Using Cuisenaire Rods and Base-Ten Blocks for Decimal Computation
Using Cuisenaire Rods for Decimal Calculation
Did you know that Cuisenaire rods can be used for decimal calculations? Read on to see how they provide a concrete tool for adding and subtracting decimals. Teachers often use base-ten blocks to introduce decimals by changing the value of the hundred flat to one.
The ten rod in a Cuisenaire set is the same size as the ten block so they are interchangeable. To avoid having all those little unit cubes to work with, use the Cuisenaire rods to represent the hundredths.
Using Cuisenaire Rods to Add and Subtract
Since students are typically around 4th or 5th grade when they begin working with decimals, they can usually understand the shift from the flat representing 100 to 1.
While you can use base-ten rods to add and subtract decimals (the hundred flat would equal 1, the ten rod would have a value of one-tenth and the unit cubes would now be worth one-hundredth) I chose to use Cuisenaire rods when working with 5th graders who were beginning to add and subtract tenths. I gave the ten rod a value of one. That made the unit rod a value of one-tenth.
Students can discover this on their own by using unit cubes to create a same-sized train and then using what they know about fractions to determine the value.
Once students identify that there are ten equal-sized pieces that make the whole (1), they should be able to transfer their understanding of fractions to label these as 1/10 and finally rename as 0.1. Allow time for students to repeat this process of using the unit rod (one-tenth) to find the value of the remaining rods.
Having shifted from whole numbers to tenths, students can now begin to use the rods for computation. Placing the 3-tenths rod together with the 5-tenths rod, for example, provides students with a concrete model to solve 0.3 + 0.5.
You can also use this model when the addends total more than 1.
As with subtracting whole numbers, students can use this manipulative to subtract tenths by finding the difference between the two lengths.
What About Multiplication?
Without a smaller unit in the Cuisenaire rod set, assigning a value of 1 to the ten rod would not be a good visual model for multiplying decimals.
You could, however, bring back the hundred flat from the base-ten blocks and assign a value of one to it. Then the ten rod would be worth one-tenth and the unit cubes would have a value of one-hundredth.
We need to think of a problem like 0.3 x 0.3 as three-tenths of three-tenths. It’s easier to show this on a representation of a hundred flat ( which is now worth one) than with the actual blocks.
Start by shading in three-tenths of the whole. You can also use slanted lines.
Next, use another color, to shade in or add lines to three-tenths in the other direction.
The product of the two numbers is the part where the two colors overlap.
In this case, that area is nine-hundredths. Students can visually see this with this model. They are taking three-tenths of three-tenths. In this example, students have broken each tenth into ten parts, hundredths and then they have taken three-tenths (three parts) of each tenth.
A Model for Division
Using base-ten blocks with the hundred flat representing one also works for division. In order to find 0.8 ÷ 0.2, students should draw a border around the total that is to be divided, in this case, 0.8. Next, determine how many 0.2s make up the total. Here, you can see that there are 4 two-tenths in eight-tenths, so 0.8 ÷ 0.2 = 4.
The next image shows how to apply this process to larger numbers, such as 1.75 ÷ 0.35. With the dividend marked, students can color in 0.35 sections. It takes five 0.35 sections to make 1.75.
I hope you have found these suggestions for decimal calculations helpful. They are an important steps in the Concrete-Representational-Abstract sequence before jumping into algorithms.
7 Good Reasons to Teach the Area Model for Multiplication
Area model can be used from beginning multiplication to algebra
Helpful for division as well as multiplication
The model works for fractions and decimal multiplication
Are you looking for a way to teach multiplication before your students learn the standard algorithm? The area model is a great tool that you can introduce your students to in third grade. One great thing about this representation is that it can be used with multiple skills. I’ve also included some ways that you can use Cuisenaire rods while developing these skills.
Building Arrays With Tiles Connects to the Area Model
When students first begin working with arrays, they often build them with cubes or counters. This work can help them see they can use repeated addition to find the total number of objects in the array. Arrays that are built with tiles or cubes, such as Cuisenaire 1 rods, can be pushed together into a rectangle. Students can begin making connections between the 3 by 4 array that they built and the rectangle they made with a length of 4 tiles and a width of 3 tiles. They will also see that the sum they found for the array using repeated addition will be the same as the product or area of the 3 by 4 rectangle. Of course, when students are drawing arrays, dots make more sense, but tiles are a great concrete tool to help students connect arrays and multiplication with area.
Students Make a Connection to a Representational Model With Tiles or Cuisenaire Rods
Students can also make a connection between the arrays they are building and the representational model. For example, four dots in a row can be represented by a rectangle with length of four units and width of one unit. Take the time to show the representation each time students build a rectangle with tiles or Cuisenaire rods. Soon students will be able to do it on their own. Once students can draw these representations, they can move beyond drawing and counting squares. As we all have probably seen, student arrays and drawings of tiles can get a little uneven. The drawings will make more sense to students once they understand the connection between the dots or tiles and the side lengths (factors).
Progression from array to area model
Students Understand Area as it Relates to Repeated Addition and Multiplication
Students learn to find the quantity in an array with repeated addition or with multiplication. Then they recognize the area of a rectangle that’s been partitioned into squares can be found in the same way. I’ve been working with a third-grade class that learned about area right after learning multiplication.
I’ve noticed two things. First, many of the students saw the connection to repeated addition and multiplication right away. This made moving to rectangles with just measurements instead of squares to count fairly quick. While not all students had their multiplication facts memorized, they understood that counting individual squares was not necessary. Second, there was no confusion of area with perimeter. The students connected area to multiplication. In the past, I’ve experienced, and you likely have, too, that students confuse area and perimeter. We try to give them catchy ways to remember which is which. I always think back to a conference I attended where Greg Tang said if we teach students to use the area model for multiplication, they will never confuse area and perimeter.
Connecting area to repeated addition and multiplication
The Area Model Creates a Great Visual for the Distributive Property of Multiplication
Once students have an understanding of the area model for multiplication, they can use it to show the distributive property which can be helpful when multiplying larger facts and multi-digit numbers. The program that my district uses teaches students that breaking apart one factor can help when multiplying numbers greater than five. As instructed by the program, students draw arrays and then divide the arrays. While that is certainly one way to go, it can require students to draw a lot of objects. This can be messy and students can miscount.
I taught students to build the rectangle with Cuisenaire rods and physically break it apart (concrete). Most students have mastered twos and fives and can skip-count for facts they don’t know yet. Next, they drew the area model represented by the rods (representational). Teaching this in third grade gives students a tool that many find helpful for multiplying larger numbers in fourth grade and beyond.
Cuisenaire rods, the area model, and the distributive property
The Area Model Can Be Used for Division
Students can combine what they know about the area model and their understanding of the relationship between multiplication and division to use it for division. Cuisenaire rods are a great concrete manipulative for representing division problems using the area model. To find the quotient, use rods the length of the divisor to create a rectangle with an area that is the same as the dividend. For example, in the equation 24 ÷ 8 = ? students would make a rectangle with an area of 24 using the 8 (brown) rods. The length of the rectangle would be 8. The width would be 3 since there are three of the 8 rods. So 24 ÷ 8 = 3.
When students draw the rectangle with the area and given side length, they can extend this concrete model to the representational stage. If the sides shown below with dotted lines are erased, students can begin to transfer this model to the traditional algorithm for division.
Division with Cuisenaire rods
What about division problems that result in a remainder?
Yes, you can use the area model and Cuisenaire rods for these problems, too. First, make the largest rectangle possible with the rod the length of the divisor. For example, for the problem 40 ÷ 7 use five of the black 7 rods. If they are not strong with multiplication facts, students can skip-count by sevens until they get as close to 40 as possible without going over. They will then see that they have a rectangle with an area of 35 with one yellow 5 rod left to make the starting dividend of 40. The visual shows that 40 ÷ 7 = 5 R 5.
Using the Area Model to Divide Larger Numbers
Understanding how to use this for division can be helpful for students who are not quite ready for the standard algorithm. Here are two examples of using it with larger numbers. I was in a fourth-grade classroom where a student was using the area model to divide 135 by 5. He set up his problem as shown below, with only the 135 and the 5 in the lower, right corner in place.
Then, he told me he would multiply 5 x 11 and see how many times he could take that out of the total, an area model to show taking out partial quotients. Finally, he added the partial quotients, shown in orange, to show that 135÷5=27.
Area Model and Partial Quotients
Here’s another approach. The dividend has been decomposed into numbers that are easily divisible by five, 100+30+5. Each is in a separate section of the rectangle. As with the other example, each partial quotient is written above the appropriate section. The partial quotients are then added to find the quotient, 27.
Area Model Division
But what if it’s not such a neatly divisible problem?
Use the same process. Instead of decomposing the dividend into the expanded form of the number, as shown above, find numbers that are multiples of the divisor. A student could start with 180 or just use 60 depending on their comfort level. For this example, I started with 120.
Area Model Division
The Area Model Can be used to Multiply Fractions and Decimals
Extend the area model to multiply fractions. Students will already be familiar with placing one factor along the width and one factor along the length of a rectangle. When multiplying larger numbers, they learned to decompose them into their place values. They will apply that concept and their early work with decomposing fractions to set up the area model for multiplying fractions. In the picture on the left, colored lines show the product of 3/4×1/2. The blue lines shade one-half of the rectangle while the red lines shade three-fourths of the rectangle in the opposite direction. You’ll notice that this results in three-fourths of the one-half or three-eighths of the rectangle being marked with both colors.
The other rectangle shows how to use the area model to multiply mixed numbers. Decompose each factor into a whole number and a fraction. Then multiply each part. Finally, add the partial products to find the total product.
Using the area model to multiply fractions
Multiplying Decimals
This model is helpful when multiplying decimals, too. Students will decompose each factor and multiply each part in the same way they did with whole numbers. Then, they will add each partial product to get the product of the two decimals.
Multiplying Decimals
Beyond Elementary School, Using the Area Model in Algebra
If students have the area model in their tool bag they will have it as an option in algebra. The model works when multiplying binomials like (a + 3)(a + 2). In the picture below a Cuisenaire ten rod represents a and the one rod represents 1. A base-ten hundred flat replaces 10 ten rods.
Area model in Algebra
Check out this area model simulation from the University of Colorado for some games to practice this skill.
I hope you’ll agree that the area model is a great tool for students to have.
The program my district uses starts third grade with lessons on rounding. This year I received inquiries from a few schools asking if it was ok to spend a few more days on rounding to the nearest ten and then a few days rounding to the nearest hundred. Every year students struggle to round and teachers resort to pulling out rhymes and their pictures of the rounding roller coaster.
While mnemonic devices can be helpful, rhymes can sometimes create more confusion. Here are two common examples found online:
Five to nine-
Climb the vine!
Zero to four-
Slide to the floor!
4 or less
Let it rest!
5 or more
Let it soar!
Neither of these rhymes mentions anything about which place value these rules apply to or what to do with the digits to the right of the place to which you are rounding.
This rhyme explains what to do with those digits but it does little to build conceptual understanding. It won’t help at all if a kid can’t remember the rhyme or which direction to look “next door.”
Find your place
Look next-door
Five or greater,
Add one more
All digits in the front stay the same
All digits behind, zero’s your name
Strengthening Number Sense
So what’s a better way to teach rounding?
Using chunks of the number line can strengthen students’ understanding of rounding. Number lines help students develop a visual image of number magnitude and order.Let’s start by rounding 27 to the nearest ten. First, we need students to identify the tens that “bookend” the number 27 by asking “What two tens does 27 fall between?” Once they identify that it falls between 20 and 30, I would draw an open number line with those two numbers identified.
I would then ask them to help me mark 27 in the correct place, using the halfway mark, 25 as a benchmark. Visually, it would be easy for them to see that 27 is closer to 30 and would therefore round to 30.
This strategy, once learned, would stay with students long past an easily forgotten rhyme. You might ask how a rounding roller coaster is different. After all, it’s just a curved number line, right? In my experience, the rounding roller coaster usually doesn’t have equal distances between each number like a number line should. If you have one that does and the middle is the halfway point, great. If you’re looking for another way to develop number sense, read on.
Bookends
In addition to lacking a strong visualization of numbers on a number line, one of the things that I see that causes difficulty for students when rounding to larger numbers is the inability to correctly identify the bookend numbers. If, for example, they are asked to round 243 to the nearest ten, they might say that the two tens that 243 falls between are 200 and 300 instead of 240 and 250. I see this become more of a struggle as they progress to larger numbers, especially when rounding to smaller place values, e.g. 1,629 rounded to the nearest ten.
You can strengthen this skill by asking students to give you the bookends for a given number when you have a few extra minutes, such as when you are lining up or waiting for dismissal. You might give a student a number like 841 and ask them for the two hundreds or tens that it falls between.
Finding Bookends Activity
Here’s a way to get students up and moving while providing some practice identifying those bookend numbers. Create sets of index cards where each set will have a pair of bookend numbers and one or more numbers that will fall between those bookend numbers, depending on the number of students in your class. Try to make them unique to avoid confusion as students try to find their group. Pass the cards out randomly and have students find the students who have the other cards in their set.
Here’s a set of 2-digit numbers to get you started! Make a copy of this Google Doc and edit for 3-digit numbers or decimals.
Read-Alouds to Support the Counting Stages and a Google Slides Freebie!
Do you use counting read-alouds to support your students as they are developing counting skills? Marilyn Burns (2015) says that “Children’s books can spark students’ imaginations in ways that exercises in textbooks or workbooks often don’t. Connecting math to literature can boost confidence for children who love books but are wary of math. And students who already love math can learn to appreciate stories in a whole new way.”Keep reading for some great counting read-aloud books to share with your students.
Counting to Answer “How Many?”
Doug Clements’ and Julie Sarama’s (2017/2019) learning trajectories work shows that around four to five years of age, children develop the ability to count five to ten objects and answer “how many?”
One Gorilla by Anthony Browne is a simple counting book that counts a different type of primate from 1 to 10. Its large numbers and simple pictures make it a perfect book for counting along.
Follow along as embellished ingredients are added to create a salad in 1 Big Salad: A Delicious Counting Book by Juana Medina.
Who doesn’t love pop-up books?! How Many Bugs in a Box? by David A. Carter is a fun counting book that combines two kid favorites- pop-ups and bugs.
Get lots of practice and fun counting everything but monkeys in Count the Monkeys by Mac Barnett and Kevin Cornell.
Ordinal Numbers
While Clements and Sarama (2017/2019) include ordinal numbers in the comparing and ordering numbers trajectory, I am adding Eric Carle’s 10 Little Rubber Ducks to this list of counting books. Here we see what happens when ten rubber ducks get lost at sea. From the first to the tenth, each has their own journey.
Read-Alouds for Counting Backward
By six, children can count backward from ten. Also around this time, children are developing the ability to count on from numbers other than one, knowing immediately the number that is one more. Their gained familiarity with the count sequence also allows them to immediately know the number just before any given number.
Clever mice outsmart a greedy snake in Ellen Stoll Walsh’s Mouse Count, embedding counting to ten and back.
Count backward On the Launch Pad in the book by Michael Dahl. This book counts down from twelve.
In Ten Little Ninjas Miranda Paul follows a familiar story path as she puts ten children to bed one by one.
Read-Alouds to Practice Skip-Counting
As children secure their knowledge of whole number counting to 100 and beyond, around age six to seven, they are developing skip-counting skills.
Two Ways to Count to Ten: A Liberian Folktale Retold by Ruby Dee is one of my favorite stories about the benefit of skip-counting. Watch as the antelope shows that physical strength isn’t always the best way to win a contest.
One is a Snail, Ten is a Crab by April Pulley Sayre and Jeff Sayre explores ways to make numbers with different combinations of animal feet, for example five is a dog and a snail. It goes on to count by tens. An extension activity might be skip-counting by other numbers shown in the story.
Making Books
There’s no shortage of lists of counting read-alouds on the internet, but hopefully, this one will remind you of a few classics you may have forgotten and introduce you to a few new favorites.
After reading counting books to your students let them make their own! They can staple paper together, number each page and draw matching representations for each number. Students can write the number, such as six, or I have six ____. You can provide sentence stems or have them pre-printed for students who need them.
You can also grab this Google Slides book freebie. Students will use their device’s camera to insert pictures that represent each quantity from one through ten. Images could also be inserted from Google Drive if a camera is not available. Finally, they will type the name of the item to complete the sentence on each page.
How do children develop subitizing skills? When I was a child, before video games, cell phones, and an abundance of television options, we played board games. Candy Land, Hi-Ho Cherry-O, Trouble, and Life were some of the games that gave us opportunities to develop our math skills from counting and one-to-one correspondence to counting and exchanging money.
Games that involved dice, playing cards, and dominoes strengthen the ability to subitize. Subitizing is being able to identify a small quantity without counting.
People who can subitize instantly recognize this as 5
or this as 6.
My sisters and I also played jacks. Subitizing allowed us to find “twosies” and“sevensies” without counting. Understanding of number is developed as students learn to subitize.
Subitizing in the Classroom
With all the digital options to occupy children’s time, games like these aren’t as much a part of childhood as they once were. Over the last several years, subitizing has made its way into primary classrooms. Teachers are providing experiences with subitizing and opportunities to practice the skill. You might hear a teacher ask “How did you know that was a five?” and a student respond “I subitized,” or “I saw a two and a three.”
Recognizing small numbers of objects, usually up to 4 or 5, instantly, such as on a die, is perceptual subitizing. Conceptual subitizing, on the other hand, is recognizing smaller subgroups and combining them. When dots are arranged in familiar patterns such as dice patterns or ten-frames, it becomes possible to subitize larger numbers.
Small paper plates are an inexpensive way to make subitizing cards. Dots can be drawn on or stickers can be used to create each set. Familiar patterned sets like those on dice are a good place to start. Adding one or two dots allows practice with counting on, while combining two smaller patterns develops conceptual subitizing skills. Give students roughly three seconds to view the cards and then ask them to share how many dots they saw. Show the card again and ask them how they saw it. One student might simply say it was like the five on a die while another may say it was a four with one more dot in the middle.
Are they really subitizing when they play?
When children have enough time to count the objects in a group, as they can when playing games, they are not subitizing. With repeated experiences with dice, cards, dominoes, and ten-frames, however, children learn the patterns and will increase their ability to recognize familiar quantities without having to count them.
Games in your math centers are one way you can provide students with opportunities to practice this important skill. Games that use dice, dominoes, or playing cards allow students to subitize while playing. You can get Roll and Coverfree!
From what I’ve been told, Cuisenaire Rods were popular in classrooms in the 1960’s and 1970’s, though they were never present in my classrooms. When I first learned about Cuisenaire Rods and began inquiring as to whether there were any in my district, I encountered teachers who had never heard of them and teachers who said, “I used to have a ton of them!” That led me to suspect that when older teachers retired and passed their classrooms on to new teachers, the Cuisenaire Rods were tucked on a shelf in the back of a closet. When discovered, they were discarded because no one knew what to do with them. I’ve been trying to change that.
I’ve encouraged schools to purchase the rods, provided professional development and included them in the math methods course I teach. Here, I will share how Iintroduce Cuisenaire Rods. I’ve done this as young as kindergarten, but it’s a good way to introduce the rods at any grade.
Getting Started
The first thing I like to have students do is sort the rods. Sometimes rods are in bins at atable and sometimes they are in individual bags. Either way works for this activity. I always allow students to explore the rods before beginning the sorting activity. After about five minutes, I ask them to sort the rods by color. They will end up with ten different-colored piles. Next, I ask them to mix the rods up and sort by size. They soon discover that they end up with the same ten different-colored piles! We discuss what this means-that all of the orange rods are the same size and all of the yellow rods are the same size.
Each time we use the rods, students are given a few minutes to explore. At the end of this time, students are instructed to build their staircase. Because students will build a staircase at the beginning of each lesson until they know the values of each color, I follow the sorting activity by doing so.
Once they each have a staircase, we wrap up by comparing rods, using colors. Number comparisons will come later. I model, for example, “The brown rod is bigger (or more) than the yellow rod” and “The red rod is smaller (or less) than the black rod.” Next, I show two rods and ask a student to “read” the rods. This is repeated as time allows.
Here’s a freebie you can use in your math centers to familiarize students with the rods and their colors.
Check back for more things to do with Cuisenaire Rods!
Recent Comments