Today’s lesson focus is on showing students that having a two-digit divisor is exactly the same as a one-digit divisor. The steps to division do not change. Students will learn how to estimate quotients by guess and check. I will start today’s lesson by showing students how they already know how to do this type of division. For example, if I were to ask students how many quarters they needed to play an arcade game that costs seventy-five cents, they would quickly be able to tell me three.
In order to show students their prior knowledge in double-digit division I will take them through an arcade and ask them at various points through our tour how many quarters they need to play a specific game. Once I start students off with a few questions of this mental strategy of division, I will show them how it translates to the traditional algorithm we have been working on.
Today we’re going to go a little field trip through an arcade. How many of you have been to an arcade before? Share with your partner some of the experiences you have had in an arcade.
I allow students a couple minutes to share stories about the arcades and then move on to explaining today’s trip.
Alright, bring it back. As we go through this arcade we will stop at various places and figure out how many quarters we need in order to play the game. We don’t have a ton of money so we won’t be able to play every game so don’t get disappointed if we skip over one you wanted to play.
I begin the video without sound and let it play for about a minute to get students engaged in the trip we are embarking on. I then stop the video and tell students we are going to figure out how many quarters are needed for this game. Although you can’t actually see the price of the games I will make them up to fit the game or ask students to think of what the price would be. I start with some easier ones.
Oh, forgot to tell you. Before we left for our trip your parents each gave you a handful of quarters, they are in your pocket. Check your pockets.
I pull out my pretend pile of quarters from my pocket and act as if I have a lot of them in my hand. I ask students to show me their quarters.
Okay, get your quarters ready we’re going to make our first stop. It looks like this game cost 75 cents to play. Think before you respond. How many quarters do we need for this game?
I do this for a few other amounts such as 50 cents, $1.25, $1.00, $2.00. Students are quickly grasping the concept of determining the number of quarters. Little do they know they are actually doing double-digit division. I then move on to relating our answers back to the traditional algorithm.
I give students a several more examples($3.00, $4.50, $2.75, $5.00) and for each I ask them to first think of the number of quarters they will need to use then I have them translate this into using the algorithm. Students use their whiteboards to complete the problems with me as I do them on the whiteboard in the front of the class.
I go through each of our steps of division each time we do the problems(Ask and Answer, Multiply, Subtract, Bring Down, Repeat). Even though the students have already determined what the quotient is for each problem I have them go through each step so they can familiarize themselves with the process so they can they use this strategy with other two-digit divisors.
Now that students are comfortable with a double-digit divisor I ask them to think about why those types of problems were relatively simple for them. Student responses focus on their knowledge of quarters and how many are in a dollar. I translate that into more of a mathematical language.
So what I hear you say is that you know your multiples of 25. You can easily multiply by 25’s just like you can easily multiply by 2’s, 3’s, 4’s, 5’s and so on. What about 14’s? Do you know your multiplies of 14?
No’s are heard around the room as expected.
Well, what do we do if we get stuck on a multiplication fact such as 9 x 6? What is our strategy we use if we need the product for that problem?
I have taught my students to start at the “bottom” of fact family if they get stuck. In this example I would expect them to start at 6 x 1, then 6 x 2, then 6 x 3 and so on until they get to 6 x 9. Although this strategy may be time consuming it is something for the student to try instead of just sitting and doing nothing. I have often seen students get stuck on a problem and just sit and stare and their mind starts to wander. If we teach them little strategies they at least have somewhere to start if they get stuck.
So if we were to figure out the multiples of 14 where should we start? The bottom is always a good place to start.
We go through the multiples of 14 up to about 10 and write them on the board. Then I ask students some division problems using our multiples of 14.
What is 98 divide by 14? What about 70 divide by 14? Well, now let’s try a more complex problem. Try this on your whiteboards. What is 728 divided by 14?
I allow students time to work and when they are finished I have a student come up to the board and explain their thinking. While they are explaining the problem I make sure they refer to the list of multiples of 14. When the student is done sharing their thinking I explain to students that although we wrote down all the multiples of 14 before starting the division problem, sometimes we can guess the quotient and then check the answer while doing the division problem.
Let me show you, let’s try this problem. What is 102 divided by 17? If we go set this problem up what is the question we will be asking ourselves to solve this division problem? (How many times does 17 go into 102?) What is another way of saying that question in terms of multiplication? (17 times something equals 102) What is a reasonable answer, let’s make a guess. Is 1 a good guess for the quotient? How about 5, is that a reasonable guess? Let’s try it.
Students complete the multiplication problem on their boards. I further their thinking by asking could we put 17 into 102 one more time or would we go over. Students were unsure if 17 could go into 102 one more time so I ask them to complete this multiplication problem on their boards to figure out the answer. They then see that 17 x 6 is actually 102.
I wrap up this portion of the lesson by explaining to students how they guessed the answer then checked their answer while completing the division problem. I give students a few more problems to complete on their own using divisors and dividends they will see on the division tic-tac-toe they will play in the closer of this lesson.
2040 ÷ 24
1320 ÷ 15
8640 ÷ 10
Students will play division tic-tac-toe with a partner. This game includes one and two-digit divisors and also three and four-digit dividends. Player 1 needs to choose a number from the dividend box and a number from the divisor box. Then they need to complete the problem and then place an X on box of their quotient. Player 1 needs to explain their thinking to Player 2. Player 2 needs to check the accuracy of Player 1. Students need to get three in a row to win. Game play can continue even after a student has received one line.
As students are playing the game with a partner I circulate the room and check for understanding while listening to students explain their thinking.