algebraically helps us to develop different ways of representing real-world situations. You may have chosen…

At the beginning of this session, we learned that finding, describing, explaining, and predicting with patterns are some of the…

Problem D1 Here is a problem in which you can use patterns to make predictions. Count the number of blocks…

Eric the Sheep Here’s a simple number puzzle that leads to a surprisingly challenging investigation. Note 3 It’s a hot…

In the toothpick problem, we used a variable to describe the relationship between the number of toothpicks and the number…

Problem A5 describes several different ways to extend the pattern given in Problem A1. When a sequence of numbers is…

Finding a Pattern Patterns are everywhere around us. We use patterns to organize what we see and hear and to…

What does algebra mean to you? For many of us, the word “algebra” conjures classroom memories of xs and ys,…

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences. Problem H1 Assume…

In this part, you’ll look at several problems that are appropriate for students in grades 6-8. For each problem, answer…

Exploring Standards The National Council of Teachers of Mathematics (NCTM, 2000) has identified measurement as a strand in its Principles…

To begin exploring what the teaching of measurement might look like in the classroom, participants in the Measurement course first…

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences. Problem H1 Assume that…

In this part, you’ll look at several problems that are appropriate for students in grades 3-5. For each problem, answer…

The National Council of Teachers of Mathematics (NCTM, 2000) has identified measurement as a strand in its Principles and Standards…

In This Part: Trend Lines In Parts A and B, you confirmed that there is a strong positive association between height…

To begin to explore what the teaching of measurement might look like in the classroom, participants in the Measurement course…

In this part, you’ll look at several short activities that are appropriate for students in grades K-2. As you read…

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences.

Problem H1 Reptiles need the sun to get them going. Many large dinosaurs had to have large fins or plates…

As this course comes to a close and you reflect on ways to incorporate your new understanding of number and…

Imagine constructing an open-topped water tank from a square metal sheet (2 m by 2 m). You would cut squares…

Determining the Relationship In the previous part, you learned that the relationship between perimeter and area is dynamic; namely, the…

Constant Perimeter If you have 72 ft. of fencing and you want to use it to make a rectangular pen…

Problem H1 The shapes below are pyramids. A pyramid is named for the shape of its base. The left shape…

Here are scenarios from two different teachers’ classrooms, each involving students’ developing ideas about number and operations. Snippets of students’…

Cross Section Method In Part A we found that we can determine the volume of rectangular prisms or boxes by…

In Part A, you examined bivariate data — data on two variables — graphed on a scatter plot. Another useful…

Volume and Nets A net is the two-dimensional representation of a three-dimensional object. For example, you can cut the net…

In This Part: A Bivariate Data Question Have you ever wondered whether tall people have longer arms than short people? We’ll…

Problem H1 A circle is inscribed in a square. What percentage of the area of the square is inside the…

Transforming a Circle Let’s further examine the formula for area of a circle, A = • r2. How do we interpret…

Circumference Perimeter — or distance around — is a measurable property of simple, closed curves and shapes. When the figure…

The National Council of Teachers of Mathematics (NCTM, 2000) has identified number and operations as a strand in its Principles…

Problem H1 If the area of each of the smallest two triangles in the tangram below is equal to 1…

Similar Figures What happens to the area of a figure if we scale it up or down (i.e., enlarge or…

Subdividing Area Let’s further examine the concept of area, using a geoboard. The unit of area on the geoboard is…

Problem H1 Conduct your own comparative study. It can be an observational study or an experimental study. The questions below…

Measuring a Surface Think of some situations that involve the measurement of area. How would you determine these areas? For…

In This Part: Exploring Standards The National Council of Teachers of Mathematics (NCTM, 2000) has identified number and operations as a…

Problem H1 At a distance of 160 m from a tower, you look up at an angle of 23 degrees…

Measuring Steepness Since early times, surveyors, navigators, and astronomers have employed triangles to measure distances that could not be measured…

In This Part: Measuring Short-Term Recall Now we’ll move on to an investigation of comparative experimental studies. We’ll begin with a…

Measuring With Shadows What other methods exist for measuring indirectly? One such method is to use shadows. For tall objects…

Similar Triangles One way to measure indirectly is to use similar triangles. In similar figures, corresponding angles are congruent, and…

In This Part: Exploring Standards The National Council of Teachers of Mathematics has identified number and operations as a strand in…

Problem H1 Draw any quadrilateral. Then draw a point anywhere inside the quadrilateral, and connect that point to each of…

In Part C, we explore the interior and exterior angles of a figure as well as the relationship between the…

Classifying by Measure Print several copies of the sample polygons (PDF) and cut out the polygons. These are based on…

In This Part: A New Raisin Question Let’s begin our investigation of comparative observational studies by returning to the raisin problem…

Since the time of the ancient Greeks (about 500 B.C.E.), angles have been measured in terms of a circle. In…

Problem H1 The meter was originally based on the size of the Earth, with the distance from the equator to…

Many statistics problems require you to make comparisons. You may be interested in comparing salaries for different professions, for example,…

As we’ve seen, the base unit for length (or distance) is the meter. Meter comes from the Greek word “metron,”…

Problem H1 The number line shown below has seven points labeled with numbers or letters. The line is not drawn…

In This Part: The Fibonacci Sequence For the final activity in this session, we’ll look at an interesting application of ratios…

In This Part: Percent as Proportion The word “percent” means “out of 100.” See Note 6 below. For example, 49% means 49 out…

Problem H1 For the allocation below, find and interpret the MAD, variance, and standard deviation. You will…

In This Part: Area Model for Multiplication In the past, you may have learned particular algorithms for the multiplication and division…

Problem H1 Show how to use Cuisenaire Rods to model 3/4 2/3. Problem H2 a. The shaded part is 5 1/4. Specify…

Rational numbers or fractions can be used in many different ways. One source of confusion, especially with fractions, is the…

In This Part: Representing Fractions With Rods Rational numbers are a “ratio” of one value to another. It’s common to think…

In This Part: Interpretation of Fractions We know that a fraction, as a “rational” number, is a ratio of two numbers.…

Problem H1 a. Find the fractional equivalent for 0.142857. b. Find the fractional equivalent for 0.142857142857142857…. Problem H2 Shigeto and Consuela were…

There are several ways to compare fractions, many of which use benchmarks or intuitive methods and do not require computation…

History The metric system was introduced in France in the 1790s as a single, universally accepted system of measurement. But…

In Part A of this session, you learned that the decimal representation for every rational number was either a terminating…

In This Part: Terminating Decimals In Part A of this session, you’ll examine the process of converting fractions to decimals, which…

Problem H1 Prime numbers have exactly two factors. Now find some numbers that have exactly three factors. What do these…

In This Part: Locating Prime Numbers In this part, we’ll continue a mathematics tradition begun by Eratosthenes of Cyrene (276-194 B.C.E.)…

In This Part: Mean Absolute Deviation (MAD) We will now focus on how to measure variation from the mean within a…

Problem H1 Imagine that you are playing the Between game with a partner. Player A picks a decimal number between…

Measuring With Precision We have learned that physical measurement involves error and that every physical measurement is an approximation. This…

Ratio and Scale Measurement is the process of quantifying properties of an object by comparing them to some standard unit.…

In This Part: Tallying Excesses and Deficits In Part B, we looked at excesses and deficits when we moved coins in…

Conservation, Transitivity, and Unit Iteration In Session 1, we established that in order to measure something, we have to (1)…

How Large Is It? This table shows the height to the shoulder and the weight of several species of buffalo.…

What is measurement? Measurement is the process of quantifying the properties of an object by expressing them in terms of…

Print this PDF and cut out the small and medium triangles to fit into the shapes. Problem C1 Using two…

Measuring Surface Area You can measure a number of attributes of rocks — for example, surface area, volume, and weight.…

In This Part: Creating a Line Plot Remember the line plots you worked with in previous sessions? If you reshuffle your…

Measurement is used in all aspects of daily life, as well as in such fields as engineering, architecture, and medicine.…

In This Part: The Venn Diagram Model The numbers 24 and 36 have certain things in common, including many common factors…

Problem H1 The number abcabc (where each letter represents one particular digit) is divisible by 7, 11, and 13 for…

You saw that you could devise tests for such numbers as 6 and 18 based on their relatively prime factors.…

In This Part: Developing Testing Rules Another way to examine the characteristics of base ten numbers is to look at the…

“Alpha math” problems, where each letter stands for one digit of a number, can help you identify some of the…

In This Part: Fair and Unfair Allocations The average for a set of data corresponds to the equal-shares allocation or fair…

Problem H1 Explain the diagrams that illustrate 124 using the following models: a. The partitive (or equal groups) model b.…

In This Part: Addition In this section, we will use manipulatives to model addition, subtraction, multiplication, and division with positive…

In This Part: Multiplication with Manipulatives Models and manipulative materials can help make multiplication and division with integers more tangible. They…

In This Part: The Mean The term average is a popular one; it is often used, and often used incorrectly.…

In This Part: Addition We will begin our look at the various meanings for each of the operations and the laws…

Problem H1 Write the base five numbers 1234five and 1.234five as base ten numbers. Problem H2 Find the base ten…

Problem H1 Determine the Five-Number Summary for each of the remaining data sets of raisin counts from Session 2, and…

In This Part: Examining Base Four In Part C, we shift our focus to the base four number system. You will…

As you saw earlier in this session, one way to represent each power of two is to write the base…

In This Part: Base Two The number systems we’ve been looking at so far in this course use 10 digits…

Problem H1 Divide the number 1 by the numbers 1 through 10 consecutively. What conjectures can you make about rational…

In This Part: The Behavior of Zero The real number system is a positional system. In such a system, the…

Infinity is infinitely large, right? But is it possible for one infinite set to be larger than another? Infinity is…

In This Part: Relating Number Sets We will continue our focus on the number line and the relationships among the…

When the Third International Mathematics and Science Study (TIMSS) was conducted in 1996, more than 60 percent of American math…

Think back to the finite system of units digits that you explored in Part A and answer the following questions…

According to veteran teacher Steve Reinhart, students can sometimes be the best instructors. In an April 2000 article in Mathematics…

Overview: In this lesson, students will learn to model situations mathematically. Specifically, students will perform investigations on the growth of…

Overview: In this lesson, students will learn to use mathematical models to represent real life situations. In particular, they will…

After exploring a number system different from our own, we are now ready to begin examining the real number system.…

Mathematical Modeling Mathematical models are useful tools for engineers who study the effects of traffic on a bridge, telephone companies…

Teacher Peggy Lynn’s mathematics classroom probably looks different from many classrooms you have seen. That’s because Peggy teaches math in…

Educational philosopher John Dewey said: “What’s in a question, you ask? Everything. It is a way of evoking stimulating response…

Overview: This lesson teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of…

Overview: This lesson teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper…

Part 1: Direct Variation Direct variation is a critical topic in Algebra 1. A direct variation represents a specific case…

In This Part: Locating the Median from Ordered Data In Part D, we used noodles to help us visualize the concept…

This slogan appeared on a basketball T-shirt several years ago: “Just elevate and decide in the air.” The phrase could…

Affective refers to those actions that result from and are influenced by emotions. Consequently, the affective domain relates to emotions,…

How does the number system you’ve been working with in Part A relate to the real number system you usually…

In This Part: Addition in Units Digit Arithmetic Stated simply, a number system is a set of objects (often numbers),…

Overview: This lesson will teach students about several of the rules regarding exponents. It uses a situation from Alice in…

Overview: This lesson provides students with an introduction to exponential functions. The class first explores the world population since 1650.…

Exponents An exponent is mathematical shorthand that means a number is multiplied by itself repeatedly. An exponential function is a…

In This Part: Five-Number Summary with Measurement Data Now we’ll look at how you can represent the Five-Number Summary graphically, using…

Educator Marilyn Burns describes algebra as “the generalization of arithmetic.” The same definition is given in The Mathematics Dictionary (4th…

“High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations…” So begins…

Overview: In this lesson, students learn about basic recursion by exploring patterns in the data they generate from two simple…

Overview: This lesson will teach students to factor trinomial expressions of the form x2 + bx + c. Students will…

Part 1: Factoring Quadratic Expressions Factoring is the process of rewriting a number or expression as a product of two…

The educational community today is focusing more and more on assessment. High school exit examinations are being used to determine…

Let’s go back to your 12 ordered noodles, arranged from shortest to longest on a new piece of paper or…

In a collaborative classroom, students are willing to share their opinions. They are enthusiastic about discussing mathematics in the interest…

In This Part: The Median Another useful summary measure for a collection of data is the median. As you learned in Session…

People who have developed expertise in particular areas are naturally able to think effectively about problems in those areas. Understanding…

“Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.” The…

In This Part: The Data Set When working with a large collection of data, it can be difficult to keep an…

Manipulatives: An old Chinese proverb states: I hear, and I forget; I see, and I remember; I do, and I…

According to the U.S. Department of Education, if the population of children in this country were evenly distributed geographically, the…

Overview: This lesson helps students learn to find the feasible region in a linear programming problem, to graph a family…

Overview: In this lesson, students compare the speed at which they write with their left hand to the speed at…

Part 1: Systems of Equations A system of equations involves the relationship between two or more functions and can be…

Overview: In this lesson, students will learn how to write an equation of a linear function when given a set…

Overview: In this lesson students will recognize patterns and represent situations using algebraic notation and variables. Time Allotment: One 50-minute…

In This Part: Understanding the Question In Parts A-C of this session, you learned several ways to organize numerical data by…

Part 1: Translating Words Into Symbols Translating words into symbols is equivalent to modeling a situation using an equation and…

The frequency histogram and grouped frequency table for the 52 time estimates contain similar information to the stem and leaf…

In This Part: Constructing a Histogram Like the line plot we explored in Session 2, the stem and leaf plot is…

In This Part: How Long Is a Minute? Let’s begin with a problem you saw in Session 1, How Long Is a…

Problem H1 For the following data sets, create line plots, frequency tables, and cumulative frequency tables. Use your results to…

In This Part: Frequency Bar Graphs The line plot is a useful graph for examining small sets of data. It’s especially…

In This Part: From Ordered Lists and Line Plots A common way to summarize data is to use numerical summaries, many…

In This Part: Making a Table As you saw in Part A, a line plot is a graphical representation of data.…

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences. Problem H1 Interview a…

Geometric Reasoning Problems, Part 2 In this part, you’ll look at several problems that are appropriate for students in grades…

Introducing van Hiele Levels The National Council of Teachers of Mathematics (NCTM, 2000) identifies geometry as a strand in its…

This section features student work from Lisa Espinosa’s unit on representation featured in Workshop video 8. Samples include: One student’s…

This student work section includes three letters written by Laura Alvarez’s students in Workshop video 7. As a culminating project,…

In This Part: Counting Raisins Let’s begin with a recap of Problem B7 from Session 1. 1. Ask a Question How…

To answer a statistical question, we collect data, which consist of measurements of a variable. When the measurements we collect…

This section includes samples of student work from Workshop video 6, in which Stanlee Brimberg’s students explored the African Burial…

This children’s book about the 1963 March on Washington for Jobs and Freedom was researched, written, and illustrated by one…

This section features drafts of “Where I’m From” poems written by Kathryn Mitchell Pierce’s students in Workshop video 4. The…

Problem H1 We used a metric tape measure to measure the lengths of 20 pine needles to the nearest millimeter:…

This section includes two samples of student work from Workshop video 3, Sally Brownfield’s unit on Indian residential schools. Samples…

In This Part: Population and Sample In data analysis, we use graphs, tables, and numerical summaries to study the variation present…

In This Part: Testing Your Measurement Bias Measurement is never perfect, and we can always expect measurement errors in our data.…

This section features sample work from Workshop video 2, in which Akiko Morimoto’s students explored works by Judith Ortiz Cofer…

This section features sample student work from Carol O’Donnell’s unit on dual identities and multiple worlds in Workshop video 1.…

In This Part: Asking Questions and Collecting Data Let’s start our exploration of statistics by focusing on the first two steps…

The word statistics may bring to mind polls and surveys, or facts and figures in a newspaper article. But statistics…

The study of geometry in high school is often associated with proof — usually an axiomatic development with a focus…

Solutions are not provided for these homework problems, since answers will vary depending on individual experiences. Problem H1 Interview…

Geometric Reasoning Problems, Part 1 In this part, you’ll look at several problems that are appropriate for students in grades…

Kristen Ferrales explains how she intentionally designs her classroom to create an effective learning environment. Teacher: Kristin Ferrales School: The Urban Assembly…

Kristen Ferrales demonstrates how having students write to prepare for discussions creates a deeper understanding of the topic and leads…

Integrating Reading and Writing This final unit presents ways to integrate the reading and writing methods covered in the previous…

Amanda Westenberg discusses her technique of using student performance data to drive instruction. Teacher: Amanda Westenberg School: Rangeview High School, Aurora, Colorado…

Steve Lazar discusses how teaching his students the specific language of his discipline gives them the tools to better discuss…

Co-teachers Addie Male and Raeann McElveen discuss how to get students to develop questions that generate good classroom discussion. Teacher: Addie…

Co-teachers Addie Male and Raeann McElveen prepare students for a Socratic seminar by using multiple sources, student-generated questions, students’ personal…

Elisabeth Shanley’s students debate the successes of the Aztec and Inca civilizations using evidence from their research. Teacher: Elisabeth Shanley School: Parker…

Watch the Video As you watch “Routes to Culture,” take notes on Mr. Muirhead’s instructional strategies, particularly how he groups students…

Introduction The previous unit focused on practices that support the reading and analysis of text for developing interpretations. This unit…

The study of geometry can include both problem solving and connections to other areas of mathematics (arithmetic, algebra, etc.). Too…

Lili Velo explains the importance of learning and practicing academic language to help students interpret historical texts, especially in a…

I think using authentic materials is essential. My students were very excited at the idea that this was a real…

Michelle Brenner’s students analyze an excerpt from Mark Twain’s The Adventures of Huckleberry Finn by gathering evidence to support their point of…

In this part, you’ll look at some activities designed for students in grades K-2. As you read each activity, answer…

As you reflect on these questions, write down your responses or discuss them as a group. Before You Watch Respond…

I encourage communication between students as much as possible. If you’re asking a friend, Can you lend me a pencil?…

Lili Velo uses historical context to teach students how to analyze historical documents. Teacher: Miroslaba “Lili” Velo School: Tennyson High School, Hayward,…

One of the things I just love about teaching languages is that we can deal with math, science, geography, social…

In LeRoy Votto’s classroom, students read, annotate, write about, and discuss Wendell Phillips’s speech “The Philosophy of the Abolitionist Movement.”…

In the previous sessions, we explored geometry as a problem-solving process. You put yourself in the position of a mathematics…

Part of what I do as a foreign language teacher is try to open students up to what goes on…

I try to create activities that allow students to express likes and dislikes. This is most important in a second…

Ann Pember’s students discuss the meaning and role of the USA Patriot Act after reading an article from the New York…

Suggested Reading Van de Walle, John A. (2001). Geometric Thinking and Geometric Concepts. In Elementary and Middle School Mathematics: Teaching…

Slicing Solids A cross section is the face you get when you make one slice through an object. Below is…

I think it’s important that we touch upon history, science, math, and all the other curricula in our Spanish class…

A real challenge to a teacher at this level is to give kids new information and find ways for them…

Nets with Regular Polygons Imagine unfolding a cube so that all of its faces are laid out as a set…

Platonic solids have the following characteristics: All of the faces are congruent regular polygons. At each vertex, the same number…

Creative activities make the language the students’ own. It’s not the teacher’s language or the textbook language; it becomes the…

Jesse Gore teaches students how to analyze evidence from primary and secondary sources in order to defend ideas. Teacher: Jesse Gore…

Problem H1 Draw any triangle on your paper. Find the three midpoints and connect them in order. Explain why the…

Right Triangles Ratios The word “trigonometry” is enough to strike fear into the hearts of many high school students. But…

When they are able to connect something that happened 2,000 years ago with something that’s happening today, they become invested…

When I design a lesson, the first thing I plan is the outcome. What kinds of things do I want…

In this lesson on the Industrial Revolution, Andrea Gambino Rhodes assigns primary and secondary source material to student groups based…

I really believe that having student-centered learning is so much more powerful than the teacher standing in front of the…

It’s all about setting the stage and comprehensible input — hearing that wait time before I expect them to start…

Students are very curious about culture. Making connections to that culture and comparisons between theirs and ours is very important…

Darby Masland uses flexible grouping to foster understanding of texts and vocabulary. Teacher: Darby Masland School: The Urban Assembly Institute of Math…

It’s very important to teach in the target language. I keep the class in German as much as possible. I’m…

Andrea Gambino Rhodes discusses the ways in which she chooses primary sources to frame her lesson to engage students, such…

When you’re teaching high school students, you have to keep the intellectual level up so that it maintains their interest…

When you start speaking to students in a target language from kindergarten on, they’re really trained to take all of…

LeRoy Votto’s students discuss their thoughts on Wendell Phillips’s speech “The Philosophy of the Abolitionist Movement.” Teacher: LeRoy Votto School: The Urban…

When the students did this activity, they became the people in the city. Having built the buildings, they took a…

Introduction Welcome to the history/social studies content-specific units on disciplinary literacy. These next four units explore disciplinary literacy in history…

A mathematics teacher likes to astound her students with tricks that can be explained through mathematics. Before the class studies…

Students in Ben Berman’s creative writing class expanded their study of figurative language by reading and considering two poems about…

Finding a “Good Copy” Here is a picture of a panther: Problem A1 Which of these pictures is a…

Students have to speak French constantly to have a high degree of ability to speak the language. It has to…

The coordinate geometry that you first worked with in Session 6 is useful for describing transformations. Before looking at how…

Andrew Spinali meets with the 7th grade content area literacy team that he established among his colleagues. They talk about…

Introduction Brenda Green, executive director of the Center for Black Literature, ends her weekly radio show by saying: “The reader…

Translation Symmetry and Frieze Patterns A translation or slide involves moving a figure in a specific direction for a specific…

I like to do content-based lessons because I think the students enjoy it. It’s language that is meaningful to them.…

Andrew Spinali has students work on sequencing techniques to help them develop theme in their writing. Teacher: Andrew Spinali School: Parker Middle…

I think from experience, I have come to realize that, short of being in the country, a film portrays the…

Determining Rotation Symmetry If you can rotate (or turn) a figure around a center point by fewer than 360° and…

In Session 5, you saw three ways to move figures around: rotation, reflection, and translation. If you can move an…

Introduction One important way we study written texts in the English language is to reproduce and revise them for personal,…

Problem H1 Find the height of an equilateral triangle with the following side lengths: a. 1 centimeter b. 2 centimeters…

I think the most valuable lesson that my kids can take away from my room is that language learning is…

Finding Missing Lengths The Pythagorean theorem is useful for finding missing lengths. If you can find a right triangle in…

What Is a Theorem A theorem in mathematics is a proven fact. A theorem about right triangles must be true…

Language is a tool of communication. If I only ask students to do what’s on the sheet of paper, that…

Calculation Area Here is a square drawn on dot paper: Problem A1 Come up with a method to find the…

Kelly Johnson helps her students determine the theme of the book they have been reading—Breath, Eyes, Memory by Edwidge Danticat—by supporting…

Problem H1 You can make your own tangram set from construction paper. Start with a large square of construction paper…

The Midline Theorem In this part, we will begin to use segment notation. Optional: About Segment Notation (see below). Let’s…

A lot of the teaching at this age level starts and ends with songs. That is a way for children…

You can solve each of the following problems in more than one way. Try to find several solutions; then pick…

Exploring Tangrams A tangram is a seven-piece puzzle made from a square. A typical tangram set contains two large isosceles…

Introduction The Real World videos in the Reading and Writing in the Disciplines course support teachers’ daily efforts to effectively…

In middle school, we have to constantly think about engaging and motivating students. I could have just shown them pictures…

Problem H1 Lines p and q are parallel. Find the measures of all of the numbered angles. Explain how you…

Now that you’ve had some experience with dynamic geometry software, discuss or reflect on the following problems. Problem D1…

A circle is the set of all points in a plane that are equidistant from a given point in the…

Jennifer Roberts asks students to compare two characters from The Cask of Amontillado by Edgar Allan Poe using evidence they have gathered…

Properties of Angles Parallel lines are two lines in the same plane that never intersect. Another way to think about…

The things that they do in class — activities, language tasks —have to be from the student, from whatever…

Dynamic geometry software like Geometer’s Sketchpad allows you to construct and manipulate geometric figures. Such software can help you discover…

Ben Berman discusses unveiling the craft of writing by teaching students how to give and receive feedback on creative pursuits.…

A quadrilateral is a polygon that has exactly four sides. Here are some definitions of special types of quadrilaterals: A…

It’s very helpful, I think, being a second-language learner myself. You know the process. You know what frustration you…

Dan Guerrero challenges his students to construct arguments using academic language and terms. Teacher: Dan Guerrero School: Tennyson High School, Hayward, CA…

I expect them to learn the culture. I expect them to appreciate the language and the culture foremost. That’s the…

Kelly Johnson discusses strategies she uses to get her students to read, discuss, and write about the emerging theme as…

Wendy Barrales has a one-on-one conference with a 6th grader who needs help applying strategies to define an unknown word.…

Definitions Definitions and proof both play essential roles in mathematics. Definition is crucial, because you need to know what something…

Polygons can be divided into groups according to certain properties. Concave polygons look like they are collapsed or have one…

Identifying Polygons Polygons are two-dimensional geometric figures with these characteristics: They are made of straight line segments. Each segment touches…

Around 95 percent of the time I speak Arabic. I feel it’s really important to speak in Arabic all the…

Problem H1 More than one feature can be combined into a triangle. Decide which of the following combinations are possible.…

Jenee Ramos engages students in reading and writing about topics that are relevant to their lives. Teacher: Jenee Ramos School: Brookline High…

In Part C, you will apply what you’ve learned about the properties of figures to a construction task. Problem…

The following education experts appear in this video: Dale Allender is an assistant professor at California State University in Sacramento, where…

By listening to them—when they are participating, when they are talking to each other—it shows me, did they understand…

A triangle has three sides, but not just any set of three lengths will make a triangle. Use linkage-strips to…

Introduction The English language arts classroom has always been the primary place where students are taught academic literacy practices. These…

Amanda Westenberg uses grouping techniques, peer revision, self-revision, and teacher feedback to address the skills needed to write a document-based…

In the following problems, you will draw a variety of triangles and take note of their differences. Note 2 Drawing Triangles…

That’s the ideal, that kids are teaching each other. When that happens, it’s awesome.… I’ll put this student who’s…

Amy Miles builds students’ close reading skills and creates an opportunity to dig into complex text to foster deeper understanding…

My classroom is very intentionally filled with tables and not desks because I feel that tables promote conversation, and…

Students learn how to apply scientific reasoning to new problems and then communicate about it with peers. Teacher: Mary Murphy School: The…

Integrating Reading and Writing The previous units highlighted ways in which reading and writing can be integrated into different aspects…

The point of the activity is to show them how to negotiate. So it is not just like, “Do you…

Jenee Ramos uses a writing workshop approach to peer revision to help students practice their writing techniques across different genres.…

Jane Cunningham engages her students with a journalism podcast project in which they create their own podcasts about a subject…

Jane Cunningham and her colleague Mike McSweeney discuss the creation of an English unit on journalism, which promotes skills such…

I cannot call it learning a language if there is no authentic opportunity for students. If you don’t use…

Students use models to explain how the relative positions of the Sun, Moon, and Earth determine Moon phases, eclipses, and…

Amy Miles uses a “power writing” exercise to build writing skills and reinforce background knowledge. Teacher: Amy Miles School: Health Sciences High…

Dr. Amanda Micsenyi raises the bar for a particularly eager student, challenging him to read additional articles about HIV/AIDS and…

Dr. Amanda Micsenyi leads her ninth grade class in a literature study of the timeline of HIV/AIDS. Students read and…

The following education experts appear in this video: Meena Balgopal is an associate professor at Colorado State University in the School…

Mary Murphy uses an engaging narrative to help students understand protein synthesis. Teacher: Mary Murphy School: The Urban School of San Francisco…

Lynn Gilbert helps students create a graphic organizer for multiple resources and various types of texts; her ultimate goal is…

Video Summary In “Engaging With Communities,” Patsy Lightbown, Professor Emeritus of Concordia University in Montreal, discusses ways of providing students…

Jennifer Roberts discusses the ways in which she incorporates the use of computers into her classroom. Teacher: Jennifer Roberts School: Point Loma…

Dan Guerrero demonstrates how he uses anecdotal evidence as a way to engage others in a discussion of ethics and…

Video Summary In “Planning for Assessment,” Understanding by Design education consultant Allison Zmuda describes the components of a good assessment…

Darby Masland explains how a class-wide focus on standards helps students meet learning goals more effectively. Teacher: Darby Masland School: The Urban…

Chemistry teacher Martin Berryman shows how to help students write strong claims based on evidence, reason, and the interpretation of…

Video Summary In “Valuing Diversity in Learners,” Professor Marjorie Hall Haley of George Mason University in Virginia talks about ways…

Amy Sheck leads students as they read and discuss a scientific paper on biofilms, design and propose an experiment, and…

Video Summary In “Rooted in Culture,” Professor Alvino Fantini from the School for International Training in Brattleboro, Vermont, discusses the…

Introduction Disciplinary literacy in science is challenging and highly rewarding. It involves more than what we traditionally think of as…

Leon Young explains how he integrates technology into the classroom to support blended learning. Teacher: Leon Young School: REALM Charter School, Berkeley,…

Video Summary In “Subjects Matter,” Patsy Lightbown, Professor Emeritus of Concordia University in Montreal, addresses the balance between form and…

Leon Young uses technology in the classroom to facilitate communication among peers and to support differentiated instruction of small learning…

Derek Boyd explains how he differentiates activities to teach specific mathematical concepts. Teacher: Derek Boyd School: MetWest High School, Oakland, CA Grade: 9…

Video Summary In “Delivering the Message,” Professor Paul Kei Matsuda of the University of New Hampshire addresses how a consideration…

The following education experts appear in this video: Jacob Foster is the director of science and technology/engineering at the Massachusetts Department…

Learning Goals How do you organize an effective written or oral presentational task that has students focus on a particular…

Introduction You have now had a number of opportunities to consider what it looks like to incorporate the disciplinary literacy…

Students in Stephanie Brown’s class work together in teams to problem solve and develop a system of equations for a…

Sarah Langer engages students with a multistep group project that uses their understanding of graph theory. Students work together to…

Laura Mourino discusses her commitment to engaging students in a variety of tasks by acknowledging individuality in learning and making…

Laura Mourino assesses formative understanding by having her 10th graders write letters answering four key questions about a unit circle.…

Derek Boyd shows how reinforcing vocabulary helps students understand and retain content. Teacher: Derek Boyd School: MetWest High School, Oakland, CA Grade: 9…

Video Summary In “Person to Person,” Professor Joan Kelly Hall of Pennsylvania State University addresses the importance of the talk…

Kelly Gay gives students an opportunity to explain their thinking, or a line of learning, about mathematical word problems as…

Stephanie Brown’s students first work individually and then in groups to construct viable mathematical arguments by critiquing each other’s reasoning.…

Kim Dinh teaches students how to determine important information to solve mathematical word problems. Teacher: Kim Dinh School: Health Sciences High and…