The Mathematics of Wallpaper

Activities

The reader will see that there are core activities presented here, but keep in mind that fourth graders need multiple opportunities to explore and practice a concept in order to know, to own it, so to speak. Therefore, I haven't included a lesson for every day of the unit, but just know you should provide many learning opportunities for your students as I will do.

Activity One: Introducing….the Number Line!

I'll begin this lesson by showing a number line on the interactive whiteboard. I will ask, "What is this object? What can you do with this thing? Where would you use it? Does it give you any information? Does it always look the same?" After our discussion, I will take my students outside to form a human number line. I will point out that each one of them represents the dot or the tick marks on the number line, and the space between each student represents the space between units on the number line. I'll have to do my best to space them evenly apart, but this is an opportunity to remind them that on a number line, the spaces between the markers are exactly the same length. Once we have our number line assembled, we'll practice some different kinds of problems. We'll begin with simply counting out loud one by one and jumping up as we say our number. On this first day, I'll keep it simple—easy addition and subtraction problems such as 4+8=12 and 24-12=12. I can have one student be the "problem solver." When I present a problem, he/she has to touch and count each person on the line. The person being counted raises their arms over their heads so we know they are part of the numbers being counted. I will also have a "recorder" who records the problem numerically on a student white board. I want to use problems that will give everyone a chance to participate, and the jobs of "problem solver" and "recorder" will be rotated. Once we've done several problems, we'll go back inside to complete the remainder of the interactive whiteboard presentation. My students will learn that the dots or ticks can represent counting up by ones, or they can represent multiples of two or ten or any number really. Other features in the presentation will include how the line theoretically goes on forever in either direction, there are negative numbers to the left of 0, and an overview of the kinds of problems one can solve using a number line (number operations, fractions, calculating intervals of time, etc.) Then on the next day we'll review our presentation and practice addition and subtraction problems on our human number line.

Activity Two: Practice with the Number Line

On this day I want my students to get used to using a number line on paper, so we will solve addition and subtraction problems with different scaling. Just to make it a little more fun, I'll let them write the number sentences that describe the movements of their number lines on their student white boards. So, they'll show the moves on a paper number line, but write the problems with white boards and markers. First, we'll use number lines marked in units of one and show how to move back and forth to solve easy problems. I will give them a variety of problems requiring the inverse operation, changing the order of the numbers. I want my fourth graders to see that you can solve problems like this: 4+4= ? and 4+ ? =8. Once they've worked on some single digit problems, I will ask if we can change the numbers (called scaling) on our number line so that we can solve two digit problems, and perhaps three digit as well. Hopefully someone will volunteer that we can mark our points on the lines by tens. Then we can work out the problems from there. It would be appropriate for my students to make up and solve their own problems, either individually, in pairs or small groups. This is a time in my classroom where I will see who is really struggling and pull them into a small group to work in a supported environment with fewer problems and/or variables. Again, this type of work can be done on successive days (but not for the entire math block) or I might send homework for reinforcement.

Activity Three: Introduction to Multiplication

I will use my recommended curriculum unit guide to introduce multiplication. This is where we will learn about arrays, also known as the area model of multiplication. Once I've taught several lessons in that unit and my students have a good understanding of the concept of multiplication, I will introduce multiplication on the number line. We'll form our human number line again, only this time we'll be skip counting to demonstrate dilation of the number line by a given numeral. For the easier facts, each student can represent one place on the line, but once we get to facts with higher values, I will give them construction paper with large numerals printed on them so we're not just doing the 1's, 2's and 3's of the multiplication table, we can model the 8's, 9's or 10's. Once we've played this for a bit, I will show my students on the interactive white board how we can use a number line to skip count and solve multiplication problems. As I said earlier, this linear model is quite different than what they are used to, so we want to have a conversation about how and why this is so.

The next part of this lesson is to give student teams strips of colored paper so they can make models of multiplication on the number line showing how each multiple results in a successively longer line. Prior to their working with their teammates, I will model with my own number lines made from colored strips of paper. These need to be prepared prior to this lesson. Using the number two in a dilation of the line, I will model how to show a jump of one, equaling 1 x 2, two jumps of 2, equaling 2 x 2, etc. I will assign one set of multiples to each team. The students may decide to use the tiles on the floor as landmarks for making even marks on their own number lines. We will post these number line sets in the classroom so they can compare them to their array models. On successive days, I'll ask my students to make up story problems or algebraic expressions (actually, a mix of types would be good) based on their number line and then students will solve each others' problems. By algebra, I'm simply referring to using a letter to represent the unknown. This could be a differentiation strategy used with students who are quite adept at math, although my state has a standard addressing the use of letters to represent unknowns, to which all of my fourth graders should be exposed.

Another activity I want to try is to make a big number line on a blacktop area outside using colored chalk. We will hop from spot to spot counting by the number called out. I know many of my students will not know their facts yet, as we will still be learning them in our number fluency program, but this is still a fun way to practice and use the number line at the same time. I could make large number cards with the multiples and hold them up as a mode of visual reinforcement. A variation on this would be to have my fourth graders hop on one foot from spot to spot, spin and jump, etc.

Activity Four: The Rules of Math Nobody Teaches

You might think I'm crazy for wanting to teach these rules and my students might think I have three heads, but I'm going to try this. It's not that I don't teach these rules now, but I haven't done a great job explaining and teaching them to mastery. I plan to directly teach the commutative property, the inverse rule, the identity rule and the distributive property. The commutative property works for addition and multiplication. It simply says that the order of the numbers in the equation doesn't matter; you will achieve the same result. So, 4+6=10 and 6+4=10, just as 4x6=24 and 6x4=24. One way to demonstrate how to remember the term commutative is to ask, "Who knows what a commuter is? Have you ever heard someone say, "I have a long commute to work?" Associate the word "commute" with movement, going somewhere. Then, have a couple of volunteers come to the front to demonstrate with number cards how they can move, or commute, and it doesn't change the answer. Then play the card game below.

As for the inverse rule, one operation is sort of the opposite of the other. Subtraction is the inverse of addition: instead of putting numbers together, you are taking them apart. For example, 7+6=13, therefore 13-6=7 and 13-7=6. It's the same for division, as division is the inverse of multiplication: 4x8=32 so, 32÷8=4 and 32÷4=8.

The way I plan to teach this is to have the students work in groups of five. Each group will have a set of cards with which to work. The cards will have numerals and the symbols for addition, subtraction, multiplication and division and they will have yarn attached so the kids can wear the numbers. Each child gets one card, so they either get a number or a symbol. One of the symbol cards will have the addition sign on one side and the subtraction card on the other side, and the other operation card will have the multiplication sign on one side and the division sign on the other. They will form number sentences (correctly) and then they will change the order in which they are standing and do a quick flip of the symbol card to make a number sentence that is the inverse of the original one. Once they've completed their set of cards, they trade with the group to the right, so every group of students gets multiple opportunities to practice. This can also be done on successive days for practice.

The identity rule states that for addition there is only one number you can add to a given number and still have the original number, and this is 0. For multiplication, you can multiply any number by 1 and still retain the original number, so for multiplication the identity is 1. Children know the identity rule, but perhaps implicitly—they haven't ever had to think about why this is so nor are they asked to explain it. For an activity, I will give one third of my students a large copy of the letter i. Another third will have large cards with the numeral 1 on one side and 0 on the other. The remaining third students will have strips with one number sentence on each side, one showing a partial addition sentence and on the other side, a multiplication sentence (like this: 4 x ? = 4, or 6 + ? = 6). They will then have to form groups of three to demonstrate the identity rule for either addition or multiplication. So a group of students should correctly form:

i = 6 + 0 = 6

Finally, I want to work on solving problems by what I like to call "chunking"; it is what I referred to earlier as the "break apart method," and what students need to know is the Distributive Rule of arithmetic. This works well for the multiplication tables with higher values, and the ones with which they struggle more on memorizing. Use problems such as 7 x 9. Following is one way to demonstrate the distributive property:

7 x 9 = 7 x (4 + 5)

= (7 x 4) + (7 x 5)

= 28 + 35

= 50 + 13

= 63

As a follow up to teaching each rule, each student will make a mini-book with the terms and the explanation and a number sentence(s) illustrating the meaning of the rule.