Money is a useful model for helping students make sense of tenths and hundredths, but students often have difficulty extending their knowledge to make sense of thousandths and ten thousandths. The Lobster Problem presents students with a problem-solving experience that helps them learn about extending decimals beyond hundredths and provides them practice with identifying decimals…

I drew a large Venn diagram on the chalkboard and labeled each of the three intersecting circles: Has taken musical instrument lessons, Doesn’t like papaya, Street address has exactly three digits. The students’ interest piqued when I wrote my initials on a 1-by-2 inch Post-it Note and placed it in the intersection of all three…

In this partner game, fifth grade students practice adding and subtracting decimals. The key to learning mathematics is understanding the “why” behind the “how”. HMH Into Math emphasizes the importance of establishing conceptual understanding and reinforces that understanding with procedural practice. The learning model asks students to first develop their reasoning before connecting their understanding to…

Teaching multiplication of fractions is, in one way, simple—the rule of multiplying across the numerators and the denominators is easy for teachers to teach and for students to learn. However, teaching so that students also develop understanding is more demanding, and Marilyn Burns tackles this in her new book Teaching Arithmetic: Lessons for Multiplying and…

After the class had experience comparing two fractions by using one-half as a benchmark, I wrote on the board: 6⁄12 “Raise your hand if you can explain why this fraction is equivalent to one-half,” I said. I waited until every student had raised a hand. While the question was trivial for most students, I planned…

Maryann Wickett and Marilyn Burns’s new book, Teaching Arithmetic: Lessons for Extending Division, Grades 4–5 (Math Solutions Publications, 2003), builds on the concepts and skills presented in Teaching Arithmetic: Lessons for Introducing Division, Grades 3–4 (Math Solutions Publications, 2002). Their new book helps students calculate with multidigit divisors and dividends (using a method that makes…

In this excerpted lesson, students divide “brownies” into halves by using either a spatial approach (focusing on the shapes of the fractional parts) or a numerical approach (calculating the number of square units in the shapes they create). In the full-length lesson of Dividing Brownies, which appears in Marilyn Burns’s Teaching Arithmetic: Lessons for Extending…

Homework: If 3 is 5% of a number, what is 30% of that number? This assignment, at ﬁrst glance, doesn’t seem very difﬁcult. But when I asked my sixth graders to solve this problem for homework, I also asked them to take notes about how they solved it. When we shared our approaches, I was…

Playing the Factor Game provides an engaging format in which students can become familiar with the factors of numbers from two to thirty by playing a two-person board game. To play Factor Game, each player chooses a number while the other player finds the sum of the available factors of that number. While playing this…

For this lesson, I planned to have the students work individually to solve a measurement problem involving fractions. Before the period began, I drew on the chalkboard a line segment that measured 22 1/2 inches. Also, I put thirty-three Uniﬁx cubes into a plastic bag. “When I snap these cubes together,” I said, “do you…

In order to introduce my students to problems that involve division with fractions, I use problem situations that draw on familiar contexts. I keep the focus of their work on making sense of the situation and explaining their strategies and solutions. Some of the problems I use don’t require that students necessarily arrive at a…

For almost two months before David Schwartz’s visit, our school took on the challenge of collecting 1,000,000 pennies.All of the considerations, from storing to rolling them, were an interesting challenge. In seven weeks, we collected 250,000 pennies, and we plan to continue at least until the end of the year to see how close we…

Before I began this lesson, I checked with a local hamburger restaurant and learned that there are about forty french fries in a single serving. I asked the class, “If a single serving of french fries has forty fries, how many friends would one thousand french fries feed?” I gave them a few minutes to…

Materials A collection of coins dated before 1990, placed in a clear plastic bag Overview of Lesson Marilyn is always on the lookout for ways to provide students experience with computing mentally. Her colleague Jane Crawford gave her the idea of presenting older students with the problem of figuring out the ages of coins. To…

Geometry comes to life in this lesson, as Rusty Bresser has fourth graders use geoboards to explore making pairs of line segments that touch exactly nine pegs, record them on dot paper, and label them as parallel, intersecting, or perpendicular. Rusty is the author of the new book Math and Literature, Grades 4–6, Second Edition…

In this lesson, Marilyn Burns shows fifth-grade students a fraction and they decide if it’s more or less than one-half and then explain their reasoning. This focus on one-half helps establish it as an important and useful benchmark. The lesson also provides practice with mental computation of whole numbers as students compare numerators and denominators.…

Collect and use data to solve a problem. Use collected data to develop statistical concepts of mean, median, and mode. References: Moira’s Birthday, by Robert Munsch (Annick, 1992) Materials Book: Moira’s Birthday Small,square sticky notes (2-by-2-inch), at least 1 per student Snap Cubes—approximately100–150 (maybe more if your class is larger than 25) Large sheets of…

This activity is excerpted from Minilessons for Math Practice, Grades 3–5, by Rusty Bresser and Caren Holtzman (Math Solutions Publications, 2006). Amy Jackson presented this lesson to her students in Florence Elementary School in San Diego, California. In this dice activity, the class works together to generate the numbers one through twelve in order. Clear…

This fifth-grade teacher knows that making connections among concepts and representations is a big idea in mathematics. She wants all of her students to be able to represent and connect number theory ideas. For example, she wants her students to represent square numbers visually by making squares on graph paper as well as to connect…

In this lesson, excerpted from A Month-to-Month Guide: Fourth-Grade Math (Math Solutions,2009), Lainie Schuster has her fourth graders start the school year with an investigation that offers them the opportunity to work in pairs to collect, represent, and analyze data. She uses the children’s book Martha Blah Blah, by Susan Meddaugh (Houghton Mifflin Company, 1996),…

Day-by-Day Math: Activities for Grades 3–6, by Susan Ohanian, is an eclectic and quirky collection of events — and the mathematical investigations, problems, or activities that are suggested by them.Each day of the year, from January 1 through December 31, lists historical events, each a lighthearted or serious moment. Some of the classroom suggestions are…

I was a little less than halfway through reading K. T. Hao’s One Pizza, One Penny to Robin Gordon’s fourth graders when I read what Ben Bear was thinking: “If Chris Croc can sell his cake, my pizzas can bring me a fortune! If each slice sells for one gold coin, and if I can sell…

I wrote ¾ on the board and asked the class, “Who can think of a fraction that has a numerator that’s greater than the numerator of the fraction I wrote on the board and also a denominator that’s greater than the denominator on the board?” A few hands sprung up. I suspected that some hadn’t heard…

While browsing through About Teaching Mathematics recently, I came upon an activity, Area and Perimeter, that had once been one of my favorites but that I hadn’t taught in quite some time. I decided to use it with ﬁfth graders in a different way than suggested in the book and have described below what I…

“Who can explain what a riddle is?” Danielle Gilligan asked her fifth graders. “It’s like a mystery you have to figure out,” Jill said. “It has clues that help you solve it,” Ramon added. “I have a book about a riddle,” Danielle said. “I’m going to read the story and stop at a certain point…

Question: Here is a set of numbers: {2, 3, 5, 7, 11, 13, 17}. How are these numbers alike? Students may offer that the numbers are all less than 20. Accept this, but push students to think about the factors of the numbers. Following are several possible responses: All have a factor of one. All…

In this lesson, fourth and fifth graders gain experience multiplying by ten and multiples of ten as they make choices about the numbers to use to reach the target amount of three hundred. I began the lesson, “Today I’d like to share a game with you. In this game you will be multiplying by ten,…

Upper elementary students beneﬁt from activities that help reinforce their understanding of factors, multiples, and prime numbers. In this two-person game, students take turns identifying factors of successive numbers, continuing until one of them can no longer contribute a new number. Directions for Playing the Game Player 1 writes down a number greater than one…

In previous lessons, students built rectangular prisms using cubic units and determined the volume of the prisms by counting cubes. Students started to devise methods for finding the volume of any rectangular prism without counting. In this lesson, students continue their work on developing a method for determining the volume of any rectangular prism. They…

In this lesson, the marbles and beans, along with the book Great Estimations by Bruce Goldstone, are used to provide students with an opportunity to explore and apply strategies for estimation. Also, collections of grid paper with different-sized grids, measuring cups of various sizes, a few unifix cubes, and balance scales and masses provide engaging…

Open-ended problems can make for excellent post-assessment. Wondering how you can design effective post-assessment tasks for your students? This lesson gives a four-step plan, including a 3–5 sample task and corresponding authentic student responses. The lesson is adapted from Math for All: Differentiating Instruction, Grades 3–5, by Linda Dacey and Jayne Bamford Lynch. Visit www.mathsolutions.com…

I began the lesson by writing on the board: 48 ÷ 3. “What does this mean?” I asked. “I’m interested in all of your ideas.” The students had a variety of ideas. “It says forty-eight divided by three,” Mason said. “It means how many times can three go into forty-eight,” Noelle added. Emma suggested a…

Guess My Number invites children to consider the structure of the number system while engaging in a logic game. Students try to guess a secret number from within a given range of possibilities. Guess My Number also presents an opportunity to reinforce mathematical symbols such as the “greater than” and “less than” signs. Through this…

A man goes into a store and says to the owner, “Give me as much money as I have with me and I will spend $10.” It is done, and the man does the same thing in a second and third store, after which he has no money left. How much did he start with?…

Cheryl began the lesson by reading Spaghetti and Meatballs for All! aloud to the class. In the story, Mr. and Mrs. Comfort invite 32 family members and friends for a reunion and set eight square tables to seat four people at each, one to a side. As guests arrive, they all have their own ideas…

Rowland Morgan’s In the Next Three Seconds . . . Predictions for the Millennium (New York: Puffin, 1997) is a collection of predictions about everyday and not-so-everyday events that will take place in the next three seconds, the next three minutes, the next three hours, days, weeks . . . all the way up to…

In this beginning lesson, students first explore arithmetic sentences to decide whether they are true or false. The lesson then introduces students to sentences that are neither true nor false but are algebraic equations, also called open sentences, such as x + 3 = 7 or 2 x= 12. The activity appears in Maryann Wickett,…

The Golden Ratio is a ratio of length to width and is approximately 1:1.618. This ratio not only appears in art and architecture, but also can be observed in nature and in the human body. The Golden Ratio is the ratio of a person’s total height to height from their feet to their navel. How…

In the previous issue of the Math Solutions newsletter (Number 23, Spring/Summer 1998), I described the Put in Order lesson that I had taught to help fifth graders learn to compare and order fractions. Throughout the year, I continued helping the class learn ways to compare fractions. As always, I learned a great deal from…

From her past experiences teaching middle school students about angles, Cathy Humphreys knew that students often have difﬁculty learning how to use protractors. Often they don’t see the need for the tool, so Cathy does not introduce protractors until after the students have had concrete experiences measuring angles several ways. When Cathy distributed protractors to…

In this partner game, fourth grade students divide by 1-digit numbers to determine if the quotient has a remainder. The key to learning mathematics is understanding the “why” behind the “how”. HMH Into Math emphasizes the importance of establishing conceptual understanding and reinforces that understanding with procedural practice. The learning model asks students to first develop…