Graphic Pages


When students are learning to multiply a one-digit number times another one-digit number, they are often given the opportunity to picture the problems using various types of arrays. However, when learning to multiply a TWO-digit number times a one-digit number, they are often only given a mechanical procedure to follow.

Problem of the Week #9 gives the opportunity to picture a TWO-digit number multiplied by a one-digit number problem using a special array. This is actually quite easy to do when you know how; but most people have never done it before, including teachers, tutors, parents, and students.

Here is the challenge: using only the “One-Dollar Bills Page” and the “Ten-Dollar Bills Page” (see below), and two pieces of blank paper, show an exact picture of 53 x 4.

Your solution must not only show the correct amount of money, but it also must picture the actual meaning of 53 x 4. It can be done with only four movements of the hand!

In an earlier post, we gave an outline for integrating assessment into a math lesson. And we said that in subsequent posts we would address several aspects in more detail. Here is a closer look at the first step – starting the lesson with a graphic representation of the concept and procedure handed out to each student.

In the post that followed, we put up a video (The Soccer Teams Graphic) that demonstrated exploring the graphic in numerical order and re-exploring it out of numerical order. Through tutoring and classroom application of the graphic pages, we have observed more effective and less effective ways of guiding a lesson using a graphic manipulative. While there are lots of manipulatives and number arrays out there, how they are implemented and integrated makes all the difference. As always, it comes down to effective teaching. Here’s an approach that works when guiding students with a graphic page and a blank piece of paper:

1. Use short phrasessay only what is needed for students to construct, count, and observe on the graphic page. AVOID explaining. “Now you see class, what you just did was…” Guide and allow for their discoveries. Short phrases move the lesson along, keep students actively engaged with the graphic, and help language learners and students who have a hard time focusing.

2. Start with an action verbcount (“Count how many…”), move (“Move your blank paper to show…”), cover (“Cover all but the first team”), uncover, and so on. This aids in fulfilling the first step. Students know right away what to do.

3. Model your directionswhen you ask them to uncover the next row of soccer players or members of a rowing team or weeks with the blank paper, do it with them at first – in a way that they can see your movements. A picture is worth a thousand words. This is helpful for everyone and especially for those learning the language – whether English or math! – and those who have a problem paying attention to aural directions. Your observations will tell you how long to continue modeling. It has proved more effective to model one step at a time rather than to give students an overview to recall and execute.

4. Observe (assess)look around to make sure everyone is engaged and following your direction on the graphic correctly. Since it involves physical activity, it is pretty easy to see who is where they need to be for the direction you gave and who is slow or looking around to see what to do. Assessing determines the speed and direction of the lesson – does a prior skill need to be addressed – who needs a little more support – who is ready for more of a challenge – and so on. Creatively using a graphic page enables you to derive multiple lessons and multiple challenge levels from the same page. Each student matters.

5. Correct student responses nowsince each student matters, something needs to be done when you see inaccurate or continually slow responses. But never give an answer. Guide individual students to correct themselves: “Show me on the chart how you got that answer.”  As they reconstruct and rethink their solution, they see for themselves what they did wrong, and therefore how to do it right. Some may require further guiding questions. It is not efficient to give them the answer and move on. You will have to retrace this same concept or procedure eventually. Do it now before it becomes an unlearned prior skill needed for another procedure the student is trying to learn.

6. Begin the transition to print say “show this many teams” as you write a “5” on the board. Don’t say the word “5.” Or say, “show this on your page” as you write on the board, “7 x 11.” By doing this, students see numbers and symbols as action prompts and begin making the connection between manipulation and computation. It prepares them for the problem pages that follow.

7. Ask questions transition to mini-word problems. “3 teams are coming to the practice tomorrow, how many water bottles will we need?” After the students have some experience with a concept and procedure, then begin to make the process conscious by asking them questions about what they did.

Our experience has been that new vocabulary and notation should be introduced gradually in conjunction with the work on the graphic pages and reiterated during practice with the partner pages that follow. It is better to solidify (talk about) concepts with the correct nomenclature after students have actually experienced using them with the graphic pages. They own something concrete then to hang those words on. We will look more closely at working with the partner pages in a later post.

Summary: Guide work on a graphic manipulative using short phrases. Model how to do it. Make sure they’re doing it correctly; help them if they aren’t. Guide the work using numbers and math notation on the board. Give further written and aural challenges at appropriate levels. Ask them questions about what they did.

We were asked if we could post the podcasts referred to in the previous assessment posting, so here is a slightly abbreviated version of the Soccer Teams podcast.

We began that posting noting the importance of designing materials that enable observation while accomplishing learning objectives. The Soccer Teams graphic page is a model of that.

1. It can be used to support students who are not succeeding with the regular text book.
2. It can be used up front to quickly orient students to a concept or procedure, increasing the likelihood that they will succeed with the regular curriculum.
3. It can also be used to efficiently cover prior skill gaps so that students are capable of handling the current lesson.

Students not doing well with multiplication or division may not understand what multiplication and division really mean. Then they try to memorize the things they don’t understand. Students can memorize, “quoth the raven…” Say it enough times and they can fill in the blank sounding like they really know something. But they may very well not know what “quoth” or “nevermore” mean or recognize a raven if they saw one.

With the Soccer Teams graphic, they are able to experience how a single group can be one and many (1 group of 11), and the feeling for magnitude between 1 x 11, 2 x 11,  7 x 11 or 12 x 11. These are some of the conceptual building blocks of multiplication. Times tables are not. By working with the graphic pages, students begin building a memory of these experiences the same way we remember our way around a new city. We don’t sit at home and memorize the streets with flash cards.  We may orient ourselves with a map, but then we go out and drive around. That’s how we learn the city.

With the graphic pages students drive around so we can observe and assess how they’re doing right from the beginning and each step of the way.

Vodpod videos no longer available.

more about “The Soccer Teams Graphic“, posted with vodpod