Place value understanding is critical when solving addition, subtraction, multiplication, and division problems. Add fractions and decimals to the mix too! This post will focus on building on parts 1 and 2 (counting and composing and decomposing numbers by place value) to add and subtract whole numbers and decimals.
Virtual Manipulatives for Place Value:
Here is one of my favorite free websites for all kinds of virtual manipulatives. You will defintely like the base ten and place value disks. Both of these include options for decimals. The 0-120 grid is also great! You can add different colors to the numbers using the paint can.: https://www.didax.com/math/virtual-manipulatives.html
With the above manipulatives, students can practice these basics concretely and pictorially; then transition to solving them mentally:
10 + single digit such as 10+7 = 17, 10 +3 = 13
Multiple of 10 + single digit such as 20 + 4 = 24, 40 + 8 = 48
Multiple of 100 + single or double digit such as 100 + 5 = 105, 200 + 30 = 230, 500 + 25 = 525
1 more, 10 more 100 more as well as 1 less, 10 less, 100 less
Add to numbers with 9’s such as 90 + 10, 290 + 10, 1900 + 100
Addition and Subtraction:
Decompose and then add or subtract
Break numbers apart by place value and follow operation (horizontal application)
Show regrouping with subtraction
Applies to decimals too
Solve in parts without “carrying” the digits. This gives students a chance to develop the full understanding of the value of the digits (vertical application)
Instead of rules about digits bigger than 5 or less than 5, rounding using a number line helps a student think about place value and where the target number falls between two benchmark numbers. Ex.: 175 comes between 100 and 200, or 175 comes between 170 and 180.
Next post will be ways to emphasize place value when solving multiplication and division problems. If you have some ideas to share, please feel free to comment.
I appreciate all of my faithful followers the past 5 years! Thank you for viewing and passing this along to other teachers or parents. I hope you all have a restful holiday break!!
Students know how to count orally, can subitize, can count on one more, know ordinal positions, and have cardinality (know that the last number counted is the number of objects). These are usually prerequisites with number sense before introducing the symbolic representation and base ten understanding. Our system is based on the repeated groupings of ten. This is a follow up from my last post (Place Value Part 1 — counting).
There are two levels of understanding place value symbols.
Place value: In the number 23, the 2 has a place value representing the tens place.
Face value: With 23, the 2’s value is 2 tens or 20.
This shows 23 and 32 are not the same amount even though they use the same digits. Conversely, this same understanding applies to decimals. For example, .4 is the same value as .40.
Here are some ways to help students develop place value competence. Below this section are activities to help teach and practice:
Organize objects into groups when counting.
If given a group of 72 objects, do students try to count them one at a time (and probably losing count somewhere along the way)? Or, can they make specific size groups such as 10’s to see they have 7 groups of 10 and 2 extra?
If given a group larger than 100 (like 134), do they note that 13 groups of ten is the same as 130?
Partition numbers into groups based on powers of ten (ones, tens, hundreds).
Students learn that 52 = 5 tens, 2 ones = 50 + 2
Students learn that 348 = 3 hundreds, 4 tens, 8 ones = 300 + 40 + 8
Students learn that .45 = 4 tenths, 5 hundredths = .4 + .05
Realize the relationship among the different places. Using the number 67 . . .
Most frequently it is represented as 6 tens, 7 ones.
But it can also be represented as 5 tens, 17 ones (which by the way is crucial to understanding the regrouping process for subtraction. Show your students a problem in which regrouping is needed for 67, with the result of 5 tens 17 ones. Ask them if 5 tens, 17 ones = 67? How many think no?).
67 can also be represented as 4 tens 27 ones, and so on.
A clear understanding of these three guiding concepts of place value are very important and you will do well to thoroughly help students experience them before embarking on number operations.
Some activities to help with the above:
Provide objects for students to count: beans, cubes, tiles, candy, popcorn kernels, etc. Depending on the size of the objects, give directions such as “take a handful” or a spoonful, or a cupful. Make it a competition with a partner, who has more? who has less? Provide different containers for students to compare which holds more or less (and thus, working a little on measurement standards and conservation of space understanding).
Use base ten ones units. Give an amount and after making piles of tens, have student trade each pile for a tens rod. How many tens? How many ones?
Build given numbers with base ten pieces. Example: “Build 47”
Match pictures with expanded form using task cards.
I recommend using place value pieces for KG and 1st graders because the pieces show the relationship (a tens rod is 10 ones units stuck together, the 100’s flat shows ten 10s and 100 units). For 2nd and up, it might be more feasible to use the place value disks because these allow students to work with larger amounts. I know I never had enough hundreds’ flats to build larger numbers. Here’s a link to place value disks on Amazon: Place value disks
Use place value number strips that layer: 2000 + 500 + 30 + 8 when laid on top of each other shows the number 2,538. See sample here at Amazon: Place value layered strips
Work on mental math thinking of adding tens and ones: 10 + 2 = 12, 30 + 5 = 35, 70 + 6 = 76. I often use this technique to show visually what we want our brain to do. These are tens and ones layered strips I made, available here for free: Digit cards 0-10 and 10-100
Practice making exchanges with place value pieces. Build a number the standard way (67 = 6 tens, 7 ones). Then exchange a ten for ten ones. Verify that 5 tens 17 ones = 67.
How many ways can you show this number? (Give a number and provide manipulative for students to check and list.)
Using base ten rods and flats, verify that 15 tens = 1 flat + 5 tens (100 + 50)
Show how number changes by changing the ones or the tens.
What is 10 more? 10 less?
Use a 100 chart. Select a number and find what ten more or ten less would be — also 1 more, 1 less. Use the chart to show ways to count by tens other than the traditional 10, 20, 30, etc. Try 27, 37, 47, 57. Practice counting forward as well as backwards.
Important tip when using base ten manipulatives(from personal experience):
When students are using base ten pieces, don’t allow random placement of the ones on their work mat. Why? This often means a student may not have the correct amount because they miscounted their “mess.” It also means the teacher has to spend more time when circulating the room to count their “mess” to check for correctness.
The ones cubes are organized!
All it takes is noticing the student(s) who likes to organize their ones pieces. Their 5 pieces look like the dots on a dice or how it might look on a ten frame, or the 9 ones are in 3 rows of 3. As the teacher, you can show others how this is so helpful to you (and them) to arrange them in some kind of order. Knowing they get some choice in the order is key. And noticing how different students show the same amount in different ways is an ego booster to most students. Finally, it really does help boost students’ understanding of number bonds when they show 5 can be 4 and 1, or 2 and 3, or 2 and 2 and 1. That’s a two for one, right?
Enjoy your place value lessons — and share some you think would also be helpful!
Yes, you can even use ten frames to teach multiplication concepts! Here are my mini ten-frames with dot cards from 1 – 10: Click HERE to get a free copy. These are helpful to use, especially if you don’t have enough tens/ones blocks . . . or you prefer manipulatives that are slightly easier to manage. These provide a strong connection to place value, and the commutiative and distributive properties.
I recommend two sets of the cards (1-9) per student. Each set has multiple copies of the same number. They can be laminated, cut, and placed in a baggie for ease in handing out and storage.
Single digits (basic facts):
For the problem 3 x 6, the ten frame is really helpful for the student to see 3 x 6 is almost like 3 x 5 with one more group of 3 added on (by being familiar with the fact that the top row on a ten frame is 5).
Because of the commutative property, I know these two facts will have the same answer. But which of these below do you think might be “easier” to solve? Students don’t often know they have a choice in how they can use the numbers to their advantage!
6 x 3 (six groups of 3)
3 x 6 (three groups of 6)
Double digit x 1 digit:
Use of these also provides a strong connection of place value and multiplication. Notice how students can see the breakdown on the 4 x 12 problem (4 groups of 12 = 4 x 10 plus 4 x 2). Great introduction to the distributive property of multiplication!
Here is where application of the commutative property also comes in handy. Which of the methods below would you rather use to solve: count by 4’s or count by 12’s? Again, show students how to use their strengths to decide which way to think about solving the problem.
Even though the number of total pieces might seem to be a little overwhelming, it definitely is worth the effort for a few lessons so students get a visual picture of the magnitude of the products.
12 x 4 (12 groups of 4)
4 x 12 (4 groups of 12): or 4 x 10 plus 4 x 2
Here are other ways to model multiplication problems with manipulatives like base ten rods or base ten disks.
Notice: 4 groups of 30 fand 4 groups of 2
4 x 32 = (4 x 30) + (4 x 2)
After students get practice using these manipulatives (concrete), then proceed to pictorial models to draw them as simply as possible. This will give them a good foundation to apply to the abstract (numbers only) problems. I always pitch for the CPA progression whenever possible!!!
I will pause a while for the summer and just post once a month until school starts up again. Take care, everyone! But please don’t be shy. Post your comments, ask your questions, etc.
In Multiplication, Part 3 I will focus on 3 strategies for double digit numbers: area model, partial products, and the bowtie method. Please also refer back to my Dec. 6th post on Number Talks for 3rd-5th grade where I mentioned these and other basic strategies for multiplication. I highly recommend helping students learn these methods BEFORE the standard algorithm because it is highly linked to number sense and place value. With these methods, students should see the magnitude of the number and increase their understanding of estimation and the ability to determine the reasonableness of their answer. Then, when they are very versed with these methods, learn the standard algorithm and compare side by side to see how they all have the same information, but in different format. Students then have a choice of how to solve. Try my “Choose 3 Ways” work mat as bell work or ticket in the door. Get it free here.
Area Model: This method can be illustrated with base ten manipulatives for a concrete experience. Remember the best methods for student learning (CPA) progresses from concrete (manipulatives) to pictorial (drawings, templates, pictures) to abstract (numbers only). Using a frame for a multiplication table, show the two factors on each corner (see examples below for 60 x 5 and 12 x 13). Then fill in the inside of the frame with base ten pieces that match the size of the factors. You must end up making a complete square or rectangle. This makes it relatively easy to see and count the parts: 60 x 3 and 5 x 3 for the first problem and (10 x 10) + (3 x 10) + (2 x 10) + (2 x 3) for the second. I’ve included a larger problem (65 x 34) in case you are curious what that looks like. The first 2 could be managed by students with materials you have in class, but I doubt you want to tackle the last one with individual students – nor do you probably have that many base ten pieces. A drawing or model would be preferred in that case. The point of the visual example is then to connect to the boxed method of the area model, which I have shown in blank form in the examples . . . and with pictures below. I also included a photo from another good strategy I saw on google images (sorry, I don’t know the author) which also shows 12 x 13 using graph paper. Continue reading →
For 1st -3rd grade students: Refer to “Number Talks Part I” (posted Nov. 12, 2016) for ways to conduct a Number Talk with KG and early 1st grade students (focusing on subitizing and number bonds). For students in 1st – 3rd grade, place extra emphasis on number bonds of 10.
Problem: 8 + 6
Problem: 15 + 9
Problem: 18 – 9
Problem: 35 + 22
Problem: 99 + 6
Problem: 200 – 48
Write a problem on the board, easel, or chart tablet with students sitting nearby to allow for focused discussion. Have the following available for reference and support: ten frame, part-part-whole template, base ten manipulatives, and a 0-100 chart. Present addition and subtraction problems to assist with recall of the following strategies. If time allows, post another similar problem so students can relate previous strategy to new problem. Students show thumbs up when they have an answer in mind. The teacher checks with a few on their answer. Then he/she asks, “How did you solve this problem?” The teacher writes how each student solved the problem.
Reference these videos on youtube.com for examples by clicking on the link: