Estimation - an introduction

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Objective

SWBAT use models to estimate an answer when adding 2, two-digit numbers.

Big Idea

Estimation allows students to reflect on the accuracy of their own solutions. It provides a perspective for critiquing the chosen strategies and outcomes when solving problems.

Warm ups

15 minutes

Today I review smiley face numbers (numbers that end in zero). I ask students if they remember smiley face numbers.

Can someone tell me a smiley face number? (10, 20, 30.. ) 

Which would be easier to add in your head - 43 + 38, or 40 + 40? Why?  

I give students the problem 36 + 18 = _________ and give them about 30 seconds to solve the problem.

Next I give students 40 + 20 = _________ and again give them 30 seconds to solve the problem.

I ask for a show of hands, "How many of you solved the first problem? How many solved the second?"  We discuss how the second one is easier, because of the smiley face numbers.

I tell them that when we change our numbers to make them the easy-to-use smiley face numbers, it is called estimation. Does anyone know what estimation is?  (It is a good guess about what we think the answer should be).

Estimation helps us to know if we are close to the answer that is correct. It helps us when we try to figure out if our answer is logical, or if we have read the street sign correctly in a math problem. Today we will be using estimation to play a game that will help us solve math problems.

Teaching the Lesson

15 minutes

I bring students to the rug to tell them that they will play a quick game where they will learn about estimating. I model how children will play with a partner.  I bring out a number line as a reference tool to be used by each set of partners. Today I want students to model with mathematics (MP4) as they use the number line to help them figure out how far they are from the smiley face number. 

Next, I draw 2 single digit playing cards (a regular card deck with face cards removed can be used). I show how I can use the two cards to create a 2-digit number, and then find the number on my number line. I model how I think about where to find the nearest smiley face number, going either up or down, counting how far away I am from the smiley face number and reminding "myself" that I want to pick the smiley face that is "closest" to my two-digit number.

I show how I can make two different choices when using my two cards. If I draw a 3 and an 8, I could make 38 and be only 2 away from 40. If I make 83, I would be 3 away from 80. I choose 38.

My partner draws 2 cards, a 4 and a 6. They can make 46, which is 4 away from 50, or 64, which is 4 away from 60. They'll want to think strategically, because if my number is closer to the nearest smiley face number, I get to keep all 4 cards.

This game has several steps that make it more rigorous, and it is critical that students understand that they will want to check both ways of constructing their two-digit number in order to be sure that they choose the one with the fewest "jumps to" (or difference from) the closest smiley face number. Otherwise they are not taking away the correct understanding of “estimating to nearest ten”.

When the draw pile in the middle is gone, players can count how many cards they have, and the person with the largest pile wins.

I help partner students up, by counting off the circle into 2s. I give each pair a number line and a pack of cards. I let students play for about 10 minutes. I circulate around during this time to see if students are using the estimation and number lines correctly. Playing the Game

Independent Practice

15 minutes

Today there will be 2 homogeneous groups working with estimation.

The group that had more difficulty with the idea of estimation during the game (those that I needed to support during the game and those that were using the actual number and not moving to the smiley face number without being reminded) will work with an adult to find the closest smiley face number to numbers that the teacher will hold up. Students put a colored chip on the nearest smiley face estimation number on their number line.

The second group, who found the smiley face numbers easily, will now use moving to smiley face numbers (estimating) to subtract with 2 digit numbers. They will need to reason abstractly and quantitatively as they work to solve the problem exactly and with smiley face numbers. (MP2) They will be given 2 2digt numbers, and they must subtract the smaller from the larger . They can turn both into smiley face numbers and then subtract, or try to subtract the original numbers and then find the closest smiley face number. (I purposely do not tell them which way to solve the problem. I want them to find out for themselves that using the smiley face numbers is easier and quicker than trying to manipulate the original numbers (MP1)). The first person that writes the correct smiley face number on their individual white board (can use paper instead), gets one colored chip. The person with the most colored chips at the end, wins the game. 

Closing

5 minutes

I ask students to return to their seats at the end of 15 minutes of small group practice. I ask them to finish the following sentence that I put on the board and to write their endings in their math notebooks.  Estimating is when I___________. It helps me because _________________.

I am expecting them to say something like estimating is when I make a good guess/ use a smiley face number/use a number ending in zero. It helps me because adding smiley face numbers is easier/counting by tens is easier.