Multiplying Fractions by a Whole Number

Fraction concepts are taken up a level in 5th grade and many of the topics are very abstract and difficult for students to understand. Multiplying fractions by a whole number doesn't need to be one of these difficult concepts! By using the CRA (Concrete-Representative-Abstract) framework your students will link what they already know about multiplication to the multiplication of fractions and will be successful in no time at all!

Concrete Multiplying Fractions by a Whole Number

Before you begin with fractions, anchor your students back into what it means to multiply. Give them a pile of blocks and ask them to model something super simple such as "4x3" your 5th graders should quickly and easily be able to model this problem. Ask your students about what they built and how they know it represents the equation. 

Go ahead and get out your pattern blocks. They are the PERFECT fraction representation. Let the hexagon represent a whole and allow your students to figure out which blocks represent 1/3 (rhombus), 1/2 (trapezoid) and 1/6 (triangles). Once your students are all set with their blocks, they are ready to begin! Don't have fraction blocks handy? Fraction circles would work just as well! 

Ask your students "If 4 groups of 3 blocks represented 4x3, how could we represent 4 x 1/3?" Allow students to use the rhombus pattern blocks to represent 4 groups of 1/3. The beauty of using blocks is that students can put these blocks back together to see both the improper fraction and mixed number that is created when fractions are multiplied together.

**This PICTURE is not a concrete model, but if you are creating this model out of fraction tiles, your students are working at the concrete level! **

Representative Multiplying Fractions by a Whole Number

Once your students are able to model 4 x 1/3 you want them to link this understanding to a representative model such as repeated addition. You may begin by asking your students to write a repeated addition equation to represent 4 x 3. This is easy for your students! Now, ask them to  use what they know about multiplication to write a repeated addition equation that represents 4 x 1/3. 

Abstract Multiplying Fractions by a Whole Number 

The concrete and representative steps of this activity allow your students to clearly understand what is going on when multiplying a fraction by a whole number. After your students have had a good deal of exposure at the concrete and representative level, give them a new equation such as 4 x 2/8 and ask your students what they *think* the product will be. You are looking for your students to make generalizations about their multiplication and fraction understandings and to be able to explain their thinking. After a student shares their thinking, ask all students to model with concrete materials or a repeated addition equation to confirm the product!

I have created a set of playing cards that includes multiplication equations, visual models, repeated addition and the resulting products. Your students can play so many traditional card games with this deck of cards - and I have included the instructions for 5 games to get you started! This resource is PERFECT for exploring the link between repeated addition and fraction multiplication. Click HERE to check it out! 


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Subtraction With Regrouping for ALL Learners

It's about that time of year when second graders begin to delve into subtraction with regrouping. Right off the bat, I want to mention that the standard algorithm is NOT a part of the second grade standards! That being said, so many of the programs that we use expect that students use the algorithm and even "exposure" sticks with our students. Luckily there is a way to teach subtraction with regrouping that honors the developmental needs of second grade students.

The C-R-A (Concrete, Representative, Abstract) approach is the way to go here! Start students with hands on materials and, at first, don't expect any notation at all in terms of the standard algorithm, and allow students to explore the crossroads of operation and place value.

Concrete Materials 

When choosing concrete materials for your students to use when adding or subtracting multi-digit numbers consider the difference between groupable and pre-grouped models. Base ten blocks are a  common tool used to represent subtraction for early learners. There is nothing wrong with using this tool. Be aware, however that base ten blocks are a pre-grouped model and are not appropriate for all learners. 

Some of your students may have difficulty understanding why you would "trade a ten for ten ones" when using base ten blocks. Consider starting your instruction using a groupable model such as linking cubes.
You can see in the example above the strength of a groupable model. When subtracting 47 from 74, sutdents were able to break a ten into ones rather than "trading" one item for another. This is more true to what actually happens when subtracting!

Representative Models

Once students understand what is happening when regrouping or unbundling in subtraction, they may be able to transition to subtracting without using a physical concrete material. One representative model that many students find success with is the place value chart. 
An especially effective way of transitioning to a representative model is to ask students to use a concrete material and to "draw what is happening to your blocks". In the 3-digit example above, students would start by building 675 with a concrete material such as base ten blocks. Students would then draw the 6 hundreds, 7 tens and 5 ones that they have built. As they manipulate their blocks, they would also reflect these changes in their drawing as well. 

As with concrete materials, there are nuances in representations as well. Your students may start by showing hundreds, tens and ones as a place value drawing because they are a more direct representation of place value. As your students become more confident, they may be able to simply draw dots in each column of the place value chart to represent the number in each place value. 

Abstract

I'll say, once again, that the standard algorithm is NOT expected in 2nd grade! That being said, programs, parents, and older brothers and sisters are all going to provide exposure to the algorithm. If you would like to introduce the algorithm in a more conceptual way, consider asking your students to "record what is happening with numbers". This can be done at either the concrete or representative stage. As students are regrouping or unbundling with blocks, they can show the changes in each place value numerically. Again, as students are using place value drawings to show unbundling, they can record these changes as they would be written in the traditional algorthim. 


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