When students enter class today they will begin to work on the Warm up projected on the screen. This resource includes five problems that use the numbers 1, 2, 3, and 4 exactly once in an expression that makes the values 1 through 5. This task is the same as what they had for homework last night, but my these expressions all have exponents in them. For example, in one of them I have one to the power of (2 x 3 x 4), so the exponent is an expression itself. The last question on the Warm Up asks:
What is the biggest number you can make with a 1,2,3,4 problem?
We will spend much of our time on this task during class.
I teach this lesson because my students are not very strong in their use of exponents, and, to help students improve their number sense. This activity gives them a little more insight into how exponents work and also their relationship to other operations.
When we go over the Warm Up I have students come up and explain the solutions to the first 5 straight forward problems. The last problem begins their exploration section: Students are always surprised by how big and how fast a number can grow with exponents.
Before we go over the last problem on the Warm Up I go back to the first problem 3^2-(4+1) and ask my students, "Will removing the parentheses would make the value of the expression bigger or smaller?" I do not expect them to be able to answer correctly until after we try it. When we try it and it ends up larger, then I would ask why they think that was. If they don't say that the one ended up being added and not subtracted I might ask what was subtracted from 9 in either case.
In Number 2 which is 1^2x3x4 or 1^24 I ask if it would be bigger if I changed the exponent on the one to be (3x4)^2 instead of 2x3x4. Two things happen here. Students will realize that no matter the exponent, if the base is one, the product will be one. But they will also see the possibility of exponents on exponents. This always inspires their curiosity. I go on to number 4 [1(3^2-4)] and ask which they think would make it the biggest, switching the 2 and the 4 to make 1(3^4-2) or switching the 1 and the 4 to make 4(3^2-1). Most of them think multiplying by four will make the biggest result and are surprised when we try it.
Now we move on to the last one and I ask what the biggest number they made was and we show it on the board. It will likely be something like 2x3x4+1. I ask the class "what if we made one of them an exponent? ... or used parentheses?" They work on making it bigger as I circulate to see what they are coming up with. When a student shows me their expression I ask what would happen if they switched certain numbers or used a different number as an exponent. As students start to use exponents more I start asking them "what if you added or multiplied in the exponent instead of the base, for example: 3^4x2+1 instead of (3x4+1)^2.
Whenever their expression shows multplying by one, I ask them how they could make that 1 stronger, make a greater impact. Realizing that multiplying by one makes no change helps reinforce their number properties anyway. I suggest that instead of multiplying by one, try adding it. I may ask questions like "will adding to the base or to the exponent make a bigger number?" I continue to make suggestions for them to try or ask them if they think switching certain numbers would make their result bigger or smaller.
Inevitably students will begin to ask for calculators which I would allow. As their numbers get far too big for their screen they will notice strange notation in their screen. If this comes up it may be a good place to incorporate a mini lesson on scientific notation or a way to transition to a larger lesson on that topic.
For today's Number Talk, I refer back to the poster we made in an earlier lesson on the properties of addition. Then I show on the board the multiplication strategies used by students in that lesson. I ask them to describe what we are doing with the numbers to make it easier.
4x20 = 4x(10x2) = (4x10)x2 = 40x2
Students may say that the numbers are being broken down and multiplied in a different order. They may not say anything about the parentheses at this point, but that's okay.
Students will probably say the numbers are being switched around and multiplied in a different order.
After seeing both of these I will ask my students how these two strategies are the same and how they are slightly different. The parentheses are an obvious difference, but they may not notice that the numbers didn't "switch around" in the top example. They will likely notice that both strategies multiply in a different order. I could ask them what happens that changes the order, which should make them realize that one uses parentheses and one actually moves the numbers. I add these descriptions to the multiplication property poster.
I tell them these are very similar to a couple of the addition properties we already looked at and I have them refer to the poster we made into find the commutative and associative properties.
I point out the Identity Property at the bottom and tell them that we discovered the Identity property of multiplication today. I ask a student to describe or read our earlier description from the poster of the identity property of addition. Then I remind them of our earlier exploration when we discovered something about the number one. I will write 3x4x2x1 on the board and ask why we changed it to 3x4x2+1 instead. Someone should be able to explain that adding one makes it bigger than multiplying by one, because multiplying by one doesn't have any effect at all. If they don't already recognize the similarity I point out by asking "does that sound an awful lot like the identity property of addition?" We add this to the poster at the bottom of the multiplication properties.