Missing Dimensions Homework (2/22)

Hi! We reviewed the homework in class today, using a systemic guess-and-check strategy. Here are a couple of screen shots:

Area Model for Multiplying Fractions by Fractions

This model should be familiar to the students from our work with multiplying decimals.  Here is a screenshot from today:

Steps:

1. The first factor: Divide the square vertically by the second denominator (5), label the fraction at the bottom (4/5), and shade the fraction with the lighter of your 2 colors.

2. The second factor: Divide the square horizontally by the first denominator (6), label the fraction on the left (1/6), and shade only the intersection of the of the both factors with the second color.

3. Finding the product: The double-shaded region represents the product (4/30). The denominator of the product is the total number of cells, shaded and unshaded (30). 

** Sometimes students shade the entire second factor, all the way across, in step 2, before finding the intersection. While this is not incorrect, I have found that it can sometimes be more confusing.

Another problem:

* For this problem, we started by shading 4/5. Since multiplication is commutative, it does matter which factor you start with..

Using Constant Difference for Subtracting Mixed Numbers

We have been working on a new strategy, constant difference, for subtracting mixed numbers in which the fractional part of the minuend is less than the fractional part of the subtrahend. In the standard algoritm, you would have to regroup from the wholes and make an improper fraction in the minuend in order to subtract the fraction. In constant difference, the goal is to add an amount to the subtrahend (the amount being subtracted) in order to make it a whole number. Then you add the same amount to the minuend (the amount you are subtracting from) in order to keep the difference constant (the same). You now have a new minuend and a new subtrahend, and a much simpler equation to solve. Below is an example taken from class:

Using an Open Array to Model Decimal x Whole Number Multiplication

Here is a screen shot for how you could use an open array to model 18 x 8.7. First you would draw an array based on how many digits are in each factor (ie two digits x two digits would be a 2 x 2 array, 3 digits x 2 digits would be a 3 x 2 array). Then you would separate each factor by place value along the sides of the array, and solve for each cell of the array by multiplying the side lengths of the cells. Finally you would add all of the partial products in each cell together.

Here is one for 56.1 x 23, which is not as nicely color-coded:

Properties of Addition and Multiplication

For help with the properties of multiplication, including a quiz, click on the image below:

For help with the properties of addition, including a quiz, click on the image below:

Using the Distributive Property to Solve Division Problems

Using the Distributive Property is a  great strategy for solving division problems - and you can use an open array as a model. When you use an open array in division, the dividend is the area, the divisor is one side length, and you are solving for the other side length (this will be your quotient). You can use the Distributive Property to break the dividend into multiples of the divisor (this is really important because it allows you to divide easily by the divisor!),  and then add the distances along the unknown side to find the quotient.  I like to make a mini array with the area next to my large array to keep track of total area because I will be breaking it apart in my large array. Here are some screen shots from today's lesson - hope they help!

Modeling Multi-digit Multiplication

Hi! Here is a screenshot from a lesson relating the open array to the standard algorithm. Hopefully, you will be able to see the relationship between the two as highlighted.