Put Math in their Hands

I often talk to teachers about the importance of concrete representations for our students.  The use of concrete tools helps all students to gain a better understanding of the function at hand.  It also creates a visual representation that will be a reference for students to use as they advance into more advanced mathematical ideas.

In this post, I want to share some simple concrete representations that teachers can help students develop as they work to make sense of a new concept.

Doubles/Doubles +1: Both the rekenrek and ten frames help students to make more sense of what a double is and how we build from that to doubles + 1.

Doubles on ten-frame:  3 + 3 =6

Doubles + 1:  3 + 4 = (3 + 3) +1 =7

Doubles on rekenrek: 6 + 6 = 12

Doubles +1:  6 + 7 = (6 + 6) + 1 = 13

The rekenrek is a great tool for doubles facts between 10 and 20.  Student can see that all of the green beads have been pulled over, which is 10.  They only need to add on the white beads.


Addition/Subtraction within 100:  For this skill to take hold, I find beaded number lines and base ten blocks to be good concrete tools to use because they carry over so nicely into an open number line representation.  Open number lines are important for students to use to build number sense.

Addition on Beaded Number Line:  17 + 35 = 52

Addition with Base Ten:  17 + 35 = 52

Subtraction on Beaded Number Line:  44 - 18 = 26

Open number lines are important, but they are often too abstract for many of our students.  Combining the use of the beaded number line or base ten blocks with the creation of an open number line helps for it all to make more sense for the students.

Multiplication:  For learning multiplication facts, creating arrays is a nice way to start.  This should help students to see how repeated addition is connected to multiplication.  Any small object will work to help students create arrays.  Using the number card templates will allow them to create a visual more quickly at times when they need a visual but don't have time to build an array.

Multiplication with Base Ten Blocks:  22 x 23 = 506

The 22 and 23 were created at the top of this image and on the left side of the problem using base ten blocks.  Multiplying a blue ten and anothe blue ten gives you an orange 100.  The same continues until we end up with 4 hundreds, 10 tens, and 6 ones for a total of 506.

For multi-digit multiplication, building the array using base-ten blocks works well to connect the arrays that student made for facts under 100 with the longer problems that they are ready for now.  Besides--the base-ten blocks transition nicely into graph paper representations and finally into the area model of multiplication for multi-digit numbers.

Decimals:  The beaded number line can be used for addition and subtraction of decimals less than 1, and it can also be used to round and compare decimals.

Rounding on the beaded number line:  0.86 is closer to 0.9 than 0.8

Adding decimals on a beaded numer line:  0.6 + o.14 = 0.74

I hope that you find some of these ideas helpful and that you find ways to incorporate them into your explanation of numbers with your students.  There are many ways that can be used for different problems.  If you have a concrete tool that works well for teaching these skills, don't think you have to switch to one of these--do what works best for you and your students.  The key is to get the concrete math in their hands!  This way it will stick in their heads.  😊