Enter two fractions, then watch as they are added together. "Why is the sum partitioned like that?"

This uses an area model approach for multiplication. Use as a tool to support students as they learn multi-digit multiplication, or take an inquiry approach by clicking "Randomize" and working slowly, carefully to figure out the two factors. What's the fewest pieces of information you need?

## Area Model of Multiplication

Use the sliders to adjust the factors, then "Slide me" to see the area model being constructed. This particular approach is meant to give additional visual support to students as they gain understanding of how tens x tens = hundreds, tens x tens = tens, and ones x ones = ones.

## Common Multiples

A simple tool for students who need support when finding a common denominator (or any other context in which they are looking for a common multiple).

## Decimal Multiplication on the Number Line

Shows one factor and the product on the number line. Adjust the second factor (by increments of 0.1) and see the result on the product. Meant to develop understanding of 1 as a threshold for bigger/smaller product.

## Deciding Multiplication or Division

This applet, while designed primarily to accompany contextual multiplication and division tasks, will work to illustrate any situation in which students are multiplying or dividing whole numbers. The main thinking behind it is that students can struggle with conceptualizing the three components of the majority of these problems: total number of objects, number of groups, and objects in each group. By providing a visual to accompany their thinking, we can help them to see the math they are doing, whether correct or incorrect based on the context. A specific way of using this would be to pair it with the Mathematical Language Routine Co-Craft Questions. Provide the beginning of the stem (like "Four friends have a recipe that makes 15 cookies"), and use this to help students consider different questions they could pose based on it.

## Dividing a Whole Number by a Fraction

Shows a quotitive approach for dividing a whole number by a proper fraction.

## Division on the Number Line

This resource is intended to get students to use proportional reasoning to estimate quotients. Use the sliders to generate random dividends and divisors, or input your own numbers. Start by dragging the numbers to the correct locations, then predict the number of "jumps" to land on the dividend.

## Division Visuals

This is designed to help students connect division algorithms to the "action" that is actually taking place. As students develop repeated subtraction or partial quotient approaches, use this to support them thinking "What is happening here?" as they represent the problem mathematically and attend to the meaning of the quantities.

## Estimating Fraction Sums and Differences

Place fractions sums and/or differences on the number line. You'll have (mostly) different denominators, so you'll have to use number sense to estimate. (These should be relatively QUICK placements!) Feedback provided.

## Estimating Sums

Use the sliders to adjust the addends, then consider the size of the sum relative to the two bounds you are given. "Would the sum be closer to ___ or ___? How do you know?" As the level of precision increases, the level of mathematical reasoning increases with it. And the GRAND purpose behind this is to get students using an estimation strategy called front-end addition. Consider building this around a central question of "How do we make our estimates more precise?"

## Factors of a Fraction

Use this applet to engage in some pre-algorithmic thinking. Enter any fraction, then use the sliders to see it arranged in various ways. Make connections to multiplication, division, addition, and (perhaps) even subtraction. Notice and describe patterns!

## Fraction Path

What if we took a hundreds chart and replaced the whole numbers with fractions? Explore this idea by clicking "Add to path," then click on a square to show its values...but try to predict the value first!

## Move the Number

This activity is meant to develop flexible, creative thinking about numbers and operations. Your goal is to move a from a starting number to a target number, but there are many ways to do this! Adjust the sliders to control the bounds of the numbers involved.

## Multiplying Fractions

Shows an array/area model for multiplying two proper fractions.

## Multiplying Fractions with Simplification

An extension of Multiplying Fractions. Shows an array/area model for multiplying two proper fractions, then allows you to simplify the product. NOTE: The simplification doesn't work perfectly, but should work well enough for simple fractions.

## Multiplying Mixed Number Fractions

Demonstrates a visual model for finding products of mixed number fractions.

## Ordering Products

A simple activity in which you put multiplication expressions in increasing order. Adjust the sliders to change the number of expressions and the number of digits in the factors, focusing more on estimation for large products.

## Ordering Quotients

A simple activity in which you put division expressions in increasing order. Adjust the slider to change the number of questions.

## Relative Number Sizes Via Area

Designed to make a more concrete connection between dividend, divisor, and quotient. Use number sense to estimate the relative size of dividend and divisor, then use that to estimate the quotient.

## Seeing Divisibility

Enter a dividend, then adjust the slider to see it "divided" into groups and, if necessary, a remainder. If you wish, display the division equation and the related multiplication equation.

## Seeing Division

Enter a division expression, then select an approach to the division (partitive or quotative).

## Seeing Factors

Designed to give an understanding of factors and prime/composite numbers. Enter an integer, then see it arranged in arrays. Move the slider to see the resulting array for various divisors.

## Subtracting Fractions

This is designed to accompany students as they begin to develop algorithmic approaches to subtracting mixed number fractions. This uses a comparison approach. (Do you see the difference, 2 5/6, in the GIF?) Go slowly and really dig into the conversation when you "regroup" (like when 5 2/6 â†’ 4 8/6) so that students develop fluency with a strong conceptual base.