Discovering Place Value
I can't tell you the number of times my colleagues and I have bemoaned the fact that our students can't seem to think while doing math. Story problems are a nightmare because the kids don't know which operations to use to solve them.
While reflecting on my own daughter's struggles with place value, I had a revelation. What if our kids can't think during math because we have trained them not to think during math?
Even Engage New York, which is a math program I really like, shows kids how to solve a problem and then asks them to repeat the steps to solve more of the same problem. Kids aren't thinking to solve these problems, they are remembering.
Remembering can be a great thing, but it isn't thinking. That is why kids struggle on a test with story problems. It is really hard to remember every skill taught in a unit.
Instead of teaching kids to remember, we need to start teaching them to think.
Here is how I would do it using place value:
First, I would take a one digit number. Let's say it is 7. I would project a place value chart up on the board and ask my students, "How do you think I can put this number 7 on the place value chart?"
Hopefully, I get some responses. What I want to guide the class towards is using seven dots in the ones place. I want them to use dots instead of the numeral because I can show addition, subtraction, multiplication, and division using the dots. I can't show anything using the numeral.
If there are any students who need more exposure to showing numbers in the ones place, give them ten place value charts and have them show the numbers 0 through 9 on them.
The next step is to show a two-digit number and repeat the process. Hopefully, you will have students suggest both all dots in the ones place and dots in both the tens and ones places. It is important that kids understand either way is correct. You want your students to be able to think flexibly around place value. We are not teaching them to memorize the "correct" way to show place value, we are introducing them to a tool they can use to think.
Again, if any students are struggling with two-digit numbers, give them more practice with place value charts and two-digit numbers. Try having them write numbers both ways - all ones and tens and ones.
Now, it is time to know your students. You can continue this process using hundreds, thousands, ten thousands, etc... or you can through up a number in the millions and see if your students can figure out how to show it on the place value chart.
What we want to do is introduce a tool, show the students how it works using numbers they are confident manipulating, and then see if they can apply the tool to bigger numbers or more complicated skills.
Once you are confident that a student understands how to use the place value chart, you can introduce tasks the student can solve using the place value chart. Here is the key: don't tell them how to use the place value chart to solve a task. That is boring and leads to remembering instead of thinking.
Here are some tasks for kids put into question form:
How could I show adding 5 and 3 on the place value chart? (Choose bigger numbers or numbers that require regrouping as your next steps.)
How ould I show subtracting 13 from 26 on the place value chart? (Choose bigger numbers or numbers that require borrowing on the place value chart as your next steps.)
How could I compare the sizes of 6 and 3 on the place value chart? (Choose bigger numbers as your next step.)
It is vitally important not to tell kids how to solve these problems. They will ask you. Thinking is difficult and we all try to avoid it. Also, if a kid solves a problem in a way that works, but isn't what you were expecting, that is okay. If a problem is solved in a way that won't work every time, give them a problem it won't work for and see what happens. It will be much more powerful if the student discovers his own error.
For example, many students solve elapsed time problems within an hour by subtracting. This will work within one hour, but it won't work across more than one hour. Don't tell the kid she solved the problem wrong. Praise her work - she did figure out how to solve the problem, but then give her a problem across two hours. Her strategy won't work anymore and she will have to come up with a new way to solve the problem.
A pretty common part of place value in fourth and fifth grade is learning that moving up and down the place value chart requires multiplying and dividing by ten. This is a pretty simple concept, but when it is taught explicitly to students, they can get confused.
Instead of telling kids and showing them that you can multiply and divide by ten to move along the place value chart, see if they can discover that fact for themselves.
Here is how it works:
Put the number 3 up on the board. Have a student show three (with three dots) on the place value chart. Then, put the number 30 up on the place value chart. Ask what the kids notice? Then, put the number 300 on the place value chart. What do they notice now?
Ask the students (individually or in small groups) to show 3,000 on their place value charts. Ask them to come up with a rule that explains what is happening on the place value chart. How can we use numbers to change 3 into 30, 30 into 300, and 300 into 3,000? The goal with this exercise is for students to figure out that they can multiply by ten to move the number to the left on the place value chart.
This is not a skill that kids will master with one try. Once a student has told you their rule, give them another set of numbers (2, 20, 200, etc...) and ask them to show you their pattern both on the place value chart and using numbers.
Once you are sure the kids are able to move up the place value chart by multiplying by tens, ask them what they would have to do to get from 4 to 400. They will need to show it both on the place value chart and using numbers. Then, try 40 to 4,000. You will want to use lots of different combinations to make sure that kids understand all they are doing is multiplying by ten repeatedly to move along the place value chart from any one place to another.
After students have a firm understanding of moving to the left along the place value chart, you will want to introduce moving to the right. Again, put up your place value chart. This time, have a student put 300 on the place value chart. Then, put 30 and 3 on the place value chart. As the students to come up with a rule that explains how you can move from 300 to 30 and from 30 to 3. The goal for this exercise is to have students realize that they need to divide by ten to move right along the place value chart.
If you are teaching fourth or fifth grade, you will want to show kids that dividing by ten is the same as multiplying by one tenth. You can do this through examples (have students divide numbers by ten and the multiply the same numbers by one tenth) or you can tell them. It depends on your class and their confidence with fractions.
Repeat the same practice you did with multiplying by ten for dividing by ten. Then, you will want to mix up problems, so students can move up and down along the place value chart with ease. This is a place where I would have them do a lot of practice. They will be applying what they have discovered themselves and not "remembering" the rules that were taught during math class.
Depending on what grade you are teaching, you may choose to introduce tenths and hundredths at this point. Nothing about their thinking changes, but it is good to give them a chance to discover this on their own. Put three on the place value chart and ask students what would happen if you divided it by ten. There are so many ways to model this and all of them are good. Hopefully, you have a student that realizes all they have to do is add another place value box to the right of the ones place and draw an arrow from ones to tenths. Again, this will be a much more powerful experience if you let the students discover this method on their own instead of telling them how to do it.
You can repeat this process with hundredths, and it will go much faster because students will have experience with tenths. Give kids some practice moving up and down the place value chart using tenths and hundredths.
At this point, I would introduce multiplication on the place value chart. I would ask the kids how they would show 6 x 2 on the place value chart. Once they are able to do this easily (they may need more problems to develop mastery), I would ask how to multiply 6 x 20. I wouldn't tell them how to do it, but let them figure it out. If any students are unsure, I give them more problems to work out. When students can solve 6 x 20, I ask them about 6 x 200. At this point, they should be able to answer any version of this problem with as many zeroes as I give them. 6 x 2,000,000 uses the same principles as 6 x 20.
When students have mastered multiplication on the place value chart, I introduce division. Honestly, the place value chart isn't my favorite tool for division. I would not require my students to show me division on the place value chart. I would remind them that it is a tool that they can use, but they don't have to show mastery on the place value chart. This brings up a good question, do the kids have to show mastery of all of these skills using the place value chart?
I would say no. The place value chart is a tool. Students can choose to use it or not. However, they will have to have a different method of showing an understanding of place value that works for all of the skills being taught during this lesson. I think the place value chart is the easiest tool to do most of these things, but that is my opinion, not a fact.
When teaching division, start with math fact division that the kids should know. However, just writing the answer doesn't show understanding. They must model the problem in some way. Once kids are able to do that consistently (or for problems with remainders) have them move on to bigger and bigger numbers. You will probably want to start with numbers with trailing zeroes. For example, 420 instead of 42 or 1,800 instead of 18. Once kids have mastered these bigger versions of numbers they are comfortable with, you can have them work on other numbers.
I hope this place value progression was helpful for you. Sign up for my weekly newsletter and you can grab this place value freebie. It has the basics you will need to teach place value in your classroom.
You may also be interested in my Place Value Progression Flip Book. It covers all of the ways a place value chart can be used from first grade to fifth grade. It is great as a review or as a differentiation tool for younger students.