They sometimes say that math is the science of patterns, and maybe nowhere is that more apparent than in our place value system. Unfortunately, we often teach place value at a very surface level and we don’t give students enough opportunities to explore those patterns. One way to build understanding is by modeling place value relationships.
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What do the standards say?
Both the Texas Essential Knowledge and Skills (TEKS) and the Common Core State Standards for Math (CCSSM) describe standards for place value relationships. Interestingly, the alignment is different. The TEKS require understanding the relationship for whole numbers in 3rd Grade and for all numbers (implying whole numbers and decimal numbers) in 4th Grade. The CCSSM shift that process a year, with whole numbers required in 4th Grade and all numbers in 5th.
TEKS 3.2(B) Describe the mathematical relationships found in the base-10 place value system through the hundred thousands place.
TEKS 4.2(A) Interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left.
CCSS.MATH.CONTENT.4.NBT.A.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
CCSS.MATH.CONTENT.5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Here’s a pictorial representation, using whole numbers.
Let’s look at some models that can help students understand the relationship.
Unfortunately, with our older students, we often forego the manipulatives. There’s a widespread misconception that manipulatives are for the younger kiddos. Nothing could be farther from the truth! Manipulatives are needed whenever new concepts are introduced, regardless of age.
When we are planning a lesson, we should always consider the prerequisite knowledge that students have and think about how we can build on that knowledge and incorporate a little problem solving. Even in 3rd Grade, students have been working with place value for several years. Hopefully, they have experienced both groupable and pregrouped models for place value through 1,200. So, basically, in 3rd Grade we should not have to teach students how to read, write, and represent numbers through 1,200. We can, then, focus on understanding the patterns and extending to larger numbers based on that understanding of patterns. Now, because 3rd-Grade students don’t have a strong understanding of multiplication yet, we might want to hold off using the “times ten” phrasing to describe the puzzle.
Here’s what a lesson might sound like:
TEACHER: Boys and girls, you have been working with place value and base-ten blocks for several years now. Today we’re going to see if we can discover and describe patterns in our place value system that can help us extend our knowledge to larger numbers. Work with your partner to build the number 222 with your base-ten blocks. Once you’ve build the number, write it in expanded form. Note: I chose a 3-digit number because we typically have very few cubes, which are used to represent thousands, in our base ten block sets.
STUDENT: That’s a funny number!
TEACHER: I know, right? I wonder why I chose a funny number like that.
After the students have built their model, I’d ask them to talk about some things they notice and wonder. You can see some sample responses in the picture below.
TEACHER: Write this number (22,222) on your whiteboard. Talk to your partner. What do you think the value of each 2 is?
Notice with that last question I am giving students the opportunity to extend their understanding by solving a problem. It’s not necessary for all students to get the correct answer, but they all get to grapple with it! As they are discussing the problem, I will circulate and listen in on the conversations so I can choose a pair or two of students to share their findings.
Keep in mind that this won’t be the last time they hear about this relationship. Basically, with any place value activities you do, you want to keep asking students to describe the relationship between the values of the digits.
To extend the understanding to decimal numbers, we have several helpful models we can use.
Students have worked with money in school since Kindergarten and quite frequently use money at home. It is a real-life model for decimals. To connect to decimals, give each pair of students some manipulative dollar bills, dimes, and pennies. Ask them to work together and see if they can describe the relationships between the dollars, dimes, and pennies. After a few minutes, ask some students to share. You are likely to hear that ten pennies make a dime, and ten dimes make a dollar. Some students might also note that it takes one hundred pennies to make a dollar. At that point, you can provide each pair of students with the money place value mat shown below. Ask them what they notice and wonder about the chart (e.g., pictures of dollar bills, dimes, and pennies; they are labeled ones, tenths, and hundredths, a decimal point that’s labeled ‘and’; etc.). Ask them to talk about why they think the dimes are labeled tenths and the pennies are labeled hundredths. If they explain that it takes ten dimes to make a dollar, rephrase it by saying So a dollar can also be broken into ten dimes. Ask students to build a money amount on their mat, such as $3.24, using manipulative bills and coins. Once they have done so, ask probing questions about equivalencies. For example, How many dimes would it take to make 3 dollars? How many pennies in 2 dimes?
An engaging game that emphasizes the patterns is called Race to a Dollar. Each player needs a money place value mat, shown below (link at the end of the post). Players use either standard or ten-sided dice to play. On a player’s turn, they roll the dice and place that many pennies on the mat. Players take turns rolling. Each time they accumulate 10 pennies, they trade them for one dime. When they accumulate 10 dimes, they trade them for one dollar. The first player to reach one dollar wins the game. You could, of course, have them play Race to Five Dollars or Race to Ten Dollars.
Students have had lots of experience using base-ten blocks to model whole numbers. To use them for decimal numbers, we need to redefine the unit, or whole. I would recommend using the base-ten blocks after using money so that students are already familiar with the concept of tenths and hundredths.
Provide students with base-ten blocks. Hold up a flat. Ask students what we have used the flat to represent with whole numbers (one hundred). Hold up a rod, asking the same question (one ten). Ask they to explain the relationship between the materials (ten tens make a hundred, or a hundred can be broken into ten tens). Then hold up the flat and ask, What if this piece represents one? Then what would the rod represent? Allow students to talk and then share their thoughts.
Now they can practice regrouping hundredths, tenths, and ones by playing Race to Five. It’s played just like Race to a Dollar, but uses base-ten blocks and a different mat.
You can use base-ten blocks to model thousandths by redefining the unit again, making the cube the unit. Introduce it using problem solving, just like we did tenths and hundredths. Now they can play Race to One. Note that this place value mat prints on legal paper.
There you go! I hope that’s enough to get you thinking about ways to teach the relationships in place value. You can download the file with the place value mats here.