Missing Factors

Throughout the multiplication unit, students have played games using array cards to become fluent with their multiplication facts. The array cards show students the number of rows and columns in an array, and then the students solve for the product or area of the array. They check their accuracy by turning the card over to find the product. 

4  x 6 = 24

This week, we played a Missing Factor game using the array cards. To play this game, students spread their cards out with the area and one dimension showing and their task was to figure out the missing factor or dimension.  They recorded the problem in two ways, as a missing factor multiplication problem and as a division problem. 4 x ___ = 24   and  24 ÷  4 = ___.   After solving, they turned the card over to find out if they found the missing factor. 

Measuring Volume

Measuring the volume of liquids and solids is a skill that takes precision and therefore takes practice. To give students more opportunity to practice, we completed a Measuring Volume Gizmo today. In the virtual lab students used beakers, graduated cylinders, and pipettes to transfer water, and then used a magnifying glass to zoom in on the meniscus and read the volume. 

There are two modes in the Measuring Volume Gizmo, Free Explore and Practice. In the Free Explore mode, students simply practice the skill of transferring water from one container into another and reading the volume in mL. However, in the Practice mode students are tested on their  measuring skills and given immediate feedback on their precision. 

By the end of the task, students felt more comfortable reading the volume of a liquid in a graduated cylinder, and were better able to measure the volume of a solid using water displacement. 

Here is one example of a task that students should be able to complete by the end of their exploration. Fill a graduated cylinder with 20 mL of water, then place a rock in the graduated cylinder. What is the volume of the rock?

(Hint: To answer correctly, be sure that you are reading from the bottom of the meniscus.) 

After the rock is placed in the graduated cylinder, the water displaces from 20 mL to 35 mL, therefore the volume of the rock, or the amount of space the rock takes up, is 15 mL. 

Division Strategies

Lately in math class, students have been exploring the idea of division and how it relates to multiplication. Students have been developing strategies that make conceptual sense to them in order to solve. Here is a chart of the strategies that came up in our closing session based on just one problem! These kids are pretty smart. Wouldn't you agree?

Doubling and Halving Math Strategy

Last week in one of our math mini lessons, students were challenged to determine if 30 x 4 = 15 x 8.  After a few minutes of finding the product of each expression, students determined that each had a product of 120.  Students were also challenged to see if they noticed anything else that these two problems had in common.  After a few moments, many students' faces lit up! Many students noticed that half of 30 is 15 and 4 doubled is 8!  After some conversation about this, students were challenged to explore this concept during the work period to *prove if this theory is true all the time. (Seeking to make a generalization or proof across all problems is one of the cornerstones of algebra. Without even knowing it, our third grade students are working on their algebraic thinking every day. How cool! Now, back to doubling and halving...) The following chart displays the two "must do" problems that they students were expected to complete during the work period. 

In closing, students determined that doubling and halving could work for all problems; however, it might not be the most efficient for all problems. For example, we discussed 11 x 23.  When halving either one of these factors (11 or 23), the mathematician will be challenged to work with fractional parts.  Is this impossible? Certainly not. Is it the most efficient way to solve? Probably not.  

Student Challenge: 

Try solving the following using the "Doubling and Halving" strategy: 16 x 25, 15 x 32, 12 x 30.

Density Gizmo

With a Density Gizmo, students continued to deepening their understanding of matter, mass, and volume this week, and explored density. They first brainstormed objects that they think would sink in water and then those that would float, and formed a hypothesis for why the objects sink or float.

After that, they did a Gizmo warm-up which familiarized them with the virtual exploration by measuring the mass of objects on a scale, and measuring the volume of objects using water displacement in a graduated cylinder.    

The essential question then focused their activity, How do mass and volume affect sinking and floating?  

During the Gizmo, students filled in a chart with each object's mass and volume, and then whether the object would sink or float when placed in the beaker.  This is a sample of their chart.

Object	Prediction (sink or float?)	Mass	Volume	Result (sink or float?)
Ping pong ball	F	3.0 g	36.0 mL	Float
Golf ball	S	45.0 g	36.0 mL	Sink
Apple	F	33.0 g	44.0 mL	Float
Chess piece	S	40.0 g	80.0 mL	Float
Penny	S	3.0 g	0.4 mL	Sink
Rock	S	200.0 g	50.0 mL	Sink

They analyzed their results and concluded that you could not predict whether an object would sink or float using the mass alone, because the mass of a ping pong ball and penny were both 3 grams and one floated while the other sank. Based on the volume alone, they concluded that you could not predict whether an object would float or sink, because the volume of a ping pong ball and golf ball were both 36 mL and one floated while the other sank.

However, mass and volume, when considered together could predict whether an object would sink or float. When an object's mass was less than the object's volume, the object floated.  When an object's mass was more than an objects volume, then the object would sink. Density refers to the mass found in a given volume of a substance. 

During our activity, we concentrated on Activity A focused on water, however students have access to Activity C, too, where they can experiment with the density of liquids like oil, gasoline, seawater, and corn syrup.  They can log on at home with the user name and password in the back of their planner. If a student moves on to the Activity C exploration, we'd love to hear their analysis, How does an object behave in different liquids?